# Fluctuating Quantum Kinetic Theory

T.R. Kirkpatrick1 and D. Belitz2,3 1Institute for Physical Science and
Technology, University of Maryland, College Park, MD 20742, USA
2 Department of Physics and Institute for Fundamental Science, University of
Oregon, Eugene, OR 97403, USA
3 Materials Science Institute, University of Oregon, Eugene, OR 97403, USA

###### Abstract

We consider a quantum Langevin kinetic equation for a system of fermions. We
first derive the Langevin force noise correlation functions in Landau’s Fermi-
liquid kinetic theory from general considerations. We then use the resulting
equation to calculate the equilibrium dynamic structure factor in the
collisionless regime at low temperatures. The result is in agreement with the
conventional many-body result. We then use the theory to derive both the
fluctuating Navier-Stokes equations for a quantum fluid and the fluctuating
hydrodynamic equations for fermions in the presence of quenched disorder. We
also discuss the modifications needed for the fluctuating hydrodynamic
equations to describe an electron fluid with long-ranged interactions, and we
prove an H-theorem for the nonlinear Landau kinetic equation.

##  I Introduction

A Langevin equation is a stochastic equation of motion that describes a subset
of degrees of freedom of a complex system. Typically, the explicit degrees of
freedom are collective (macroscopic) variables that are assumed to change
slowly with time compared to the remaining, microscopic, degrees of freedom.
The fast microscopic degrees of freedom are not described explicitly, but
rather taken into account via a random force (“Langevin force” or “noise”) in
the macroscopic equations of motion for the collective variables. These ideas
for an effective or reduced theory for complex systems were first developed by
Paul Langevin for the problem of Brownian motion, i.e., a classical particle
coupled to a classical heat bath.Langevin (1908) Ever since they have been
widely used to study problems in both classical and quantum statistical
mechanics.

One generalization of Langevin’s ideas was their application to classical
many-body systems such as fluids, which act as their own heat bath. Examples
are the equations of fluctuating hydrodynamics by Landau and Lifshitz,Landau
and Lifshitz (1959) and some of the models analyzed by Hohenberg and Halperin
in their discussion of critical dynamics.Hohenberg and Halperin (1977) In
these applications the noise correlation is determined as follows: One assumes
that the linearized hydrodynamic equations, with Langevin forces added to the
velocity and heat equations, correctly describe dynamic fluctuations about an
equilibrium state, and that the Langevin forces are delta-correlated in time.
The amplitudes of the noise correlations are then determined by the
requirement that the dynamic equations yield the correct equal-time
correlation functions, which in classical statistical mechanics are related to
thermodynamic quantities. The same reasoning has been used to develop a
fluctuating classical kinetic theory.Dorfman et al. (2021) Here the reduced
description is in terms of the fluctuating one-particle distribution function.

Other generalizations aimed at describing quantum mechanical systems. The
first description of a single quantum particle coupled to a quantum-mechanical
heat bath in terms of a Langevin equation was given by Ford, Kac, and
Mazur;Ford et al. (1965) this work was later extended by others.Caldeira and
Leggett (1983); Ford and Kac (1987); Ford et al. (1988); Olavo et al. (2012)
Some key result were as follows. First, the most general quantum Langevin
equation (QLE) can be realized with specific models. Second, the objects
described by the QLE are Heisenberg operators. Third, the operator-valued
Langevin force is not delta-correlated in time, i.e., the random force is not
Markovian. The operator nature of the terms in the QLE naturally leads one to
consider symmetrized and antisymmetrized correlation functions for both the
physical quantities of interest and for the operator-valued Langevin force.
This in turn is related to the fact that in the quantum limit there are two
types of fundamental correlation functions: dynamic response functions and
dynamic fluctuation functions.Lifshitz and Pitaevskii (1981a); Forster (1975)

This raises the question of how to apply Langevin’s ideas to quantum many-body
systems. Quantum mechanically, the statics and the dynamics couple. As a
result, the equal-time correlation functions in general depend on the dynamics
of the system, are not related to thermodynamic quantities, and are in general
not known. This means that an important ingredient of classical fluctuating
hydrodynamics is no longer applicable. On the other hand, the static response
functions are in general related to thermodynamic quantities, at least for
long wavelengths, even in the quantum limit. This is true in particular for
the static response function that naturally occurs in quantum kinetic theory,
and it is this observation that we will exploit in this paper. In a previous
paper (Ref. Belitz and Kirkpatrick, 2021, to be referred to as Paper I) we
showed that Landau Fermi-liquid (LFL) theory is fully consistent with
hydrodynamics at all temperatures. The present paper relates to Paper I the
same way Boltzmann-Langevin theoryBixon and Zwanzig (1969); Dorfman et al.
(2021) relates to the Boltzmann equation, or flucuating hydrodynamics to
Navier-Stokes theory. In contrast to Paper I, which solved the kinetic
equation and explicitly determined the hydrodynamic frequencies and modes, the
current paper will focus on deriving the relevant hydrodynamic equations.
Specifically, we will use a fluctuating quantum kinetic theory for the one-
particle distribution operator to express the relevant dynamic response
function in terms of known kinetic operators and an anti-symmetrized Langevin
force correlation function. We will then use the known static response
function to determine the amplitude of this noise correlation function. The
fluctuation-dissipation theorem then allows us to determine the symmetrized
Langevin force correlation. From these two results all of the relevant
physical information can be obtained.

The long-term goal of developing this formalism is to provide a method for
describing the behavior of quantum many-body systems in non-equilibrium states
where conventional quantum many-body theory is difficult to use.

The outline of this paper is as follows. In Sec. II we formulate the
fluctuating quantum kinetic theory and show how to derive the correlation
functions for the operator-valued Langevin force. In Sec. III we consider
three applications. In Sec. III.1 we calculate the dynamic structure factor
for a fermionic system in the collisionless regime. In Sec. III.2 we derive
the fluctuating Navier-Stokes equations for a quantum fluid and recover the
results stated in Ref. Lifshitz and Pitaevskii, 1981a. In Sec. III.3 we derive
the corresponding fluctuating hydrodynamic equations for a quantum system with
quenched disorder. We conclude with a discussion and some additional remarks
in Sec. IV. Five appendices deal with technical details and some related
issues. In Appendix A we list some relevant linearized collision operators,
Appendix B contains technical details of the derivation of the Langevin force
correlation, in Appendix C we prove an $H$-theorem for the nonlinear
Boltzmann-Landau kinetic equation, Appendix C lists some useful thermodynamic
identities, and Appendix E sketches the changes necessary for an application
of the theory to electrons with a long-ranged Coulomb interaction.

##  II Derivation of a fluctuating quantum kinetic theory

###  II.1 A fluctuating Boltzmann-Landau equation

Consider the $\mu$-space or single-particle phase spaceEhrenfest and Ehrenfest
(1990) spanned by the momentum ${\bm{p}}$ and the position ${\bm{x}}$ of a
particle. If we perform a Fourier transform from the real-space variable
${\bm{x}}$ to a momentum variable ${\bm{k}}$, the one-particle phase-space
distribution operator, ${\hat{f}}({\bm{p}},{\bm{k}})$, and its fluctuation,
$\delta{\hat{f}}({\bm{p}},{\bm{k}})$, are defined by

${\hat{f}}({\bm{p}},{\bm{k}})={\hat{a}}^{\dagger}_{{\bm{p}}-{\bm{k}}/2}\,{\hat{a}}_{{\bm{p}}+{\bm{k}}/2}=\delta_{{\bm{k}},0}\,f_{\text{eq}}({\bm{p}})+\delta{\hat{f}}({\bm{p}},{\bm{k}})$
(1a) Here ${\hat{a}}^{\dagger}_{\bm{p}}$ and ${\hat{a}}_{\bm{p}}$ are creation
and annihilation operators, respectively, for fermions with momentum
${\bm{p}}$, and
$f_{\text{eq}}({\bm{p}})=\langle{\hat{a}}^{\dagger}_{\bm{p}}\,{\hat{a}}_{\bm{p}}\rangle=\frac{1}{\exp(\xi_{\bm{p}}/T)+1}$
(1b)

is the equilibrium Fermi-Dirac distribution. $\langle\ldots\rangle$ denotes a
quantum-mechanical expectation value plus a statistical mechanics average, and
$\xi_{p}=\epsilon_{p}-\mu$ with $\epsilon_{p}$ the equilibrium single-particle
energy (which depends only on $p=|{\bm{p}}|$) and $\mu$ the chemical
potential. Throughout this paper, by ‘particles’ we mean quasiparticles in the
sense of LFL theory.Lifshitz and Pitaevskii (1981a) As in Paper I we consider
spinless fermions for simplicity, and we use units such that
$\hbar=k_{\text{B}}=1$. We further define an operator-valued function
$\hat{\phi}$ by

$\delta{\hat{f}}({\bm{p}},{\bm{k}})=w({\bm{p}})\,{\hat{\phi}}({\bm{p}},{\bm{k}})$
(2a) where $\displaystyle w({\bm{p}})$ $\displaystyle=$
$\displaystyle-\partial
f_{\text{eq}}({\bm{p}})/\partial\epsilon_{p}=\frac{1}{T}\,f_{\text{eq}}({\bm{p}})\left[1-f_{\text{eq}}({\bm{p}})\right]$
(2b) $\displaystyle=$ $\displaystyle\frac{1}{4T\cosh^{2}(\xi_{p}/2T)}\ ,$

For later reference we define a scalar product $\langle\ldots|\ldots\rangle$
in the space of functions of the momentum ${\bm{p}}$ with $w$ as the weight
function:

$\langle
A({\bm{p}})|B({\bm{p}})\rangle=\frac{1}{V}\sum_{\bm{p}}w({\bm{p}})\,A({\bm{p}})\,B({\bm{p}})\
,$ (3a) with $V$ the system volume.the  The functions $A$ and $B$ can be
number valued or operator valued. In particular, $N_{0}=\langle 1|1\rangle$
(3b) is the normalization of the weight function. Also useful is an average
with respect to the weight function $w$ defined asave  $\langle
A({\bm{p}})\rangle_{w}=\langle A({\bm{p}})|1\rangle/N_{0}\ .$ (3c)

Within LFL theory a small change $\delta{\hat{f}}$ of the distribution leads
to a change $\delta\epsilon$ of the single-particle energy given byLifshitz
and Pitaevskii (1981a, b)

$\delta{\hat{\epsilon}}({\bm{p}},{\bm{x}})=\frac{1}{N_{0}V}\sum_{{\bm{p}}^{\prime}}w({\bm{p}}^{\prime})F({\bm{p}},{\bm{p}}^{\prime})\,\delta{\hat{f}}({\bm{p}}^{\prime},{\bm{x}})\
,$ (4)

where $F({\bm{p}},{\bm{p}^{\prime}})$ is Landau’s interaction function that
parameterizes the interaction between the fermions and we consider a real-
space variable ${\bm{x}}$ instead of the momentum variable ${\bm{k}}$. The
distribution in Eq. (1a) thus corresponds to a single-particle energy

${\hat{\epsilon}}({\bm{p}},{\bm{x}})=\epsilon_{p}+\delta{\hat{\epsilon}}({\bm{p}},{\bm{x}})\
,$ (5)

which is operator valued and spatially varying.

Going back to Fourier space, we assume that the time evolution of the dynamic
fluctuation operator $\hat{\phi}$ is governed by a fluctuating Boltzmann-
Landau kinetic equationdyn

$\left[\partial_{t}+L_{\bm{k}}({\bm{p}})\right]{\hat{\phi}}({\bm{p}},{\bm{k}},t)={\hat{F}}_{\text{L}}({\bm{p}},{\bm{k}},t)$
(6a) or, after a temporal Fourier transform with $\omega$ as the frequency,
$\left[-i\omega+L_{\bm{k}}({\bm{p}})\right]{\hat{\phi}}({\bm{p}},{\bm{k}},\omega)={\hat{F}}_{\text{L}}({\bm{p}},{\bm{k}},\omega)$
(6b) Here $L_{\bm{k}}({\bm{p}})=L_{\bm{k}}^{(1)}({\bm{p}})-\Lambda({\bm{p}})$
(6c) is a linearized kinetic operator. It is comprised of a linearized
collision operator $\Lambda$ and
$L_{\bm{k}}^{(1)}({\bm{p}})=i{\bm{k}}\cdot{\bm{v}}_{p}\,\left[1+I({\bm{p}})\right]$
(6d) where
$I({\bm{p}})=\frac{1}{N_{0}V}\sum_{{\bm{p}}^{\prime}}w({\bm{p}}^{\prime})\,F({\bm{p}}^{\prime},{\bm{p}})\,R_{{\bm{p}}\to{\bm{p}}^{\prime}}$
(6e) with $R$ a replacement operator defined by
$R_{{\bm{p}}\to{\bm{p}}^{\prime}}A({\bm{p}})=A({\bm{p}}^{\prime})$.
${\bm{v}}_{p}={\bm{\nabla}}_{\bm{p}}\epsilon_{p}$ is the quasiparticle
velocity which obeys the integral equationLifshitz and Pitaevskii (1981a)
${\bm{v}}_{p}=\frac{1}{m}\,{\bm{p}}-\frac{1}{N_{0}V}\sum_{{\bm{p}}^{\prime}}w({\bm{p}}^{\prime})\,F({\bm{p}},{\bm{p}})\,{\bm{v}}_{p^{\prime}}\
.$ (6f)

where $m$ is the bare particle mass. Near the Fermi surface,
$\epsilon_{p}=\mu+v_{\text{F}}^{*}(p-p_{\text{F}})$, and hence
${\bm{v}}_{p}={\bm{p}}/m^{*}$ where $p_{\text{F}}$ is the Fermi momentum and
$v_{\text{F}}^{*}=p_{\text{F}}/m^{*}$ is the Fermi velocity with $m^{*}$ the
quasiparticle effective mass. The form of the linearized collision operator
$\Lambda$ depends on the scattering processes considered. In Appendix A we
give examples of collision operators that describe fermion-fermion scattering,
fermion-phonon scattering, and fermion-impurity scattering, respectively.

${\hat{F}}_{\text{L}}$ in Eqs. (6a, 6b) is the fluctuating $\mu$-space
Langevin force operator. We assume that it is Gaussian distributed with zero
mean. The latter property follows because the average of ${\hat{\phi}}$ must
be governed by the averaged kinetic equation. The second moment, which then
completely determines the Gaussian distribution, we will determined below.

###  II.2 Response and correlation functions

As mentioned in the Introduction, there are two different correlation
functions for the variable of interest, which is $\hat{\phi}$.Forster (1975);
Zubarev (1960) One is the $\mu$-space dynamic response function, or commutator
correlation function, $\chi^{\prime\prime}$, defined by

$\frac{1}{2}\Big{\langle}\left[{\hat{\phi}}({\bm{p}}_{1},{\bm{k}}_{1},\omega_{1}),{\hat{\phi}}({\bm{p}}_{2},{\bm{k}}_{2},\omega_{2})\right]_{-}\Big{\rangle}=2\pi\delta(\omega_{1}+\omega_{2})\,V\delta_{{\bm{k}}_{1}+{\bm{k}}_{2},0}\,\chi^{\prime\prime}({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}}_{1},\omega_{1})\
.$ (7a) The other is the $\mu$-space fluctuation function, or anticommutator
correlation function, $\Phi$, defined by
$\frac{1}{2}\Big{\langle}\left[{\hat{\phi}}({\bm{p}}_{1},{\bm{k}}_{1},\omega_{1}),{\hat{\phi}}({\bm{p}}_{2},{\bm{k}}_{2},\omega_{2})\right]_{+}\Big{\rangle}=2\pi\delta(\omega_{1}+\omega_{2})\,V\delta_{{\bm{k}}_{1}+{\bm{k}}_{2},0}\,\Phi({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}}_{1},\omega_{1})\
.$ (7b)

Here $[{\hat{A}},{\hat{B}}]_{\mp}={\hat{A}}\mp{\hat{B}}$.
$\chi^{\prime\prime}$ and $\Phi$ are related by the fluctuation-dissipation
theorem,Callen and Welton (1951); Forster (1975); Pines and Nozières (1989)
which in the present context takes the form

$\Phi({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}},\omega)=\coth(\omega/2T)\,\chi^{\prime\prime}({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}},\omega)\
.$ (8)

$\chi^{\prime\prime}$ is the spectrum of the related causal functionForster
(1975)

$\chi({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}},z)=\int\frac{d\omega}{\pi}\,\frac{\chi^{\prime\prime}({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}},\omega)}{\omega-z}$
(9a) and can be obtained from the latter via the relation
$\chi^{\prime\prime}({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}},\omega)=\lim_{\epsilon\to
0}{\text{Im}}\,\chi({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}},\omega+i\epsilon)$ (9b)

The static $\mu$-space response function can be written as the minus first
frequency moment of $\chi^{\prime\prime}$,

$\displaystyle\chi({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}})$ $\displaystyle=$
$\displaystyle\chi({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}},z=0)$ (10)
$\displaystyle=$
$\displaystyle\int\frac{d\omega}{\pi}\,\frac{\chi^{\prime\prime}({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}},\omega)}{\omega}\
.$

###  II.3 The Langevin force correlation

We are interested in the correlations of the Langevin noise operator
${\hat{F}}$. Temporal and spatial translational invariance imply that they
must have the form

$\frac{1}{2}\Big{\langle}\left[{\hat{F}}_{\text{L}}({\bm{p}}_{1},{\bm{k}}_{1},\omega_{1}),{\hat{F}}_{\text{L}}({\bm{p}}_{2},{\bm{k}}_{2},\omega_{2})\right]_{\pm}\Big{\rangle}=2\pi\delta(\omega_{1}+\omega_{2})\,V\delta_{{\bm{k}}_{1}+{\bm{k}}_{2},0}\,\Psi_{\pm}({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}}_{1},\omega_{1})\
.$ (11)

By using the kinetic equation (6b) in Eq. (7a) we obtain a formal relation
between $\Psi_{-}$ and the dynamic response function,

$\chi^{\prime\prime}({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}},\omega)=\frac{1}{\left[-i\omega+L_{\bm{k}}({\bm{p}}_{1})\right]\left[i\omega+L_{-{\bm{k}}}({\bm{p}}_{2})\right]}\,\Psi_{-}({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}},\omega)\
.$ (12)

We now make an ansatz that assumes that $\Psi_{-}$ is linear in $\omega$,

$\Psi_{-}({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}},\omega)=\omega\,\psi({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}})\
.$ (13)

By using Eq. (10) we then obtain an expression for $\Psi_{-}$ in terms of the
static $\mu$-space response function,

$\Psi_{-}({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}},\omega)=\frac{\omega}{2T}\left[L_{\bm{k}}({\bm{p}}_{1})+L_{-{\bm{k}}}({\bm{p}}_{2})\right]T\chi({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}})\
.$ (14)

This result depends on the collision operator via the kinetic operator
$L_{\bm{k}}({\bm{p}})$, Eq. (6c). The static response function does not depend
on the collision operator, see Appendix B. As a result, the streaming and
interaction terms in $L_{\bm{k}}({\bm{p}})$ do not contribute, and our final
result is

$\Psi_{-}({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}},\omega)=\frac{-\omega}{2T}\left[\Lambda({\bm{p}}_{1})+\Lambda({\bm{p}}_{2})\right]T\chi({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}})\
,$ (15a) where $\chi$ is the solution of the integral equation
$\displaystyle\chi({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}})+\frac{1}{N_{0}V}\sum_{{\bm{p}}^{\prime}}w({\bm{p}}^{\prime})\,F({\bm{p}}_{1},{\bm{p}}^{\prime})\,\chi({\bm{p}}^{\prime},{\bm{p}}_{2};{\bm{k}})$
$\displaystyle=$ (15b)
$\displaystyle\hskip-110.0pt=V\delta_{{\bm{p}}_{1},{\bm{p}}_{2}}\,\frac{1}{w({\bm{p}}_{1})}\
.$ See Appendix B for a derivation of these results. Equation (15b) determines
the static response function. Finally, Eq. (8) provides a corresponding
expression for $\Psi_{+}$,
$\Psi_{+}({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}},\omega)=\coth\left(\frac{\omega}{2T}\right)\Psi_{-}({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}},\omega)$
(15c)

The latter result gives an a posteriori motivation for the ansatz (13): In the
high-temperature limit, where $(\omega/2T)\coth(\omega/2T)\to 1$, Eq. (15c) is
structurally identical to the corresponding result for the classical
fluctuating Boltzmann equation, see Chapter 8 in Ref. Dorfman et al., 2021.

For explicit calculations within LFL theory an explicit interaction model is
often used that performs an angular-momentum expansion of the interaction
function $F({\bm{p}}_{1},{\bm{p}}_{2})$, truncated at first order,

$F({\bm{p}}_{1},{\bm{p}}_{2})=F_{0}+F_{1}\,\frac{{\bm{p}}_{1}\cdot{\bm{p}}_{2}}{\langle{\bm{p}}^{2}\rangle_{w}}\
.$ (16a) The operator $I({\bm{p}})$ from Eq. (6e) then can be written
$I({\bm{p}})=\frac{F_{0}}{\langle 1|1\rangle}\,|1\rangle\langle
1|+\frac{F_{1}}{\langle{\bm{p}}|{\bm{p}}\rangle}\,|{\bm{p}}\rangle\langle{\bm{p}}|\
,$ (16b)

the integral equation (15b) becomes separable,Lan  and we have

$\displaystyle\chi({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}})$ $\displaystyle=$
$\displaystyle
V\delta_{{\bm{p}}_{1},{\bm{p}}_{2}}\,\frac{1}{w({\bm{p}}_{1})}-\frac{1}{N_{0}V}\,\frac{F_{0}}{1+F_{0}}$
(17) $\displaystyle\hskip
10.0pt-\frac{1}{N_{0}V}\,\frac{{\bm{p}}_{1}\cdot{\bm{p}}_{2}}{\langle{\bm{p}}^{2}\rangle_{w}}\,\frac{F_{1}}{1+F_{1}/3}\
.$

This yields the correct result for the static density response function, viz.,
$N_{0}/(1+F_{0})=(\partial n/\partial\mu)_{T,V}$, and the static momentum
response function, $nm^{*}(1+F_{1}/3)=nm$. Note that in general Eq. (17)
implies nontrivial correlations in momentum space. These do not occur in
classical fluids in equlibrium, where only the
$\delta_{{\bm{p}}_{1},{\bm{p}}_{2}}$ term is present. If the collision
operator conserves both particle number and momentum (as is the case, e.g., if
the only collisions considered are fermion-fermion collisions), then neither
of the Landau parameters $F_{0}$ and $F_{1}$ contribute and we have

$\Psi_{-}({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}},\omega)=\frac{-\omega}{2T}\left[\Lambda({\bm{p}}_{1})+\Lambda({\bm{p}}_{2})\right]\,V\delta_{{\bm{p}}_{1},{\bm{p}}_{2}}\,\frac{T}{w({\bm{p}}_{1})}\
.$ (18)

The absence of Landau interaction parameters in the noise correlation reflects
the fact that the interactions included in this model are purely deterministic
and do not contribute to dissipative processes. See also Appendix D, where we
prove an H-theorem for the nonlinear Landau kinetic equation.

##  III Applications of the kinetic theory

###  III.1 The equilibrium structure factor in the collisionless regime

In this section we use the fluctuating kinetic equation derived in the
previous section to calculate the dynamic structure factor
$S({\bm{k}},\omega)$ of a clean neutral Fermi liquid in the collisionless
regime, which is always realized at sufficiently low temperatures and
sufficiently large wave numbers. In this limit, the streaming and interaction
terms in the kinetic operator $L_{\bm{k}}({\bm{p}})$ dominate over the
collision operator, and $-\Lambda({\bm{p}})$ is replaced by a positive
infinitesimal constant $\epsilon$ that regularizes the free-fermion
propagator. The quasiparticle interaction is described by the Landau function
$F({\bm{p}}_{1},{\bm{p}}_{2})$. In order to make contact with well-known
results from conventional many-body theoryFetter and Walecka (1971) we
restrict the calculation to a model where the latter is a constant,
$F({\bm{p}}_{1},{\bm{p}}_{2})=F_{0}$. Equation (6f) then implies
${\bm{v}}_{p}={\bm{p}}/m$, and the kinetic equation (6b) takes the form

$\displaystyle{\hat{\phi}}({\bm{p}},{\bm{k}},\omega)$ $\displaystyle=$
$\displaystyle
G_{0}({\bm{p}},{\bm{k}},\omega)\,{\hat{F}}_{\text{L}}({\bm{p}},{\bm{k}},\omega)$
(19)
$\displaystyle-\frac{{\bm{p}}\cdot{\bm{k}}}{m}\,G_{0}({\bm{p}},{\bm{k}},\omega)\,\frac{F_{0}}{N_{0}}\,\delta{\hat{n}}({\bm{k}},\omega)\
.$

Here

$G_{0}({\bm{p}},{\bm{k}},\omega)=\frac{i}{\omega+i\epsilon-{\bm{p}}\cdot{\bm{k}}/m}$
(20)

is the free-fermion Green function, and

$\delta{\hat{n}}({\bm{k}},\omega)=\frac{1}{V}\sum_{\bm{p}}\delta{\hat{f}}({\bm{p}},{\bm{k}},\omega)=\langle
1|{\hat{\phi}}({\bm{p}},{\bm{k}},\omega)\rangle$ (21)

is the number-density fluctuation operator.

The structure factor is given in terms of the density-density anticommutator
correlation function via the relationcor

$\frac{1}{2}\big{\langle}\left[\delta{\hat{n}}({\bm{k}}_{1},\omega_{1}),\delta{\hat{n}}({\bm{k}}_{2},\omega_{2})\right]_{+}\big{\rangle}=2\pi\delta(\omega_{1}+\omega_{2})\,V\delta_{{\bm{k}}_{1}+{\bm{k}}_{2},0}\,S({\bm{k}}_{1},\omega_{1})\
.$ (22a) Via the fluctuation-dissipation theorem, Eq. (8), it is related to
the density-density response function
$\chi^{\prime\prime}_{nn}({\bm{k}},\omega)=\frac{1}{V^{2}}\sum_{{\bm{p}}_{1},{\bm{p}}_{2}}\chi^{\prime\prime}({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}},\omega)=\tanh(\omega/2T)S({\bm{k}},\omega)\
.$ (22b)

From Eqs. (19) and (21) we obtain an explicit expression for
$\delta{\hat{n}}({\bm{k}},\omega)$,

$\delta{\hat{n}}({\bm{k}},\omega)=\frac{1}{1+F_{0}J_{1}({\bm{k}},\omega)}\,\frac{1}{V}\sum_{\bm{p}}w({\bm{p}})\,G_{0}({\bm{p}},{\bm{k}},\omega)\,{\hat{F}}_{\text{L}}({\bm{p}},{\bm{k}},\omega)\
,$ (23a) where $\displaystyle J_{1}({\bm{k}},\omega)$ $\displaystyle=$
$\displaystyle\frac{-1}{N_{0}V}\sum_{\bm{p}}w({\bm{p}})\,\frac{{\bm{p}}\cdot{\bm{k}}/m}{\omega+i\epsilon-{\bm{p}}\cdot{\bm{k}}/m}$
(23b) $\displaystyle=$ $\displaystyle J_{1}(-{\bm{k}},-\omega)^{*}\ .$

By using Eq. (23a) in (22a) we see that the structure factor can be expressed
in terms of the symmetrized Langevin force correlation $\Psi_{+}$. Using Eq.
(11) we find

$S({\bm{k}},\omega)=\frac{1}{1+F_{0}J_{1}({\bm{k}},\omega)}\,\frac{1}{1+F_{0}J_{1}(-{\bm{k}},-\omega)}\,\frac{1}{V^{2}}\sum_{{\bm{p}}_{1},{\bm{p}}_{2}}w({\bm{p}}_{1})\,w({\bm{p}}_{2})\,G_{0}({\bm{p}}_{1},{\bm{k}},\omega)\,G_{0}({\bm{p}}_{2},-{\bm{k}},-\omega)\,\Psi_{+}({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}},\omega)\
.$ (24)

We now perform the zero-temperature limit in the terms where we have not done
so yet. Then we have

$\displaystyle-
G_{0}({\bm{p}}_{1},{\bm{k}},\omega)\,G_{0}({\bm{p}}_{2},-{\bm{k}},-\omega)\left[\Lambda({\bm{p}}_{1})+\Lambda({\bm{p}}_{2}\right]\delta_{{\bm{p}}_{1},{\bm{p}}_{2}}$
$\displaystyle\to$
$\displaystyle\delta_{{\bm{p}}_{1},{\bm{p}}_{2}}2\pi\delta(\omega-{\bm{p}}_{1}\cdot{\bm{k}}/m)$
(25a) $\displaystyle\omega\coth(\omega/2T)$ $\displaystyle\to$
$\displaystyle\omega\,{\text{sgn\,}}\omega$ (25b)

Finally, the imaginary part of $J_{1}$ is

${\text{Im}}\,\,J_{1}({\bm{k}},\omega)=\frac{\pi}{N_{0}V}\sum_{\bm{p}}w({\bm{p}})\,\frac{{\bm{p}}\cdot{\bm{k}}}{m}\,\delta(\omega-{\bm{p}}\cdot{\bm{k}}/m)\
.$ (26)

Combining all of this we obtain

$S({\bm{k}},\omega)=N_{0}\,\frac{{\text{sgn\,}}\omega\,{\text{Im}}\,J_{1}({\bm{k}},\omega)}{\left(1+F_{0}\,{\text{Re}}\,J_{1}({\bm{k}},\omega)\right)^{2}+\left(F_{0}\,{\text{Im}}\,\,J_{1}({\bm{k}},\omega)\right)^{2}}$
(27)

We recognize this as the standard result obtained from many-body theory in the
so-called ring or random-phase approximation.Fetter and Walecka (1971) See
also Ref. Belitz and Kirkpatrick, 2012, where the same result was obtained by
field-theoretic methods. Its physical contents include the well known $\ell=0$
zero-sound excitation and the particle-hole continuum that were discussed in
Paper I.

###  III.2 Fluctuating Navier-Stokes equations for a fermionic quantum fluid

We now consider the hydrodynamic regime, where the collision operator
dominates over the operator $L_{\bm{k}}^{(1)}$. It is realized in the limit of
small frequencies, $\omega\tau\ll 1$, and wave numbers, $k\ell\ll 1$, with
$\tau$ a generic collision time and $\ell$ a generic mean-free path. As in the
previous subsection, we consider a neutral Fermi liquid; at the end of this
subsection we will discuss the modifications necessary for discussing metals.
For simplicity we will use the simple quasiparticle model interaction given in
Eq. (16a). Within this model, ${\bm{v}}_{p}={\bm{p}}/m^{*}$ and
$\epsilon_{p}={\bm{p}}^{2}/2m^{*}$. We ignore a $p$-independent contribution
to $\epsilon_{p}$ that also depends on the FL interaction,Fetter and Walecka
(1971) as we did in Paper I.

The quantum mechanical generalization of the Boltzmann equation for this case
was first formulated by Uehling and Uhlenbeck.Uehling and Uhlenbeck (1933) A
crucial aspect of any kinetic theory for a fluid, quantum or classical, is
that the linearized collision operator has five (or $d+2$ in $d$ spatial
dimensions) zero eigenvalues, and corresponding zero eigenfunctions, that
represent the conservation of particle number, momentum, and energy.

The relevant hydrodynamic variables are the density fluctuation
$\delta{\hat{n}}$, the three components of the fluid velocity fluctuation
$\delta{\hat{u}}_{i}$ ($i=x,y,z$), and the temperature fluctuation
$\delta{\hat{T}}$. With the scalar product defined in Eq. (3a) they can be
expressed as follows:

$\displaystyle\delta{\hat{n}}({\bm{k}},\omega)$ $\displaystyle=$
$\displaystyle\langle
a_{1}({\bm{p}})|\hat{\bm{\phi}}({\bm{p}},{\bm{k}},\omega)\rangle\ ,$ (28a)
$\displaystyle\delta{\hat{u}}_{i}({\bm{k}},\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{nm}\langle
a_{i}({\bm{p}})|\hat{\bm{\phi}}({\bm{p}},{\bm{k}},\omega)\rangle\ ,$ (28b)
$\displaystyle\delta{\hat{T}}({\bm{k}},\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{c_{V}}\,\langle
a_{5}({\bm{p}})|\hat{\bm{\phi}}({\bm{p}},{\bm{k}},\omega)\rangle\ ,$ (28c)
where $c_{V}$ is the specific heat at constant volume. Also of interest is the
entropy density fluctuation
$\delta{\hat{s}}({\bm{k}},\omega)=\frac{1}{T}\left\langle\epsilon_{p}-\mu\Big{|}{\hat{\phi}}({\bm{p}},{\bm{k}},\omega)\right\rangle\
,$ (28d) the pressure fluctuation
$\delta{\hat{p}}({\bm{k}},\omega)=\left(\frac{\partial p}{\partial
n}\right)_{T,V}\delta{\hat{n}}({\bm{k}},\omega)+\left(\frac{\partial
p}{\partial T}\right)_{N,V}\delta{\hat{T}}({\bm{k}},\omega)\ ,$ (28e) and the
energy density fluctuation
$\delta{\hat{e}}({\bm{k}},\omega)=\langle\epsilon_{p}|{\hat{\phi}}({\bm{p}},{\bm{k}},\omega)\rangle\
,$ (28f)

The five functions $a_{\alpha}({\bm{p}})$ are

$\displaystyle a_{1}({\bm{p}})$ $\displaystyle=$ $\displaystyle 1\ ,$ (29a)
$\displaystyle a_{i}({\bm{p}})$ $\displaystyle=$ $\displaystyle
p_{i}\qquad(i=x,y,z)\ ,$ (29b) $\displaystyle a_{5}({\bm{p}})$
$\displaystyle=$ $\displaystyle\epsilon_{p}-\langle\epsilon_{p}\rangle_{w}\ .$
(29c)

The validity of Eqs. (28a, 28b, 28e, 28f) is obvious. For a derivation of Eq.
(28c) see Appendix A 2 in Paper I, and for (28d) see Eq. (3.18) in Paper I.

The five functions $a_{\alpha}({\bm{p}})$ are eigenfunctions of the collision
operator for the five zero eigenvalues that represent the conservation of
particle number, momentum, and energy,

$\Lambda({\bm{p}})\,|a_{\alpha}({\bm{p}})\rangle=0\ .$ (30)

They span the hydrodynamic subspace ${\cal L}_{0}$. This, combined with the
fact that the collisions dominate the physics, implies that one can derive
hydrodynamic equations by using a projection operator ${\cal P}$ defined by

${\cal P}=\sum_{\alpha}\frac{|a_{\alpha}({\bm{p}})\rangle\langle
a_{\alpha}({\bm{p}})|}{\langle
a_{\alpha}({\bm{p}})|a_{\alpha}({\bm{p}})\rangle}\ .$ (31a) that projects onto
${\cal L}_{0}$. The projector onto the complementary space ${\cal L}_{\perp}$
is ${\cal P}_{\perp}=\openone-{\cal P}\ ,$ (31b)

where $\openone$ is the unit operator.

####  III.2.1 The projected $\mu$-space distribution

Now consider the kinetic equation (6b) and operate from the left with ${\cal
P}$. As in the classical Boltzmann-Langevin equation,Dorfman et al. (2021) the
fluctuating force has no overlap with the five zero eigenfunctions
$a_{\alpha}({\bm{p}})$ of the collision operator, Eqs. (29, 30). This can be
seen explicitly from Eq. (18): The function $\Psi_{-}$ has no overlap with
either $a_{\alpha}({\bm{p}}_{1})$ or $a_{\alpha}({\bm{p}}_{2})$. Since the
(zero) first moment of ${\hat{F}}_{\text{L}}$ and the second moment $\Psi_{-}$
completely determine the distribution we thus have

${\cal P}|{\hat{F}}_{\text{L}}({\bm{p}},{\bm{k}},\omega)\rangle=0\ .$ (32a)
Equation (6b) then implies $\left[-i\omega{\cal P}+{\cal
P}L_{\bm{k}}^{(1)}({\bm{p}})\right]|{\hat{\phi}}({\bm{p}},{\bm{k}},\omega)\rangle=0\
.$ (32b)

By using Eq. (31b) this can be written

$\left[-i\omega+{\cal P}L_{\bm{k}}^{(1)}({\bm{p}})\right]{\cal
P}|{\hat{\phi}}({\bm{p}},{\bm{k}},\omega)\rangle=-{\cal
P}L_{\bm{k}}^{(1)}({\bm{p}}){\cal
P}_{\perp}|{\hat{\phi}}({\bm{p}},{\bm{k}},\omega)\rangle$ (33)

Our goal is to obtain a closed equation for the projected $\mu$-space
distribution ${\cal P}|{\hat{\phi}}\rangle$. To this end we multiply Eq. (6b)
from the left with ${\cal P}_{\perp}$, which yields

$\left[-i\omega+{\cal
P}_{\perp}L_{\bm{k}}^{(1)}({\bm{p}})-\Lambda({\bm{p}})\right]{\cal
P}_{\perp}|{\hat{\phi}}({\bm{p}},{\bm{k}},\omega)\rangle=-{\cal
P}_{\perp}L_{\bm{k}}^{(1)}({\bm{p}}){\cal
P}|{\hat{\phi}}({\bm{p}},{\bm{k}},\omega)\rangle+{\cal
P}_{\perp}|{\hat{F}}_{\text{L}}({\bm{p}},{\bm{k}},\omega)\rangle\ .$ (34)

Here we have used Eq. (30), which implies ${\cal
P}_{\perp}\Lambda({\bm{p}})=\Lambda({\bm{p}}){\cal
P}_{\perp}=\Lambda({\bm{p}})$.

We are interested in hydrodynamic equations that are valid for small wave
numbers up to $O(k^{2})$, with the fluctuating force included to lowest order
in $k$. In a neutral Fermi liquid the hydrodynamic frequency scales either
linearly or quadratically with $k$, while the collision operator acting on the
orthogonal space ${\cal L}_{\perp}$ scales as a constant.col  Therefore, the
first two terms in brackets on the left-hand side of Eq. (34) are small
corrections to $\Lambda$ and we can approximate ${\cal P}_{\perp}{\hat{\phi}}$
by

${\cal
P}_{\perp}|{\hat{\phi}}({\bm{p}},{\bm{k}},\omega)\rangle\approx\Lambda_{\perp}^{-1}({\bm{p}})L_{\bm{k}}^{(1)}({\bm{p}}){\cal
P}|{\hat{\phi}}({\bm{p}},{\bm{k}},\omega)\rangle-\Lambda_{\perp}^{-1}({\bm{p}})|{\hat{F}}_{\text{L}}({\bm{p}},{\bm{k}},\omega)\rangle\
.$ (35)

Here $\Lambda_{\perp}^{-1}({\bm{p}})={\cal
P}_{\perp}\left(\Lambda({\bm{p}})\right)^{-1}{\cal P}_{\perp}$, and we have
used the projector property ${\cal P}_{\perp}^{2}={\cal P}_{\perp}$.
Corrections are of relative $O(k)$. By using Eq. (35) in (33) we obtain a
closed formal equation for ${\cal P}{\hat{\phi}}$,

$\left[-i\omega+{\cal P}L_{\bm{k}}^{(1)}({\bm{p}})\right]{\cal
P}|{\hat{\phi}}({\bm{p}},{\bm{k}},\omega)\rangle=-{\cal
P}L_{\bm{k}}^{(1)}({\bm{p}})\Lambda_{\perp}^{-1}({\bm{p}})L_{\bm{k}}^{(1)}({\bm{p}}){\cal
P}|{\hat{\phi}}({\bm{p}},{\bm{k}},\omega)\rangle+{\cal
P}L_{\bm{k}}^{(1)}({\bm{p}})\Lambda_{\perp}^{-1}({\bm{p}})|{\hat{F}}_{\text{L}}({\bm{p}},{\bm{k}},\omega)\rangle\
.$ (36)

####  III.2.2 The fluctuating Navier-Stokes equations

We can now derive the five hydrodynamic equations by multiplying Eq. (36) from
the left with $\langle a_{\alpha}({\bm{p}})|$. For the remainder of this
subsection we will write the hydrodynamic quantities as functions of the real-
space position ${\bm{x}}$ rather than the wave vector ${\bm{k}}$.

##### Density equation

Multiplying Eq. (36) from the left with $\langle 1|$ yields the continuity
equation for the density,

$-i\omega\,\delta{\hat{n}}({\bm{x}},\omega)+n{\bm{\nabla}}\cdot\delta{\hat{\bm{u}}}({\bm{x}},\omega)=0\
.$ (37)

The number current density given by $n{\hat{\bm{u}}}$ is exact, and neither
the collision operator nor the fluctuating force enter the density equation.

##### Fluid velocity equation

The equation for the velocity, which we obtain by multiplying Eq. (36) from
the left with $\langle{\bm{p}}|$, is more involved. The projector $P_{\perp}$
acting on functions that are even (odd) in ${\bm{p}}$ yields functions that
also are even (odd) in ${\bm{p}}$, and the inverse collision operator
$\Lambda^{-1}({\bm{p}})$ is isotropic in momentum space. As a result, the
angular integrations make sure that the contributions from the various modes
on the right-hand side of Eq. (36) do not couple, and we find

$-i\omega\,nm\,\delta{\hat{u}}_{i}({\bm{x}},\omega)+\partial_{j}\,{\hat{\tau}}_{ij}({\bm{x}},\omega)+\partial_{j}\left({\hat{\tau}}_{\text{L}}\right)_{ij}({\bm{x}},\omega)=0\
.$ (38a) Here $\displaystyle{\hat{\tau}}_{ij}({\bm{x}},\omega)$
$\displaystyle=$ $\displaystyle\delta_{ij}\,\delta{\hat{p}}({\bm{x}},\omega)$
(38b)
$\displaystyle\hskip-50.0pt-\eta\left[\partial_{i}\,\delta{\hat{u}}_{j}({\bm{x}},\omega)+\partial_{j}\,{\hat{u}}_{i}({\bm{x}},\omega)-\frac{2}{3}\,\delta_{ij}\,{\bm{\nabla}}\cdot{\hat{\bm{u}}}({\bm{x}},\omega)\right]$
$\displaystyle+\zeta\,\delta_{ij}\,{\bm{\nabla}}\cdot{\hat{\bm{u}}}({\bm{x}},\omega)$
is the stress tensor. Its reactive part is given in terms of the pressure
fluctuation $\delta{\hat{p}}$, Eq. (28e). The dissipative part has the form
familiar from the classical Navier-Stokes equations.Forster (1975); Landau and
Lifshitz (1959) $\eta$ is the shear viscosity given by
$\eta=-{\hat{k}}_{\perp}^{i}{\hat{k}}^{j}\,{\hat{k}}_{\perp}^{l}{\hat{k}}^{m}\left\langle\sigma_{ij}({\bm{p}})|\Lambda^{-1}({\bm{p}})|\sigma_{lm}({\bm{p}})\right\rangle\
.$ (38c) Here $\hat{\bm{k}}$ is the unit vector in the direction of
${\bm{k}}$, and $\hat{\bm{k}}_{\perp}$ is either of the two vectors
perpendicular to ${\bm{k}}$. The $\mu$-space momentum current $\sigma_{ij}$ is
$|\sigma^{ij}({\bm{p}})\rangle=|\sigma_{1}^{ij}({\bm{p}})\rangle-\delta^{ij}\,|\sigma_{2}({\bm{p}})\rangle$
(38d) with
$|\sigma_{1}^{ij}({\bm{p}})\rangle=|p^{i}v_{p}^{j}\rangle-\delta^{ij}\frac{1}{3}\,|{\bm{p}}\cdot{\bm{v}}_{p}\rangle\
.$ (38e) and $|\sigma_{2}({\bm{p}})\rangle=\frac{-1}{3}\,{\cal
P}_{\perp}|{\bm{p}}\cdot{\bm{v}}_{p}\rangle\ .$ (38f) $\zeta$ is a
contribution to the bulk viscosity given by
$\zeta=-\langle\sigma_{2}({\bm{p}})|\Lambda^{-1}({\bm{p}})|\sigma_{2}({\bm{p}})\rangle\
.$ (38g) Note that within the simple model characterized by Eq. (16a),
${\bm{p}}\cdot{\bm{v}}_{p}=2\epsilon_{p}\in{\cal L}_{0}$, and hence
$\sigma_{2}({\bm{p}})$ vanishes, but it is nonzero in general. For an
expression of $\sigma_{2}$ in terms of thermodynamic quantities see Paper I.
Finally,
$\left({\hat{\tau}}_{\text{L}}\right)_{ij}({\bm{x}},\omega)=\left\langle\sigma_{ij}({\bm{p}})|\Lambda^{-1}({\bm{p}})|{\hat{F}}_{\text{L}}({\bm{p}},{\bm{x}},\omega)\right\rangle$
(38h)

is the fluctuating part of the stress tensor. In deriving Eqs. (38) we have
made use of various thermodynamic relations that are derived in Appendix A of
Paper I; the most important ones are also given in Appendix E.

##### Temperature or heat equation

Finally, multiplying Eq. (18) from the left with $\langle a_{5}({\bm{p}})|$
yields the heat equation. The calculation yields

$\displaystyle-i\omega\,c_{V}\delta{\hat{T}}({\bm{x}},\omega)$
$\displaystyle+$ $\displaystyle T\left(\frac{\partial p}{\partial
T}\right)_{N,V}{\bm{\nabla}}\cdot{\hat{\bm{u}}}({\bm{x}},\omega)$ (39a)
$\displaystyle\hskip-60.0pt-\kappa\,{\bm{\nabla}}^{2}\delta{\hat{T}}({\bm{x}},\omega)+{\bm{\nabla}}\cdot{\hat{\bm{q}}}_{\text{L}}({\bm{x}},\omega)=0\
.$ Here $\kappa=\frac{-1}{T}\,{\hat{k}}_{i}{\hat{k}}_{j}\left\langle
v_{p}^{i}\,\psi_{5}^{\text{L}(0)}({\bm{p}})|\Lambda^{-1}({\bm{p}})|v_{p}^{j}\,\psi_{5}^{\text{L}(0)}({\bm{p}})\right\rangle$
(39b) is the heat conductivity with
$\psi_{5}^{\text{L}(0)}({\bm{p}})=a_{5}({\bm{p}})-\frac{T}{n}\left(\frac{\partial
p}{\partial T}\right)_{N,V}a_{1}({\bm{p}})$ (39c) the heat mode to zeroth
order in the wave vector ${\bm{k}}$; see Paper I for the identification of the
heat mode and Appendix C for a relation between the present approach and the
kinetic-theory approach in Paper I. The final term in Eq. (39a) is the
fluctuating heat current, given by
${\hat{\bm{q}}}_{\text{L}}({\bm{x}},\omega)=-\langle{\bm{v}}_{p}\,\psi_{5}^{\text{L}(0)}({\bm{p}})|\Lambda^{-1}({\bm{p}})|{\hat{F}}_{\text{L}}({\bm{p}},{\bm{x}},\omega)\rangle\
.$ (39d)

Alternatively, we can write the heat equation in terms of the time derivative
of either the energy density fluctuation $\delta{\hat{e}}$ or the entropy
density fluctuation $\delta{\hat{s}}$ instead of $\delta{\hat{T}}$. From Eqs.
(39a) and (37), and using the identity (81), we find the heat equation in the
formForster (1975)

$\displaystyle-i\omega\,\delta{\hat{e}}({\bm{x}},\omega)$ $\displaystyle+$
$\displaystyle(e+p){\bm{\nabla}}\cdot\delta{\hat{\bm{u}}}({\bm{x}},\omega)-\kappa\,{\bm{\nabla}}^{2}\delta{\hat{T}}({\bm{x}},\omega)$
(40) $\displaystyle+$
$\displaystyle{\bm{\nabla}}\cdot{\hat{\bm{q}}}_{\text{L}}({\bm{x}},\omega)=0\
.$

Finally, using Eq. (37) to eliminate the
${\bm{\nabla}}\cdot\delta{\hat{\bm{u}}}$ term in (40) yields it in the
formLandau and Lifshitz (1959); Lifshitz and Pitaevskii (1981a)

$-i\omega\,Tn\,\delta{\hat{s}}_{n}({\bm{x}},\omega)-\kappa\,{\bm{\nabla}}^{2}\delta{\hat{T}}({\bm{x}},\omega)+{\bm{\nabla}}\cdot{\hat{\bm{q}}}_{\text{L}}({\bm{x}},\omega)=0\
,$ (41a) where
$\delta{\hat{s}}_{n}=\frac{1}{n}\,\delta{\hat{s}}-\frac{s}{n^{2}}\,\delta{\hat{n}}=\frac{1}{Tn}\left(\delta{\hat{e}}-\frac{e+p}{n}\,\delta{\hat{n}}\right)$
(41b)

is the fluctuation of the entropy per particle (see Eq. (3.18) in Paper I).
Equations (39a), (40), and (41) are pairwise equivalent.

##### Fluctuating force correlations

We still need to determine the correlations of the fluctuating forces in the
hydrodynamic equations. From Eqs. (11), (15c), and (18), and repeatedly using
the fact that $\Lambda({\bm{p}})$ and $\Lambda^{-1}({\bm{p}})$ are self-
adjoint with respect to the scalar product $\langle\ldots|\ldots\rangle$, we
find for the anticommutator correlation of the fluctuating stress tensor, Eq.
(38h),

$\frac{1}{2}\Big{\langle}\left[\left({\hat{\tau}}_{\text{L}}\right)_{ij}({\bm{x}}_{1},\omega_{1}),\left({\hat{\tau}}_{\text{L}}\right)_{kl}({\bm{x}}_{2},\omega_{2})\right]_{+}\Big{\rangle}=2\pi\delta(\omega_{1}+\omega_{2})\,\delta({\bm{x}}_{1}-{\bm{x}}_{2})\,\omega_{1}\coth(\omega_{1}/2T)\left[\eta(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})+(\zeta-\frac{1}{3}\,\eta)\delta_{ij}\delta_{kl}\right]\
.$ (42a) Similarly, the anticommutator correlation of the fluctuating heat
current, Eq. (39d), is
$\frac{1}{2}\big{\langle}[q_{\text{L}}^{i}({\bm{x}}_{1},\omega_{1}),q_{\text{L}}^{j}({\bm{x}}_{2},\omega_{2})]_{+}\big{\rangle}=\delta^{ij}2\pi\delta(\omega_{1}+\omega_{2})\,\delta({\bm{x}}_{1}-{\bm{x}}_{2})\,\omega_{1}T\coth(\omega_{1}/2T)\,\kappa\
.$ (42b) and the cross correlations vanish,
$\frac{1}{2}\Big{\langle}\left[\left({\hat{\tau}}_{\text{L}}\right)_{ij}({\bm{x}}_{1},\omega_{1}),q_{\text{L}}^{k}({\bm{x}}_{2},\omega_{2})\right]_{+}\Big{\rangle}=0\
.$ (42c)

The corresponding commutator correlations are given by the same expressions
with $\coth(\omega_{1}/2T)$ replaced by $1$.

The equations (42) are identical to those given in §88 of Ref. Lifshitz and
Pitaevskii, 1981a without a derivation.

###  III.3 Fluctuating hydrodynamic equations for a disordered Fermi liquid

We now consider fermions in the presence of quenched disorder. In this
situation particle number and energy are still conserved, but the fermion
momentum is not. For a derivation of hydrodynamic equations for such a system
we can still use Eq. (36), but the projector ${\cal P}$ now projects on a
space that is spanned by $a_{1}({\bm{p}})$ and $a_{5}({\bm{p}})$ only.
Accordingly, ${\cal P}|{\bm{p}}\rangle={\cal P}|{\bm{v}}_{p}\rangle=0$ and
$\Lambda({\bm{p}})$ has only two zero eigenvalues with eigenvectors
$a_{1}({\bm{p}})$ and $a_{5}({\bm{p}})$. In addition to the fermion-impurity
scattering $\Lambda({\bm{p}})$ can contain parts that describe other
scattering processes, in particular fermion-fermion scattering, as long as
they conserve energy. In this subsection we use a model where only the LFL
parameter $F_{0}$ is nonzero.

####  III.3.1 Fluctuating hydrodynamic equations

##### Density equation

Multiplying Eq. (36) from the left with $\langle a_{1}({\bm{p}})|$ yields an
equation for the density fluctuation. The streaming/interaction term on the
left-hand side does not contribute, but the dissipative and fluctuating terms
both do. A calculation that is analogous to the one in Sec. III.2.2 yields

$\displaystyle-i\omega\,\delta{\hat{n}}({\bm{x}},\omega)$ $\displaystyle=$
$\displaystyle
D_{11}{\bm{\nabla}}^{2}\,\delta{\hat{n}}({\bm{x}},\omega)+D_{12}\,{\bm{\nabla}}^{2}\,\delta{\hat{T}}({\bm{x}},\omega)$
(43a) $\displaystyle\hskip
50.0pt-{\bm{\nabla}}\cdot{\bm{j}}_{\text{L}}({\bm{x}},\omega)\ .$ Here
$D_{11}=\frac{-1}{m^{2}}\,\frac{1}{(\partial
n/\partial\mu)_{T,V}}\,\langle{\hat{\bm{k}}}\cdot{\bm{p}}|\Lambda^{-1}({\bm{p}})|{\hat{\bm{k}}}\cdot{\bm{p}}\rangle\
,$ (43b) and
$D_{12}=\frac{-1}{m^{2}}\,\frac{1}{T}\,\langle{\hat{\bm{k}}}\cdot{\bm{p}}|\Lambda^{-1}({\bm{p}})|({\hat{\bm{k}}}\cdot{\bm{p}})a_{5}({\bm{p}})\rangle$
(43c) are transport coefficients and
${\bm{j}}_{\text{L}}({\bm{x}},\omega)=-\,\langle{\bm{v}}_{p}|\Lambda^{-1}({\bm{p}})|{\hat{F}}({\bm{p}},{\bm{x}},\omega)\rangle$
(43d)

is a fluctuating force that dimensionally is a number current density. Note
that the absence of momentum conservation completely changes the structure of
the density equation compared to the continuity equation in Sec. III.2.2.

##### Heat equation

Multiplying Eq. (36) from the left with $\langle a_{5}({\bm{p}})|$ yields

$\displaystyle-i\omega\,c_{V}\delta{\hat{T}}({\bm{x}},\omega)$
$\displaystyle=$ $\displaystyle\frac{T}{(\partial
n/\partial\mu)_{N,V}}\,D_{12}\,{\bm{\nabla}}^{2}\delta{\hat{n}}({\bm{x}},\omega)$
(44a)
$\displaystyle\hskip-50.0pt+{\bar{\kappa}}\,{\bm{\nabla}}^{2}\delta{\hat{T}}({\bm{x}},\omega)-{\bm{\nabla}}\cdot{\bm{\bar{q}}}({\bm{x}},\omega)\
.$ Here ${\bar{\kappa}}=\frac{-1}{T}\,{\hat{k}}_{i}{\hat{k}}_{j}\left\langle
v_{p}^{i}\,a_{5}({\bm{p}})|\Lambda^{-1}({\bm{p}})|v_{p}^{j}\,a_{5}({\bm{p}})\right\rangle$
(44b) is a modified heat conductivity, and
${\bm{\bar{q}}}_{\text{L}}({\bm{x}},\omega)=-\langle
a_{5}({\bm{p}}){\bm{v}}_{p}|\Lambda^{-1}({\bm{p}})|{\hat{F}}({\bm{p}},{\bm{x}},\omega)\rangle$
(44c)

is the relevant fluctuating force.

##### Fluctuating force correlations

For the anticommutator correlations of the fluctuating forces we find

$\displaystyle\frac{1}{2}\big{\langle}[v_{\text{L}}^{i}({\bm{x}}_{1},\omega_{1}),v_{\text{L}}^{j}({\bm{x}}_{2},\omega_{2})]_{+}\big{\rangle}$
$\displaystyle=$
$\displaystyle\delta^{ij}2\pi\delta(\omega_{1}+\omega_{2})\,\delta({\bm{x}}_{1}-{\bm{x}}_{2})\,\omega_{1}\coth(\omega_{1}/2T)\,\frac{1}{Tn^{2}}\,D_{11}\
,$ (45a)
$\displaystyle\frac{1}{2}\big{\langle}[{\bar{q}}_{\text{L}}^{i}({\bm{x}}_{1},\omega_{1}),{\bar{q}}_{\text{L}}^{j}({\bm{x}}_{2},\omega_{2})]_{+}\big{\rangle}$
$\displaystyle=$
$\displaystyle\delta^{ij}2\pi\delta(\omega_{1}+\omega_{2})\,\delta({\bm{x}}_{1}-{\bm{x}}_{2})\,\omega_{1}\coth(\omega_{1}/2T)\,{\bar{\kappa}}\
,$ (45b)
$\displaystyle\frac{1}{2}\big{\langle}[{\bar{q}}_{\text{L}}^{i}({\bm{x}}_{1},\omega_{1}),v_{\text{L}}^{j}({\bm{x}}_{2},\omega_{2})]_{+}\big{\rangle}$
$\displaystyle=$
$\displaystyle\delta^{ij}2\pi\delta(\omega_{1}+\omega_{2})\,\delta({\bm{x}}_{1}-{\bm{x}}_{2})\,\omega_{1}\coth(\omega_{1}/2T)\,\frac{1}{n}\,D_{12}\
.$ (45c)

The corresponding commutator correlations are again given by the same
expressions with $\coth(\omega_{1}/2T)$ replaced by $1$.

####  III.3.2 Onsager relations

We now show that the results in Sec. III.3.1 are consistent with irreversible
thermodynamics and the Onsager reciprocal relations.

In the absence of momentum conservation the relevant currents are the number
current density ${\bm{j}}_{n}$ and the heat or entropy current density
${\bm{j}}_{s}$.Callen (1985); Mahan (1981) They are driven by gradients of the
chemical potential and the temperature, which defines Onsager coefficients
$L_{ij}$ via the linear-response relations

$\displaystyle{\bm{j}}_{n}$ $\displaystyle=$
$\displaystyle\frac{-L_{11}}{T}\,{\bm{\nabla}}\mu-\frac{L_{12}}{T^{2}}\,{\bm{\nabla}}T\
,$ (46a) $\displaystyle{\bm{j}}_{s}$ $\displaystyle=$
$\displaystyle\frac{-L_{21}}{T}\,{\bm{\nabla}}\mu-\frac{L_{22}}{T^{2}}\,{\bm{\nabla}}T\
.$ (46b)

The relevant Onsager relation consists of the statement $L_{21}=L_{12}$. The
dynamic equations associated with these currents are

$\displaystyle\partial_{t}n$ $\displaystyle=$
$\displaystyle-{\bm{\nabla}}\cdot{\bm{j}}_{n}\ ,$ (47a) $\displaystyle
T\,\partial_{t}s$ $\displaystyle=$
$\displaystyle-{\bm{\nabla}}\cdot{\bm{j}}_{s}\ .$ (47b)

Equation (47a) is the continuity equation for the number density, see Eq.
(B1a) in Paper I. Equation (47b) expresses energy conservation. In order to
relate it to Eq. (B4a) in Paper I we note that the heat mode $\psi_{5}^{(0)}$
in that paper is a linear combination of the entropy density and the number
density fluctuations.ent

Ignoring fluctuations, the hydrodynamic equation (43a) yields

${\bm{j}}_{n}=-D_{11}{\bm{\nabla}}n-D_{12}{\bm{\nabla}T}\ .$ (48)

In order to relate this to Eq. (46a) we express ${\nabla\mu}$ in terms of
${\bm{\nabla}}n$ and ${\bm{\nabla}}T$ via

$\displaystyle{\bm{\nabla}}\mu$ $\displaystyle=$
$\displaystyle\left(\frac{\partial\mu}{\partial
n}\right)_{T,V}{\bm{\nabla}}n+\left(\frac{\partial\mu}{\partial
T}\right)_{N,V}{\bm{\nabla}}T$ (49) $\displaystyle=$
$\displaystyle\frac{1}{(\partial
n/\partial\mu)_{T,V}}{\bm{\nabla}}n-\langle\epsilon_{p}-\mu\rangle_{w}\frac{1}{T}\,{\bm{\nabla}}T\
,\quad$

where we have used Eq. (82). Comparing coefficients in Eqs. (46a) and (48)
yields

$\displaystyle L_{11}$ $\displaystyle=$
$\displaystyle\frac{-T}{m^{2}}\,\langle{\hat{\bm{k}}}\cdot{\bm{p}}|\Lambda^{-1}({\bm{p}})|{\hat{\bm{k}}}\cdot{\bm{p}}\rangle\
,$ (50) $\displaystyle L_{12}$ $\displaystyle=$
$\displaystyle\frac{-T}{m^{2}}\,\langle{\hat{\bm{k}}}\cdot{\bm{p}}|\Lambda^{-1}({\bm{p}})|({\hat{\bm{k}}}\cdot{\bm{p}})(\epsilon_{p}-\mu)\rangle\
.$ (51)

In order to obtain the remaining two Onsager coefficients we note that entropy
density, temperature, and number density fluctuations are related by (see Eq.
(3.18) in Paper I)

$T\delta s=c_{V}\delta T-T(\partial\mu/\partial T)_{N,V}\delta n\ .$ (52a)
With Eqs. (47) this yields
$c_{V}\partial_{t}T=-{\bm{\nabla}}\cdot{\bm{j}}_{s}-T\left(\frac{\partial\mu}{\partial
T}\right)_{N,V}{\bm{\nabla}}\cdot{\bm{j}}_{n}\ .$ (52b)

By using Eqs. (44), neglecting the fluctuating force term, this allows us to
express ${\bm{j}}_{s}$ in terms of density and temperature gradients,

$\displaystyle{\bm{j}}_{s}$ $\displaystyle=$
$\displaystyle\left[T\left(\frac{\partial\mu}{\partial
T}\right)_{N,V}D_{11}-T\left(\frac{\partial\mu}{\partial
n}\right)_{T.V}D_{12}\right]{\bm{\nabla}}n$ (53)
$\displaystyle+\left[T\left(\frac{\partial\mu}{\partial
T}\right)_{N,V}D_{12}-{\bar{\kappa}}\right]{\bm{\nabla}}T\ .$

By comparing the coefficients in Eqs. (53) and (46b), and using Eqs. (49) and
(82) we find

$\displaystyle L_{21}$ $\displaystyle=$ $\displaystyle L_{12}\ ,$ (54)
$\displaystyle L_{22}$ $\displaystyle=$
$\displaystyle\frac{-T}{m^{2}}\langle({\hat{\bm{k}}}\cdot{\bm{p}})(\epsilon_{p}-\mu)|\Lambda^{-1}({\bm{p}})|({\hat{\bm{k}}}\cdot{\bm{p}})(\epsilon_{p}-\mu)\rangle\
.$ (55)

Equation (54) is the required Onsager reciprocal relation. Note that the heat
current that appears in Eqs. (51) and (55) is the one associated with the
entropy density, Eq. (28d). It is different from the heat current that
determines the thermal conductivity in a fluid with momentum conservation
(which is associated with the entropy per particle, see Paper I), and also
from the one that determines the transport coefficient $\bar{\kappa}$, Eq.
(44b).

##  IV Discussion, and Outlook

We conclude with a discussion of some aspects of the kinetic theory and its
applications as formulated in Secs. II and III, as well an outline of some
applications and generalizations that we leave for future work.

###  IV.1 General aspects of the kinetic theory

We have considered a kinetic theory that provides an effective, or reduced,
description of a fermionic many-body system in terms of the $\mu$-space
distribution function. In general, the latter depends on all of the
$N$-particle distribution functions. The Boltzmann equation closes the system
by factorizing the two-particle distribution. In the Landau-Boltzmann equation
the simple product of single-particle distribution functions is generalized to
account for the Pauli principle, see Eq. (73). This procedure is controlled in
the limit of low density (e.g., for a classical dilute gas), but not in
general. The hydrodynamic description in Sec. III.2 further reduces the number
of explicitly considered degrees of freedom by projecting onto the space of
the five hydrodynamic quantities. This is controlled in the limit of low
frequencies and small wave numbers. Either description involves an averaging
process that ignores fluctuations of, and correlations between, the
hydrodynamic modes. The Langevin/Landau-Lifshitz concept of a fluctuating
force is designed to take these into account, and fluctuating hydrodynamics is
capable of describing mode-mode coupling effects that are not included in the
averaged equations. For an example in the context of a non-equilibrium
classical fluid, see the long-range correlations originally obtained by means
of kinetic theory and mode-coupling theoryKirkpatrick et al. (1982a, b, c)
that were also derived by using fluctuating hydrodynamics.Ronis and Procaccia
(1982)

A remarkable aspect of LFL theory is the fact that the entropy has the same
functional form as in a noninteracting Fermi gas, see Eq. (75a) and Eq. (A.11)
in Paper I. This is usually motivated by the assumption that the
classification of energy levels is the same with or without
interactions.Lifshitz and Pitaevskii (1981a) An equivalent statement is that
the equilibrium single-particle distribution in a Fermi liquid has the same
functional form as in a non-interacting Fermi gas, i.e., it is given by the
Fermi-Dirac distribution. The equivalency follows from the observation that
maximizing the entropy subject to the constraints of the conservation of
particle number and energy must yield the equilibrium distribution. An
argument for the single-particle distribution having the Fermi-Dirac form can
be made as follows. The usual derivation of the Fermi-Dirac distribution
within Statistical Mechanics considers the grand canonical potential of the
particles in a given quantum state with energy $\epsilon_{p}$ that can
exchange particles with all other states.Landau and Lifshitz (1980) In the
case of fermions the state can be occupied by at most one particle. Therefore,
the energy of $N_{p}$ particles in that state is $N_{p}\epsilon_{p}$ (i.e.,
either zero or $\epsilon_{p}$) whether or not the fermions in the system as a
whole interact. (Note that this is not true for bosons.) Therefore, the
derivation of the average occupation number, i.e., the Fermi-Dirac
distribution, is unchanged from the noninteracting case. The energy
$\epsilon_{p}$ depends on the interactions in a complicated way, but the
distribution function depends on them only via $\epsilon_{p}$. This fixes the
functional form of the entropy up to a constant, and the latter is determined
by the Third Law.

###  IV.2 Outlook

We finally list some problems that would be interesting to investigate using
the formalism developed in this paper.

  1. 1.

The static $\mu$-space response function given in Eq. (17), which we have used
to specify the Langevin force operator correlations, does not include long-
range correlations that are present in certain static response functions in
the quantum limit. For example, it is known that the static spin
susceptibility $\chi_{s}({\bm{k}})$ is a nonanalytic function of the wave
number, which reflects long-range correlations in the zero-temperature quantum
system,Belitz et al. (1997) and this long-range behavior fundamentally changes
the nature of some quantum phase transitions.Brando et al. (2016) The static
number density and momentum response functions, on the other hand, do not
display this long-ranged behavior.Belitz et al. (2002) These effects will
become relevant once the spin degrees of freedom, which we have not
considered, are included in the theory.

  2. 2.

As already mentioned, in classical statistical mechanics it is well known that
in non-equilibrium steady states equal-time correlation functions and static
response functions are generically of extraordinarily long range; see Refs.
Dorfman et al., 1994, 2021 and references therein. They also display a well
defined kind of generalized rigidity.Kirkpatrick et al. (2021) It would be of
great interest to consider quantum systems in a corresponding non-equiblibrium
state and examine the same functions in both the collisionless regime and the
hydrodynamic regime.

  3. 3.

In order to derive the fluctuating hydrodynamic equations for a disordered
fermion system in Sec. III.3 we assumed that the disorder average has already
been performed as a first step. An interesting extension would be to derive
hydrodynamic equations for a fixed impurity configuration. This could be done
by working in real space and taking the fermion-impurity collision operator,
Eq. (57), to be proportional to a fixed impurity density $n_{i}({\bm{x}})$.
The hydrodynamic equations would then implicitly depend on $n_{i}({\bm{x}})$
via a dependence of the transport coefficients on $n_{i}({\bm{x}})$. In order
to derive physical quantities one would then have to average over both the
noise fluctuations and the static disorder fluctuations.

  4. 4.

Another aspect of the model considered in Sec. III.3 is that both fermion
number and energy are conserved in collisions. This is the case, for instance,
if fermion-fermion scattering and fermion-impurity scattering are present.
Another interesting class of problems conserves only the fermion number; this
can be realized, for instance, by adding electron-phonon scattering. At the
level of the hydrodynamic description the only soft mode would then be the
diffusive number density mode. The relevant equation can be derived by using
the techniques from Sec. III.2 and III.3 and projecting on the mode $a_{1}$
only. In order to describe heat transport in such a system one would have to
consider the energy transport in the phonon system as well as the energy
carried by the electrons and take into account that the energy in the combined
system is still conserved.

##  Appendix A Some linearized collision operators

In Appendix C of Paper I we gave expressions for collision operators
describing fermion-fermion scattering, fermion-phonon scattering, and fermion-
impurity scattering respectively. They remain valid in the present context if
we replace the distribution function $\phi$ by the operator-valued function
$\hat{\phi}$, and we list them here again for completeness. Remarks concerning
their derivations from the more general expressions given in Ref. Lifshitz and
Pitaevskii, 1981b can be found in Paper I.

The linearized fermion-fermion collision operator can be written

$\displaystyle\Lambda_{\text{f-f}}({\bm{p}})\,\hat{\phi}({\bm{p}})$
$\displaystyle=$
$\displaystyle\frac{1}{1-f_{\text{eq}}({\bm{p}})}\,\frac{1}{V^{3}}\sum_{{\bm{p}}^{\prime}\\!,{\bm{p}}_{1},{\bm{p}}_{1}^{\prime}}W({\bm{p}},{\bm{p}}_{1};{\bm{p}}^{\prime},{\bm{p}}_{1}^{\prime})\,\delta(\epsilon_{p}+\epsilon_{p_{1}}-\epsilon_{p^{\prime}}-\epsilon_{p_{1}^{\prime}})\,\delta({\bm{p}}+{\bm{p}}_{1}-{\bm{p}}^{\prime}-{\bm{p}}_{1}^{\prime})$
(56a) $\displaystyle\times
f_{\text{eq}}({\bm{p}}_{1})[1-f_{\text{eq}}({\bm{p}}^{\prime})][1-f_{\text{eq}}({\bm{p}}_{1}^{\prime})]\,[\hat{\phi}({\bm{p}}^{\prime})+\hat{\phi}({\bm{p}}_{1}^{\prime})-\hat{\phi}({\bm{p}})-\hat{\phi}({\bm{p}}_{1})]\
.$ Here $W$ is the probability for two fermions in momentum states ${\bm{p}}$
and ${\bm{p}}_{1}$ to be scattered into momentum states ${\bm{p}}^{\prime}$
and ${\bm{p}}_{1}^{\prime}$. Time reversal symmetry implies
$W({\bm{p}},{\bm{p}}_{1};{\bm{p}}^{\prime},{\bm{p}}_{1}^{\prime})=W({\bm{p}}^{\prime},{\bm{p}}_{1}^{\prime};{\bm{p}},{\bm{p}}_{1})\
.$ (56b)

The fermion-impurity collision operator is

$\Lambda_{\text{f-i}}({\bm{p}})\,\hat{\phi}({\bm{p}})=\frac{1}{V}\sum_{{\bm{p}^{\prime}}}W({\bm{p}}^{\prime},{\bm{p}})\,\delta(\epsilon_{p^{\prime}}-\epsilon_{p})\left[\hat{\phi}({\bm{p}}^{\prime})-\hat{\phi}({\bm{p}})\right]\
,$ (57)

with $W$ another scattering probability. Here the fermion particle number and
energy are conserved, but the momentum is not.

The fermion-boson collision operator can be written

$\displaystyle\Lambda_{\text{f-b}}({\bm{p}})\,\hat{\phi}({\bm{p}})$
$\displaystyle=$
$\displaystyle\frac{1}{f_{\text{eq}}({\bm{p}})(1-f_{\text{eq}}({\bm{p}}))}\,\frac{1}{V^{2}}\sum_{{\bm{p}}^{\prime}\\!,{\bm{k}}}\delta({\bm{p}}^{\prime}-{\bm{p}}-{\bm{k}})\,n_{\text{eq}}({\bm{k}})\Bigl{[}W({\bm{p}}^{\prime};{\bm{p}},{\bm{k}})\,f_{\text{eq}}({\bm{p}})(1-f_{\text{eq}}({\bm{p}}^{\prime}))\,\delta(\epsilon_{p^{\prime}}-\epsilon_{p}-\omega_{k})$
(58) $\displaystyle\hskip
100.0pt+W({\bm{p}}^{\prime},-{\bm{k}};{\bm{p}})\,f_{\text{eq}}({\bm{p}}^{\prime})(1-f_{\text{eq}}({\bm{p}}))\,\delta(\epsilon_{p^{\prime}}-\epsilon_{p}+\omega_{k})\Bigr{]}\left[\hat{\phi}({\bm{p}}^{\prime})-\hat{\phi}({\bm{p}})\right]\
.$

Here $n_{\text{eq}}({\bm{k}})=1/(\exp(\omega_{k}/T)-1)$ is the equilibrium
Bose-Einstein distribution function, $\omega_{k}$ is the energy of a boson
with wave number $k$, and $W$ is again a transition probability. The fermion
particle number is still conserved, but the fermion momentum and energy are
not. Umklapp processes can be taken into account if desirable.

Finally, a quantum version of the Bhatnagar-Gross-Krook collision operator in
classical kinetic theoryBhatnagar et al. (1954); Dorfman et al. (2021) is
useful for explicit calculations it uses a single relaxation time $\tau(T)$
and builds in the five conservation laws for the particle number, momentum,
and energy. It is given by

$\Lambda_{\text{f-f}}^{\text{BGK}}({\bm{p}})=\frac{-1}{\tau}\,{\cal
P}_{\perp}\ ,$ (59a) where ${\cal
P}_{\perp}=\openone-\sum_{\alpha=1}^{5}\frac{|a_{\alpha}({\bm{p}}\rangle\langle
a_{\alpha}({\bm{p}})|}{\langle
a_{\alpha}({\bm{p}})|a_{\alpha}({\bm{p}})\rangle}$ (59b)

is the projection operator defined in Sec. III.2 that enforces the five
conservation laws.

In general, the linearized collision operator $\Lambda({\bm{p}})$ is a sum of
several of these individual collision operators.

##  Appendix B The $\mu$-space static response function, and the Langevin
force correlation

In this appendix we show how to derive Eqs. (15).

We can obtain a kinetic equation for the dynamic response function by adding a
term $\delta\mu({\bm{p}},{\bm{x}},t)$ to the chemical potential that depends
on momentum, space, and time, and acts as an external field conjugate to the
operator $\delta{\hat{f}}$. The single-particle energy from Eq. (5) then
becomes

${\hat{\epsilon}}({\bm{p}},{\bm{x}})=\epsilon_{p}+\delta{\hat{\epsilon}}({\bm{p}},{\bm{x}})-\delta\mu({\bm{p}},{\bm{x}},t)\
.$ (60)

In the kinetic equation this produces a new term, and instead of Eq. (6b) we
obtain

$\left[-i\omega+L_{\bm{k}}({\bm{p}})\right]w({\bm{p}}){\hat{\phi}}({\bm{p}},{\bm{k}},\omega)-i{\bm{k}}\cdot{\bm{v}}_{p}\,\delta{\tilde{\mu}}({\bm{p}},{\bm{k}},\omega)={\hat{F}}_{\text{L}}({\bm{p}},{\bm{k}},\omega)\
,$ (61)

where
$\delta{\tilde{\mu}}({\bm{p}},{\bm{k}},\omega)=w({\bm{p}})\delta\mu({\bm{p}},{\bm{k}},\omega)$.
The dynamic response function infinitesimally above the real frequency axis is
given by the functional derivative

$\chi({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}},\omega+i\epsilon)=\delta\langle{\hat{\phi}}({\bm{p}}_{1},{\bm{k}},\omega)\rangle/\delta{\tilde{\mu}}({\bm{p}}_{2},{\bm{k}},\omega)\
.$ (62)

Equation (61) yields

$\left[-i\omega+L_{\bm{k}}({\bm{p}}_{1})\right]\chi({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}},\omega+i\epsilon)=V\delta_{{\bm{p}}_{1},{\bm{p}}_{2}}\,\frac{i{\bm{k}}\cdot{\bm{v}}_{p_{1}}}{w({\bm{p}}_{1})}\
,$ (63a) which has the formal solution
$\chi({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}},\omega+i\epsilon)=-\left[\omega+iL_{\bm{k}}^{(1)}({\bm{p}}_{1})-i\Lambda({\bm{p}}_{1})\right]^{-1}\,V\delta_{{\bm{p}}_{1},{\bm{p}}_{2}}\,\frac{{\bm{k}}\cdot{\bm{v}}_{p_{1}}}{w({\bm{p}}_{1})}\
,$ (63b)

with $L_{\bm{k}}^{(1)}$ from Eq. (6d). Using Eq. (9b) in (10), and performing
the frequency integration before taking the imaginary part (note that
$iL_{\bm{k}}^{(1)}$ is real, whereas $i\Lambda$ is imaginary), yields

$\chi({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}})=-{\text{Im}}\,\left(L_{\bm{k}}({\bm{p}}_{1})\right)^{-1}V\delta_{{\bm{p}}_{1},{\bm{p}}_{2}}\,\frac{{\bm{k}}\cdot{\bm{v}}_{p_{1}}}{w({\bm{p}}_{1})}\
.$ (64a) Alternatively we can obtain the static response by putting $\omega=0$
in Eq. (63b),
$\chi({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}})=\left(L_{\bm{k}}({\bm{p}}_{1})\right)^{-1}V\delta_{{\bm{p}}_{1},{\bm{p}}_{2}}\,\frac{i{\bm{k}}\cdot{\bm{v}}_{p_{1}}}{w({\bm{p}}_{1})}\
.$ (64b)

Equations (64a) and (64b) are equivalent due to the spectral representation of
the response function, see Eqs. (9) and (10). Since the static response cannot
depend on transport coefficients,sta  the collision operator implicit in this
expression cannot contribute and we can evaluate Eqs. (64) for
$\Lambda({\bm{p}}_{1})=-i\epsilon$. The static response function thus obeys an
integral equation

$L_{\bm{k}}^{(1)}({\bm{p}}_{1})\,\chi({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}})=V\delta_{{\bm{p}}_{1},{\bm{p}}_{2}}\,\frac{i{\bm{k}}\cdot{\bm{v}}_{p_{1}}}{w({\bm{p}}_{1})}\
.$ (65)

With $L_{\bm{k}}^{(1)}$ from Eqs. (6) this is Eq. (15b).

We still need to show how to obtain Eq. (15a) from (14). To this end we write

$\chi({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}})=V\delta_{{\bm{p}}_{1},{\bm{p}}_{2}}\,\frac{1}{w({\bm{p}}_{1})}+\delta\chi({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}})$
(66)

From Eq. (15b) we see that $\delta\chi$ obeys the integral equation

$\left[1+I({\bm{p}}_{1})\right]\delta\chi({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}})=\frac{-1}{N_{0}}\,F({\bm{p}}_{1},{\bm{p}}_{2})\
,$ (67a) with $I({\bm{p}})$ from Eq. (6e). Since $\delta\chi$ and $F$ are both
symmetric under interchange of the two ${\bm{p}}$-variables, Eq. (67a) can
also be written
$\left[1+I({\bm{p}}_{2})\right]\delta\chi({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}})=\frac{-1}{N_{0}}\,F({\bm{p}}_{1},{\bm{p}}_{2})\
.$ (67b)

Simple algebra that makes use of Eqs. (67) then yields

$\left[L_{\bm{k}}^{(1)}({\bm{p}}_{1})+L_{-{\bm{k}}}^{(1)}({\bm{p}}_{2})\right]\chi({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}})=0.$
(68)

Equation (14) is therefore equivalent to Eq. (15a).

##  Appendix C The $\mu$-space distribution, and the hydrodynamic modes

In order to illustrate the connection between the approaches in Paper I and
the current paper, we consider the kinetic equation (6c) and its formal
solution

$|{\hat{\phi}}({\bm{p}},{\bm{k}},\omega)\rangle=\left[-i\omega+L_{\bm{k}}({\bm{p}})\right]^{-1}|{\hat{F}}_{\text{L}}({\bm{p}},{\bm{k}},\omega)\rangle\
.$ (69)

Now we insert a complete set of modes between the propagator and the
fluctuating force on the right-hand side. As part of that complete set we
choose the five hydrodynamic modes $\psi_{\alpha}({\bm{k}},{\bm{p}})$
($\alpha=1,\ldots,5$) that are defined as the eigenfunctions of the kinetic
operator $L_{\bm{k}}({\bm{p}})$ with eigenvalues $\omega_{\alpha}({\bm{k}})$,
see Eq. (3.1) in Paper I. For the part of ${\hat{\phi}}$ in the space spanned
by the hydrodynamic modes we then have (note that the left and right
eigenfunctions are not identical, see Paper I)

$|{\hat{\phi}}_{\text{hyd}}({\bm{p}},{\bm{k}},\omega)\rangle=\sum_{\alpha}\,|\psi_{\alpha}^{(\text{R})}({\bm{k}},{\bm{p}})\rangle\,\left[-i\omega+\omega_{\alpha}({\bm{k}})\right]^{-1}\langle\psi_{\alpha}^{(\text{L})}({\bm{k}},{\bm{p}})|{\hat{F}}_{\text{L}}({\bm{p}},{\bm{k}},\omega)\rangle\
.$ (70)

Here we have assumed that the hydrodynamic modes are normalized to unity.
Equation (70) is equivalent to an explicit solution of the hydrodynamic
equations in Sec. III.2. The eigenfrequencies $\omega_{\alpha}({\bm{k}})$ and
the eigenfunctions $\psi_{\alpha}$ have been determined explicitly in Paper I.
To zeroth order in an expansion of the hydrodynamic modes in powers of the
wave number $k=|{\bm{k}}|$, the $\psi_{\alpha}$ are linear combinations of the
zero eigenfunctions $a_{\alpha}({\bm{p}})$ of the collision operator, see Eq.
(3.6) in Paper I, and therefore have no overlap with the fluctuating force.
However, their contributions to first order in the $k$-expansion are elements
of the complementary space ${\cal L}_{\perp}$, see Eqs. (3.28) in Paper I. The
leading contribution to the right-hand side of Eq. (70) is thus of $O(k)$.
This is consistent with the fact that the Langevin forces in fluctuating
hydrodynamics are currents, see Sec. III.2.2 and Ref. Landau and Lifshitz,
1959.

##  Appendix D $H$-theorem for the nonlinear Boltzmann-Landau kinetic equation

In this appendix we derive an $H$-theorem for the fermionic quantum kinetic
equation; see ch. 10.3.3 in Ref. Dorfman et al., 2021 for a closely related
discussion. Let
$f_{\bm{p}}({\bm{x}},t)=\langle{\hat{f}}({\bm{p}},{\bm{x}},t)\rangle$ be the
average non-equilibrium distribution function, $\delta
f_{\bm{p}}=f_{\bm{p}}-f_{\text{eq}}({\bm{p}})$ its deviation from the
equilibrium distribution, and

$\epsilon_{\bm{p}}({\bm{x}},t)=\langle{\hat{\epsilon}}({\bm{p}},{\bm{x}},t)\rangle=\epsilon_{p}+\frac{1}{N_{0}V}\sum_{{\bm{p}}^{\prime}}F({\bm{p}},{\bm{p}}^{\prime})\,\delta
f_{{\bm{p}}^{\prime}}({\bm{x}},t)$ (71)

the average single-particle energy. The nonlinear Boltzmann-Landau kinetic
equation for $f_{\bm{p}}$ isLifshitz and Pitaevskii (1981b)

$\partial_{t}f_{\bm{p}}+({\bm{\nabla}_{\bm{x}}}f_{\bm{p}})\cdot{\bm{\nabla}}_{\bm{p}}\epsilon_{\bm{p}}-({\bm{\nabla}}_{\bm{p}}f_{\bm{p}})\cdot{\bm{\nabla}}_{\bm{x}}\epsilon_{\bm{p}}=C(f)$
(72)

For the collision integral we assume the one appropriate for fermion-fermion
collisions (Ref. Lifshitz and Pitaevskii, 1981b, see Eq. (56a) for the
linearized version)

$\displaystyle C(f)$ $\displaystyle=$
$\displaystyle\frac{1}{V^{3}}\sum_{{\bm{p}}^{\prime}\\!,{\bm{p}}_{1},{\bm{p}}_{1}^{\prime}}W({\bm{p}},{\bm{p}}_{1};{\bm{p}}^{\prime},{\bm{p}}_{1}^{\prime})\,\delta(\epsilon_{p}+\epsilon_{p_{1}}-\epsilon_{p^{\prime}}-\epsilon_{p_{1}^{\prime}})\,\delta({\bm{p}}+{\bm{p}}_{1}-{\bm{p}}^{\prime}-{\bm{p}}_{1}^{\prime})$
(73)
$\displaystyle\times\left[f_{{\bm{p}}^{\prime}}f_{{\bm{p}}_{1}^{\prime}}(1-f_{\bm{p}})(1-f_{{\bm{p}}_{1}})-f_{\bm{p}}f_{{\bm{p}}_{1}}(1-f_{{\bm{p}}^{\prime}})(1-f_{{\bm{p}}_{1}^{\prime}})\right]\
.$

The transition rate $W$ is positive and has the symmetry properties

$\displaystyle
W({\bm{p}},{\bm{p}}_{1};{\bm{p}}^{\prime},{\bm{p}}_{1}^{\prime})$
$\displaystyle=$ $\displaystyle
W(-{\bm{p}},-{\bm{p}}_{1};-{\bm{p}}^{\prime},-{\bm{p}}_{1}^{\prime})$ (74a)
$\displaystyle=$ $\displaystyle
W({\bm{p}}^{\prime},{\bm{p}}_{1}^{\prime};{\bm{p}},{\bm{p}}_{1})$ (74b)
$\displaystyle=$ $\displaystyle
W({\bm{p}}_{1},{\bm{p}};{\bm{p}}_{1}^{\prime},{\bm{p}}^{\prime})\ ,$ (74c)

which express invariance under spatial inversions, time reversal, and
interchange of particles. The local $h$-function is usually defined as minus
the entropy density

$h({\bm{x}},t)=\frac{1}{V}\sum_{\bm{p}}\left[f_{\bm{p}}\ln
f_{\bm{p}}+(1-f_{\bm{p}})\ln(1-f_{\bm{p}})\right]\ .$ (75a) The global
$H$-function is $H(t)=\int d{\bm{x}}\,h({\bm{x}},t)\ .$ (75b)

The $H$-theorem consists of the statement

$dH/dt\leq 0\ ,$ (76)

with the equal sign holding if and only if $f_{\bm{p}}$ is the equilibrium
distribution $f_{\text{eq}}({\bm{p}})$, i.e., the Fermi-Dirac distribution.

In order to prove the $H$-theorem we start by using Eq. (72) to show that the
local $h$-function obeys a continuity equation

$\partial_{t}h+{\bm{\nabla}}_{\bm{x}}\cdot{\bm{j}}_{h}=I(f)\ ,$ (77a) with
${\bm{j}}_{h}({\bm{x}},t)=\frac{1}{V}\sum_{\bm{p}}({\bm{\nabla}}_{\bm{p}}\epsilon_{\bm{p}})\left[f_{\bm{p}}\ln
f_{\bm{p}}+(1-f_{\bm{p}})\ln(1-f_{\bm{p}})\right]$ (77b) the $h$-current
density, and
$I(f)=\frac{1}{V}\sum_{\bm{p}}C(f)\ln\left(\frac{f_{\bm{p}}}{1-f_{\bm{p}}}\right)$
(77c)

a source term. To arrive at this result we have integrated by parts in
momentum space and assumed that any surface terms vanish. One now can show
that $I(f)\leq 0$ by proceeding as in the case of the classical Boltzmann
equation:Cercignani (1988) Substitute Eq. (73) into Eq. (77c), interchange
${\bm{p}}$ and ${\bm{p}}_{1}$ as well as ${\bm{p}}^{\prime}$ and
${\bm{p}_{1}}^{\prime}$, add the two expressions and divide by $2$, using Eq.
(74c). Next interchange ${\bm{p}}$ and ${\bm{p}}^{\prime}$ as well as
${\bm{p}}_{1}$ and ${\bm{p}}_{1}^{\prime}$, again add the two expressions and
divide by $2$, using Eq. (74b). The result is

$\displaystyle I(f)$ $\displaystyle=$
$\displaystyle\frac{-1}{4V^{4}}\sum_{{\bm{p}},{\bm{p}}_{1},{\bm{p}}^{\prime},{\bm{p}}_{1}^{\prime}}W({\bm{p}},{\bm{p}}_{1};{\bm{p}}^{\prime},{\bm{p}}_{1}^{\prime})\,\delta(\epsilon_{p}+\epsilon_{p_{1}}-\epsilon_{p^{\prime}}-\epsilon_{p_{1}^{\prime}})\,\delta({\bm{p}}+{\bm{p}}_{1}-{\bm{p}}^{\prime}-{\bm{p}}_{1}^{\prime})$
(78) $\displaystyle\times
f_{\bm{p}}f_{{\bm{p}}_{1}}(1-f_{{\bm{p}}^{\prime}})(1-f_{{\bm{p}}_{1}^{\prime}})\left[\frac{f_{{\bm{p}}^{\prime}}f_{{\bm{p}}_{1}^{\prime}}(1-f_{\bm{p}})(1-f_{{\bm{p}}_{1}})}{f_{\bm{p}}f_{{\bm{p}}_{1}}(1-f_{{\bm{p}}^{\prime}})(1-f_{{\bm{p}}_{1}^{\prime}})}-1\right]\ln\left(\frac{f_{{\bm{p}}^{\prime}}f_{{\bm{p}}_{1}^{\prime}}(1-f_{\bm{p}})(1-f_{{\bm{p}}_{1}})}{f_{\bm{p}}f_{{\bm{p}}_{1}}(1-f_{{\bm{p}}^{\prime}})(1-f_{{\bm{p}}_{1}^{\prime}})}\right)$

But

$(x-1)\ln x\geq 0\qquad\forall x>0$ (79)

with the equal sign holding if and only if $x=1$ (this is a special case of
Jensen’s inequality). Accordingly, $I(f)\leq 0$. Integrating Eq. (77a) over
all of space, and again ignoring surface terms, yields Eq. (76).

It remains to be shown that $dH/dt=0$ if and only if
$f_{\bm{p}}=f_{\text{eq}}({\bm{p}})$. This follows in exact analogy to the
classical case:Cercignani (1988) Suppose $f_{\bm{p}}$ is the Fermi-Dirac
distribution, Eq. (1b). Then $\ln[f_{\bm{p}}/(1-f_{\bm{p}})]=-\xi_{\bm{p}}/T$
is a collision invariant, and hence $I(f_{\text{eq}})=0$. Now suppose
$I(f)=0$. Then by Eq. (79) $\ln[f_{\bm{p}}/(1-f_{\bm{p}})]$ must be a
collision invariant, and thus a linear combination of the five conserved
quantities $\epsilon_{p}$ (energy), $1$ (particle number), and ${\bm{p}}$
(momentum). The latter has a zero coefficient by Galilean invariance, and
therefore $\ln[f_{\bm{p}}/(1-f_{\bm{p}})]=-\epsilon_{p}/T+\mu$, with minus the
inverse temperature and the chemical potential serving as the coefficients in
the linear combination. $I(f)=0$ thus implies that $f$ is given by Eq. (1b).

##  Appendix E Some useful identities

In this appendix we list, for completeness, some thermodynamic identities that
were derived in Paper I and that we have used throughout the main text.

Two common averages are, see Eqs. (A19a) and (A23a) in Paper I,

$\displaystyle\langle v_{p}^{i}|p^{j}\rangle$ $\displaystyle=$
$\displaystyle\delta^{ij}n\ ,$ (80a) $\displaystyle\langle
v_{p}^{i}|p^{j}a_{5}({\bm{p}})\rangle$ $\displaystyle=$
$\displaystyle\delta^{ij}T(\partial p/\partial T)_{N,V}\ .$ (80b)

The average energy density $e$, the average pressure $p$, the average density
$n$, and the temperature derivative of the pressure are related by

$n\langle\epsilon_{p}\rangle_{w}+T(\partial p/\partial T)_{N,V}=e+p\ .$ (81)

For a derivation, see Eq. (3.16) in Paper I. Another relation between
$\langle\epsilon_{p}\rangle_{w}$ and a thermodynamic derivative is

$\langle\epsilon_{p}\rangle_{w}=\mu-T(\partial\mu/\partial T)_{N,V}\ ,$ (82)

see Eq. (A7) in Paper I.

##  Appendix F Fluctuating Navier-Stokes equations for a clean metal

In this appendix we briefly discuss the changes that occur in Sec. III.2 if
the kinetic theory is applied to the electron fluid in a metal, i.e., a
fermion system with a long-ranged Coulomb interaction. We will assume that the
metal is clean in the sense that there is no electron-impurity scattering.

A long-ranged Coulomb interaction results in an additional contribution $4\pi
N_{0}e^{2}/k^{2}$ to the Landau interaction parameter $F_{0}$. In the fluid
velocity equation (38a) this leads to an additional term

$ik\,\frac{4\pi ne^{2}}{k^{2}}\,\delta{\hat{n}}({\bm{k}},\omega)\ .$

Combined with the density equation (37) this singular term leads to nonzero
eigenfrequencies $\omega=\pm\omega_{p}+O(k^{2})$ in the longitudinal (parallel
to ${\bm{k}}$) channel, with $\omega_{p}=(4\pi ne^{2}/m)^{1/2}$ the plasmon
frequency. This in turn implies that the frequency in Eq. (34) can no longer
be neglected compared to the collision operator, as both are of $O(1)$ with
respect to $k$.

In order to capture these effects it is advantageous to introduce, instead of
Eq. (29b), longitudinal and transverse momentum modes as

$\displaystyle a_{2}({\bm{p}})$ $\displaystyle=$
$\displaystyle{\hat{\bm{k}}}\cdot{\bm{p}}\ ,$ (83a) $\displaystyle
a_{3}({\bm{p}})$ $\displaystyle=$
$\displaystyle{\hat{\bm{k}}_{\perp}^{(1)}}\cdot{\bm{p}}\ ,$ (83b)
$\displaystyle a_{4}({\bm{p}})$ $\displaystyle=$
$\displaystyle{\hat{\bm{k}}_{\perp}^{(2)}}\cdot{\bm{p}}\ ,\ $ (83c)

where $\hat{\bm{k}}$ and ${\hat{\bm{k}}}_{\perp}^{(1,2)}$ are three pairwise
orthogonal unit vectors in the direction of ${\bm{k}}$ and the two directions
perpendicular to ${\bm{k}}$, respectively. Repeating the procedure from Sec.
III.2.2 we find that the transverse fluid velocity equation is unchanged, and
the transverse stress tensor is still given in terms of a transverse shear
viscosity that is unchanged from the short-range case, Eq. (38c),

$\eta_{\perp}=-\left\langle(\hat{\bm{k}}_{\perp}\cdot{\bm{p}})(\hat{\bm{k}}\cdot{\bm{v}}_{p})\big{|}\Lambda^{-1}({\bm{p}})\big{|}(\hat{\bm{k}}\cdot{\bm{v}}_{p})(\hat{\bm{k}}_{\perp}\cdot{\bm{p}})\right\rangle\
.$ (84a) The dissipative term in the equation for the longitudinal fluid
velocity, on the other hand, is given in terms of
$\eta_{\parallel}^{\pm}=-\left\langle(\hat{\bm{k}}_{\perp}\cdot{\bm{p}})(\hat{\bm{k}}\cdot{\bm{v}}_{p})\big{|}(\pm
i\omega_{p}+\Lambda({\bm{p}}))^{-1}\big{|}(\hat{\bm{k}}\cdot{\bm{v}}_{p})(\hat{\bm{k}}_{\perp}\cdot{\bm{p}})\right\rangle\
.$ (84b)

This transport coefficient, which can be interpreted as a high-frequency shear
viscosity, has both a real part and an imaginary part, and thus contributes
both to the ${\bm{k}}$-dependence of the plasmon frequency and to the plasmon
damping, see Paper I.

Similarly, the fluctuating stress tensor, Eq. (38h), has transverse parts

$\left({\hat{\tau}}_{\text{L}}\right)_{\perp}^{(1,2)}({\bm{x}},\omega)=\left\langle({\hat{\bm{k}}}_{\perp}^{(1,2)}\cdot{\bm{p}})({\hat{\bm{k}}}\cdot{\bm{p}})|\Lambda^{-1}({\bm{p}})|{\hat{F}}_{\text{L}}({\bm{p}},{\bm{x}},\omega)\right\rangle$
(85a) and a longitudinal part
$\left({\hat{\tau}}_{\text{L}}\right)_{\parallel}^{\pm}({\bm{x}},\omega)=\left\langle({\hat{\bm{k}}}\cdot{\bm{p}})^{2}|(\pm
i\omega_{p}+\Lambda^{-1}({\bm{p}}))^{-1}|{\hat{F}}_{\text{L}}({\bm{p}},{\bm{x}},\omega)\right\rangle$
(85b)

These different contributions to the fluctuating force are not correlated with
each other. The transverse noise correlation is given in terms of the ordinary
zero-frequency shear viscosity, as in Eq. (42a), while the longitudinal
correlation depends on the dissipative (i.e., real) part of the high-frequency
shear viscosity, Eq. (84b).

We finally mention that in a two-dimensional metal the Coulomb contribution to
the interaction parameter $F_{0}$ is proportional to $1/k$, and consequently
the plasmons are soft with a frequency $\omega_{p}\propto k^{1/2}$. As a
result, the longitudinal and transverse shear viscosities in the long-
wavelength limit are identical.

## References

  * Langevin (1908) P. Langevin, C.R. Hebd. Acad. Sci. 146, 530 (1908), For an English translation see D. S. Lemons and A. Gythiel, Am. J. Phys. 65, 1079 (1997).
  * Landau and Lifshitz (1959) L. D. Landau and E. M. Lifshitz, _Fluid Mechanics_ (Pergamon, Oxford, 1959), third revised english edition ed.
  * Hohenberg and Halperin (1977) P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977).
  * Dorfman et al. (2021) J. R. Dorfman, H. van Beijeren, and T. R. Kirkpatrick, _Contemporary Kinetic Theory of Matter_ (Cambridge University Press, Cambridge, UK, 2021).
  * Ford et al. (1965) G. W. Ford, M. Kac, and P. Mazur, J. Math. Phys. 6, 504 (1965).
  * Caldeira and Leggett (1983) A. O. Caldeira and A. J. Leggett, Physica A 121, 587 (1983).
  * Ford and Kac (1987) G. W. Ford and M. Kac, J. Stat. Phys. 46, 803 (1987).
  * Ford et al. (1988) G. W. Ford, J. T. Lewis, and R. F. O’Connell, Phys. Rev. A 37, 4419 (1988).
  * Olavo et al. (2012) L. S. F. Olavo, L. C. Lapas, and A. Figueiredo, Ann. Phys. 327, 1391 (2012).
  * Lifshitz and Pitaevskii (1981a) E. M. Lifshitz and L. P. Pitaevskii, _Statistical Physics, Part 2_ (Pergamon, Oxford, 1981a), 2nd ed.
  * Forster (1975) D. Forster, _Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions_ (Benjamin, Reading, MA, 1975).
  * Belitz and Kirkpatrick (2021) D. Belitz and T. R. Kirkpatrick (2021), eprint arXiv:2112.14246.
  * Bixon and Zwanzig (1969) M. Bixon and R. Zwanzig, Phys. Rev. 187, 267 (1969).
  * Ehrenfest and Ehrenfest (1990) P. Ehrenfest and T. Ehrenfest, _The Conceptual Foundations of the Statistical Approach in Statistical Mechanics_ (Dover, New York, 1990), this is a translation of the article “Begriffliche Grundlagen der statistischen Auffassung in der Mechanik”, Enzyklopädie der mathematischen Wissenschaften, vol. IV/4, Art. 32, Teubner (Leipzig 1907-1914).
  * (15) Throughout this paper we assume the thermodynamic limit, $V\to\infty$, so $(1/V)\sum_{\bm{p}}$ should be considered as synonymous with $\int d{\bm{p}}/(2\pi)^{d}$ in $d$ spatial dimensions.
  * (16) The subscript $w$ distinguishes this average from the one defined after Eq. (1a). The scalar product, Eq. (3a), also uses an angular bracket notation, but there can be no confusion between it and either average.
  * Lifshitz and Pitaevskii (1981b) E. M. Lifshitz and L. P. Pitaevskii, _Physical Kinetics_ (Butterworth-Heinemann, Oxford, 1981b).
  * (18) $\hat{\phi}$ is defined in terms of the Heisenberg operators ${\hat{a}}^{\dagger}$ and ${\hat{a}}$, and hence its time evolution is given by the Heisenberg equations with the full Hamiltonian of the many-body system. The idea behind the kinetic theory is to replace this exact, but entirely formal, time evolution by a kinetic equation with a fluctuating force. See Sec. IV for further discussion of this point.
  * Zubarev (1960) D. N. Zubarev, Usp. Fiz. Nauk 71, 71 (1960), [Sov. Phys. Usp. 3, 320 (1960)].
  * Callen and Welton (1951) H. B. Callen and T. A. Welton, Phys. Rev. 83, 34 (1951).
  * Pines and Nozières (1989) D. Pines and P. Nozières, _The Theory of Quantum Liquids_ (Addison-Wesley, Redwood City, CA, 1989).
  * (22) The integral equation (15b) remains separable if all terms in the expansion in Legendre polynomials are included. If desirable, $\chi({\bm{p}}_{1},{\bm{p}}_{2};{\bm{k}})$ can thus be given in closed form with all Landau parameters $F_{\ell}$ included.
  * Fetter and Walecka (1971) A. L. Fetter and J. D. Walecka, _Quantum Theory of Many-Particle Systems_ (McGraw-Hill, New York, 1971).
  * (24) The dynamic structure factor defined in Eq. (22a) is sometimes called the “symmetrized structure factor”, or “symmetrized form factor”, as opposed to the “structure factor” or “van Hove function” defined as the density-density correlation without the anticommutator. The response function $\chi^{\prime\prime}$ is also known as the “dissipation function” or “absorptive response function”, and another related correlation function is Kubo’s correlation function. See Refs. Martin, 1965; Forster, 1975; Pines and Nozières, 1989; Fetter and Walecka, 1971.
  * Belitz and Kirkpatrick (2012) D. Belitz and T. R. Kirkpatrick, Phys. Rev. B 85, 125126 (2012).
  * Uehling and Uhlenbeck (1933) E. A. Uehling and G. E. Uhlenbeck, Phys. Rev. 43, 552 (1933).
  * (27) In general the collision operator also has a wave number and frequency dependence that we neglect.
  * Callen (1985) H. B. Callen, _Thermodynamics and an Introduction to Thermostatistics_ (Wiley, New York, 1985).
  * Mahan (1981) G. D. Mahan, _Many-Particle Physics_ (Plenum, New York, 1981).
  * (30) Note that in a fluid with a conserved momentum the heat mode is given by the entropy per particle (or per mass), rather than the entropy per volume, see Paper I and Eq. (49.4) in Ref. Landau and Lifshitz, 1959 ($s$ in this reference is the entropy per mass and $\rho$ is the mass density).
  * Kirkpatrick et al. (1982a) T. R. Kirkpatrick, E. G. D. Cohen, and J. R. Dorfman, Phys. Rev. A 26, 950 (1982a).
  * Kirkpatrick et al. (1982b) T. R. Kirkpatrick, E. G. D. Cohen, and J. R. Dorfman, Phys. Rev. A 26, 972 (1982b).
  * Kirkpatrick et al. (1982c) T. R. Kirkpatrick, E. G. D. Cohen, and J. R. Dorfman, Phys. Rev. A 26, 995 (1982c).
  * Ronis and Procaccia (1982) D. Ronis and I. Procaccia, Phys. Rev. A 26, 1812 (1982).
  * Landau and Lifshitz (1980) L. D. Landau and E. M. Lifshitz, _Statistical Physics Part 1_ (Butterworth Heinemann, Oxford, 1980).
  * Belitz et al. (1997) D. Belitz, T. R. Kirkpatrick, and T. Vojta, Phys. Rev. B 55, 9452 (1997).
  * Brando et al. (2016) M. Brando, D. Belitz, F. M. Grosche, and T. R. Kirkpatrick, Rev. Mod. Phys. 88, 025006 (2016).
  * Belitz et al. (2002) D. Belitz, T. R. Kirkpatrick, and T. Vojta, Phys. Rev. B 65, 165112 (2002).
  * Dorfman et al. (1994) J. R. Dorfman, T. R. Kirkpatrick, and J. V. Sengers, Ann. Rev. Phys. Chem. 45, 213 (1994).
  * Kirkpatrick et al. (2021) T. R. Kirkpatrick, D. Belitz, and J. R. Dorfman, J. Chem. Phys. B 125, 7499 (2021).
  * Bhatnagar et al. (1954) P. L. Bhatnagar, E. P. Gross, and M. Krook, Phys. Rev. 94, 511 (1954).
  * (42) This is known to be true classically, where the static response function and the equal-time correlation function are the same apart from a factor of $T$ and the equal-time correlation function is known; in the Boltzmann limit of a vanishing fluid density it is just a delta function in both momentum and real space.Dorfman et al. (2021) The same mechanism that causes the static response to be independent of the collision operator must be operative here since the kinetic equation in Sec. II.1 has the same structure as a classical kinetic equation.
  * Cercignani (1988) C. Cercignani, _The Boltzmann equation and its applications_ (Springer, New York, 1988).
  * Martin (1965) P. C. Martin, in _Statistical Mechanics of Equilibrium and Non-Equilibrium_ , edited by J. Meixner (North Holland, Amsterdam, 1965), p. 100\.