like in our hydrodynamic formalism. Furthermore, we can write the electric
current as $\vec{J}=nq\langle\vec{v}\rangle$ for some uniform charge density
$n$, which gives us the DC conductivity

$\vec{J}=\frac{nq^{2}\tau}{m}\vec{\mathbb{E}}\qquad\Rightarrow\qquad\sigma_{\text{DC}}=\frac{nq^{2}\tau}{m}$
(4.81)

An applied external electric field generally heats up the system, as it
happens also in hydrodynamics. This means that if the system reaches a steady
state the heat produced must be dissipated away, otherwise the system melts.
We can see this effect also in the Drude model: from (4.80) we arrive to an
equation for the kinetic energy of the gas particles

$\frac{m}{2}\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\langle\vec{v}\rangle^{2}=q\langle\vec{v}\rangle\vec{\mathbb{E}}-\frac{m}{\tau}\langle\vec{v}\rangle^{2}$
(4.82)

which implies that if the velocity is time-independent (steady state), then
the energy provided by the electric field is lost to impurities with a rate
proportional to $\langle\vec{v}\rangle^{2}$. The last term is effectively an
energy relaxation rate $\Gamma_{\varepsilon}$ like the one we introduced in
our hydrodynamic formalism.

###  4.4 Dissipative corrections and optical conductivity

This final section is based on work still in progress, and is not published
research yet.

The obvious next step forward is to start discussing dissipative corrections.
However, this becomes quickly very complicated with the setup discussed in
this chapter, because having two different vectors at order zero in
derivatives, $v^{i}$ and $\mathbb{E}^{i}$, means that we can construct many
tensor structures $\sim\mathcal{O}(10^{2})$ that can appear in the
constitutive relations at order one, each with its own associated transport
coefficient555Even without order-zero electric fields, boost-agnostic
hydrodynamics has 30 transport coefficients at order one in derivatives
[101]..

To make our life simpler, we can then demote the electric field to order one
in derivative counting, hence dropping all the terms that depend on the
electric field in the thermodynamics (4.48) and ideal fluid (4.54).
Furthermore, we also demote to order one the relaxation rate $\Gamma_{P}$ and
$\Gamma_{\varepsilon}$, while $\Gamma_{\mathbb{P}}$ disappears, since we do
not have an order-zero polarization any more. We remark that this does not
modify the result of (4.61), since hydrostatic constraints are exact results
which do not depend on the choice of derivative counting for the external
fields.

With these simplifications, the most interesting term we may wish to study are
the dissipative corrections to the $\mathrm{U(1)}$ current. This is because
usually it receives derivative corrections in the form
$J_{i}\supset\sigma\left(\mathbb{E}_{i}-\partial_{i}\mu\right)$, but now that
we modified the hydrostatic constraint (4.61) we expect to find

$J_{i}\supset\sigma\left(\mathbb{E}_{i}-\partial_{i}\mu-\Gamma\frac{P_{i}}{n}\right)$
(4.83)

where we dropped the subindex $P$ from the relaxation rate, since there is now
only one decay rate. In particular, we are interested in studying how this
correction to the constitutive relations affects the usual linear-response
charge transport.

To understand the physics, consider a relativistic constitutive relation
expanded at small velocity. If we add dissipative correction to the current
$J^{\mu}=J^{\mu}_{(0)}+J^{\mu}_{(1)}$ we find that second law of
thermodynamics becomes

$\partial_{t}s+\partial_{i}\left(sv^{i}-\frac{\mu}{T}J^{i}_{(1)}\right)+J^{i}_{(1)}\partial_{i}\frac{\mu}{T}=-\frac{\Gamma_{\varepsilon}\varepsilon}{T}+\frac{\Gamma(P\cdot
v)}{T}+\frac{E\cdot J_{(1)}}{T}\geq 0$ (4.84)

The first two terms in the LHS are the divergence of the entropy current, thus
we rewrite it as

$\partial_{\mu}S^{\mu}=-\frac{\Gamma_{\varepsilon}}{T}+\Gamma\frac{P\cdot
v}{T}+\frac{J^{i}_{(1)}}{T}\left(E_{i}-T\partial_{i}\frac{\mu}{T}\right)\geq
0$ (4.85)

If we now impose the relations we found for ideal fluids (4.3.1) to cancel the
two relaxation terms, we are left with a quantity on the RHS that does not
vanish in equilibrium if we use our modified condition (4.61).

To address the problem, the relation between energy and momentum relaxation
must receive dissipative derivative corrections. Namely, we identify

$\Gamma_{\varepsilon}=\Gamma P\cdot v+\Gamma\frac{P\cdot
J_{(1)}}{n}=\Gamma\frac{P_{i}\left(nv^{i}+J^{i}_{(1)}\right)}{n}=\Gamma\frac{P\cdot
J}{n}$ (4.86)

where in the last step we introduced the total free current, meaning that if
polarization was present, it would not appear there. Obviously, on hydrostatic
solutions the new term proportional to $J_{(1)}$ vanishes and we find again
(4.3.1).

Indeed, with this identification, we see that the divergence of the entropy
current takes the desired form

$\partial_{\mu}S^{\mu}=\frac{J^{i}_{(1)}}{T}\left(E_{i}-T\partial_{i}\frac{\mu}{T}-\Gamma\frac{P_{i}}{n}\right)\geq
0$ (4.87)

and the RHS is positive definite only if the dissipative part of the current
is of the form (4.83)

$J^{i}_{(1)}=\sigma\left(E_{i}-T\partial_{i}\frac{\mu}{T}-\Gamma\frac{P_{i}}{n}\right)$
(4.88)

From this result, we can now compute the optical conductivities on a
background with zero velocity and zero electric field using the standard
Martin-Kadanoff method 2.2.1. We find that

$\sigma(\omega)=\frac{n^{2}-i\chi_{\pi\pi}\sigma\omega}{\Gamma\chi_{\pi\pi}-i\chi_{\pi\pi}\omega}$
(4.89)

with $\chi_{\pi\pi}$ the momentum-momentum susceptibility. Notice, in
particular, that in DC at $\omega=0$ the incoherent conductivity $\sigma$
drops out and we find

$\sigma_{\text{DC}}=\frac{n^{2}}{\Gamma\chi_{\pi\pi}}$ (4.90)

in agreement with (4.79).

There are ways to reintroduce the transport coefficient $\sigma$ in the DC
conductivity, by considering non-hydrostatic momentum relaxations
$\Gamma=\Gamma_{(0)}+\Gamma_{(1)}$ and modifying the relation between
$\Gamma_{\varepsilon}$ and $\Gamma$ appropriately. This gives rise to
interesting results that renormalize the incoherent conductivity $\sigma$
similarly to what is found in [157], however we will not discuss them further.

###  4.5 Summary, discussion and outlook

In this chapter we characterized the stationary flows of a charged fluid
immersed in an electric field, which is also subject to the effect of
relaxations that remove energy and momentum from the system, thus behaving as
sinks connected with external baths.

Since momentum relaxation breaks boost, we employed a boost-agnostic formalism
and computed the constitutive relations up to order one in derivatives using
the hydrostatic generating functional. Subsequently, we added by hand the
relaxation terms and studied the minimal corrections to the hydrostatic
conditions (4.38) consistent with the equations of motion. We found that only
one single order-zero constraint had to be modified, taking the new form
(4.61), and we also obtained a kinematic relation between energy and momentum
relaxation (4.3.1).

As for future perspectives, it would be fascinating to see if it is possible
to implement relaxation rates as in our setup from a more formal and
geometrical approach, so to include the effect of relaxations in a (suitably
generalized) generating functional. This could be done using the modified
diffeomorphisms of [257], following the approach of [157], from the Schwinger-
Keldysh effective actions as in [49], or, finally, following the thermodynamic
approach of [100] to include a fixed vector in the theory.

In the line of Section 4.4, it would also be interesting to systematically
study dissipative corrections to discuss how the modified hydrostatic relation
(4.61) changes the AC thermoelectric transport.

Finally, we could discuss a similar setup in holographic models [258],
following e.g. the result of [158], which finds a contribution due to the
velocity in the spatial current as in (4.83).

##  Chapter 5 Onsager relations in relaxed hydrodynamics

> “Nature does nothing uselessly.”
>
> Aristotle, _Politics_

###  5.1 Introduction

A common feature of all hydrodynamic theories is that a charged fluid in an
external electric field gives rise to optical conductivities which are
divergent in the DC limit $\omega\rightarrow 0$ [14]. The physical reason for
this divergence is very simple: in hydrodynamics momentum is a conserved
quantity, therefore an external electric field accelerates the fluid
increasing momentum without bound, since there are no mechanism to dissipate
it. On a more technical level, this can be traced back to the exact
translation symmetry of the theory, or more directly to the existence of a
conserved momentum operator that couples to the charged degrees of freedom.

Exactly like in Chapter 4, the standard approach to avoid this issue is to
include a small effective phenomenological relaxation parameter
$\tau_{m}^{-1}$ which breaks translation invariance and relaxes the total
momentum of the fluid, so that a steady state with finite DC conductivities
can be reached. One assumption behind this approach is that, although momentum
is no longer conserved, it can still be a relevant hydrodynamic charge if the
parameter $\tau_{m}$ is large enough [244, 162, 259, 192, 50, 157, 158, 164],
i.e. if the associated gapped mode is closer to the real line in frequency
space compared to the other non-hydrodynamic modes (see the discussion about
the hydrodynamic modes in Section 2.2.5 and in the Introduction 1).

As we will see better in the next chapter, there are circumstances in which
momentum relaxation is not enough to obtain finite DC conductivities. This is
particularly relevant for the case of anomalous hydrodynamics, i.e. chiral
fluids in the presence of a $\mathrm{U(1)}$ axial anomaly. In this case,
momentum relaxation $\tau_{m}$ is not enough to obtain finite DC
conductivities, and one also needs to relax energy and axial charge [260, 257,
16, 261, 262]. Furthermore, from a physical perspective, even without chiral
anomalies, energy and charge relaxations can be important ingredients of the
low energy dynamics of condensed matter systems, as argued in [263, 264, 127,
265, 126] from a kinetic theory perspective.

On the other hand, from a computational perspective there are two main
approaches to obtain the retarded response functions (thus the conductivities)
from hydrodynamics, as reviewed in Section 2.2: the Martin-Kadanoff [175], see
Section 2.2.1 and the variational method [74, 68], Section 2.2.2. In
particular, the former is based on thermodynamics and linear response theory,
and works by perturbing the fluid with thermodynamic sources, while the latter
is rooted in field theory and uses metric and gauge field fluctuations as
sources.

However, because relaxation rates break the Lorentz and $U(1)$ symmetry, the
variational approach is usually difficult to implement in quasihydrodynamics,
since there is no unique proper way to covariantize the decay terms. On the
other hand, the canonical approach is more robust against modifications of
hydrodynamics, and is applicable also to quasihydrodynamic models, however it
is lacking under other aspects. Specifically it cannot be used to compute all
the Green functions in the theory [74, 88] and, if the thermodynamics becomes
more complicated (e.g., in the presence of order zero electromagnetic fields),
it might be hard to identify the susceptibilities and the correct conjugate
sources.

In this chapter we are going to explore models of linearized
quasihydrodynamics to compute Green functions using both the Martin-Kadanoff
and the variational method. In particular, we find that it is possible to use
the variational method in the presence of relaxations too. We start by
including extra source terms in the linearised equations of motion, and what
we find is that the coefficients of these extra terms are uniquely fixed by
the requirement of time-reversal covariance of the response functions alone,
i.e. Onsager reciprocal relations.

###  5.2 Martin-Kadanoff for quasihydrodynamics

We now focus specifically on a simple charged relativistic fluid with
conserved stress-energy tensor and a $\mathrm{U(1)}$ current. We linearize the
theory and add the most general relaxations terms to the equations of motion;
subsequently we impose Onsager relations on the Green functions, positivity of
entropy production, and linear stability from the study of the modes. What we
find is a set of constraints and relations that must be obeyed very generally
by all fluids.

As we discussed in Section 2.2, we can write the equations of linearized
hydrodynamics in the form

$\partial_{t}\phi_{a}(t,\boldsymbol{\mathbf{k}})+M_{ab}(\boldsymbol{\mathbf{k}})\phi_{b}(t,\boldsymbol{\mathbf{k}})=0$
(5.1)

where we Fourier transformed in space, while keeping the explicit time
dependence. Here, $\phi_{a}$ are the fluctuations of the (almost-)conserved
charges, while $M_{ab}$ is the hydrodynamic matrix that depends on the
constitutive relations and equations of motion. We also rewrite here the
definition of the susceptibility matrix as

$\chi_{ab}=\frac{\partial\phi_{a}}{\partial\lambda_{b}}$ (5.2)

in which $\lambda_{a}$ are the sources canonically conjugate to the charges.

Following the arguments of Section 2.2.1 we can use these expressions to
compute the Green functions using the Martin-Kadanoff method 2.55.
Furthermore, we have also discussed how time-reversal covariance of the
microscopic theory, namely the requirement that

$G_{ab}^{R}(\omega,\boldsymbol{\mathbf{k}})=\eta_{a}\eta_{b}G^{R}_{ab}(\omega,-\boldsymbol{\mathbf{k}})$
(5.3)

imposes a constraint on the hydrodynamic matrix

$\chi SM^{T}(-\boldsymbol{\mathbf{k}})=M(\boldsymbol{\mathbf{k}})\chi S$ (5.4)

in which we set the magnetic field to zero and remind here that
$S=\text{diag}(\eta_{1},\eta_{2},\dots)$ is the matrix of the time-reversal
eigenvalues $\eta_{a}$.

Consider now a relativistic fluid in the Landau frame in $(3+1)$-dimensional
flat spacetime and without external electromagnetic fields, so that the order
one constitutive relations are given in (2.26), and the non-linear
conservation equations are simply
$\partial_{\mu}T^{\mu\nu}=\partial_{\mu}J^{\mu}=0$. We now linearize these
equations around a global equilibrium state with constant energy and charge
density, and zero spatial velocity

$u^{\mu}=(1,\delta
v^{i})\qquad\varepsilon=\varepsilon_{0}+\delta\varepsilon\qquad n=n_{0}+\delta
n$ (5.5)

The linearised equations of motion take the following form

$\displaystyle\partial_{t}\delta\varepsilon+\left(\varepsilon_{0}+P_{0}\right)\partial_{i}\delta
v^{i}$
$\displaystyle=-\left(\frac{\delta\varepsilon}{\tau_{\varepsilon\varepsilon}}+\frac{\delta
n}{\tau_{\varepsilon n}}\right)$ (5.6a) $\displaystyle\partial_{t}\delta
n+n_{0}\partial_{i}\delta v^{i}-\sigma
T_{0}\partial_{i}^{2}\delta\frac{\mu}{T}$
$\displaystyle=-\left(\frac{\delta\varepsilon}{\tau_{n\varepsilon}}+\frac{\delta
n}{\tau_{nn}}\right)$ (5.6b)
$\displaystyle\partial_{t}\delta\pi^{i}+\partial^{i}\delta
P-\eta\left(\partial_{j}^{2}\delta
v^{i}+\frac{1}{3}\partial^{i}\partial_{j}\delta
v^{j}\right)-\zeta\partial_{i}\partial_{j}\delta v^{j}$
$\displaystyle=-\frac{1}{\tau_{m}}\delta\pi^{i}$ (5.6c)

where the linear momentum is $\delta\pi^{i}=(\varepsilon_{0}+P_{0})\delta
v^{i}$ for a relativistic fluid. These correspond to the (non-)conservation of
energy, charge and momentum respectively. In the above equations we also
included on the RHS the most general relaxation terms at order one in
fluctuations and order zero in derivatives111The reason for this is that we
take the relaxation rates $\tau$s to be large, namely
$\tau^{-1}\sim\mathcal{O}(\partial)$ in the linearized regime. compatible with
isotropy and rotational invariance (hence only a single $\tau_{m}$ appears,
instead of a momentum-relaxation matrix). Specifically, $\tau_{m}$ is the
standard momentum relaxation, while $\tau_{\varepsilon\varepsilon}$ and
$\tau_{nn}$ are commonly known as energy and charge relaxation respectively.
The other two off-diagonal relaxations $\tau_{\varepsilon n}$ and
$\tau_{n\varepsilon}$, however, are mostly new (see [16]) and represent mixed
relaxations which do not have units of time: $\tau_{\varepsilon n}$
parametrizes the energy loss due to charge fluctuations and vice versa for
$\tau_{n\varepsilon}$. Notice that in principle, because of the relaxations,
the constitutive relations might have more and different transport
coefficients [49], however because we are assuming the $\tau$s are very large
these corrections are taken to be very small.

In the context of hydrodynamics, _relaxation_ usually means that the
fluctuation of some conserved charge decays exponentially to equilibrium, e.g.
consider the simple charge conservation equation on a uniform background

$\partial_{t}\delta n=-\frac{1}{\tau_{nn}}\delta n$ (5.7)

Real time solutions to this equation take the form

$\delta n(t)=\delta n(0)e^{-t/\tau_{nn}}$ (5.8)

However, even a simple example like this one, cannot be properly
covariantized, for example writing $u_{\mu}J^{\mu}/\tau$ would not give the
proper relaxation term at non-zero velocity. Indeed, $J^{0}$ represent the
charge density in the laboratory frame, while $n=-u_{\mu}J^{\mu}$ is the
charge density in the fluid rest frame, thus
$\partial_{t}J^{0}=\frac{1}{\tau}u_{\mu}J^{\mu}$ would not relax the right
charge. Furthermore, if we try to covariantize in this way, we will never be
able to write a momentum relaxation term, since the comoving momentum is zero
by definition.

We then realize that to write any relaxation in a covariant non-linear form we
need some extra information beyond the fluid data alone. This can be done e.g.
in the boost-agnostic formalism by using the laboratory-frame clock form
$\tau_{\mu}=(1,\boldsymbol{\mathbf{0}})$, in the relativistic case following
[266] by considering a second four-velocity vector $U^{\mu}$ which pins the
fluid velocity, or introducing extra fields as in [100, 157, 156].

The second important thing to observe is that relaxations act as constraints
on the system, by selecting a specific background. This is seen very clearly
for the momentum relaxation, in which $1/\tau_{m}\delta\pi^{i}$ naturally
selects the equilibrium solution with zero velocity as the only equilibrium
solution of the system. This should be contrasted with the case without
momentum relaxation, in which any equilibrium with constant non-zero velocity
is a valid equilibrium, thanks to the boost symmetry.

However, even without relaxations, although the theory has Lorentz symmetry,
the equilibrium solution does not [74, 167], and because we are interested in
computing two-point functions about some global thermodynamic equilibrium any
choice of the background will break the symmetry. For these reasons we are
agnostic on possible non-linear completions of the theory, thus we focus only
on the linearized version of hydrodynamics, which is enough for the purpose of
computing the Green functions. Then, we simply write the relaxation terms as
the most general expressions, following the EFT prescription, of all the
scalars and vectors in the linearized theory. At the end, we expect that any
covariant version of the non-linear relaxations should lead to the same
linearized expressions we work with, as we will see in the next chapter with
specific examples.

Having clarified the reasoning behind our choice of relaxations, we proceed by
imposing Onsager relations in (5.4) on our linearized hydrodynamic theory
(5.6). We take $\phi(t,\boldsymbol{\mathbf{k}})=\left(\delta\varepsilon,\delta
n,\delta\pi^{i}\right)$ and, setting $\boldsymbol{\mathbf{k}}=(k_{x},0,0)$
without loss of generality, the matrix $M_{ab}$ becomes

$M=\\\
\begin{pmatrix}\frac{1}{\tau_{\varepsilon\varepsilon}}&\frac{1}{\tau_{\varepsilon
n}}&ik_{x}&0&0\\\
k_{x}^{2}\sigma\beta_{\varepsilon}+\frac{1}{\tau_{n\varepsilon}}&k_{x}^{2}\sigma\beta_{n}+\frac{1}{\tau_{nn}}&\frac{ik_{x}n_{0}}{P_{0}+\varepsilon_{0}}&0&0\\\
ik_{x}\frac{\partial P}{\partial\varepsilon}&ik_{x}\frac{\partial P}{\partial
n}&\frac{k_{x}^{2}(3\zeta+4\eta)}{3(P_{0}+\varepsilon_{0})}+\frac{1}{\tau_{p}}&0&0\\\
0&0&0&\frac{k_{x}^{2}\eta}{P_{0}+\varepsilon_{0}}+\frac{1}{\tau_{m}}&0\\\
0&0&0&0&\frac{k_{x}^{2}\eta}{P_{0}+\varepsilon_{0}}+\frac{1}{\tau_{m}}\end{pmatrix}$
(5.9)

where
$\beta_{\varepsilon}=\frac{\partial\mu}{\partial\varepsilon}-\frac{\mu_{0}}{T_{0}}\frac{\partial
T}{\partial\varepsilon}$ and $\beta_{n}=\frac{\partial\mu}{\partial
n}-\frac{\mu_{0}}{T_{0}}\frac{\partial T}{\partial n}$. Then we find that
Onsager relations are obeyed iff the relaxation rates satisfy the condition

$\frac{\chi_{\varepsilon\varepsilon}}{\tau_{n\varepsilon}}-\frac{\chi_{\varepsilon
n}}{\tau_{\varepsilon\varepsilon}}+\frac{\chi_{n\varepsilon}}{\tau_{nn}}-\frac{\chi_{nn}}{\tau_{\varepsilon
n}}=0$ (5.10)

where the susceptibilities are

$\chi_{n\varepsilon}=\chi_{\varepsilon
n}=\frac{\partial\varepsilon}{\partial\mu}=T_{0}\frac{\partial n}{\partial
T}+\mu_{0}\frac{\partial n}{\partial\mu}\qquad\chi_{nn}=\frac{\partial
n}{\partial\mu}\qquad\chi_{\varepsilon\varepsilon}=T_{0}\frac{\partial\varepsilon}{\partial
T}+\mu_{0}\frac{\partial\varepsilon}{\partial\mu}$ (5.11)

The above thermodynamic derivatives are in the grand canonical ensemble, and
should be understood at fixed $\mu$ or $T$ respectively.

Notice that this result is not in contrast with the previous chapter, in
particular the $\tau_{\varepsilon\varepsilon}$ that appears here is not
related to the $\Gamma_{\varepsilon}$ introduced in (4.55). This is because,
due to the relation (4.3.1)

$\Gamma_{\varepsilon}=\Gamma_{P}P^{i}v_{i}$ (5.12)

between energy and momentum relaxations on hydrostatic solutions,
$\Gamma_{\varepsilon}$ vanishes when linearizing around a zero-velocity
background.

The constraint (5.10) is very general and must be obeyed by any fluid which
preserves microscopic time-reversal invariance. In particular, we can see that
if the mixed relaxations are zero $\tau_{n\varepsilon}^{-1}=\tau_{\varepsilon
n}^{-1}=0$, then it implies $\tau_{\varepsilon\varepsilon}=\tau_{nn}$, i.e.
energy and charge must relax at the same rate as was already observed in
[257].

We can also study the effect of generalized relaxations on the second law of
thermodynamics. The entropy current, at the order we are interested in, is
just given by the canonical entropy current

$S^{\mu}=\frac{1}{T}\left(Pu^{\mu}+T^{\mu\nu}u_{\nu}-\mu J^{\mu}\right)$
(5.13)

We can linearize the entropy current and compute the divergence. Using the
linearized equations of motion (5.6) and following the usual steps, we arrive
at the result

$T_{0}\partial_{\mu}\delta
S^{\mu}=\delta\varepsilon\left(\frac{\mu_{0}}{\tau_{n\varepsilon}}-\frac{1}{\tau_{\varepsilon\varepsilon}}\right)+\delta
n\left(\frac{\mu_{0}}{\tau_{nn}}-\frac{1}{\tau_{\varepsilon
n}}\right)+\mathcal{O}(\partial^{2},\delta^{2})$ (5.14)

The RHS should be positive definite on any solution of the hydrodynamic
equations of motion. In this case, because the fluctuations of
$\delta\varepsilon$ and $\delta n$ are not positive definite, we must impose
that the coefficients multiplying the fluctuations vanish, namely

$\frac{1}{\tau_{\varepsilon\varepsilon}}=\frac{\mu_{0}}{\tau_{n\varepsilon}}\qquad\qquad\frac{1}{\tau_{\varepsilon
n}}=\frac{\mu_{0}}{\tau_{nn}}$ (5.15)

These are two further constraints on the relaxation rates that stem from the
local form of the second law of thermodynamics, together with the constraint
(5.10) required by Onsager relations. For a system that is constrained by both
second law and Onsager relations there are then three constraints and four
total relaxation rates, which means that the whole relaxation dynamics is
actually characterized by a single parameter (in addition to momentum decay
$\tau_{m}$). Furthermore, because of the structure of these constraints, not
all relaxation parameters must be positive.

There is an intuitive thermodynamic argument that we can give, following
[260], to explain why the mixed relaxations are related by the positivity of
entropy production in (5.15). Consider a fluid element which undergoes a
microscopic scattering process that relaxes the energy. Then, because of the
scattering, the fluid will lose some energy $\delta\varepsilon$ in a time
$\tau_{\varepsilon\varepsilon}$, on average. However, if it loses energy, this
implies that it is also losing charge by $\delta n=\frac{\partial
n}{\partial\varepsilon}\delta\varepsilon=\delta\varepsilon/\mu_{0}$ in the
same interval. The argument in reverse, for a scattering that relaxes the
charge, tells us that the system must also lose energy as
$\delta\varepsilon=\mu_{0}\delta n$. We are thus let to the expressions (5.15)
that come from the second law.

Indeed, both the above argument based on thermodynamics and the second law
hold only for closed systems. However, if the relaxation rates parametrize the
fact that our system is open and loses energy and charge to the environment,
then the second law might fail if applied to the fluid alone, as we will see
in the next chapter. The total entropy should still increase, but we have to
account for the entropy of the fluid plus of its environment for this to
happen.

In (5.14) we computed the entropy production at first order in fluctuations,
the order in which the relaxation rates first appear. However, linearized
hydrodynamics is usually discussed at order two in fluctuations, from the
point of view of the entropy current, since only at order
$\mathcal{O}(\delta^{2})$ we can bound the dissipative transport coefficients.
Therefore, we check the second law at order two in fluctuations, and we find

$\displaystyle T_{0}\partial_{\mu}\delta s^{\mu}$
$\displaystyle=\frac{\delta\varepsilon\delta
T}{T_{0}\tau_{\varepsilon\varepsilon}}+\frac{\delta n\delta
T}{T_{0}\tau_{\varepsilon n}}+\frac{\delta\mu\delta
n}{\tau_{nn}}+\frac{\delta\mu\delta\varepsilon}{\tau_{n\varepsilon}}-\mu_{0}\frac{\delta
T\delta n}{T_{0}\tau_{nn}}-\mu_{0}\frac{\delta
T\delta\varepsilon}{T_{0}\tau_{n\varepsilon}}$
$\displaystyle\quad+\frac{1}{\tau_{m}}(\varepsilon_{0}+P_{0})\delta
v^{2}+\sigma\left(\frac{\mu_{0}}{T_{0}}\partial\delta
T-\partial\delta\mu\right)^{2}$
$\displaystyle\quad+\eta\left(\delta\sigma^{ij}\right)^{2}+\zeta\left(\partial_{i}\delta
v^{i}\right)^{2}+\mathcal{O}(\delta^{3},\partial^{3})$ (5.16)

where $\delta\sigma^{ij}$ are the fluctuations of the shear tensor. Thus, we
find that positivity of entropy production at order two in fluctuations gives
the usual constraints on the transport coefficients $\sigma\geq 0$, $\eta\geq
0$ and $\zeta\geq 0$, furthermore, as expected, we find $\tau_{m}\geq 0$ for
the momentum relaxation rate.

Then, we can focus only on the first line in the above formula, the one
depending on the energy and charge relaxations. We rewrite it in terms of
fluctuations of temperature and chemical potential alone, by expanding the
energy and charge fluctuations as

$\delta\varepsilon=\frac{\partial\varepsilon}{\partial T}\delta
T+\frac{\partial\varepsilon}{\partial\mu}\delta\mu\qquad\delta
n=\frac{\partial n}{\partial T}\delta T+\frac{\partial
n}{\partial\mu}\delta\mu$ (5.17)

and arrive at the following expression

$\displaystyle T_{0}\partial_{\mu}\delta s^{\mu}$
$\displaystyle=\delta\mu\delta
T\left(\frac{\partial\varepsilon}{\partial\mu}\frac{1}{T_{0}\tau_{\varepsilon\varepsilon}}+\frac{\partial
n}{\partial\mu}\frac{1}{T_{0}\tau_{\varepsilon n}}+\frac{\partial n}{\partial
T}\frac{1}{\tau_{nn}}+\frac{\partial\varepsilon}{\partial
T}\frac{1}{\tau_{n\varepsilon}}-\frac{\mu_{0}}{T_{0}}\frac{\partial
n}{\partial\mu}\frac{1}{\tau_{nn}}\right.$
$\displaystyle\left.\quad-\frac{\mu_{0}}{T_{0}}\frac{\partial\varepsilon}{\partial\mu}\frac{1}{\tau_{n\varepsilon}}\right)+\frac{\delta
T^{2}}{T_{0}}\left(\frac{\partial\varepsilon}{\partial
T}\frac{1}{\tau_{\varepsilon\varepsilon}}+\frac{\partial n}{\partial
T}\frac{1}{\tau_{\varepsilon n}}-\frac{\partial n}{\partial
T}\frac{\mu_{0}}{\tau_{nn}}-\frac{\partial\varepsilon}{\partial
T}\frac{\mu_{0}}{\tau_{n\varepsilon}}\right)$
$\displaystyle\quad+\delta\mu^{2}\left(\frac{\partial
n}{\partial\mu}\frac{1}{\tau_{nn}}+\frac{\partial\varepsilon}{\partial\mu}\frac{1}{\tau_{n\varepsilon}}\right)+\ldots$
(5.18)

where the dots represent the positive definite terms we already discussed.
Employing both Onsager relations (5.10) and the constraints (5.15) we can
simplify the above expression to

$T_{0}\partial_{\mu}\delta S^{\mu}=\delta\mu^{2}\left(\frac{\partial
n}{\partial\mu}\frac{1}{\tau_{nn}}+\frac{\delta\varepsilon}{\delta\mu}\frac{1}{\tau_{n\varepsilon}}\right)+\dots$
(5.19)

which can finally be rearranged in terms of susceptibilities as

$\frac{\delta\mu^{2}}{\tau_{nn}}\left(\chi_{\varepsilon\varepsilon}\chi_{nn}-\chi_{\varepsilon
n}^{2}\right)\lessgtr 0$ (5.20)

The sign of the inequality depends on the sign of
$\partial\varepsilon/\partial T$. The susceptibility matrix is positive semi-
definite, which means that its determinant is also non-negative and so is the
quantity in the bracket above. Thus, we see that the sign of $\tau_{nn}$ is
constrained by the sign of $\partial\varepsilon/\partial T$: when
$\partial\varepsilon/\partial T\geq 0$ then $\tau_{nn}\geq 0$ and vice versa.
It is important to observe that this condition is not of equality-type, like
(5.15) and (5.10), therefore, even including terms which are order two in
fluctuations in the second law, we still have a one-parameter family of
relaxations222Notice that we could also include relaxation terms which are
order two in fluctuations directly in the equations of motion, to cancel
$\mathcal{O}(\delta^{2})$ terms that appear in the positivity of entropy
production..

Finally, we can also check if linear stability, namely the requirement that
hydrodynamic modes must decay and not grow with time, imposes further
constraints on the relaxation rates. In $d+1$ spacetime dimensions we find
$d+2$ modes, one for each equation of motion, which at
$\boldsymbol{\mathbf{k}}=0$ are given by

$\omega_{1}=-\frac{i}{\tau_{m}}\qquad\omega_{2}=-\frac{i}{2}\left(\frac{1}{\tau_{\varepsilon\varepsilon}}+\frac{1}{\tau_{nn}}\right)\pm\frac{i}{2}\sqrt{\left(\frac{1}{\tau_{\varepsilon\varepsilon}}-\frac{1}{\tau_{nn}}\right)^{2}+\frac{4}{\tau_{n\varepsilon}\tau_{\varepsilon
n}}}$ (5.21)

where $\omega_{1}$ has multiplicity $d$. Stability requires that the imaginary
part of the modes should be negative, thus the first mode simply tells us that
$\tau_{m}\geq 0$, exactly the same constraint that we find from the second law
of thermodynamics. The two other $\omega_{2}$ modes instead can be propagating
or not, and their stability depends on the thermodynamics and the relations
between the $\tau$s. However, if we also impose all the other constraints
(5.15) and (5.10), then we can rewrite $\omega_{2}$ as a null mode and a
decaying mode

$\omega_{2}=0\qquad\omega_{2}=-i\left(\frac{1}{\tau_{nn}}+\frac{1}{\tau_{\varepsilon\varepsilon}}\right)=-\frac{i}{\tau_{nn}}\left(\frac{\frac{\partial\varepsilon}{\partial
T}-\mu_{0}\frac{\partial n}{\partial T}}{\frac{\partial\varepsilon}{\partial
T}}\right)$ (5.22)

In particular, the stability of the second mode depends again on the relative
sign of thermodynamic derivatives and $\tau_{nn}$, as in (5.20).

Albeit it is not the main goal of this chapter, to conclude the section we can
ask about possible non-linear completion of our generalized relaxations (we
will see more in the next chapter). In particular, we can focus on relaxations
which obey both the second law of thermodynamics and Onsager relations, and
see how these constraints restrict the possible non-linear expressions.

To proceed, we change the basis of fluctuations from the charges $\delta n$
and $\delta\varepsilon$ to the conjugate sources $\delta T$ and $\delta\mu$,
specifically

$\frac{\delta\varepsilon}{\tau_{\varepsilon\varepsilon}}+\frac{\delta
n}{\tau_{\varepsilon n}}=\frac{\delta T}{\tau_{\varepsilon
T}}+\frac{\delta\mu}{\tau_{\varepsilon\mu}}\qquad\qquad\frac{\delta\varepsilon}{\tau_{n\varepsilon}}+\frac{\delta
n}{\tau_{nn}}=\frac{\delta T}{\tau_{nT}}+\frac{\delta\mu}{\tau_{n\mu}}$ (5.23)

We can now impose Onsager relations and positivity of entropy production on
this new basis, obtaining the constraints

$\frac{1}{\tau_{\varepsilon
T}}=0\qquad\frac{1}{\tau_{nT}}=0\qquad\tau_{\varepsilon\mu}=\frac{\tau_{nn}}{\mu_{0}}\frac{\frac{\partial\varepsilon}{\partial
T}}{\frac{\partial n}{\partial\mu}\frac{\partial\varepsilon}{\partial
T}-\frac{\partial n}{\partial
T}\frac{\partial\varepsilon}{\partial\mu}}\qquad\tau_{n\mu}=\frac{\tau_{nn}\frac{\partial\varepsilon}{\partial
T}}{\frac{\partial n}{\partial\mu}\frac{\partial\varepsilon}{\partial
T}-\frac{\partial n}{\partial T}\frac{\partial\varepsilon}{\partial\mu}}$
(5.24)

Consider now a pair of differentiable functions333This $\Gamma_{\varepsilon}$
must not be confused with the one of the previous chapter (5.12).
$\Gamma_{\varepsilon}$ and $\Gamma_{n}$ which, upon linearization, give rise
to generalized relaxations

$\displaystyle\delta\Gamma_{\varepsilon}$
$\displaystyle=\frac{\partial\Gamma_{\varepsilon}}{\partial\mu}\delta\mu+\frac{\partial\Gamma_{\varepsilon}}{\partial
T}\delta T=\frac{\delta\mu}{\tau_{\varepsilon\mu}}+\frac{\delta
T}{\tau_{\varepsilon T}}$ (5.25a) $\displaystyle\delta\Gamma_{n}$
$\displaystyle=\frac{\partial\Gamma_{n}}{\partial\mu}\delta\mu+\frac{\partial\Gamma_{n}}{\partial
T}\delta T=\frac{\delta\mu}{\tau_{n\mu}}+\frac{\delta T}{\tau_{nT}}$ (5.25b)

If we want this to be consistent, then the linearized relaxations must satisfy
integrability conditions

$\frac{\partial}{\partial
T}\left(\frac{1}{\tau_{\varepsilon\mu}}\right)=\frac{\partial}{\partial\mu}\left(\frac{1}{\tau_{\varepsilon
T}}\right)\qquad\qquad\frac{\partial}{\partial
T}\left(\frac{1}{\tau_{n\mu}}\right)=\frac{\partial}{\partial\mu}\left(\frac{1}{\tau_{nT}}\right)$
(5.26)

because of the commutativity of the second derivatives of
$\Gamma_{\varepsilon}$ and $\Gamma_{n}$. This leads to

$\frac{\partial}{\partial\mu}\left(\frac{1}{\tau_{\varepsilon
T}}\right)=0\qquad\qquad\frac{\partial}{\partial\mu}\left(\frac{1}{\tau_{nT}}\right)=0$
(5.27)

thus, the most general $\tau_{nn}$ compatible with Onsager relations, second
law of thermodynamics and non-linear completion takes the form

$\frac{1}{\tau_{nn}(T,\mu)}=\frac{f(\mu)\frac{\partial\varepsilon}{\partial
T}}{\frac{\partial n}{\partial\mu}\frac{\partial\varepsilon}{\partial
T}-\frac{\partial n}{\partial T}\frac{\partial\varepsilon}{\partial\mu}}$
(5.28)

where $f$ is an arbitrary function of $\mu$. The remaining relaxations are
uniquely fixed in terms of $\tau_{nn}$ from (5.15) and (5.10).

###  5.3 Variational approach and Onsager relations

In the previous section we considered a charged relativistic fluid in the
presence of the most general linear relaxations, and found a set of very
generic constraints that the relaxation parameters must obey to satisfy
positivity of entropy production, Onsager relations and linear stability. Now
we investigate the same problem, but from the perspective of the variational
method, and give a prescription for how to obtain Onsager reciprocal Green
functions in flat spacetime using this approach.

To proceed we must find the current and stress-energy tensor which solve the
linearized equations of motion on a curved background
$g_{\mu\nu}=\eta_{\mu\nu}+\delta h_{\mu\nu}$ and with a gauge field
perturbation $\delta A_{\mu}$. The linearized equations of motion for an order
one fluid in the Landau frame then take the form (2.2.2)

$\displaystyle\tau^{\mu}\partial_{\mu}\delta\varepsilon+\left(\varepsilon_{0}+P_{0}\right)\nabla^{(0)}_{\mu}\delta
v^{\mu}=-\left(\frac{1}{\tau_{\varepsilon\varepsilon}}\delta\varepsilon+\frac{1}{\tau_{\varepsilon
n}}\delta n\right)$ (5.29a) $\displaystyle\tau^{\mu}\partial_{\mu}\delta
n+n_{0}\nabla^{(0)}_{\mu}\delta
v^{\mu}-\sigma\left(P^{\mu\nu}\nabla^{(0)}_{\mu}\partial_{\nu}\delta\mu-\frac{\mu_{0}}{T_{0}}P^{\mu\nu}\nabla^{(0)}_{\mu}\partial_{\nu}\delta
T-\nabla^{(0)}_{\mu}\delta E^{\mu}\right)$
$\displaystyle=-\left(\frac{1}{\tau_{n\varepsilon}}\delta\varepsilon+\frac{1}{\tau_{nn}}\delta
n\right)$ (5.29b) $\displaystyle P^{\mu\nu}\partial_{\nu}\delta P-\zeta
P^{\mu\nu}\nabla_{\nu}^{(0)}\nabla_{\rho}^{(0)}\delta
v^{\rho}+\left(\varepsilon_{0}+P_{0}\right)P^{\mu}_{\
\rho}\left(\tau^{\nu}\nabla_{\nu}^{(0)}\delta v^{\rho}\right)-2\eta
P^{\mu\rho}\nabla_{\sigma}^{(0)}\delta\sigma^{\sigma}_{\ \rho}$
$\displaystyle=-\frac{1}{\tau_{m}}\left(\varepsilon_{0}+P_{0}\right)\delta
v^{\mu}+n_{0}\delta E^{\mu}-P_{0}P^{\mu}_{\ \nu}\nabla_{\rho}^{(0)}\delta
h^{\rho\nu}-\left(\varepsilon_{0}+P_{0}\right)P^{\mu}_{\
\rho}\tau^{\nu}\delta\Gamma_{\nu\sigma}^{\rho}\tau^{\sigma}$ (5.29c)

where $P^{\mu\nu}=\eta^{\mu\nu}+\tau^{\mu}\tau^{\nu}$ is the projector
orthogonal to the background velocity
$\tau_{\mu}=(-1,\boldsymbol{\mathbf{0}})$ and $\nabla^{(0)}_{\mu}$ is the
covariant derivative computed with respect to the background metric (if, like
in the present case, we expand around Minkowski metric, then
$\nabla^{(0)}_{\mu}=\partial_{\mu}$).

As we did in the previous section, we also included decay terms on the RHS. We
remark that because the relaxation rates break Lorentz symmetry they cannot be
uniquely written in a covariant form. This means that, depending on the
arbitrary choice of non-linear covariantizations, it is possible to find
different source terms associated to the relaxations, i.e. terms in which a
source fluctuation, $\delta h_{\mu\nu}$ or $\delta A_{\mu}$, multiplies a
relaxation parameter. In order to avoid making this choice, in (5.29) we have
decided to keep the relaxation terms as they appear in flat spacetime, without
any possible source contribution.

One can then solve the above linear equations to express the fluctuations of
the hydrodynamic fields in terms of the sources, plug the results into the
definitions of the generators $\mathcal{J}^{\mu}$ and $\mathcal{T^{\mu\nu}}$
in (2.59) in the presence of (linear) background source fields, and compute
the complete set of Green functions using (2.60), reducing to flat spacetime,
zero gauge field at the end. Proceeding in this way, we find that not all
correlators are time-reversal covariant, and some of them violate Onsager
relations by terms explicitly dependent on the relaxation rates, e.g.

$\left.\langle T^{tt}T^{xx}\rangle-\langle
T^{xx}T^{tt}\rangle\right|_{\boldsymbol{\mathbf{k}}=0}=\\\
=-\frac{\left(\left(\varepsilon_{0}+P_{0}\right)\left(\tau_{n\varepsilon}\tau_{\varepsilon
n}-\tau_{nn}\tau_{\varepsilon\varepsilon}\right)\right)+i\tau_{nn}\tau_{n\varepsilon}\left((\varepsilon_{0}+P_{0})\tau_{\varepsilon
n}+n_{0}\tau_{\varepsilon\varepsilon}\right)\omega}{\tau_{nn}\tau_{\varepsilon\varepsilon}+\tau_{n\varepsilon}\tau_{\varepsilon
n}\left(i+\tau_{nn}\omega\right)\left(i+\tau_{\varepsilon\varepsilon}\omega\right)}$
(5.30)

The above expression is non-zero even after we impose (5.15) and (5.10) on the
relaxation parameters, furthermore some of the two-point functions are
different when computed from the variational and Martin-Kadanoff approach.

All these issues are related to the presence of relaxations in our
quasihydrodynamic model. Therefore, we modify the linearized theory (5.29) by
including the most general set of extra source terms. Because we want to make
contact with the flat spacetime, zero gauge field limit of the previous
section, we will take these extra source terms to be proportional to the
fluctuations $\delta h_{\mu\nu}$ and $\delta A_{\mu}$, so that they vanish in
the appropriate limit. Furthermore, because the relaxation terms contain only
ideal fluid data (the $\tau$ parameters are never multiplied by derivatives of
the hydrodynamic fields), we restrict ourselves to extra source terms without
any derivatives. Schematically, we write

$\text{sources of
\eqref{eqn:ch5:equations_of_motion_curved_space}}\quad\rightarrow\quad\text{sources
of \eqref{eqn:ch5:equations_of_motion_curved_space}}+c_{a}^{\mu\nu}\delta
h_{\mu\nu}+r_{a}^{\mu}\delta A_{\mu}$ (5.31)

and, with this new set of equations of motion, we compute again all the
correlators in the theory using the variational approach, which will now
depend on these extra coefficients $c_{a}^{\mu\nu}$ and $r_{a}^{\mu}$. In the
expression above we take $a=(\varepsilon,n,x,y,z)$ to identify the
corresponding equation of motion (energy, charge or momentum along $x$, $y$ or
$z$).

From the above prescription we add in total 70 source terms, 14 for each
equation, however we can use symmetry arguments to reduce the number of
independent extra source terms. Specifically, our theory does not have any
scalar parameter that breaks parity, hence all the $c_{a}^{\mu\nu}$ and
$r_{a}^{\mu}$ must be parity even. Furthermore, the background flat spacetime
theory has rotational invariance, which, together with $\mathcal{P}$-symmetry,
allows us to impose parity with respect to a single axis
$\mathcal{P}_{i}:i\rightarrow-i$, with $i=x,y,z$ and reduce the number of non-
zero coefficients to only 13. Finally, because we take $\tau_{m}$ to be
isotropic, we also assume isotropy of the sources too, which reduces the total
number of independent non-vanishing coefficients to 9.

We can now take the full set of Green functions, that depends on these extra
parameters $c_{a}^{\mu\nu}$ and $r_{a}^{\mu}$, and impose Onsager reciprocal
relations (5.4), trying to fix the free coefficients in such a way that the
correlators preserve time-reversal covariance of the microscopic theory. The
expressions are very large, for this reason we first impose Onsager relations
at $\boldsymbol{\mathbf{k}}=0$ and then at $\omega=k_{y}=k_{z}=0$. We found
that this is enough to fix all nine parameters in terms of the relaxations
$\tau$s, and subsequently we checked that these expressions lead to Green
functions which obey Onsager relations at arbitrary $\omega$ and
$\boldsymbol{\mathbf{k}}$.

Of the nine extra source terms, we found that five of them are zero, while the
non-vanishing ones are

energy:
$\displaystyle-\left(\frac{\delta\varepsilon}{\tau_{\varepsilon\varepsilon}}+\frac{\delta
n}{\tau_{\varepsilon n}}\right)-c_{\varepsilon}^{tt}\delta
h_{tt}-r^{t}_{\varepsilon}\delta A_{t}$ (5.32a) charge:
$\displaystyle-\left(\frac{\delta\varepsilon}{\tau_{n\varepsilon}}+\frac{\delta
n}{\tau_{nn}}\right)-c_{n}^{tt}\delta h_{tt}-r^{t}_{n}\delta A_{t}$ (5.32b)

The value of the non-vanishing extra coefficients is given in terms of the
susceptibilities and relaxation rates

$\displaystyle c_{\varepsilon}^{tt}$
$\displaystyle=\frac{1}{2}\left(\frac{\chi_{\varepsilon n}}{\tau_{\varepsilon
n}}+\frac{\chi_{\varepsilon\varepsilon}}{\tau_{\varepsilon\varepsilon}}\right)$
(5.33a) $\displaystyle r^{t}_{\varepsilon}$
$\displaystyle=\frac{\chi_{\varepsilon\varepsilon}}{\tau_{n\varepsilon}}+\frac{\chi_{\varepsilon
n}}{\tau_{nn}}=2c_{n}^{tt}$ (5.33b) $\displaystyle r_{n}^{t}$
$\displaystyle=\frac{\chi_{\varepsilon
n}}{\tau_{n\varepsilon}}+\frac{\chi_{nn}}{\tau_{nn}}$ (5.33c)

With these corrections the correlators computed using the variational approach
agree with the ones obtained via Martin-Kadanoff, up to the usual contact
terms. Furthermore, we have not found other constraints on the relaxation
parameters other than the one found already in the Martin-Kadanoff framework
(5.10).

It is worth pointing out that the value of the coefficients is not important,
what matters is that the final equations of motion (5.32) have the right form
after we substitute in the explicit values of $r_{a}^{\mu}$ and
$c_{a}^{\mu\nu}$. In particular, we explained above that we decided to write
the relaxation terms as they appear in flat space, agnostic on possible
covariant completions. However, starting from a specific covariant expression
for the relaxation terms, we could find relaxations-dependent source terms
already in the undeformed equations of motion (5.29). Then, adding all
possible relaxations as in (5.32), we expect different values for the
coefficients $r_{a}$ and $c_{a}$, but such that they lead to the same final
result.

Finally, we can also consider positivity of entropy production for the
linearized hydrodynamic theory with background source, particularly taking
into account the new metric and gauge field fluctuation terms discovered. In
this case, the divergence of the entropy current gives

$\displaystyle T_{0}\nabla_{\mu}^{(0)}\delta S^{\mu}$
$\displaystyle=\delta\varepsilon\left(\frac{\mu_{0}}{\tau_{n\varepsilon}}-\frac{1}{\tau_{\varepsilon\varepsilon}}\right)+\delta
n\left(\frac{\mu_{0}}{\tau_{nn}}-\frac{1}{\tau_{\varepsilon n}}\right)$
$\displaystyle\quad-\left(r_{\varepsilon}^{t}-\mu_{0}r_{n}^{t}\right)\delta
A_{t}-\left(c_{\varepsilon}^{tt}-\mu_{0}c_{n}^{tt}\right)\delta
h_{tt}+\mathcal{O}(\partial^{2},\delta^{2})$ (5.34a)

However, it is easy to check that with the expressions in (5.33), positivity
of entropy production is satisfied if the relaxation rates obey the Martin-
Kadanoff constraints (5.15), specifically the new terms proportional to
$\delta A_{t}$ and $\delta h_{tt}$ in the equation above identically vanish
and do not lead to further constraints.

###  5.4 Summary, discussion and outlook

In this chapter we studied relativistic hydrodynamics in the presence of
relaxation terms in the linearized regime (5.6). We added all the possible
relaxations up to order zero in derivatives and order one in fluctuations, and
computed all the constraints the relaxation rates must satisfy that arise from
microscopic time-reversal symmetry (5.10), positivity of entropy production
(5.15) and linear stability (5.21).

Subsequently, we included the same set of relaxations in a theory placed on a
weakly curved spacetime $\delta h_{\mu\nu}$ and in the presence of a linear
gauge field $\delta A_{\mu}$. The response functions computed from the
variational approach in general are not Onsager reciprocal and differ from the
one obtained from the Martin-Kadanoff procedure. To amend the problem, we
added all the possible terms linear in the external sources to the equations
of motion (5.31), each multiplied with a different unknown coefficient. What
we found is that the simple requirement of time-reversal invariance of the
theory allows us to uniquely fix all the newly added transport coefficients in
terms of relaxation rates and susceptibilities (5.33).

Although we tested this approach only for a relativistic charged fluid on a
flat spacetime without background magnetic fields, we expect the procedure
outlined in Section 5.3 to hold very generally, for fluids with different
symmetry content (boost-agnostic, Carrollian, Galilean, …), with different
degrees of freedom (such as extra scalar fields), and for different
backgrounds (curved spacetime or constant magnetic fields).

The outlook for this paper is closely connected to the previous Chapter 4: it
would be interesting to study the generalized relaxations we introduced in a
broader context, particularly looking for realizations beyond the linearized
regime. Then, we could try to understand if the relaxation can enter the
constitutive relations by modifying certain terms as in [2], inducing new
transport coefficients [49] or renormalizing existing ones [157].

Finally, in the next chapter we are going to use certain results obtained here
to study the anomalous DC transport in Weyl semimetals. We will see how
generalized relaxations can be obtained from a kinetic theory perspective,
however it remains an open question how to arrive at the same result from
holography or Schwinger-Keldysh formalism.

##  Chapter 6 Anomalous hydrodynamics and Weyl semimetals

> “I usually solve problems by letting them devour me.”
>
> Franz Kafka, _Letter to Max Brod_

###  6.1 Introduction

One of the most peculiar features of quantum field theories is the presence of
anomalies: the breakdown of some classical symmetry due to quantum
fluctuations. One of the most well-known examples is the so-called
chiral/axial anomaly (or ABJ anomaly, after Adler, Bell and Jackiw [267,
268]), namely the classical system enjoys a chiral symmetry, so that left- and
right-handed fermions are separately conserved, but the quantum system is
anomalous and there is mixing between left- and right-handed particles in the
presence of external electromagnetic fields [269, 270, 271].

More than a decade ago, following hints from holography [272, 273], it was
discovered that quantum anomalies could leave an imprint on the macroscopic
dynamics of fluids, not only by modifying the equations of motion for the
anomalous current, but also by allowing for extra transport coefficients in
the constitutive relations [34, 274, 275]. The anomalous terms are non-
dissipative and parity-odd, since they are proportional to the external
magnetic fields and vorticity. See some of the reviews [271, 276, 270] and
original works [34, 277, 35, 278, 279] for more details, also about the role
of the mixed-gravitational anomaly and the generating functional.

What was quickly realized is that anomalous hydrodynamics could play an
important role, not only for the description of heavy-ion collisions, but also
to understand the transport properties of Weyl semimetals (WSMs) [16]. WSMs
are a novel class of $3d$ topological quantum materials that host gapless
chiral excitations. In these compounds the valence and conductance bands have
linear crossing points (the Weyl nodes) close to the Fermi surface, and the
system enjoys an accidental chiral symmetry, such that the Weyl nodes must
always come in pairs of opposite chirality [280, 281]. If the Weyl cones lie
on top of each other, then the material is a Dirac semimetal, but in the
presence of parity or time-reversal symmetry breaking the cones will be
separated in momentum space, in which case we speak of a WSM. Because of their
chiral symmetry, many of their properties are strongly influenced by the
presence of the axial anomaly, thus allowing us to study QFT anomalies in a
controlled tabletop setting, contrary to what happens with heavy-ion-
collisions experiments. For some reviews on WSMs, see e.g., [282, 283, 284].

Figure 6.1: Figure taken from [285]. Observation of longitudinal NMR in a
sample of doped $\mathrm{NbP}$ Weyl semimetal.

In particular, one of the most prominent features expected in WSMs due to the
anomaly is a large longitudinal negative magnetoresistance (NMR), first argued
in [286]. This means that, contrary to ordinary metals in which the
conductivity decreases with the applied magnetic field [287], in WSMs we
expect the longitudinal electrical conductivity (measured parallel to the
applied external magnetic field) to be strongly enhanced by the presence of a
background electric field. Indeed, longitudinal NMR has been observed both for
the electric and thermal transport in different compounds [285, 288, 289, 290,
291, 292]. In practice, real materials can have multiple source of
longitudinal NMR due to other effects, and distinguishing the various
contributions from experiments is generally difficult.

The NMR can be predicted from many different approaches, based on
semiclassical models (such as diffusion or chiral kinetic theory [293]) or on
microscopic quantum models [294, 17, 295, 296, 297, 298, 299, 300, 301], but
also from hydrodynamic arguments [16, 260, 257]. Furthermore, like other Dirac
materials, WSMs could enjoy an extended hydrodynamic regime, as observed in
[144, 154, 132]. Consequently, anomalous hydrodynamics might apply reasonably
well to study the macroscopic transport dynamics of these systems.

In this chapter, we are going to discuss some questions related to anomalous
fluids and possible applications to models of WSMs. First, following [4], we
review the usual computations of the anomalous optical thermoelectric
conductivities and show that the NMR obtained in previous works is not
physical, since that different hydrodynamic frames lead to different results.
Specifically, the NMR depends quadratically on the applied magnetic field,
which means that order-one hydrodynamics with an order-one in derivatives
magnetic field is not appropriate to study the magnetic-field dependence of
the conductivities. We discuss this issue and suggest a way to properly
compute the NMR in Section 6.2.

Next, in Section 6.3 we discuss some problems related to the DC values of the
thermoelectric conductivities. We already discussed in the previous chapter
that hydrodynamics leads to finite DC conductivities only if there is some
effective mechanism to relax momentum [14], but in the presence of an axial
anomaly energy and axial charge should relax too for the conductivities to be
finite in the $\omega\rightarrow 0$ limit [260, 257]. However, we show that
previous models of anomalous relaxed hydrodynamics necessarily predict that
electric charge, energy and momentum should all relax at the same rate of the
axial charge, which we deem unphysical. Therefore, to build a better model, we
employ the generalized relaxations introduced in Chapter 5 and obtain a theory
of relaxed hydrodynamics which conserves the electric charge of the system.

Finally, in Section 6.4, we show how these relaxations can be obtained from
kinetic theory, using an appropriate Relaxation-Time-Approximation (RTA)
ansatz for the scattering of electrons with impurities and phonons in which
the microscopic relaxation rate depends on the quasiparticle energy.

###  6.2 Anomalous hydrodynamics and transport

Remember that hydrodynamics, as an effective field theory, presents an
ambiguity in the choice of the hydrodynamic variables at derivative order, see
Section 2.1.4. By changing the definitions of temperature $T$, fluid velocity
$u^{\mu}$ and chemical potential $\mu$, we can change how the constitutive
relations look. This choice however is not physical, but a simple re-
parametrization: all frames, in the hydrodynamic regime, should give the same
values for any observable, therefore the conductivities, i.e. the retarded
Green functions, should take the same form in all hydrodynamic frames.
Nevertheless, the choice of frame can have huge impact on certain aspects of
the theory (such as stability and causality) at high-frequency/high-momenta,
see the discussion in the Introduction 1.

In this section we consider order-one hydrodynamics with a $\mathrm{U(1)}$
anomalous chiral current, and we show that the optical thermoelectric
conductivities computed in the presence of a background order-one magnetic
field $B\sim\mathcal{O}(\partial)$ are explicitly frame-dependent, which means
that the contribution of the anomaly to the conductivities is not a physical
effect and should be discarded.

To resolve this problem, one could work with hydrodynamics at order two in
derivatives, however this would just shift the issue to the
$\mathcal{O}(B^{3})$ subleading corrections to the conductivities. Thus, to
avoid the issue altogether, we suggest working with an external magnetic field
which is order zero in derivatives $B\sim\mathcal{O}(1)$, so that the
expressions for the conductivities can be trusted at any value of $B$ without
resorting to order-two hydrodynamics.

####  6.2.1 Standard approach to anomalous hydrodynamics

We consider now a hydrodynamic theory with a conserved stress-energy tensor
and a single $\mathrm{U(1)}$ anomalous chiral current in $(3+1)$-dimensional
spacetime. For the moment, we will not consider the effects of the
gravitational anomaly or other setups, however in later sections we will
instead consider a theory with symmetry
$\mathrm{U(1)}_{V}\times\mathrm{U(1)}_{A}$, which is the relevant symmetry
content to describe WSMs.

The equations of hydrodynamics for a covariant anomalous current111For a short
discussion on covariant and consistent anomalies we refer to Appendix B. and
stress-energy tensor in an external electromagnetic field are

$\displaystyle\partial_{\mu}T^{\mu\nu}$
$\displaystyle=F^{\nu\lambda}J_{\lambda}$ (6.1a)
$\displaystyle\partial_{\mu}J^{\mu}$ $\displaystyle=cE^{\mu}B_{\mu}$ (6.1b)

where $c$ is the anomaly coefficient and its explicit value depends on the
microscopic theory. For the moment, following the standard approach [34, 275,
277], we are considering both $E_{\mu}$ and $B_{\mu}$ to be order one in
derivatives $\mathcal{O}(\partial)$.

To write the constitutive relations for the anomalous fluid we will follow the
original work [34]. While there are now other methods, based on the
equilibrium generating functional [35, 277, 302], which are easier to
generalize to arbitrary dimensions or different anomalies (while also
providing insights on the appearance of the gravitational anomaly at order one
in derivatives [278]), the original approach gives a clearer picture on the
issue of hydrodynamic frames.

To begin, consider the constitutive relations in the Landau frame, and add the
most generic parity-odd terms which are order one in derivatives and that
depend on the external magnetic field $B^{\mu}$ and vorticity
$\Omega^{\mu}=\epsilon^{\mu\nu\rho\sigma}u_{\nu}\partial_{\rho}u_{\sigma}$.
Thus, we arrive at

$\displaystyle T^{\mu\nu}$ $\displaystyle=\varepsilon
u^{\mu}u^{\nu}+P\Delta^{\mu\nu}+\xi^{\varepsilon}_{B}\left(u^{\mu}B^{\nu}+u^{\nu}B^{\mu}\right)+\xi^{\varepsilon}_{\Omega}\left(u^{\mu}\Omega^{\nu}+u^{\nu}\Omega^{\mu}\right)$
$\displaystyle\quad-\eta\Delta^{\mu\alpha}\Delta^{\nu\beta}\sigma_{\alpha\beta}-\zeta\Delta^{\mu\nu}\partial_{\alpha}u^{\alpha}+\mathcal{O}(\partial^{2})$
(6.2a) $\displaystyle J^{\mu}$
$\displaystyle=nu^{\mu}+\sigma\Delta^{\mu\nu}\left(E_{\nu}-T\partial_{\nu}\frac{\mu}{T}\right)+\xi_{\Omega}\Omega^{\mu}+\xi_{B}B^{\mu}+\mathcal{O}(\partial^{2})$
(6.2b)

where $\sigma_{\mu\nu}$ is the shear tensor. The quantities $\xi$ are the
anomalous parity-odd dissipationless transport coefficients. Like $\sigma$,
$\zeta$ and $\eta$ their values depend on the order-zero thermodynamic
variables $\xi(\mu,T)$, however, unlike the standard dissipative transport
coefficients, this dependence is almost entirely fixed by the anomaly and the
choice of hydrodynamic frame.

From a physical perspective, the $\xi_{\Omega}$ and $\xi_{B}$ terms represent
the well-known Chiral Vortical Effect, Chiral Magnetic Effect (or Chiral
Separation Effect, if multiple currents are present) discussed in [303, 304,
279].

####  6.2.2 Positivity of entropy production

The constitutive relations must be constrained by the second law of
thermodynamics. To begin, we write the total entropy current $S^{\mu}$ as the
sum of the canonical entropy current (that is related to the standard non-
anomalous fluid) and a second contribution $S^{\mu}_{\text{n.c.}}$ that is
related to the anomaly

$S^{\mu}=\frac{1}{T}\left(Pu^{\mu}-T^{\mu\nu}u_{\nu}-\mu
J^{\mu}\right)+S^{\mu}_{\text{n.c.}}$ (6.3)

This second term is needed to ensure positivity of entropy production holds,
and we can express it in a generic form as

$S^{\mu}_{\text{eq}}=\xi^{s}_{B}B^{\mu}+\xi^{s}_{\Omega}\Omega^{\mu}$ (6.4)

Subsequently, we can compute the divergence of the entropy current, using the
equations of motion and the constitutive relations in the usual way, to arrive
at the following expression

$\displaystyle\partial_{\mu}S^{\mu}$
$\displaystyle=\frac{\zeta}{T}\theta^{2}+\frac{\eta}{T}\sigma_{\mu\nu}\sigma^{\mu\nu}+\sigma\left(E_{\mu}-T\partial_{\mu}^{\perp}\frac{\mu}{T}\right)^{2}$
$\displaystyle\quad+\left(\frac{\partial\xi^{s}_{B}}{\partial
T}+\frac{\mu}{T}\frac{\partial\xi^{s}_{B}}{\partial\mu}-\frac{\xi^{\varepsilon}_{B}}{T^{2}}\right)B^{\mu}\left(\partial_{\mu}^{\perp}T+Ta_{\mu}\right)$
$\displaystyle\quad+\left(\frac{\partial\xi^{s}_{\Omega}}{\partial
T}+\frac{\mu}{T}\frac{\partial\xi^{s}_{\Omega}}{\partial\mu}-\frac{\xi^{\varepsilon}_{\Omega}}{T^{2}}\right)\Omega^{\mu}\left(\partial_{\mu}^{\perp}T+Ta_{\mu}\right)$
$\displaystyle\quad-\left(\frac{\partial\xi^{s}_{B}}{\partial\mu}-\frac{\xi_{B}}{T}\right)B^{\mu}\left(E_{\mu}-T\partial_{\mu}^{\perp}\left(\frac{\mu}{T}\right)\right)$
$\displaystyle\quad-\left(\frac{\partial\xi^{s}_{\Omega}}{\partial\mu}-\frac{\xi_{\Omega}}{T}\right)\Omega^{\mu}\left(E_{\mu}-T\partial_{\mu}^{\perp}\left(\frac{\mu}{T}\right)\right)$
$\displaystyle\quad+\left(\xi^{s}_{B}-T\frac{\partial\xi^{s}_{B}}{\partial
T}-\mu\frac{\partial\xi^{s}_{B}}{\partial\mu}\right)B\cdot
a+\left(2\xi^{s}_{\Omega}-T\frac{\partial\xi^{s}_{\Omega}}{\partial
T}-\mu\frac{\partial\xi^{s}_{\Omega}}{\partial\mu}\right)\Omega\cdot a$
$\displaystyle\quad+\left(\frac{\partial\xi^{s}_{B}}{\partial\mu}-\frac{c\mu}{T}\right)E\cdot
B+\left(\frac{\partial\xi^{s}_{\Omega}}{\partial\mu}-\xi^{s}_{B}\right)E\cdot\Omega\geq
0$ (6.5)

where we defined $\partial_{\mu}^{\perp}=\Delta_{\mu\nu}\partial^{\nu}$.

When studying the positivity of entropy production, we can encounter two
different kinds of conditions: inequality type constraints, which are related
to the terms with positive definite signature in the first line, thus
requiring $\sigma,\zeta,\eta\geq 0$, and equality type constraints, which
originate from terms without definite sign and therefore their coefficients
must vanish. This is for example what we found in Section 2.1 when discussing
$\chi_{T}$, see (2.25).

We can employ the equations of motion of the ideal fluid to relate the
divergences of the vorticity and the magnetic field to other quantities as

$\displaystyle\partial_{\mu}\Omega^{\mu}$
$\displaystyle=-\frac{2}{\varepsilon+P}\Omega^{\mu}\left(\partial_{\mu}P-nE_{\mu}\right)$
(6.6a) $\displaystyle\partial_{\mu}B^{\mu}$
$\displaystyle=-2\Omega^{\mu}E_{\mu}+\frac{1}{\varepsilon+P}\left(nE_{\mu}B^{\mu}-B^{\mu}\partial_{\mu}P\right)$
(6.6b)

which we can use to simplify the constraints that stem from the second law of
thermodynamics. The requirement that $\partial_{\mu}S^{\mu}\geq 0$ means that
we should set to zero all the anomaly-related terms, which corresponds to

$\displaystyle 0$
$\displaystyle=\partial_{\mu}\xi^{s}_{\Omega}-\frac{2\partial_{\mu}P}{\varepsilon+P}\xi^{s}_{\Omega}-\xi_{\Omega}\partial_{\mu}\frac{\mu}{T}+\left(\frac{2\partial_{\mu}P}{\varepsilon+P}-a_{\mu}-\partial_{\mu}\right)\xi^{\varepsilon}_{\Omega}$
(6.7a) $\displaystyle 0$
$\displaystyle=\partial_{\mu}\xi^{s}_{B}-\frac{\partial_{\mu}P}{\varepsilon+P}\xi^{s}_{B}-\xi_{B}\partial_{\mu}\frac{\mu}{T}+\left(\frac{\partial_{\mu}P}{\varepsilon+P}-a_{\mu}-\partial_{\mu}\right)\xi^{\varepsilon}_{B}$
(6.7b) $\displaystyle 0$
$\displaystyle=\frac{2n\xi^{s}_{\Omega}}{\varepsilon+P}-2\xi^{s}_{B}+\frac{\xi_{\Omega}}{T}+2\xi^{\varepsilon}_{B}-2\xi^{\varepsilon}_{\Omega}\frac{n}{\varepsilon+P}$
(6.7c) $\displaystyle 0$
$\displaystyle=\frac{n\xi^{s}_{B}}{\varepsilon+P}+\frac{\xi_{B}}{T}-c\frac{\mu}{T}-\xi^{\varepsilon}_{B}\frac{n}{\varepsilon+P}$
(6.7d)

The above set of equations is not closed: there are six different $\xi$s, but
only four equations. This is because we kept $\xi^{\varepsilon}_{B}$ and
$\xi^{\varepsilon}_{\Omega}$ non-zero in the above expressions, contrary to
what is done in [34]. It is then clear that we need some other constraints to
uniquely determine the $\xi$s, which come from the choice of frame. In
particular, the Landau frame matching conditions require us to set
$\xi^{\varepsilon}_{B}=\xi^{\varepsilon}_{\Omega}=0$, then we can integrate
the above differential equations to obtain, up to integration constants
related to the gravitational anomaly and CPT-violating terms [275], the
following expressions

$\displaystyle\xi^{\varepsilon}_{B}$ $\displaystyle=0$ $\displaystyle\xi_{B}$
$\displaystyle=c\left(\mu-\frac{1}{2}\frac{n\mu^{2}}{\varepsilon+p}\right)$
(6.8a) $\displaystyle\xi^{\varepsilon}_{\Omega}$ $\displaystyle=0$
$\displaystyle\xi_{\Omega}$
$\displaystyle=c\left(\mu^{2}-\frac{2}{3}\frac{n\mu^{3}}{\varepsilon+p}\right)$
(6.8b)

in agreement with [34].

Another possibility is to look for a frame in which the entropy current
receives no corrections [305], or one might prefer to work in the
thermodynamic frame, that keeps the definition of temperature, fluid velocity
and chemical potential the canonical ones (2.30).

To move from one frame to another, as we discussed in Section 2.1.4, we have
to change the definitions of the fluid fields by derivative corrections which
vanish in equilibrium. For the purpose of this discussion, i.e. frame
transformations that shift around the anomaly-related terms, the only frame
transformations we are interested in are of the form

$u^{\mu}\rightarrow u^{\mu}+f_{B}(\mu,T)B^{\mu}+f_{\Omega}(\mu,T)\Omega^{\mu}$
(6.9)

which act on the constitutive relations by changing the values of the
anomalous coefficients as

$\displaystyle\xi^{\varepsilon}_{B,\Omega}$
$\displaystyle\rightarrow\xi^{\varepsilon}_{B,\Omega}+(\varepsilon+P)f_{B,\Omega}(\mu,T)$
(6.10a) $\displaystyle\xi_{B,\Omega}$
$\displaystyle\rightarrow\xi_{B,\Omega}+nf_{B,\Omega}(\mu,T)$ (6.10b)

Notice that the above frame transformations are allowed because both $B^{\mu}$
and $\Omega^{\mu}$ are $\mathcal{O}(\partial)$ and orthogonal to $u^{\mu}$. In
particular, we can use

$f_{B}=\frac{c\mu^{2}}{2\left(\varepsilon+P\right)}\qquad\qquad
f_{\Omega}=\frac{c\mu^{3}}{3\left(\varepsilon+P\right)}$ (6.11)

to move between the constitutive relations in the Landau frame (6.8) and the
thermodynamic frame, in which the anomalous transport coefficients take the
simplest form

$\displaystyle\xi^{\varepsilon}_{B}$ $\displaystyle=\frac{1}{2}c\mu^{2}$
$\displaystyle\xi_{B}$ $\displaystyle=c\mu$ (6.12a)
$\displaystyle\xi^{\varepsilon}_{\Omega}$ $\displaystyle=\frac{1}{3}c\mu^{3}$
$\displaystyle\xi_{\Omega}$ $\displaystyle=\frac{1}{2}c\mu^{2}$ (6.12b)

####  6.2.3 Frame-dependent conductivities

Having discussed the constitutive relations, we now study how the choice of
frame affects the value of conductivities, at order one in derivatives. To
proceed, we can analyse more carefully the general argument from Section 2.2
that is used to solve for the retarded Green functions in hydrodynamics.
Consider the linearized hydrodynamic equations for a fluid order one in
derivatives, schematically

$D_{ab}\phi_{b}=\lambda_{b}+\mathcal{O}(\partial^{3})$ (6.13)

where $D_{ab}$ is an operator which is at least order one in derivatives,
$\phi_{a}$ is a vector of the fluctuating hydrodynamic fields
$\phi_{a}=\left(\delta T,\delta\mu,\delta v^{i}\right)$, and $\lambda_{a}$ are
the external sources, like gauge fields, metric fluctuations, but also their
derivatives. If we consider order-one fluids, then both $\lambda_{a}$ and
$D_{ab}$ contain, at most, terms with two derivatives.

In Fourier space we can introduce a formal parameter $\epsilon$ to count the
derivative order of the various terms. With the setup we are using we have
$\omega,\boldsymbol{\mathbf{k}},B\sim\epsilon$. Then, a generic Green function
is obtained by inverting the operator $D$ and extracting the corresponding
entry on the RHS (2.55), so that schematically we arrive at

$G^{R}\sim\frac{a(\omega,\boldsymbol{\mathbf{k}})+\mathcal{O}(\epsilon^{2})}{b(\omega,\boldsymbol{\mathbf{k}})+\mathcal{O}(\epsilon^{3})}$
(6.14)

where $a$ and $b$ are functions that depend on the correlator of interest. The
numerator is thus determined only up to order one in the counting parameter
$\epsilon$, as a consequence of the fact that the constitutive relations for
the currents are known only up to order one in derivatives.

We can study the consequences of this argument on the anomalous transport:
consider the constitutive relations for the chiral fluid in a generic frame
(6.2). To compute the longitudinal magnetoresistance we need to select a
background with non-zero magnetic field $B$, which we take along $\hat{z}$,
$F^{12}=B$, moreover we will have $T$ and $\mu$ constant and zero spatial
velocity. With this choice of background the equilibrium configuration for the
fluid is given by

$\displaystyle T^{\mu\nu}$
$\displaystyle=\begin{pmatrix}\varepsilon&0&0&\xi_{B}^{\varepsilon}B\\\
0&P&0&0\\\ 0&0&P&0\\\ \xi_{B}^{\varepsilon}B&0&0&P\end{pmatrix}$ (6.15a)
$\displaystyle J^{\mu}$ $\displaystyle=\left(n,0,0,\xi_{B}B\right)$ (6.15b)

We can now add linearized fluctuations to the hydrodynamic fields and sources
$F^{0z}=\delta\mathbb{E}^{z}$, and solve the equations of motion (6.1) in
Fourier space to obtain the optical conductivities using the Martin-Kadanoff
method as discussed in Section 2.2.1.

Generally, we find that the fluctuations of the hydrodynamic fields depend on
the sources as, e.g.

$\delta
u_{z}\bigr{|}_{\boldsymbol{\mathbf{k}}=\boldsymbol{\mathbf{0}}}=\left[\frac{-in+\epsilon^{2}\alpha
B^{2}+\mathcal{O}(\epsilon^{3})}{(\varepsilon+p)\omega\epsilon+\mathcal{O}(\epsilon^{3})}\right]\delta\mathbb{E}_{z}$
(6.16)

with $\alpha$ some thermodynamic function. From the argument above, we know
that the $\mathcal{O}(\epsilon^{2})$ dependence on the numerator should be
discarded: it is not physical and is an artefact of our equations and choice
of the background.

Indeed, we can compute the leading-order dependence on the magnetic field of
the longitudinal electric conductivity (similar results hold for the thermal
and thermoelectric ones too) and check directly that this expression depends
explicitly on the hydrodynamic frame, proving that it is not physical

$\displaystyle\sigma(\omega)$ $\displaystyle=\sigma+\frac{in^{2}}{\omega
w}+\frac{iB^{2}}{\omega w^{2}(\frac{\partial\varepsilon}{\partial
T}\frac{\partial
n}{\partial\mu}-\frac{\partial\varepsilon}{\partial\mu}\frac{\partial
n}{\partial
T})}\left[\left(w\frac{\partial\xi_{B}}{\partial\mu}-n\frac{\partial\xi_{B}^{\varepsilon}}{\partial\mu}\right)\right.$
$\displaystyle\quad\left(w\left(c\frac{\partial\varepsilon}{\partial
T}-\frac{\partial n}{\partial
T}\xi_{B}\right)-\frac{\partial\varepsilon}{\partial
T}n\xi_{B}+2\frac{\partial n}{\partial
T}n\xi_{B}^{\varepsilon}\right)-\left(w\frac{\partial\xi_{B}}{\partial
T}-n\frac{\partial\xi_{B}^{\varepsilon}}{\partial T}\right)$
$\displaystyle\left.\quad\left(w\left(c\frac{\partial\varepsilon}{\partial\mu}-\frac{\partial
n}{\partial\mu}\xi_{B}\right)-\frac{\partial\varepsilon}{\partial\mu}n\xi_{B}+2\frac{\partial
n}{\partial\mu}n\xi_{B}^{\varepsilon}\right)\right]$ (6.17)

where $w=\varepsilon+P$ is the enthalpy density. The $\mathcal{O}(B^{2})$ term
is explicitly frame dependent, as it depends on the values of the anomalous
transport coefficients $\xi$s and their derivatives.

We can compute the above formula in the Landau frame at
$\xi^{\varepsilon}_{B}=0$ to recover the results of [260], or we can plug in
the thermodynamic-frame transport coefficients (6.12) to get a different
result [257].

Again, we emphasize that this observed frame dependence, and thus the $B^{2}$
correction in the conductivity, are the signature that we are incorrectly
keeping terms which are higher order in derivatives than our constitutive
relations allow us to. The $B^{2}$ term should be discarded, it is not
physical, which means that the conductivities do not depend on the anomaly at
all, at this order in derivatives.

There are other issues with the thermoelectric conductivities computed in
different frames at order one in derivatives, which hint to the fact that the
magnetic-field dependent terms should be ignored. For example, in general the
correlators are not Onsager reciprocal $\alpha\neq\bar{\alpha}$, moreover
computing the conductivities using the gradient of the chemical potential as a
source, instead of $\delta\mathbb{E}_{z}$, leads to different results.

Notice that in anomalous hydrodynamics the equilibrium around which we
fluctuate also depend on the frame (6.15), while usually the equilibrium
configuration is frame independent. Although even in standard hydrodynamics
response functions computed in different frames can lead to apparently non-
equivalent expressions, they can always be related to one another by mapping
the transport coefficients. In the anomalous case (6.2.3) such map does not
exist, because the parity-odd anomalous transport coefficients take a specific
value for each given frame, see (6.8) and (6.12).

####  6.2.4 Killing the ambiguity

It should be clear by now that the longitudinal conductivities in order-one
anomalous hydrodynamics do not have any dependence on the magnetic field. To
restore the NMR, i.e. to observe the effect of the anomaly on transport, one
way is to work at order two in derivatives [257, 306, 307, 308]. Then the
$\mathcal{O}(B^{2})$ term in the conductivity is an honest physical effect and
can be trusted, however the same ambiguity would now arise at
$\mathcal{O}(B^{3})$.

To avoid these problems altogether we suggest to simply consider the magnetic
field to be order zero in derivatives $B\sim\mathcal{O}(1)$, so that it is an
integral part of the thermodynamics. With this choice, we can study the
anomalous transport without resorting to order-two hydrodynamics, and
furthermore we are allowed to trust our results at large values of $B$, since
we do not have to truncate the results at leading order in $B^{2}$. Morally,
we can think of this procedure as a partial re-summation at all orders of the
hydrodynamic theory with $B\sim\mathcal{O}(\partial)$ (partial in the sense
that we are not including derivatives of $B$ in the re-summation).

When $B\sim\mathcal{O}(1)$ it cannot be used in frame transformations (6.9),
then the equilibrium configuration on a background with a fixed magnetic field
will be unique for all frames and the conductivities will be well-defined.

A theory with order-zero magnetic field has been studied in [179] for the case
of a single $\mathrm{U(1)}$ axial current. We can then start from their
constitutive relations to compute the thermoelectric transport of an anomalous
fluid. We will focus only on the ideal fluid: on one hand, because the
expressions are simpler (and even the conductivities of an ideal fluid can
quickly become complicated with relaxation terms, as we will see), on the
other hand, because this is already enough to observe anomalous transport
phenomena.

We decompose the stress-energy tensor and the current with respect to
$u^{\mu}$ in the usual way (2.4)

$\displaystyle T^{\mu\nu}$
$\displaystyle=\mathcal{E}u^{\mu}u^{\nu}+\mathcal{P}\Delta^{\mu\nu}+\mathcal{Q}^{\mu}u^{\nu}+\mathcal{Q}^{\nu}u^{\mu}+\mathcal{T}^{\mu\nu}$
(6.18a) $\displaystyle J^{\mu}$
$\displaystyle=\mathcal{N}u^{\mu}+\mathcal{J}^{\mu}$ (6.18b)

and, because we are interested in the ideal fluid, we can use the hydrostatic
generating functional. We define the hydrodynamic fields in terms of the
metric, gauge field and Killing vector $V^{\mu}$ in agreement with (2.30),
which at order one and higher defines the thermodynamic frame.

The non-anomalous part of the constitutive relations can be obtained from the
standard generating functional

$W[g,A]=\int\mathop{}\\!\mathrm{d}^{3+1}x\ \sqrt{-g}P(T,\mu,B^{2})$ (6.19)

where the equilibrium pressure now depends also on $B^{2}$. Varying with
respect to metric and gauge fluctuations as explained in Section 2.1.7, we
arrive at the constitutive relations

$\displaystyle\mathcal{E}$ $\displaystyle=-P+sT+\mu n$ (6.20a)
$\displaystyle\mathcal{P}$ $\displaystyle=P-\frac{2}{3}\chi_{B}B^{2}$ (6.20b)
$\displaystyle\mathcal{Q}^{\mu}$
$\displaystyle=-\chi_{B}\epsilon^{\mu\nu\rho\sigma}u_{\nu}E_{\rho}B_{\sigma}$
(6.20c) $\displaystyle\mathcal{T}^{\mu\nu}$
$\displaystyle=\chi_{B}\left(B^{\mu}B^{\nu}-\frac{1}{3}\Delta^{\mu\nu}B^{2}\right)$
(6.20d) $\displaystyle\mathcal{N}$ $\displaystyle=n$ (6.20e)
$\displaystyle\mathcal{J}^{\mu}$
$\displaystyle=\epsilon^{\mu\nu\rho\sigma}u_{\nu}\nabla_{\rho}\mathfrak{m}_{\sigma}+\epsilon^{\mu\nu\rho\sigma}u_{\nu}a_{\rho}\mathfrak{m}_{\sigma}$
(6.20f)

where we introduced the magnetization and magnetic susceptibility as

$\mathfrak{m}^{\mu}=\chi_{B}B^{\mu}\qquad\qquad\chi_{B}=2\frac{\partial
P}{\partial B^{2}}$ (6.21)

This completely characterizes the ideal fluid constitutive relations without
anomaly with an order-zero magnetic field.

Finally, we need to introduce the parity-odd contributions due to the anomaly.
The details can be found in [179, 277]. In $3+1$-dimensions, the anomalous
generating functional for the covariant current takes the form

$W_{\text{anom}}=\int\mathrm{d}^{3+1}x\ \sqrt{-g}\ \frac{c}{3}\mu\
B^{\mu}A_{\mu}-\frac{c}{24}\int\mathrm{d}^{4+1}x\
\sqrt{-G}\epsilon^{mnopq}A_{m}F_{no}F_{pq}\ .$ (6.22)

The first term is responsible for generating the consistent-anomaly
constitutive relations [309, 310], the latter term instead is the anomaly-
inflow Chern-Simons functional [311], whose boundary contributions are the
Bardeen-Zumino currents that make the total current covariant [271], see
Appendix B. By varying this functional with respect to metric and gauge
fluctuations we find the two anomalous transport coefficients for the
covariant current as

$\xi^{\varepsilon}_{B}=\frac{1}{2}c\mu^{2}\qquad\qquad\xi_{B}=c\mu$ (6.23)

These coefficients agree with the thermodynamic frame, obviously: they are
obtained from the same generating functional (6.22) in which the magnetic
field has been promoted to be $B\sim\mathcal{O}(1)$. Nevertheless, even if the
contribution to the constitutive relations due to the anomaly is the same as
in the thermodynamic frame, the longitudinal optical conductivities computed
on a background at constant $B$ are different

$\displaystyle\sigma(\omega)$
$\displaystyle=\frac{i}{\omega}\left[\frac{n^{2}}{(p+\varepsilon)}+\Xi
B^{2}\right]+\mathcal{O}(\omega^{0})$ (6.24a) $\displaystyle\alpha(\omega)$
$\displaystyle=\frac{i}{\omega}\left[\frac{ns}{(p+\varepsilon)}-\mu\Xi
B^{2}\right]+\mathcal{O}(\omega^{0})$ (6.24b)
$\displaystyle\bar{\kappa}(\omega)$
$\displaystyle=\frac{i}{\omega}\left[\frac{s^{2}T}{(p+\varepsilon)}+\frac{\mu^{2}\Xi
B^{2}}{T}\right]+\mathcal{O}(\omega^{0})$ (6.24c) where
$\Xi=\frac{c^{2}s^{2}T^{2}(\frac{\partial n}{\partial
T}\mu-\frac{\partial\varepsilon}{\partial
T})}{(p+\varepsilon)(\frac{\partial\varepsilon}{\partial\mu}\frac{\partial
n}{\partial T}-\frac{\partial\varepsilon}{\partial T}\frac{\partial
n}{\partial\mu})+B^{2}c^{2}\mu^{2}(\frac{\partial\varepsilon}{\partial
T}-\frac{\partial n}{\partial T}\mu)}$ (6.24d)

This is because now the expressions must not be truncated at
$\mathcal{O}(B^{0})$, but can be taken to be valid also for large values of
$B$. In the above expressions, all thermodynamic quantities are now functions
of $T$, $\mu$ and $B^{2}$.

The conductivities (6.24) are finally well-defined: the anomalous
magnetoresistance contribution is a physical effect, they do not change upon
frame transformations, and are Onsager reciprocal. Furthermore, they obey the
standard Ward identities that hold also for non-anomalous hydrodynamics
(2.74).

With this simple computation we correct a derivative-counting mistake that was
unnoticed and easy to miss in the literature. In particular, we characterize
the full magnetic-field dependence of the longitudinal magnetoresistance for
an anomalous fluid in arbitrary background magnetic field. It is possible, by
following [179], to also include order-one corrections to such anomalous
fluid. What we expect to find is that, contrary to what is usually believed,
other transport coefficients should enter the conductivities together with
$\sigma$222We still expect $\sigma$ to be the only relevant transport
coefficient when expanding at small magnetic field..

###  6.3 The problem with DC conductivities

####  6.3.1 Weyl semimetals, hydrodynamics and transport

In this section we consider a hydrodynamic model for WSMs. We focus on the
simplest case of a WSM with two Weyl nodes of opposite chirality separated in
momentum space by a vector $b^{\mu}$, thus the low-energy excitations are
massless Weyl fermions.

Classically, we would expect two separate conserved currents
$\mathrm{U(1)}_{L}\times\mathrm{U(1)}_{R}$, for the left- and right-handed
particles, however due to quantum fluctuations these currents become
anomalous. It is then convenient to describe the system in terms of vector and
axial currents

$J^{\mu}=J^{\mu}_{+}+J^{\mu}_{-}\qquad\qquad
J^{\mu}_{5}=J^{\mu}_{+}-J^{\mu}_{-}$ (6.25)

so that the vector electric current, protected by the gauge symmetry, is
conserved, while the axial one is anomalous. The hydrodynamic equations of
motion for the covariant currents are simply

$\displaystyle\partial_{\mu}T^{\mu\nu}$
$\displaystyle=F^{\nu\lambda}J_{\lambda}$ (6.26a)
$\displaystyle\partial_{\mu}J^{\mu}$ $\displaystyle=0$ (6.26b)
$\displaystyle\partial_{\mu}J^{\mu}_{5}$ $\displaystyle=cE\cdot B$ (6.26c)

where $c$ is again the axial anomaly coefficient.

In our setup we are considering a single fluid with two different species of
particles, which is the simplest and more realistic setup for interacting Weyl
nodes. However, if the two Weyl nodes are very far separated in momentum space
$b^{\mu}\gg T$, one could also consider the Weyl nodes to give rise to two
separate, weakly-interacting fluids at different temperatures and chemical
potentials, such that each fluid conserves energy and momentum [16].

Using the insight of the previous section, we study the WSM in the
hydrodynamic regime using ideal fluid dynamics with an order zero magnetic
field $B\sim\mathcal{O}(1)$. Then the constitutive relations (neglecting
magnetization terms which are irrelevant for the longitudinal transport) read

$\displaystyle T^{\mu\nu}$ $\displaystyle=\varepsilon
u^{\mu}u^{\nu}+P\Delta^{\mu\nu}+\xi_{\varepsilon}\left(B^{\mu}u^{\nu}+B^{\nu}u^{\mu}\right)$
(6.27a) $\displaystyle J^{\mu}$ $\displaystyle=nu^{\mu}+\xi B^{\mu}$ (6.27b)
$\displaystyle J^{\mu}_{5}$ $\displaystyle=n_{5}u^{\mu}+\xi_{5}B^{\mu}$
(6.27c)

where $n_{5}=n_{5}(T,\mu,\mu_{5},B^{2})$ is the axial charge density and, like
every other thermodynamic quantity, in general depends on the temperature, the
two chemical potentials, and the external magnetic field. The $\xi$s are the
anomalous transport coefficients and take the value [271]

$\xi=c\mu_{5}\qquad\xi_{5}=c\mu\qquad\xi_{\varepsilon}=c\mu\mu_{5}$ (6.28)

Interestingly, there are no contributions due to the mixed axial-gravitational
anomaly in the anomalous transport coefficients multiplying the magnetic
field, which would appear as a term proportional to the temperature. This
fact, as we will see, has the important consequence that the DC thermal and
thermoelectric conductivities are not anomalous.

From this setup we can now simply compute the full longitudinal thermoelectric
matrix (2.66) in linear response theory. We consider a background with zero
spatial velocity, constant $T$, $\mu$ and $\mu_{5}$, and an external magnetic
field along the $\hat{z}$ axis. Around this background we add fluctuations and
linear sources parallel to the magnetic field

$\displaystyle T$ $\displaystyle\rightarrow T+\delta T-Tz\delta\zeta_{z}$
$\displaystyle\mu$ $\displaystyle\rightarrow\mu+\delta\mu$
$\displaystyle\mu_{5}$ $\displaystyle\rightarrow\mu_{5}+\delta\mu_{5}$
$\displaystyle u^{\mu}$
$\displaystyle\rightarrow(1,\delta\boldsymbol{\mathbf{v}})$ $\displaystyle
F^{0z}$ $\displaystyle\rightarrow\delta\mathbb{E}^{z}$ (6.29)

where $\delta\zeta^{i}$ and $\delta\mathbb{E}^{i}$ are, as usually, the
thermal and electric source respectively, see Section 2.2.4. Finally, the
linearized canonical heat current is defined as $\delta Q^{i}=\delta
T^{0i}-\mu\delta J^{i}-\mu_{5}\delta J^{i}_{5}$.

####  6.3.2 DC limits and relaxation terms

Unexpectedly, when we compute the full thermoelectric matrix at non-zero
frequency using the prescription above, we find that the DC limits of the
conductivities diverge.

We have already discussed in Chapter 4 and 5 the importance of momentum
relaxation to obtain finite DC conductivities: the electric field keeps adding
momentum to the fluid without bound, so a mechanism to balance this effect and
dissipate momentum on a timescale $\tau_{m}$ is needed. In the case of
anomalous fluids, however, we also need energy and axial charge relaxation.
The former comes from the fact that the covariant current
$\boldsymbol{\mathbf{J}}\propto\boldsymbol{\mathbf{B}}$ is not zero in the
background with a constant magnetic field, due to the anomalous current,
subsequently the joule heating term
$\boldsymbol{\mathbf{E}}\cdot\boldsymbol{\mathbf{J}}\sim\boldsymbol{\mathbf{E}}\cdot\boldsymbol{\mathbf{B}}$
is not zero either and the energy of the fluid keeps increasing without bound
even at zero velocity, unless energy relaxation is present. Finally, axial
charge decay is also needed, to balance the effect of the anomaly itself
$\boldsymbol{\mathbf{E}}\cdot\boldsymbol{\mathbf{B}}$, which otherwise would
lead to an infinite axial charge. These terms act as soft cut-off for the
associated conserved charge, rendering the conductivities finite.

From a physical perspective, the presence of these relaxations in WSMs can
also be argued from a microscopic point of view. Momentum relaxation is a
feature of all metals, due to the presence of impurities and phonons that can
take away momentum [115, 114, 121]. A finite axial charge relaxation time
$\tau_{5}$ is also expected for Weyl semimetals, because the chiral symmetry
is not exact, but only approximate. In particular, the electronic bands are
not linear at arbitrary-high momenta, and instead they flatten, leading to an
effective mass for the chiral particles which destroy the anomaly.
Furthermore, arguments from topology requires WSMs to have multiple Weyl cones
in each Brillouin zone [281], such that the total Berry curvature is zero, but
this allows particles of opposite chirality to interact with each other via
impurities (the so-called inter-valley scattering) and this leads to a finite
axial charge depletion rate $\tau_{5}$. Lastly, energy relaxation
$\tau_{\varepsilon}$ can happen in the presence of phonons, that can take away
energy from the electron fluid [114, 121].

Although finite momentum, energy and axial charge relaxations can be present
in Weyl semimetals, they are caused by different microscopic phenomena, as we
have just discussed, hence we would anticipate the various relaxation rates to
have different values in real materials. Specifically, we expect electric
charge to be exactly conserved (it is protected by the gauge symmetry), while
energy relaxation should be a subleading effect compared to momentum and axial
charge relaxation, which can also be rather strong effects
$\tau_{\varepsilon}\gg\tau_{5},\tau_{m}$.

Keeping in mind this discussion, we now study some possible approaches to
relaxations in anomalous fluids. Specifically, we constrain our model from
three fundamental and phenomenological assumptions: the fluid must have finite
DC conductivities in real samples, electric charge is conserved (or, at least,
the decay rate $\tau^{-1}$ can be tuned to be arbitrary small), and finally
the Green functions must obey Onsager relations.

##### Option 1: Canonical charge relaxation

The first and simplest case is the one used in [260, 257, 262], and is the
most straightforward generalization of the momentum relaxation term in [14].
Namely, this corresponds to relaxing the total charge densities in the frame
of the laboratory. In this approach we modify the linearized equations of
motion to take the following form

$\displaystyle\partial_{\mu}\delta T^{\mu 0}$
$\displaystyle=\delta(F^{0\lambda}J_{\lambda})-\frac{1}{\tau_{\varepsilon}}\delta
T^{00}\leavevmode\nobreak\ ,$ (6.30a) $\displaystyle\partial_{\mu}\delta
T^{\mu i}$
$\displaystyle=\delta(F^{i\lambda}J_{\lambda})-\frac{1}{\tau_{m}}\delta
T^{0i}\leavevmode\nobreak\ ,$ (6.30b) $\displaystyle\partial_{\mu}\delta
J^{\mu}$ $\displaystyle=-\frac{1}{\tau_{n}}\delta J^{0}\leavevmode\nobreak\ ,$
(6.30c) $\displaystyle\partial_{\mu}\delta J^{\mu}_{5}$ $\displaystyle=c\delta
E\cdot B-\frac{1}{\tau_{5}}\delta J^{0}_{5}\leavevmode\nobreak\ .$ (6.30d)

Notice that we also included a possible electric-charge relaxation term, not
present in the original works, and which is not necessary to obtain finite DC
results.

First, we notice that all the longitudinal DC conductivities computed from
this model are anomalous, meaning that they all depend on $c^{2}B^{2}$ in a
non-trivial way. Specifically the scaling is not simply quadratic in the
magnetic field, since it also appears in the denominator (the linear $B^{2}$
dependence is recovered only in the limit of small magnetic field). Although
the expressions are too large to report here, we can write the results at zero
axial chemical potential $\mu_{5}=n_{5}=\frac{\partial
n_{5}}{\partial\mu}=\frac{\partial n_{5}}{\partial T}=0$ obtaining

$\displaystyle\sigma$ $\displaystyle=\frac{n^{2}\frac{\partial
n_{5}}{\partial\mu_{5}}\tau_{m}+B^{2}c^{2}\left(sT\tau_{5}-n\mu\tau_{m}\right)}{\frac{\partial
n_{5}}{\partial\mu_{5}}(P+\varepsilon)-B^{2}c^{2}\mu^{2}}\leavevmode\nobreak\
,$ (6.31a) $\displaystyle\alpha$ $\displaystyle=\frac{s^{2}\frac{\partial
n_{5}}{\partial\mu_{5}}T\tau_{m}}{\frac{\partial
n_{5}}{\partial\mu_{5}}(P+\varepsilon)-B^{2}c^{2}\mu^{2}}\leavevmode\nobreak\
,$ (6.31b) $\displaystyle\bar{\kappa}$
$\displaystyle=\frac{s\left(n\frac{\partial
n_{5}}{\partial\mu_{5}}-B^{2}c^{2}\mu\right)\tau_{m}}{\frac{\partial
n_{5}}{\partial\mu_{5}}(P+\varepsilon)-B^{2}c^{2}\mu^{2}}\leavevmode\nobreak\
.$ (6.31c)

Notice that in the non-chiral regime, only axial charge and momentum
relaxation are needed to obtain finite DC conductivities. This is expected,
since at $\mu_{5}=0$ there is no background current
$\boldsymbol{\mathbf{J}}=0$, see (6.28), and thus no Joule heating happens.

There is however one problem with the above approach: in general, on a
background with $\mu_{5}\neq 0$, $\alpha$ and $\bar{\kappa}$ depend only on
$\tau_{m}$, while $\sigma$ and $\bar{\alpha}$ also on $\tau_{5}$ and
$\tau_{\varepsilon}$. Thus, when we impose Onsager relations
$\alpha=\bar{\alpha}$, we find that time-reversal invariance implies that all
relaxation rates must be equal
$\tau_{\varepsilon}=\tau_{5}=\tau_{n}=\tau_{m}$. This result is against our
working assumptions (electric charge is not conserved) and contradicts
phenomenological expectations, for which we expect the relaxation rates to be
different, since they should come from various microscopic processes.

Hence, we are forced to look for other approaches to obtain finite
conductivities, while also preserving charge conservation.

##### Option 2: Normal charge relaxation

Looking at the relaxation terms of the previous paragraph, we see that the
linearized charge densities observed in the laboratory are always composed of
two terms: one is the normal charge density, respectively $\delta\varepsilon$,
$\delta n$, $\delta n_{5}$, $(P+\varepsilon)\delta v^{i}$, while the second
one is the anomalous contribution that comes from the effect of Lorentz boost
on the anomaly-related terms $\xi B^{\mu}$.

However, we know that the anomalous part of the fluid does not produce
entropy, create drag or heat [266, 312, 313] and has superfluid-like
behaviour333Of course, it is not a real superfluid, since it carries entropy,
but it shares some phenomenological similarities, such as a persistent
dissipationless current.. Particularly, a chiral fluid in a constant magnetic
field in the presence of impurities will relax the momentum until its
equilibrium velocity is zero, but the anomalous momentum will keep flowing
past the impurities. Furthermore, even arguments from kinetic theory suggest
that only the normal part of the fluid should relax [314, 315].

Then, following these lines of reasoning, we propose that the relaxation rates
of the previous section should be modified in order to relax only the normal
fluid components

$\displaystyle\frac{\delta T^{00}}{\tau_{\varepsilon}}$
$\displaystyle=\frac{1}{\tau_{\varepsilon}}\left(\delta\varepsilon+2c\mu\mu_{5}\boldsymbol{\mathbf{B}}\cdot\delta\boldsymbol{\mathbf{v}}\right)$
$\displaystyle\longrightarrow$
$\displaystyle\frac{\delta\varepsilon}{\tau_{\varepsilon}}$ (6.32a)
$\displaystyle\frac{\delta T^{0i}}{\tau_{m}}$
$\displaystyle=\frac{1}{\tau_{m}}\left[(\varepsilon+P)\delta
v^{i}+B^{i}c(\mu_{5}\delta\mu+\mu\delta\mu_{5})\right]$
$\displaystyle\longrightarrow$
$\displaystyle\frac{\varepsilon+P}{\tau_{m}}\delta v^{i}$ (6.32b)
$\displaystyle\frac{\delta J^{0}}{\tau_{n}}$
$\displaystyle=\frac{1}{\tau_{n}}\left(\delta
n+c\mu_{5}\boldsymbol{\mathbf{B}}\cdot\delta\boldsymbol{\mathbf{v}}\right)$
$\displaystyle\longrightarrow$ $\displaystyle\frac{\delta n}{\tau_{n}}$
(6.32c) $\displaystyle\frac{\delta J^{0}_{5}}{\tau_{5}}$
$\displaystyle=\frac{1}{\tau_{5}}\left(\delta
n_{5}+c\mu\boldsymbol{\mathbf{B}}\cdot\delta\boldsymbol{\mathbf{v}}\right)$
$\displaystyle\longrightarrow$ $\displaystyle\frac{\delta n_{5}}{\tau_{5}}$
(6.32d)

This partially solves the problems of the previous approach, namely with this
choice momentum relaxation ($\tau_{m}\geq 0$ for stability) now decouples from
the other relaxations and is arbitrary, however Onsager relations (2.63) still
require that all other relaxation rates must take the same value
$\tau_{n}=\tau_{5}=\tau_{\varepsilon}$, which is again in strong disagreement
with phenomenological and theoretical arguments.

The conductivities computed from this approach are qualitatively and
quantitatively different from the previous case: all conductivities now depend
on energy, momentum and axial charge relaxation, however in DC only the
electric conductivity remains anomalous, and with a simple $B^{2}$ dependence,
while the other ones take the standard hydrodynamic form (2.73). This, as we
have already argued above, is due to the fact that the linearized canonical
heat current does not have any anomaly-related term, and that the anomalous
transport coefficients $\xi$s do not contain $T$-dependent contributions
induced by the mixed-gravitational anomaly444We remind that the thermal
conductivity measured in experiments $\kappa$ is related to the one obtained
from linear response $\bar{\kappa}$ via
$\kappa=\bar{\kappa}-T\alpha^{2}/\sigma$, see the discussion around (2.67).
Thus, even if $\bar{\kappa}$ is not anomalous in DC, the same is not true for
$\bar{\kappa}$, which inherits a magnetoresistance from $\sigma$..

Finally, at zero chemical potential $\mu_{5}=n_{5}=\frac{\partial
n_{5}}{\partial\mu}=\frac{\partial n_{5}}{\partial T}=0$ we find that the DC
electric conductivity is

$\sigma=\frac{n^{2}\tau_{m}}{P+\varepsilon}+\frac{B^{2}c^{2}\tau_{5}}{\frac{\partial
n_{5}}{\partial\mu_{5}}}\leavevmode\nobreak\ .$ (6.33)

while the other conductivities take on their standard hydrodynamic form that
depends only on $\tau_{m}$.

This approach already leads to an interesting result, specifically a different
prediction for the thermoelectric transport compared to other studies. In
older works [276, 16, 257, 300, 316, 317, 318] the anomaly seems to appear
also in the thermoelectric conductivity from hydrodynamics, lattice
simulations and kinetic theory approaches. This difference can be understood
because the authors either relax the total charge, use different assumptions
on the model (e.g. having two separate fluids for each Weyl cone) or simply
are not in the hydrodynamic regime. This prediction is then open to
experimental verification [288, 291, 292] and could be used as a probe of
single-fluid hydrodynamic anomalous transport, although practically it can be
difficult to disentangle the various contribution to the magnetoresistance and
to isolate the effect of the anomaly.

##### Option 3: Generalised relaxations

In the previous section we managed to decouple momentum relaxation from the
other decay rates, however electric charge must still relax for the system to
obey Onsager relations, and moreover the relaxations must all be equal. To
improve the model, we propose to use the generalized relaxation of Chapter 5.

Following the previous chapter, we then consider the most general set of
relaxations

$\begin{rcases*}\text{energy:}\hskip
36.98866pt\frac{1}{\tau_{\varepsilon\varepsilon}}\delta\varepsilon+\frac{1}{\tau_{\varepsilon
n}}\delta n+\frac{1}{\tau_{\varepsilon n_{5}}}\delta n_{5}\\\
\text{charge:}\hskip
36.98866pt\frac{1}{\tau_{n\varepsilon}}\delta\varepsilon+\frac{1}{\tau_{nn}}\delta
n+\frac{1}{\tau_{nn_{5}}}\delta n_{5}\\\ \text{axial
charge:}\quad\frac{1}{\tau_{n_{5}\varepsilon}}\delta\varepsilon+\frac{1}{\tau_{n_{5}n}}\delta
n+\frac{1}{\tau_{n_{5}n_{5}}}\delta
n_{5}\end{rcases*}=\hat{\tau}\cdot\varphi\leavevmode\nobreak\ ,$ (6.34)

where $\varphi=\left(\delta\varepsilon,\delta n,\delta n_{5}\right)$ and
$\hat{\tau}$ is the $3\times 3$ matrix of inverse relaxation times. We can
again employ Onsager relations (2.63) on these relaxed equations of motion to
find the constraints that the relaxation times must obey for the system to
preserve time-reversal invariance (5.4). Thus, we find

$\displaystyle 0$ $\displaystyle=\frac{\chi_{nn_{5}}}{\tau_{\varepsilon
n_{5}}}+\frac{\chi_{nn}}{\tau_{\varepsilon n}}-\frac{\chi_{\varepsilon
n_{5}}}{\tau_{nn_{5}}}+\frac{\chi_{\varepsilon
n}}{\tau_{\varepsilon\varepsilon}}-\frac{\chi_{\varepsilon
n}}{\tau_{nn}}-\frac{\chi_{\varepsilon\varepsilon}}{\tau_{n\varepsilon}}\leavevmode\nobreak\
,$ (6.35a) $\displaystyle 0$
$\displaystyle=\frac{\chi_{n_{5}n_{5}}}{\tau_{\varepsilon
n_{5}}}+\frac{\chi_{nn_{5}}}{\tau_{\varepsilon n}}-\frac{\chi_{\varepsilon
n_{5}}}{\tau_{n_{5}n_{5}}}+\frac{\chi_{\varepsilon
n_{5}}}{\tau_{\varepsilon\varepsilon}}-\frac{\chi_{\varepsilon
n}}{\tau_{n_{5}n}}-\frac{\chi_{\varepsilon\varepsilon}}{\tau_{n_{5}\varepsilon}}\leavevmode\nobreak\
,$ (6.35b) $\displaystyle 0$
$\displaystyle=\frac{\chi_{n_{5}n_{5}}}{\tau_{nn_{5}}}-\frac{\chi_{nn_{5}}}{\tau_{n_{5}n_{5}}}+\frac{\chi_{nn_{5}}}{\tau_{nn}}-\frac{\chi_{nn}}{\tau_{n_{5}n}}+\frac{\chi_{\varepsilon
n_{5}}}{\tau_{n\varepsilon}}-\frac{\chi_{\varepsilon
n}}{\tau_{n_{5}\varepsilon}}\leavevmode\nobreak\ ,$ (6.35c)

or in matrix formulation

$\hat{\tau}\cdot\hat{\chi}-\hat{\chi}\cdot\hat{\tau}^{T}=0\leavevmode\nobreak\
,$ (6.36)

where $\hat{\chi}$ is the $3\times 3$ susceptibility matrix for the energy,
electric and axial charge, while $T$ denotes the transpose matrix. In
agreement with the previous case, these constraints tell us that if we set the
off-diagonal relaxations to zero, then Onsager relations imply
$\tau_{nn}=\tau_{n_{5}n_{5}}=\tau_{\varepsilon\varepsilon}$.

The claim is that we now have enough parameters to satisfy all our
requirements, namely: finite DC conductivities, charge conservation and
Onsager relations (microscopic time-reversal invariance). To begin the
analysis of the parameter space, first we set to zero the charge relaxations
$\tau_{nn}^{-1}=\tau_{nn_{5}}^{-1}=\tau_{n\varepsilon}^{-1}=0$ so that the
electric charge is exactly conserved. Thus, starting from 9 relaxation (plus
momentum $\tau_{m}$, which we decoupled in the previous section) we are left
with only 6 non-zero relaxations. There are 3 Onsager conditions, which reduce
the number of independent parameters to only 3, which we assume to be
$\tau_{\varepsilon\varepsilon},\tau_{n_{5}n_{5}}$ and
$\tau_{n_{5}\varepsilon}$, and express the other relaxations rates as
functions of them.

This setup is guaranteed to lead to conductivities which are Onsager
reciprocal, however the only way to check if they are also finite in DC is to
perform the computation explicitly, and we find that they are indeed finite.
Notice that this is not trivial: from Onsager relations alone one might be
tempted to set to zero also $\tau_{\varepsilon n}^{-1}=\tau_{n_{5}n}^{-1}=0$,
leaving $\tau_{n_{5}n_{5}}$ as the only free parameter555From 9 initial
relaxations, 5 are set to zero
$\tau_{nn}^{-1}=\tau_{nn_{5}}^{-1}=\tau_{n\varepsilon}^{-1}=\tau_{\varepsilon
n}^{-1}=\tau_{n_{5}n}^{-1}=0$, thus only 4 are non-vanishing, but with 3
Onsager conditions we are left with a 1-parameter family of solutions.,
however the conductivities computed on this setup are not DC finite.

From a qualitative perspective the conductivities computed with generalized
relaxations are similar to the ones analysed in the previous section: all
conductivities depend on all relaxation rates, however in DC only the electric
conductivity is anomalous, while the other conductivities are not.
Furthermore, at zero axial chemical potential we have the freedom to also
conserve energy, namely $\tau_{\varepsilon
n}^{-1}=\tau_{\varepsilon\varepsilon}^{-1}=\tau_{\varepsilon n_{5}}^{-1}=0$.
Following the argument below (6.31), this is expected: energy relaxation is
needed to balance the Joule heating term
$\boldsymbol{\mathbf{E}}\cdot\boldsymbol{\mathbf{J}}$ with
$\boldsymbol{\mathbf{J}}\propto\boldsymbol{\mathbf{B}}$, however at
$\mu_{5}=0$ the electric current is zero and not anomalous, thus we expect no
Joule heating.

Remarkably, with the above setup we cannot also impose positivity of entropy
production. Following the approach of Chapter 5 on the linearized second law
of thermodynamics, we find the following set of constraints

$\displaystyle 0$
$\displaystyle=\frac{1}{\tau_{\varepsilon\varepsilon}}-\frac{\mu}{\tau_{n\varepsilon}}-\frac{\mu_{5}}{\tau_{n_{5}\varepsilon}}\leavevmode\nobreak\
,$ (6.37a) $\displaystyle 0$ $\displaystyle=\frac{1}{\tau_{\varepsilon
n_{5}}}-\frac{\mu}{\tau_{nn_{5}}}-\frac{\mu_{5}}{\tau_{n_{5}n_{5}}}\leavevmode\nobreak\
,$ (6.37b) $\displaystyle 0$ $\displaystyle=\frac{1}{\tau_{\varepsilon
n}}-\frac{\mu}{\tau_{nn}}-\frac{\mu_{5}}{\tau_{n_{5}n}}\leavevmode\nobreak\ .$
(6.37c)

However, it is not possible to have electric charge conservation
$\tau_{nn}^{-1}=\tau_{nn_{5}}^{-1}=\tau_{n\varepsilon}^{-1}=0$, positivity of
entropy production (6.37) and time-reversal covariance (6.35) with generalized
relaxations. Indeed, charge conservation reduces the number of relaxation
parameters from 9 to 6, Onsager relations and positivity of entropy production
each correspond to three equations, but the only solution to the whole set of
equations is the trivial one that sets all the relaxations to zero. We remark
that this conclusion, however, is not too problematic, since generalized
relaxations can be understood in terms of the fluid interacting with the
environment and thus describing an open system (it can lose energy, charge and
also entropy).

Finally, we can briefly comment that in the presence of charged impurities, or
multiple gates, one might have an arbitrary small effective electric charge
relaxation. However, this can be easily incorporated using generalized
relaxations as an extra parameter that can be tuned at will.

To conclude, generalized relaxations allow us to obtain finite DC
conductivities, which obey Onsager relations and that conserve the electric
charge. Furthermore, the six non-zero relaxations are constrained by only
three equations, meaning that the parameter space can accommodate for
different phenomenological timescales for which e.g., energy relaxation can be
parametrically smaller than axial charge decay rate.

###  6.4 Kinetic theory

In Chapter 5 we introduced the generalized relaxations with arguments uniquely
based on the EFT prescription, while above we argued how they are necessary to
properly describe the DC transport in models of Weyl semimetals. However, we
still have to show that generalized relaxation can actually happen in
condensed matter systems. This is exactly what we are going to do in this
section: we will show that if charge relaxations are present at all in the
system, then we should always expect generalized relaxations that take exactly
the form described in Chapter 5, and obey Onsager relations (6.35). We will do
this by using suitable energy-dependent modifications of the Relaxation Time
Approximation (RTA for short), which are supported by microscopic collision
integrals with impurities and phonons.

Notice that Onsager constraints (6.35) and the structure of generalized
relaxations (6.34) do not depend on the anomaly when we relax only the non-
anomalous charges, thus they hold true for all fluids (even if their relevance
becomes more apparent in anomalous hydrodynamics). This means that we can use
non-chiral kinetic theory to justify their presence. Although not necessary,
we will consider a linear dispersion relation such as in Weyl semimetals, so
that we can perform certain computations analytically. First, we will focus on
a single-current ideal fluid, and we will explain how to obtain generalized
relaxations from modifications of the RTA in this simpler case, matching the
expressions of the previous Chapter 5. After that we will move to the case
relevant to Weyl semimetals, namely that of two chiral currents as in Section
6.3, and we will discuss possible scattering mechanisms there.

####  6.4.1 A primer on kinetic theory and hydrodynamics

Consider the relativistic Boltzmann equation for the one-particle distribution
function
$f(\boldsymbol{\mathbf{x}},\boldsymbol{\mathbf{p}},t)=f_{\boldsymbol{\mathbf{p}}}$
in the absence of external forces666External forces correspond, in the
hydrodynamic picture, to the Maxwell term in the stress-energy tensor
conservation equation.

$\partial_{t}f_{\boldsymbol{\mathbf{p}}}+\boldsymbol{\mathbf{p}}\cdot\boldsymbol{\mathbf{\nabla}}f_{\boldsymbol{\mathbf{p}}}=I_{\text{coll}}[f_{\boldsymbol{\mathbf{p}}}]$
(6.38)

with $\boldsymbol{\mathbf{p}}$ the quasiparticle momentum and
$I_{\text{coll}}$ the collision integral, which is a functional of
$f_{\boldsymbol{\mathbf{p}}}$.

In the simplest case, which is often used to construct hydrodynamics from
kinetic theory, we consider only a single species of particles (in our case,
dressed electrons) for which 2-to-2 particles scatterings dominate. Then
$I_{\text{coll}}=I_{\text{ee}}$ and corresponds to the usual textbook
electron-electron collision integral [114, 90, 319]. Under standard
assumptions about the microscopic scattering, such as unitarity, conservation
of energy and momentum, time-reversal and parity symmetry, and the hypothesis
of molecular chaos, the electron-electron collision integral is written as

$I_{\text{ee}}=\int\mathop{}\\!\mathrm{d}^{3}\boldsymbol{\mathbf{p}}_{2}\mathop{}\\!\mathrm{d}^{3}\boldsymbol{\mathbf{p}}_{1}^{\prime}\mathop{}\\!\mathrm{d}^{3}\boldsymbol{\mathbf{p}}_{2}^{\prime}W_{\boldsymbol{\mathbf{p}}_{1}^{\prime}\boldsymbol{\mathbf{p}}_{2}^{\prime}\rightarrow\boldsymbol{\mathbf{p}}\boldsymbol{\mathbf{p}}_{2}}\left(f_{\boldsymbol{\mathbf{p}}_{1}^{\prime}}f_{\boldsymbol{\mathbf{p}}_{2}^{\prime}}[1+af_{\boldsymbol{\mathbf{p}}}][1+af_{\boldsymbol{\mathbf{p}}_{2}}]\right.\\\
\left.-f_{\boldsymbol{\mathbf{p}}}f_{\boldsymbol{\mathbf{p}}_{2}}[1+af_{\boldsymbol{\mathbf{p}}_{1}^{\prime}}][1+af_{\boldsymbol{\mathbf{p}}_{2}^{\prime}}]\right)$
(6.39)

where $a=\pm 1$ for bosons/fermions and $a=0$ for classical particles, while
$W_{\boldsymbol{\mathbf{p}}_{1}^{\prime}\boldsymbol{\mathbf{p}}_{2}^{\prime}\rightarrow\boldsymbol{\mathbf{p}}\boldsymbol{\mathbf{p}}_{2}}$
is the scattering rate.

From here, it can be shown that the collision integral vanishes when
$f_{\boldsymbol{\mathbf{p}}}$ satisfies the detailed balance equation

$\log\left(\frac{f_{\boldsymbol{\mathbf{p}}_{1}^{\prime}}}{1+af_{\boldsymbol{\mathbf{p}}_{1}^{\prime}}}\right)+\log\left(\frac{f_{\boldsymbol{\mathbf{p}}_{2}^{\prime}}}{1+af_{\boldsymbol{\mathbf{p}}_{2}^{\prime}}}\right)=\log\left(\frac{f_{\boldsymbol{\mathbf{p}}}}{1+af_{\boldsymbol{\mathbf{p}}}}\right)+\log\left(\frac{f_{\boldsymbol{\mathbf{p}}_{2}}}{1+af_{\boldsymbol{\mathbf{p}}_{2}}}\right)$
(6.40)

which is understood as a conservation equation
($\boldsymbol{\mathbf{p}}_{1}^{\prime}$ and
$\boldsymbol{\mathbf{p}}_{2}^{\prime}$ are after the collision,
$\boldsymbol{\mathbf{p}}$ and $\boldsymbol{\mathbf{p}}_{2}$ before). Hence,
the logarithm solves the detailed balance when it is proportional to the
conserved quantities in the system: energy, momentum and particle number. The
solution is then given by the local thermodynamic equilibrium (LTE) form of
$f_{\boldsymbol{\mathbf{p}}}$ as a Fermi-Dirac (for $a=-1$) distribution

$f_{\boldsymbol{\mathbf{p}}}^{\text{eq}}=\frac{1}{1+e^{(\varepsilon_{p}-\boldsymbol{\mathbf{u}}\cdot\boldsymbol{\mathbf{p}}-\mu)/T}}$
(6.41)

where the Lagrange multipliers $T$, $\mu$ and $\boldsymbol{\mathbf{u}}$ are
functions of space. The LTE distribution function solves the collision
integral, however it does not vanish on the LHS of the Boltzmann equation
(only global thermodynamic equilibrium solves the full equation), nonetheless
in the hydrodynamic regime the non-vanishing of the LHS corresponds to small
gradient corrections. For this reason, the ideal fluid corresponds in the
kinetic-theory picture to LTE (6.41), while higher derivative corrections in
hydrodynamics take us away from LTE.

Given the LTE form $f_{\boldsymbol{\mathbf{p}}}^{\text{eq}}$ and the
dispersion relation $\varepsilon(p)=\varepsilon_{p}$ we can compute
thermodynamic quantities in local equilibrium, by integrating over momentum
space and summing over particles and holes contributions

$\displaystyle\varepsilon^{\text{eq}}(T,\mu)$
$\displaystyle=\sum_{p,h}\int\frac{\mathop{}\\!\mathrm{d}^{3}\boldsymbol{\mathbf{p}}}{(2\pi)^{3}}\varepsilon_{p}f_{\boldsymbol{\mathbf{p}}}^{\text{eq}}$
(6.42a) $\displaystyle n^{\text{eq}}(T,\mu)$
$\displaystyle=\sum_{p,h}\int\frac{\mathop{}\\!\mathrm{d}^{3}\boldsymbol{\mathbf{p}}}{(2\pi)^{3}}f_{\boldsymbol{\mathbf{p}}}^{\text{eq}}$
(6.42b)

Finally, to obtain the hydrodynamic equations of motions we simply multiply
the Boltzmann equation by $\varepsilon_{p}$, $\boldsymbol{\mathbf{p}}$ or 1
and then integrate over the quasiparticle momentum. In particular, given the
electron-electron collision integral (6.39), it can be shown that it vanishes
when integrated over conserved quantities, independently on the form of the
distribution function $f_{\boldsymbol{\mathbf{p}}}$. Specifically

$\int\frac{\mathop{}\\!\mathrm{d}^{3}\boldsymbol{\mathbf{p}}}{(2\pi)^{3}}A(\boldsymbol{\mathbf{x}},\boldsymbol{\mathbf{p}})I_{\text{ee}}[f_{\boldsymbol{\mathbf{p}}}]=0\qquad\qquad\text{for}\quad
A=\\{\varepsilon_{p},\boldsymbol{\mathbf{p}},1\\}$ (6.43)

Thus, multiplying with $\varepsilon_{p}$, $\boldsymbol{\mathbf{p}}$, or 1 and
integrating the Boltzmann equation over momentum space, we recover the
equations of ideal hydrodynamics.

If one wishes to go beyond order zero in the hydrodynamic derivative expansion
the distribution function receives out-of-equilibrium corrections, which can
be computed perturbatively (e.g. in Knudsen number) [91, 90]

$f_{\boldsymbol{\mathbf{p}}}=f_{\boldsymbol{\mathbf{p}}}^{\text{eq}}+\delta
f_{\boldsymbol{\mathbf{p}}}$ (6.44)

However, the non-equilibrium distribution function
$f_{\boldsymbol{\mathbf{p}}}$ does not solve the electron-electron collision
integral, hence to make progress we linearize it and employ the standard RTA
[114]777We are not using the relativistic form of the RTA [93, 320] since this
form is enough for the present discussion.

$I_{\text{ee}}\approx-\frac{f_{\boldsymbol{\mathbf{p}}}-f_{\boldsymbol{\mathbf{p}}}^{\text{eq}}}{\tau}=-\frac{\delta
f_{\boldsymbol{\mathbf{p}}}}{\tau}$ (6.45)

with $\tau$ some constant that represents the timescale it takes for the
system to relax to equilibrium.

Importantly, the RTA is only a crude model for the real linearized collision
integral, and does not inherit all of its properties [319]: the only property
that survives this approximation is the vanishing of $I_{\text{coll}}$ when
$f_{\boldsymbol{\mathbf{p}}}$ takes the Fermi-Dirac form (6.41),
$I_{\text{coll}}[f^{\text{eq}}_{\boldsymbol{\mathbf{p}}}]=0$. Indeed, the RTA
model does not vanish when integrated in momentum space against conserved
quantities (6.43), this means that the equations of hydrodynamics receive
corrections as decay terms at order one in derivatives

$\displaystyle\partial_{t}\varepsilon+\dots$
$\displaystyle=-\frac{\varepsilon-\varepsilon^{\text{eq}}}{\tau}$ (6.46a)
$\displaystyle\partial_{t}n+\dots$
$\displaystyle=-\frac{n-n^{\text{eq}}}{\tau}$ (6.46b)

Then, enforcing the equations of hydrodynamics to hold true, we are led to the
requirements $\varepsilon=\varepsilon^{\text{eq}}$ and $n=n^{\text{eq}}$,
which are part of the Landau matching conditions used to fix the hydrodynamic
frame888Notice that, using the method developed in [93], it is possible to
obtain hydrodynamics from kinetic theory in any frame..

Because in what follows we will only be interested in perfect fluids to match
the discussion of the previous section, $f_{\boldsymbol{\mathbf{p}}}$ will
always be in LTE. Thus, we drop the _eq_ superscript from now on and simply
call $f_{\boldsymbol{\mathbf{p}}}$ the LTE form of the Fermi-Dirac
distribution (6.41). Similarly, we will avoid using _eq_ to define the
equilibrium energy and charge, since they are always assumed to be in LTE.

####  6.4.2 Adding momentum relaxation

We will now focus on ideal fluid dynamics and, following [314, 126], we will
see how to modify kinetic theory to incorporate a momentum relaxation effect
in the equations of hydrodynamics.

Given the Fermi-Dirac distribution (6.41) we can expand it at small velocity

$f_{\boldsymbol{\mathbf{p}}}\approx
f^{(0)}-(\boldsymbol{\mathbf{p}}\cdot\boldsymbol{\mathbf{u}})\frac{\partial
f^{(0)}}{\partial\varepsilon_{p}}\qquad\text{with}\qquad
f^{(0)}=\frac{1}{1+e^{(\varepsilon_{p}-\mu)/T}}$ (6.47)

where $f^{(0)}$ indicates the zero velocity distribution function. We can
compute the energy and charge density for massless fermions by employing a
linear dispersion relation $\varepsilon_{p}=p$ (we are setting the Fermi
velocity to unity). Because of the linearization at small velocity, the second
term linear in $\boldsymbol{\mathbf{u}}$ does not contribute to any
thermodynamic quantity, since it vanishes when integrated in momentum space,
see the integrals in the Appendix A. Then we find

$\displaystyle\varepsilon$
$\displaystyle=\frac{15\mu^{4}+30\pi^{2}T^{2}\mu^{2}+7\pi^{4}T^{4}}{120\pi^{2}}$
(6.48a) $\displaystyle n$
$\displaystyle=\mu\left(\frac{\mu^{2}}{6\pi^{2}}+\frac{T^{2}}{6}\right)$
(6.48b)

while the pressure is $P=\varepsilon/3$.

To include momentum relaxation effects in the hydrodynamic picture, we now
assume the RTA ansatz given by

$I_{\text{imp}}\approx-\frac{f_{\boldsymbol{\mathbf{p}}}-f^{(0)}}{\tau_{m}}$
(6.49)

The form of the RTA is such that it vanishes when integrated against the
quasiparticle energy and the identity, while it gives the momentum relaxation
term we seek when integrated against the quasi-momentum
$\boldsymbol{\mathbf{p}}$

$\partial_{t}(\varepsilon+P)v^{i}+\dots=-\frac{(\varepsilon+P)v^{i}}{\tau_{m}}$
(6.50)

This RTA is physically very different from the standard one (6.45): first,
here both $f_{\boldsymbol{\mathbf{p}}}$ and $f^{(0)}$ are LTE solutions, hence
they both make the electron-electron collision integral (6.39) vanish, while
this is not true in (6.45). Furthermore, this RTA has physical effects and
leads to momentum relaxation, while the electron-electron RTA model was only
used to impose Landau matching conditions. We remark that momentum relaxation
behaves as an extra constraint on the theory, and acts in such a way that of
all the possible equilibrium solutions with different constant velocities
(which are related by boosts), only the zero-velocity equilibrium is actually
picked by the system.

The expression in (6.49) can be justified from electron-impurity scatterings.
Consider a collision integral that takes the following form

$I_{\text{coll}}=I_{\text{ee}}+I_{\text{imp}}$ (6.51)

The first term is the electron-electron scattering of (6.39): in the
hydrodynamic regime it dominates (its associated timescale is very short) and
its role is to force the distribution function to be Fermi-Dirac. The second
term instead describes interaction of electrons with impurities and is assumed
to be a weak correction (so that a quasihydrodynamic description still
applies, and momentum is still a relevant degree of freedom). The electron-
impurity collision integral takes the simple form [114]

$I_{\text{imp}}=\int\mathop{}\\!\mathrm{d}^{3}\boldsymbol{\mathbf{p}}^{\prime}W_{\boldsymbol{\mathbf{p}}\rightarrow\boldsymbol{\mathbf{p}}^{\prime}}[f_{\boldsymbol{\mathbf{p}}}-f_{\boldsymbol{\mathbf{p}}^{\prime}}]\delta(\varepsilon_{p}-\varepsilon_{p^{\prime}})$
(6.52)

where again
$W_{\boldsymbol{\mathbf{p}}\rightarrow\boldsymbol{\mathbf{p}}^{\prime}}$ is
the scattering rate. Following the usual approach of the detailed balance
(6.40), we should look for solutions to this collision integral (among the
spectrum of all the LTE solutions, required by the vanishing of
$I_{\text{ee}}$). However, these are exactly the zero-velocity distribution
functions $f^{(0)}$ we introduced in (6.47), therefore a good ansatz for an
RTA model of this collision integral should vanish when
$f_{\boldsymbol{\mathbf{p}}}=f^{(0)}$, and we are led to (6.49).

####  6.4.3 Adding charge relaxations

Having understood how to include momentum relaxation in our hydrodynamic model
from kinetic theory, we can now follow the same logic to include energy and
charge relaxation in the system.

In particular, the momentum-relaxing RTA suggests we can add charge relaxation
by including an additional collision integral into $I_{\text{coll}}$ that
depends on $\bar{f}^{(0)}$, so that the system relaxes to a reference LTE
solution with fixed energy $\bar{\varepsilon}$ and charge $\bar{n}$
respectively. With this new quantity at hand, the most generic RTA is

$I_{\text{coll}}-I_{\text{ee}}\approx-\frac{f_{\boldsymbol{\mathbf{p}}}-f^{(0)}}{\tau^{\prime}_{m}}-\frac{f_{\boldsymbol{\mathbf{p}}}-\bar{f}^{(0)}}{\tau_{n}}=-\frac{f_{\boldsymbol{\mathbf{p}}}-f^{(0)}}{\tau_{m}}-\frac{f^{(0)}-\bar{f}^{(0)}}{\tau_{n}}$
(6.53)

where $\bar{f}^{(0)}$ has the same form of $f^{(0)}$

$\bar{f}^{(0)}=\frac{1}{1+e^{(\varepsilon_{p}-\bar{\mu})/\bar{T}}}$ (6.54)

but now computed at fixed reference values of temperature $\bar{T}$ and
chemical potential $\bar{\mu}$. From now on, we will assume that all barred
thermodynamic quantities are computed with respect to $\bar{f}^{(0)}$ and are
thus functions of $\bar{T}$ and $\bar{\mu}$. In the second equality we
redefined $\tau_{m}$ to separate the RTA in terms which induce different
relaxations: the former is zero on the charge and energy equations, and only
gives momentum relaxation, while the latter relaxes energy and charge at the
same rate $\tau_{n}$ and vanishes in the momentum equation, see again the
integrals in the Appendix A. With the collision integral (6.53) the equations
of hydrodynamics become

$\displaystyle\partial_{t}\varepsilon+\dots$
$\displaystyle=-\frac{\varepsilon-\bar{\varepsilon}}{\tau_{n}}$ (6.55a)
$\displaystyle\partial_{t}(P+\varepsilon)v^{i}+\dots$
$\displaystyle=-\frac{(P+\varepsilon)v^{i}}{\tau_{m}}$ (6.55b)
$\displaystyle\partial_{t}n+\dots$ $\displaystyle=-\frac{n-\bar{n}}{\tau_{n}}$
(6.55c)

When linearizing around $\bar{\varepsilon}$ and $\bar{n}$ (which are the
reference equilibrium values), we immediately recover the one-current analogue
of equations (6.32), while also obtaining the Onsager constraint
$\tau_{\varepsilon}=\tau_{n}$ for free, directly from kinetic theory.

We can again justify the RTA in (6.53) from a microscopic model. The total
collision integral now takes the form

$I_{\text{coll}}=I_{\text{ee}}+I_{\text{imp}}+I_{\text{ph}}$ (6.56)

We have already discussed the role of $I_{\text{ee}}$, that dominates and
forces the system to be in LTE, and of $I_{\text{imp}}$, which relaxes
momentum via scattering with impurities. Provided that our solution is in LTE,
these first two collision integrals vanish in the energy and charge equation.
The third term instead is the collision integral that describes scattering
between electrons and phonons. It is given by [114]

$\displaystyle I_{\text{ph}}$
$\displaystyle=\int\mathop{}\\!\mathrm{d}^{3}\boldsymbol{\mathbf{q}}W_{\boldsymbol{\mathbf{p}}^{\prime}\boldsymbol{\mathbf{q}}\rightarrow\boldsymbol{\mathbf{p}}}\left[f_{\boldsymbol{\mathbf{p}}^{\prime}}(1-f_{\boldsymbol{\mathbf{p}}})n_{\boldsymbol{\mathbf{q}}}-f_{\boldsymbol{\mathbf{p}}}(1-f_{\boldsymbol{\mathbf{p}}^{\prime}})(1+n_{\boldsymbol{\mathbf{q}}})\right]\delta\left(\varepsilon_{p}-\varepsilon_{p^{\prime}}-\omega_{q}\right)$
$\displaystyle+\int\mathop{}\\!\mathrm{d}^{3}\boldsymbol{\mathbf{q}}W_{\boldsymbol{\mathbf{p}}^{\prime}\rightarrow\boldsymbol{\mathbf{p}}\boldsymbol{\mathbf{q}}}\left[f_{\boldsymbol{\mathbf{p}}^{\prime}}(1-f_{\boldsymbol{\mathbf{p}}})(1+n_{\boldsymbol{\mathbf{q}}})-f_{\boldsymbol{\mathbf{p}}}(1-f_{\boldsymbol{\mathbf{p}}^{\prime}})n_{\boldsymbol{\mathbf{q}}}\right]\delta\left(\varepsilon_{p}+\omega_{k}-\varepsilon_{p^{\prime}}\right)$
(6.57)

where $\omega_{q}$ and $n_{\boldsymbol{\mathbf{q}}}$ are respectively the
energy and one-particle distribution functions for the phonons, while
$W_{\boldsymbol{\mathbf{p}}^{\prime}\boldsymbol{\mathbf{q}}\rightarrow\boldsymbol{\mathbf{p}}}$
is, as usual, the scattering rate. The first term represents the case in which
a quasi-electron of momentum $\boldsymbol{\mathbf{p}}$ emits a phonon with
momentum $\boldsymbol{\mathbf{q}}$ and is scattered to a state with momenta
$\boldsymbol{\mathbf{p}}^{\prime}$, and its inverse process, in which an
electron of momentum $\boldsymbol{\mathbf{p}}^{\prime}$ absorbs the phonon. In
this first case, the momenta obey the conservation equation
$\boldsymbol{\mathbf{p}}=\boldsymbol{\mathbf{p}}^{\prime}+\boldsymbol{\mathbf{k}}+\boldsymbol{\mathbf{b}}$,
where $\boldsymbol{\mathbf{b}}$ accounts for Umklapp processes, and the
electron $\varepsilon_{p}$ is scattered to lower-energy states. On the other
hand, the second term represents the case in which an electron of momentum
$\boldsymbol{\mathbf{p}}$ absorbs a phonon of momentum
$\boldsymbol{\mathbf{q}}$, and its inverse process of emission, so that
momentum conservation reads
$\boldsymbol{\mathbf{p}}+\boldsymbol{\mathbf{k}}=\boldsymbol{\mathbf{p}}^{\prime}+\boldsymbol{\mathbf{b}}$
and the electron $\varepsilon_{p}$ is of lower energy. Notice that the above
collision integral identically vanishes when integrated in momentum space
against the identity, thanks to the fact that electron-phonon scatterings
conserves the electric charge.

Furthermore, the above electron-phonon collision integral also vanishes when
both electrons and phonons are in global thermodynamic equilibrium [114].
Following [264], we assume that the temperature is high enough so that the
phonons are in a global thermodynamic equilibrium state defined by $\bar{T}$
and $\bar{\mu}$

$n_{\boldsymbol{\mathbf{q}}}=\bar{n}^{(0)}=\frac{1}{e^{\left(\omega_{q}-\bar{\mu}\right)/\bar{T}}-1}$
(6.58)

and behave as a thermal bath for the electrons. Having fixed the phonon
distribution, then $I_{\text{ph}}$ vanishes only when
$f_{\boldsymbol{\mathbf{p}}}$ is also in global thermodynamic equilibrium with
the phonons, i.e. only when $f_{\boldsymbol{\mathbf{p}}}=\bar{f}^{(0)}$.

If we now wish to write the electron-phonon collision integral in a RTA form,
this should vanish when $f_{\boldsymbol{\mathbf{p}}}$ is equal to the global
thermodynamic equilibrium induced by the phonon bath $\bar{f}^{(0)}$, and we
are thus led to the second term in (6.53).

####  6.4.4 Generalized relaxations from RTA

We have shown how to introduce energy and charge relaxation in the system,
without mixed relaxations, hence it is time to discuss how to include
generalized relaxations from a kinetic theory perspective. In the RTA (6.53),
the first term is related to momentum relaxation and will not be affected by
this discussion, however we will modify the second term in such a way to
obtain generalized relaxations.

To begin, we simply need to assume that the scattering rate $\tau$ is a
function of the momentum or, to be more specific, of the quasiparticle energy
$\tau=\tau(\varepsilon_{p})$, assuming isotropy. From a microscopic
perspective, this is believed to be universally true for most scattering
mechanism [114, 319]: the RTA is a very simple approximation of the true
collision integral that tries to capture the complicated non-linear scattering
dynamics in a single constant parameter $\tau$, however in realistic cases the
effective scattering rate does depend on the energy of the quasiparticles
[321, 322] and, as we will see, this induces generalized relaxations on the
equations of hydrodynamics.

Consider the second term in (6.53) due to electron-phonon scattering, and
assume that $\tau_{n}(\varepsilon_{p})$ can be expanded in the generic power-
series form

$\displaystyle I_{\text{ph}}$
$\displaystyle\approx\sum_{N\geq-2}\varepsilon_{p}^{N}\frac{f^{(0)}-\bar{f}^{(0)}}{\tau_{N+2}}=$
$\displaystyle=\frac{1}{\varepsilon_{p}^{2}}\frac{f^{(0)}-\bar{f}^{(0)}}{\tau_{0}}+\frac{1}{\varepsilon_{p}}\frac{f^{(0)}-\bar{f}^{(0)}}{\tau_{1}}+\frac{f^{(0)}-\bar{f}^{(0)}}{\tau_{2}}+\varepsilon_{p}\frac{f^{(0)}-\bar{f}^{(0)}}{\tau_{3}}+\dots$
(6.59)

The first two terms, with negative powers of $\varepsilon_{p}$, are the only
ones with $N<0$ that lead to finite quantities when integrated in momentum
space for the linear dispersion relation we are considering
$\varepsilon_{p}=p$. We define with $M_{N+2}$ the $N$-th energy-moment of
$f^{(0)}$

$M_{N+2}=\sum_{p,h}\int\frac{\mathop{}\\!\mathrm{d}^{3}\boldsymbol{\mathbf{p}}}{(2\pi)^{3}}\varepsilon_{p}^{N}f^{(0)}$
(6.60)

such that, in particular, $M_{2}=n$ and $M_{3}=\varepsilon$. Then, inserting
(6.4.4) in the Boltzmann equation, multiplying with the identity and the
quasiparticle energy, and subsequently integrating over momentum space, we
obtain the equations of relaxed hydrodynamic. The expressions for energy and
charge are

$\displaystyle\partial_{t}\varepsilon+\dots$
$\displaystyle=-\frac{M_{1}-\bar{M}_{1}}{\tau_{0}}-\frac{n-\bar{n}}{\tau_{1}}-\frac{\varepsilon-\bar{\varepsilon}}{\tau_{2}}-\frac{M_{4}-\bar{M}_{4}}{\tau_{3}}+\dots$
(6.61a) $\displaystyle\partial_{t}n+\dots$
$\displaystyle=-\frac{M_{0}-\bar{M}_{0}}{\tau_{0}}-\frac{M_{1}-\bar{M}_{1}}{\tau_{1}}-\frac{n-\bar{n}}{\tau_{2}}-\frac{\varepsilon-\bar{\varepsilon}}{\tau_{3}}+\dots$
(6.61b)

where the dots on the RHS represent higher moments. The $M_{N}$ are
thermodynamic functions of $\mu$ and $T$ that can be analytically computed,
e.g. the first few terms for massless relativistic fermions are

$\displaystyle M_{0}$ $\displaystyle=\frac{\mu}{2\pi^{2}}$ (6.62a)
$\displaystyle M_{1}$
$\displaystyle=\frac{T^{2}}{4\pi^{2}}\left(\frac{\mu^{2}}{T^{2}}+\frac{\pi^{2}}{3}\right)$
(6.62b) $\displaystyle M_{4}$
$\displaystyle=\frac{T^{4}\mu}{30\pi^{2}}\left(\pi^{2}+\frac{\mu^{2}}{T^{2}}\right)\left(7\pi^{2}+3\frac{\mu^{2}}{T^{2}}\right)$
(6.62c)

while $M_{2}$ and $M_{3}$ are given in (6.48). In (6.61), the non-bar
quantities are dynamical functions of $\mu$ and $T$ that obey the equations of
hydrodynamics, while the bar quantities are understood as constraints without
dynamics.

As they appear now, they do not resemble the generalized relaxations of
Chapter 5. This is because, as we have already discussed (see Chapter 5), the
generalized relaxations added to the equations of linearized hydrodynamics are
agnostic on the full non-linear form of the relaxations. For this reason, if
we want to match the expressions in (6.61) with the relaxations in (5.6), we
must first linearize the equations. Clearly, the equilibrium solution is
chosen so that all the collision integrals, and thus the relaxations terms,
vanish. Consequently, the equilibrium will be given by the zero-velocity state
with $\varepsilon=\bar{\varepsilon}$ and $n=\bar{n}$, to which we add linear
fluctuations as $\varepsilon=\bar{\varepsilon}+\delta\varepsilon$ and
$n=\bar{n}+\delta n$ to obtain the linearized equations of hydrodynamics

$\displaystyle\partial_{t}\delta\varepsilon+\dots$
$\displaystyle=-\frac{\delta\varepsilon}{\tau_{\varepsilon\varepsilon}}-\frac{\delta
n}{\tau_{\varepsilon n}}$ (6.63a) $\displaystyle\partial_{t}\delta n+\dots$
$\displaystyle=-\frac{\delta\varepsilon}{\tau_{n\varepsilon}}-\frac{\delta
n}{\tau_{nn}}$ (6.63b)

where in writing the above non-conservation equations we defined the usual
macroscopic relaxation times in terms of the microscopic $\tau_{N}$ of
(6.4.4). For example, considering the first few terms shown in (6.61), they
are