# Hydrodynamics of thermal active matter

Jay Armas<EMAIL_ADDRESS>Akash Jain<EMAIL_ADDRESS>Ruben Lier<EMAIL_ADDRESS>Institute for Theoretical Physics, University of Amsterdam, 1090 GL Amsterdam,
The Netherlands  Dutch Institute for Emergent Phenomena, 1090 GL Amsterdam,
The Netherlands  Institute for Advanced Study, University of Amsterdam, Oude
Turfmarkt 147, 1012 GC Amsterdam, The Netherlands

(August 28, 2024)

###### Abstract

Active matter concerns many-body systems comprised of living or self-driven
agents that collectively exhibit macroscopic phenomena distinct from
conventional passive matter. Using Schwinger-Keldysh effective field theory,
we develop a novel hydrodynamic framework for thermal active matter that
accounts for local temperature variations and the ensuing stochastic effects.
This framework provides a deeper understanding of energy balance, second law
of thermodynamics, and thermostated steady states in active matter, while also
addressing the systematic violations of fluctuation-dissipation theorem and
detailed balance. We use our framework of active hydrodynamics to develop
effective field theory actions for active superfluids and active nematics that
offer a first-principle derivation of various active transport coefficients
and feature activity-induced phase transitions.

###### pacs:

Valid PACS appear here

##  1 Let there be life

The second law of thermodynamics posits that matter should move towards
disorder, finally reaching a state of thermal equilibrium that maximises
entropy. To postpone this ultimate fate of demise, living matter produces an
excessive amount of entropy that is released to its surroundings as heat, thus
allowing it to maintain its ordered state [1]. Living organisms achieve this
by burning fuel at the cellular level, primarily adenosine-triphosphate (ATP),
to maintain themselves at an operating temperature sufficiently higher than
the ambient temperature, ensuring an uninterrupted outflux of entropy to the
environment. These metabolic processes of creating and releasing entropy allow
living organisms to undertake otherwise statistically unfavourable activities,
like self-replication, adaptation, self-organisation, and spontaneous motion,
and may be viewed as a _local_ violation of the second law of thermodynamics.
At macroscopic scales, this dynamics results in entirely new phases of matter,
called _active matter_ [2], of which living systems are the most prominent
example, exhibiting collective behaviour distinct from their non-living
passive counterparts.

The inherent non-equilibrium nature of active matter maintained via the
entropy exchange with the environment and the self-driven nature of the
microscopic constituents, leads to novel collective behaviour in the form of
pattern formation, non-equilibrium phase transitions, breaking of fluctuation-
dissipation theorems, and new forms of mechanical/elastic responses. These
properties are manifested in a plethora of systems including active liquid
crystals, active solids, and active gels. In particular, active liquid
crystals with polar order exhibit a disordered/ordered flocking transition
even in two spatial dimensions [3, 4, 5, 6, 7], while with nematic order
activity can give rise to nematic turbulence with spectacular spatially
modulated patterns at mesoscopic scales [8, 9, 10, 11, 12, 13]. In turn,
active solids may display odd-elastic responses that violate mechanical
reciprocity [14] and may become susceptible to instabilites due to the
presence of noise [15]. Besides being realised in biological systems, active
matter may also be engineered using motile particles with inbuilt batteries,
light-activated beads, mechanically/electrically driven systems [16], or
designed using metamaterials [17] and colloidal particles [18, 19].

To arrive at macroscopic descriptions of phases of matter, one can use a
bottom-up reductionist approach, starting from the microscopic constituents
and their mutual interactions. This approach is often quite difficult and only
really manageable for weakly interacting systems, using techniques such as
kinetic theory and agent-based modelling [20, 21].111An alternative bottom-up
approach for strongly-interacting systems is holography [22, 23, 24], though
the application of such methods to non-equilibrium open systems remains
challenging.  However, most collective behaviour in nature turn out to be
largely agnostic to the underlying microscopic dynamics and are effectively
described by a handful of collective variables, such as local temperature and
density, and a few transport parameters, such as viscosity and conductivity.
This allows one to take a top-down emergent approach, commonly known as
_hydrodynamics_ , to directly obtain effective theories for phases of matter
based on the laws of thermodynamics and the underlying symmetries, such as
rotations, translations, or number conservation, and the relevant symmetry
breaking patterns. In this context, the most important guiding principle for
passive phases of matter is the _local second law of thermodynamics_ , which
requires that the local rate of entropy production must be non-negative
everywhere throughout the system. Despite being just an inequality, this
requirement is immensely constraining for hydrodynamic models as it needs to
be satisfied for every possible configuration admitted by the system.

The lack of a local second law of thermodynamics allows for a wider range of
transport properties in hydrodynamic models of active matter. In fact, active
systems can bypass all and any of the usual constraints on transport imposed
by the local second law, e.g. viscosity is not necessarily positive. The
standard prescription to derive these novel effects is to incorporate a fuel
source into the framework of hydrodynamics, e.g. in models of polar/nematic
gels and viscoelasticity [25, 26, 27, 5, 28, 29, 30]. Fuel consumption leads
to novel contributions to the hydrodynamic constitutive relations, such as odd
strains arising from interactions between non-reciprocal springs [14, 31, 32,
33] and active nematic stresses in liquid crystals responsible for nematic
turbulence [34, 12, 13]. In contrast with passive systems, active hydrodynamic
constitutive relations induced by fuel consumption do not obey the usual
fluctuation-dissipation theorems (FDTs) nor Onsager’s reciprocity relations
[35, 36, 37, 38] that are rooted in microscopic time-reversal dynamics.

The absence of standard fluctuation-dissipation statistics, microscopic time-
reversibility, and a local second law of thermodynamics suggests a more
phenomenological approach to active hydrodynamics based solely on the relevant
symmetries for a given phase of matter. This viewpoint has been pursued in
various works, for instance in the widely-studied Toner-Tu hydrodynamic model
[3, 4] of active matter with polar order (see also [2] for a review of many
hydrodynamic models of active matter). However, most active systems still
operate relatively close to equilibrium, suggesting that instead of abandoning
entirely the framework of passive hydrodynamics, one should depart from it
slowly as the strength of activity is increased; see e.g. [39]. In light of
this discussion, our aim is to develop a systematic framework for active
hydrodynamics where the effects of activity are introduced perturbatively by
means of a control parameter.

In pursuit of this goal, we are immediately faced with a challenge: any
discussion of local entropy, producing or otherwise, cannot be made systematic
without the introduction of the _local temperature_ in active hydrodynamics.
Despite their successes, none of the active hydrodynamic models developed
previously are suitable for describing temperature as a local hydrodynamic
variable. Neglecting the role of local temperature in such models may be
sensible when the microscopic active agents are themselves too large to be
affected by temperature fluctuations. On the other hand, active matter
comprised of microscopic agents such as bacteria or molecular motors is
susceptible to Brownian motion [40, 41, 42] and warrants a hydrodynamic
description that accounts for local temperature variations.

The inclusion of local temperature in active hydrodynamics turns out to be
more of a conceptual development than an incremental one. To set up a
hydrodynamic model, one must first specify the global equilibrium state, or a
global steady state for active matter, around which the hydrodynamic variables
locally fluctuate. However, with a fuel source constantly injecting energy
into the system, the local temperature rises indefinitely and such a steady
state can never be achieved. The solution, as one may naturally guess, is to
also incorporate an energy sink within the hydrodynamic model that can take
away the excess energy. To model this, we take inspiration from a large body
of literature developing hydrodynamic models with momentum and charge sinks
[43, 44, 45, 46, 47, 48]. However, energy sinks are conceptually different
from momentum and charge sinks. In these latter cases, such sinks do not
extract entropy from the system and thus may also occur in passive systems,
whereas energy sinks cause entropy to be released to the environment.
Interactions between the fuel source and the energy sink define the global
steady state temperature of the active system, in turn determining the rate at
which entropy is released to the environment.

Since active systems operate around non-equilibrium steady states, their
dynamics cannot be well-described by deterministic models and one must also
account for stochastic thermal noise. To this end, and to systematise the role
of fuel source and energy sink in active hydrodynamics, we use the methodology
of _Schwinger-Keldysh effective field theory_ (SK-EFT) [49, 50, 51, 52, 53,
54, 55]. SK-EFT is a systematic symmetry-based effective action approach to
hydrodynamics that has stochastic noise built into the framework. It features
a discrete Kubo-Martin-Schwinger (KMS) symmetry that is responsible for
imposing FDTs, Onsager’s relations, and the local second law of thermodynamics
in passive systems. We propose a suitably generalised _active KMS symmetry_
that allows for systematic violations of each of these requirements in terms
of the rates of fuel consumption and heat loss, perturbatively controlled by
the strength of activity. The proposal stems from the physical requirement
that when an active system runs out of fuel and all external forces are turned
off, dubbed the _famine state_ , it must behave as an ordinary passive system.
The active KMS symmetry can be understood as a field theoretic realisation of
the principle of microscopic reversibility when subjected to external work
[56, 57] as we demonstrate in our work.

We apply this new framework of active hydrodynamics to active superfluids and
active nematics, while sketching potential extensions to several other active
phenomena. We systematically study how active contributions may arise in the
hydrodynamic constitutive relations, and cause violations of FDTs, Onsager’s
relations, and the local second law of thermodynamics. Depending on the choice
of parameters, these models admit activity-induced phase transitions between
the ordered and disordered states and present the perfect opportunity for
approaching active phase transitions [58] from a Wilsonian renormalisation
group perspective.

A few recent works used EFT techniques for modelling certain aspects of active
matter. In [27], a SK-EFT model for active nematics was constructed by
including a fuel source, as done previously for conventional models of active
hydrodynamics [5, 25]. In [59], a complimentary EFT approach to non-
equilibrium systems based on the Fokker-Planck equation and stationary
probability distributions was presented and applied to systems without time-
reversal symmetry. However, these works have not addressed the aspects of
active matter that drive this work, i.e. energy balance, local temperature
variations, ambient temperature, or violations of the local second law of
thermodynamics. Furthermore, [59] discussed a version of FDT applicable to
states described by non-thermal stationary distributions, generalising
previous work of [60, 61]. In contrast, we find that FDTs are violated for
active matter in a thermostated steady state.

The structure of this work is as follows. In section 2, we develop the general
hydrodynamic framework for active matter starting from the principle of energy
balance and the existence of a famine state. In particular, we introduce the
notion of active KMS symmetry in SK-EFTs and discuss how it relates to the
systematic violations of the local second law of thermodynamics, FDTs, and
detailed balance. Then, in section 3, we consider a simple toy model of active
superfluids without momentum conservation, and discuss the salient features of
our model such as active transport coefficients and activity-induced phase
transitions. In section 4, we extend this construction to active nematics. We
conclude in section 5 with a discussion of future directions. The paper is
accompanied by two appendices, where we discuss the details of the SK
formalism and the detailed construction of SK effective actions respectively.

##  2 A field theory for activity

###  2.1 Activity engines

Consider a physical system that burns fuel from a reservoir for energy that it
uses to stay active. For instance, this may be ATP used by cells, food
consumed by a bacterial population, or internal batteries of motile particles.
Burning fuel produces heat proportional to the chemical energy differential
$\Delta E$ at rate $r_{{\mathsf{F}}}$. The _energy balance_ can be summarised
as

$\displaystyle\partial_{t}{\epsilon}+\partial_{i}\epsilon^{i}=\ell
r_{{\mathsf{F}}}\Delta E-\ell
r_{{\mathsf{E}}}{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T_{{\mathsf{E}}}\leavevmode\nobreak\ ,$ (1)

where $\epsilon$ and $\epsilon^{i}$ are the energy density and flux, while
${\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}r_{{\mathsf{E}}}$ denotes the rate of entropy loss to the environment
of ambient temperature $T_{{\mathsf{E}}}$, and
${\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}$ is the Boltzmann constant. The parameter $\ell$ serves as a
bookkeeping tool to keep track of any contributions tied to the in/outflow of
energy. We will assume $\ell$ to be small, so that we remain in a regime where
energy conservation is still approximately applicable. Depending on the
underlying symmetry structure, we may also need to account for conservation
equations, e.g. momentum, mass, or particle conservation, and Goldstone
equations for spontaneously broken symmetries. We shall return to these
considerations in detail later.

A homogeneous active steady-state is achieved when the energy
$r_{{\mathsf{F}}}\Delta E$ received from the fuel consumption in eq. 1 is
balanced by the heat loss
$r_{{\mathsf{E}}}{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T_{{\mathsf{E}}}$. Assuming the rate of fuel consumption to be
proportional to the chemical energy differential $\Delta E$ and that of
entropy loss to the thermal gradient $\Delta T=T-T_{{\mathsf{E}}}$, where $T$
is the system temperature, we may write down

$\displaystyle r_{{\mathsf{F}}}=\ell\gamma_{{\mathsf{F}}}\Delta
E\leavevmode\nobreak\ ,\qquad
r_{{\mathsf{E}}}=\ell\gamma_{{\mathsf{E}}}{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}\Delta T+\ldots\leavevmode\nobreak\ ,$ (2)

where ellipsis denote further derivative corrections. For simplicity, we will
assume $r_{{\mathsf{F}}}$ to not admit any derivative corrections throughout
this work, which amounts to the physical assumption that the fluid does not
backreact on the fuel burning process. Eqs. 2 and 1 together lead to the
steady-state temperature $T_{0}$ to be

$T_{0}=T_{{\mathsf{E}}}\left(1+\frac{\gamma_{{\mathsf{F}}}}{\gamma_{{\mathsf{E}}}}\frac{\Delta
E^{2}}{{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}^{2}T_{{\mathsf{E}}}^{2}}\right)\leavevmode\nobreak\ ,$ (3)

which is above the ambient temperature as the system is subjected to a
chemical gradient $\Delta E\neq 0$. It will be useful to identify some
dimensionless measure for quantifying the strength of activity. There are two
natural candidates: an external measure $\hat{\aleph}=\ell\Delta
E/({\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T_{{\mathsf{E}}})$ that measures the amount of energy being injected by
the fuel relative to the environment, and a dynamical measure
$\aleph=\ell(T-T_{{\mathsf{E}}})/T_{{\mathsf{E}}}$ that measures the operating
temperature of the active system relative to the environment. In the steady
state (3), the two are related by
$\aleph=\ell\gamma_{{\mathsf{F}}}/\gamma_{{\mathsf{E}}}\,\hat{\aleph}^{2}$.

When an active system runs out of fuel, i.e. $\Delta E=0$, it cools down to a
global thermal equilibrium state with $T_{0}=T_{{\mathsf{E}}}$. We refer to
this lack of fuel/food as the _famine state_. This state is effectively
described by ordinary passive hydrodynamics, but with short-lived energy or
heat fluctuations with the characteristic relaxation rate

$\Gamma_{\epsilon}=\frac{\ell^{2}{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}^{2}T_{{\mathsf{E}}}\gamma_{{\mathsf{E}}}}{c_{v}}\leavevmode\nobreak\
,$ (4)

where $c_{v}=\partial\epsilon/\partial T$ the heat capacity. This story is
analogous to the recently developed hydrodynamic framework for approximate or
weakly explicitly broken symmetries, featuring approximately conserved relaxed
charges [43, 44, 45, 46, 47, 48]. In this instance, the approximate symmetry
is time-translations, associated with the approximately conserved energy
density.

Instead of the activity being supported by a chemical engine, we may also
consider other engines of activity, such as an external thermal gradient,
electric fields, or mechanical forcing. If we remain agnostic of the details
of the engine itself, many such physical scenarios merely amount to
reinterpretations of $r_{{\mathsf{F}}}$ and $\Delta E$. For example, consider
a 2d active system driven by electric fields or mechanical forcing transverse
to the plane [18, 19, 62, 16]. In these cases, $r_{{\mathsf{F}}}$ and $\Delta
E$ may be interpreted as the transverse components of the charge flux/momentum
density and electric fields/acceleration respectively. The source of driving
may also be inhomogeneous in time and/or space, modelled by a non-constant
profile for $\Delta E$, e.g. $\Delta E\propto\sin(\Omega t)$ for an
oscillatory activity engine with frequency $\Omega$. However, we will only
consider homogeneous activity engines throughout this work for simplicity.
Lastly, we may also consider external electric fields or mechanical forcing
within the dimensionality of the active system, but this additionally requires
one to introduce momentum sinks to balance the momentum imparted by the
external fields. We will comment on this case towards the end of this paper.

###  2.2 Active KMS symmetry

Hydrodynamics describes the evolution of conserved charges in a physical
system and is characterised by the constitutive relations for the conserved
fluxes expressed in terms of the conserved densities, and possibly order
parameters of spontaneously broken symmetries. In addition to symmetries, the
construction of constitutive relations for a passive system is guided by
physical requirements such as the local second law of thermodynamics [63] and
the existence of local thermal equilibrium [64, 65]. However, active systems
may freely dump entropy into the environment thereby violating the local
second law. They also operate around non-equilibrium steady-states and
typically do not admit stable thermal equilibrium states. To help us traverse
this uncharted territory, we look towards _Schwinger-Keldysh (SK)
hydrodynamics_ [49, 50, 51, 52, 53, 54, 55], which is a recently-developed
effective field theory framework for hydrodynamics that, in principle, applies
arbitrarily far from equilibrium.

The primary ingredient in SK hydrodynamics is the Kubo-Martin-Schwinger (KMS)
symmetry. It is an incarnation of microscopic reversibility and ensures that
the thermal correlators of hydrodynamic operators satisfy the fluctuation-
dissipation theorems (FDTs) and Onsager’s reciprocity relations [35, 36, 37,
38]. For two-point symmetric and retarded correlators, denoted
$G^{\mathrm{S}}$ and $G^{\mathrm{R}}$ respectively, these statements read in
Fourier space

$\displaystyle G^{\mathrm{S}}_{\mathcal{O}\mathcal{O}^{\prime}}$
$\displaystyle=\frac{1}{i\omega\beta_{0}}\Big{(}G^{\mathrm{R}}_{\mathcal{O}\mathcal{O}^{\prime}}\mp
G^{\mathrm{R}*}_{\mathcal{O}\mathcal{O}^{\prime}}\Big{)}\leavevmode\nobreak\
,$ $\displaystyle G^{\mathrm{R}}_{\mathcal{O}\mathcal{O}^{\prime}}$
$\displaystyle=\pm
G^{\mathrm{R}}_{\mathcal{O}^{\prime}\mathcal{O}}\leavevmode\nobreak\ ,$ (5)

where
$\beta_{0}=1/({\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T_{0})$, and $\mathcal{O}$, $\mathcal{O}^{\prime}$ represent the
hydrodynamic operators of interest such as the conserved densities and fluxes.
The upper/lower signs in section 2.2 apply when the time-reversal eigenvalues
of the two operators are the same/opposite. Analogous statements apply for
higher-point functions [38].

In the SK formalism, FDTs, Onsager’s relations, and their higher-point
generalisations, are all realised via a discrete symmetry of the effective
field theory, known as the KMS symmetry. In simplest terms, we introduce a
pair of external sources, denoted ${\sf s}_{r}$, ${\sf s}_{a}$, that can be
used to compute the expectation values of the operators $\mathcal{O}$ as well
as their symmetric and retarded correlators, i.e.

$\displaystyle\langle\mathcal{O}\rangle$
$\displaystyle=\frac{-i\delta}{\delta{\sf s}_{a}}\ln{\cal
Z}\leavevmode\nobreak\ ,$ $\displaystyle
G^{\mathrm{S}}_{\mathcal{O}\mathcal{O}^{\prime}}$
$\displaystyle=\frac{-i\delta}{\delta{\sf
s}^{\prime}_{a}}\frac{-i\delta}{\delta{\sf s}_{a}}\ln{\cal
Z}\leavevmode\nobreak\ ,$ $\displaystyle
G^{\mathrm{R}}_{\mathcal{O}\mathcal{O}^{\prime}}$
$\displaystyle=\frac{\delta}{\delta{\sf
s}^{\prime}_{r}}\frac{-i\delta}{\delta{\sf s}_{a}}\ln{\cal
Z}\leavevmode\nobreak\ ,$ (6)

and similarly for higher-point correlation functions. Here
$\mathcal{Z}=\int\mathcal{D}\psi\mathcal{D}\psi_{a}\exp(i\int\mathrm{d}t\,L)$
is the SK generating functional of the theory, with the Lagrangian $L$, and
$\psi$, $\psi_{a}$ collectively denote the physical and stochastic dynamical
fields.222The physical dynamical fields $\psi$ are not given a label $r$
because they may generically differ from the $r$-type “average” fields used in
the SK formalism; see appendix A. Heuristically, ${\sf s}_{r}$ can be
understood as the true background fields and ${\sf s}_{a}$ as the associated
thermal noise. The KMS symmetry can be stated as the invariance of the theory
under a discrete transformation

$\displaystyle{\sf s}_{r}$
$\displaystyle\xrightarrow{\mathrm{{KMS}}}\eta_{\Theta}\,{\sf
s}_{r}\leavevmode\nobreak\ ,$ $\displaystyle{\sf s}_{a}$
$\displaystyle\xrightarrow{\mathrm{{KMS}}}\eta_{\Theta}\,\hat{\sf
s}_{a}\equiv\eta_{\Theta}\\!\left({\sf s}_{a}+i\beta_{0}\partial_{t}{\sf
s}_{r}\right)\leavevmode\nobreak\ ,$ (7)

where $\eta_{\Theta}$ denotes the time-reversal eigenvalue. Throughout this
paper, the right-hand sides of KMS transformation are understood to be
evaluated at $(-t,\vec{x})$. The KMS symmetry is realised on the dynamical
fields as
$\psi,\psi_{a}\xrightarrow{\mathrm{{KMS}}}\eta_{\Theta}\psi,\eta_{\Theta}\hat{\psi}_{a}$,
where the explicit form of $\hat{\psi}_{a}$ depends on the model under
consideration and will be discussed later in explicit examples. Depending on
the application in mind, the KMS symmetry can analogously be defined for other
kinds of reversibility symmetries involving combinations of spatial-parity and
charge-conjugation, such as PT, CT, or CPT.

To describe an active system, we need two new ingredients in the SK framework:
a fuel source and a heat sink, modelled using the external _fuel_ and
_environment sources_ $\Phi^{{{\mathsf{F}}},{{\mathsf{E}}}}_{r}$ and the
associated noise partners $\Phi^{{{\mathsf{F}}},{{\mathsf{E}}}}_{a}$.
Physically, these may be interpreted as the pair of external sources coupled
to the rate operators $r_{{{\mathsf{F}}},{{\mathsf{E}}}}$ introduced in
section 2.1, and for a homogeneous and non-stochastic fuel source and
environment, take the value

$\displaystyle\Phi_{r}^{{\mathsf{F}}}=-\Delta
E\,t,\quad\Phi^{{\mathsf{E}}}_{r}={\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T_{{\mathsf{E}}}\,t,\qquad\Phi_{a}^{{\mathsf{F}}}=\Phi^{{{\mathsf{E}}}}_{a}=0\leavevmode\nobreak\
.$ (8)

For inhomogeneous profiles of $\Delta E$ or $T_{{\mathsf{E}}}$, we can instead
identify $\Phi_{r}^{{\mathsf{F}}}=-\int\mathrm{d}t\,\Delta E$ and
$\Phi_{r}^{{\mathsf{E}}}=\int\mathrm{d}t\,T_{{\mathsf{E}}}$.

To set up the active KMS symmetry, let us recall the hydrostatic principle
which states that a passive system always flows towards global thermal
equilibrium when coupled to time-independent non-stochastic background fields,
i.e $\partial_{t}{\sf s}_{r}={\sf s}_{a}=0$. A crucial observation in this
regard is that the external noise fields in such configurations remain zero
under KMS transformation (2.2). While this need not generically apply to the
fuel/environment fields, we do expect the system to flow to global thermal
equilibrium in the famine state, $\Delta E=T_{0}-T_{{\mathsf{E}}}=0$, i.e.
$\Phi^{{\mathsf{E}}}_{r}={\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T_{0}t$,
$\Phi^{{\mathsf{F}}}_{r},\Phi^{{{\mathsf{F}}},{{\mathsf{E}}}}_{a}=0$.
Requiring that the KMS transformation leaves the noise fields vanishing in and
only in the famine state, fixes the KMS transformation to

$\displaystyle\Phi^{{\mathsf{F}}}_{r}$
$\displaystyle\xrightarrow{\mathrm{{KMS}}}-\Phi^{{\mathsf{F}}}_{r}\leavevmode\nobreak\
,$ $\displaystyle\Phi^{{\mathsf{E}}}_{r}$
$\displaystyle\xrightarrow{\mathrm{{KMS}}}-\Phi^{{\mathsf{E}}}_{r}\leavevmode\nobreak\
,$ $\displaystyle\Phi^{{\mathsf{F}}}_{a}$
$\displaystyle\xrightarrow{\mathrm{{KMS}}}-\hat{\Phi}^{{\mathsf{F}}}_{a}\equiv-\Phi^{{\mathsf{F}}}_{a}-i\beta_{0}\partial_{t}\Phi^{{\mathsf{F}}}_{r}\leavevmode\nobreak\
,$ $\displaystyle\Phi^{{\mathsf{E}}}_{a}$
$\displaystyle\xrightarrow{\mathrm{{KMS}}}-\hat{\Phi}^{{\mathsf{E}}}_{a}\equiv-\Phi^{{\mathsf{E}}}_{a}-i\beta_{0}\partial_{t}\Phi^{{\mathsf{E}}}_{r}+i\leavevmode\nobreak\
,$ (9)

with time-reversal eigenvalues $-1$. Note the additional “$i$” shift in the
KMS transformation of $\Phi^{{\mathsf{E}}}_{a}$. This is the _active KMS
transformation_ , suitable for describing out-of-equilibrium systems coupled
to a thermal bath. The fluctuations
$\delta\Phi^{{\mathsf{E}}}_{r}=\Phi^{{\mathsf{E}}}_{r}-{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T_{{\mathsf{E}}}t$,
$\delta\Phi^{{\mathsf{E}}}_{a}=\Phi^{{\mathsf{E}}}_{a}$ satisfy the standard
KMS symmetry (2.2) in the famine state, so the correlators in this state
satisfy FDTs and Onsager’s relations. Departing from the famine state, the
background field configuration itself breaks the KMS symmetry and the system
has to settle in a non-equilibrium steady-state that causes violations of FDTs
and Onsager’s relations in section 2.2. Note that the effective field theory
still realises the active KMS symmetry (2.2), but it relates states with
$\Phi_{a}^{{{\mathsf{F}}},{{\mathsf{E}}}}=0$ to states with
$\Phi_{a}^{{{\mathsf{F}}},{{\mathsf{E}}}}\neq 0$. This is philosophically
similar to how applying external magnetic fields to a rotationally-invariant
theory leads to anisotropic low-energy observables and rotations relate states
with different orientations of external magnetic fields. Except that in our
case, the active KMS symmetry maps physical steady states without noise in an
active system to auxiliary stochastic states with nonzero noise.

As a final comment, we note that while the environment fields
$\Phi^{{\mathsf{E}}}_{r,a}$ are essential for our construction, the role of
fuel fields $\Phi^{{\mathsf{F}}}_{r,a}$ may instead be played by any of the
other background fields ${\sf s}_{r,a}$ relevant to the system under
consideration. The only requirement for driving activity is that
$\partial_{t}{\sf s}_{r}\neq 0$. This may be achieved, e.g., via a second heat
bath with a temperature different from $T_{{{\mathsf{E}}}}$. Alternatively,
one may consider background electric fields or mechanical driving, however,
this introduces anisotropy in the system and also requires the introduction of
a momentum sink to counter the momentum imparted by the external fields [62].
Lastly, apart from representing a fuel source such as ATP or food, the fuel
fields $\Phi^{{\mathsf{F}}}_{r,a}$ can also be used for modelling internal
batteries of motile particles, or, in two dimensions, electric fields or
mechanical driving transverse to the plane of the system.

###  2.3 Energy balance, unitarity, and the second law of thermodynamics

Just like the two copies of background sources, all the global symmetries in
the SK framework are also doubled. The “$r$-type symmetries” ensure that the
classical equations of motion are invariant under symmetry transformations,
while the “$a$-type symmetries” are responsible for imposing the respective
conservation equations. For instance, the symmetries relevant for energy
balance are the doubled time-translations. The $r$-type time-translations act
on all the fields as usual, i.e. $f(t)\to f(t+\chi^{t}_{r})$, for a constant
parameter $\chi^{t}_{r}$. Whereas, the $a$-type time-translations only act on
the $a$-type noise fields and mix them with the physical fields. Collectively
denoting the background sources as
$\underline{\mathsf{s}}_{r,a}=(\mathsf{s}_{r,a},\ell\Phi^{{\mathsf{F}}}_{r,a},\ell\Phi^{{\mathsf{E}}}_{r,a})$,
we have

$\displaystyle\underline{\mathsf{s}}_{a}\to\underline{\mathsf{s}}_{a}+\chi^{t}_{a}\partial_{t}\underline{\mathsf{s}}_{r}\leavevmode\nobreak\
,$ (10)

for a constant parameter $\chi_{a}^{t}$. The transformations of the noise
dynamical fields $\psi_{a}$ are more involved, depending on the particulars of
the system under consideration, and will be discussed later. Using the
standard Noether procedure, eq. 10 implies the balance of energy

$\displaystyle\partial_{t}{\epsilon}+\partial_{i}\epsilon^{i}=-\mathcal{O}\partial_{t}{\sf
s}_{r}-\ell r_{{\mathsf{F}}}\partial_{t}\Phi^{{\mathsf{F}}}_{r}-\ell
r_{{\mathsf{E}}}\partial_{t}\Phi^{{\mathsf{E}}}_{r}\leavevmode\nobreak\ ,$
(11)

which extends eq. 1 to account for the energy imparted or removed by time-
dependent external fields.

The SK framework also features a set of constraints arising from the unitarity
of the microscopic time-evolution operator [51, 55]. These are summarised as

$\displaystyle L\big{|}_{f_{a}\to 0}=0\leavevmode\nobreak\ ,\quad
L\big{|}_{f_{a}\to-f_{a}}=-L^{*},\quad\operatorname{Im}L\geq
0\leavevmode\nobreak\ ,$ (12)

where $f_{a}$ represents all the $a$-type fields. To implement the second
condition later in the text, it is useful to introduce the SK-unitarity
operator

$\displaystyle(\ldots)^{\dagger}=(\ldots)^{*}\big{|}_{f_{a}\to-
f_{a}}\leavevmode\nobreak\ ,$ (13)

so that $L^{\dagger}=-L$. In particular, $f_{a}^{\dagger}=-f_{a}^{*}$.

The KMS symmetry, energy balance, and the unitarity constraints, together
conspire to give rise to the local second law of thermodynamics for passive
systems [66]. This is the statement that there exists an entropy density $s$
and associated flux $s^{i}$ such that entropy is locally produced, i.e.
$\partial_{t}s+\partial_{i}s^{i}\geq 0$. However for active systems, owing to
the additional “$i$” in the active KMS symmetry (2.2), one finds that the
local second law modifies to

$\partial_{t}s+\partial_{i}s^{i}+\ell{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}r_{{\mathsf{E}}}\geq 0\leavevmode\nobreak\ .$ (14)

As we would physically expect, this says that the entropy of an active system,
plus the entropy lost to the environment, is locally produced. We will see how
this works in specific examples later, while a general derivation of the
active second law from the active KMS symmetry is presented in section A.4.

The passive statement of the local second law is obviously restored for a
passive system where $r_{{\mathsf{E}}}=0$. It is also restored in the famine
state when the source of activity is absent. To wit, when the possible sources
of driving are absent, i.e.
$\partial_{t}\mathsf{s}_{r},\partial_{t}\Phi^{{\mathsf{F}}}_{r}=0$, and
$\partial_{t}\Phi^{{\mathsf{E}}}_{a}={\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T_{{\mathsf{E}}}$ is constant, eqs. 11 and 14 together imply that
$\partial_{t}s_{\text{fam}}+\partial_{i}s^{i}_{\text{fam}}\geq 0$, with
$s_{\text{fam}}=s-\epsilon/T_{{\mathsf{E}}}$ and
$s^{i}_{\text{fam}}=s^{i}-\epsilon^{i}/T_{{\mathsf{E}}}$. Note that this does
not work when the environment temperature $T_{{\mathsf{E}}}$ is varying in
space or time. The spatial or temporal gradients of $T_{{\mathsf{E}}}$ act as
thermal engines that drive the system away from the famine state. We shall
only consider a homogeneous and stationary environment in this paper.

###  2.4 Microscopic reversibility and detailed balance

Let us take a quick detour to see how the (active) KMS symmetry relates to
microscopic reversibility and detailed balance. Given that the system starts
from an initial state $\psi(t_{\text{i}})=\uppsi_{\text{i}}$ at time
$t_{\text{i}}$, the conditional probability distribution for it to follow a
path $\psi(t)$ until a final time $t_{\text{f}}$ is given by a path integral
over the noise fields

$\mathbb{P}(\psi|\uppsi_{\text{i}},t_{\text{i}})=\frac{1}{\mathcal{N}}\displaystyle\int{\mathcal{D}\psi_{a}}\exp\\!\left(i\int_{t_{\text{i}}}^{t_{\text{f}}}{\\!\\!\mathrm{d}t}\,L\big{(}\psi,\psi_{a},\underline{\mathsf{s}}_{r},\underline{\mathsf{s}}_{a}\big{)}\right)\leavevmode\nobreak\
,$ (15)

with boundary conditions $\psi_{a}(t_{{\text{i}},{\text{f}}})=0$. Using this,
we can also obtain the conditional probability distribution for the system to
transition to the final state $\psi(t_{\text{f}})=\uppsi_{\text{f}}$ by
integrating over all the paths

$\displaystyle\mathbb{P}(\uppsi_{\text{f}},t_{\text{f}}|\uppsi_{\text{i}},t_{\text{i}})=\int_{\uppsi_{\text{i}}}^{\uppsi_{\text{f}}}\mathcal{D}\psi\,\mathbb{P}(\psi|\uppsi,t_{\text{i}})\leavevmode\nobreak\
.$ (16)

The normalisation constant $\mathcal{N}$ in eq. 15 is fixed such that the
probabilities add up, i.e.
$\int\mathrm{d}\uppsi_{\text{f}}\,\mathbb{P}(\uppsi_{\text{f}},t_{\text{f}}|\uppsi_{\text{i}},t_{\text{i}})=1$.

Under KMS transformation,
$L\xrightarrow{\mathrm{{KMS}}}L-i\beta_{0}\partial_{t}\Omega$, where we have
kept the possible temporal boundary term but the spatial boundary terms may be
ignored. It is convenient to include a time-translation along with the time-
reversal transformation while implementing KMS on a finite time-interval,
$\Theta f(t)=\eta_{\Theta}f(-t+t_{\text{i}}+t_{\text{f}})$, so that the
interval maps to itself. Repeating the KMS transformation brings $L$ back to
itself, which requires that $\Omega$ does not contain any $a$-type noise
fields and is even under time-reversal. Note that the definition of $\Omega$
is ambiguous because $L$ can be redefined with arbitrary boundary terms. This
may typically be fixed by requiring that $L$ is invariant under all the
relevant continuous symmetries without leftover boundary terms.333In a
nutshell, this means that the kinetic terms in the Lagrangian take the form
$f(\psi)\partial_{t}\psi_{a}$ instead of $\psi_{a}\partial_{t}f(\psi)$. The
terms of the latter kind are typically only invariant under $a$-type global
symmetries up to boundary contributions; see e.g. section 3.1. With this
choice, $\Omega(\uppsi,t)=\Omega(\uppsi,\underline{\mathsf{s}}_{r}(t))$ can be
interpreted as the grand canonical free energy distribution of states.

The KMS symmetry implies a relation between the probabilities of the original
and time-reversed processes. Let us denote
$\mathbb{P}_{\Theta}(\eta_{\Theta}\psi|\eta_{\Theta}\uppsi_{\text{f}},t_{\text{i}})$
as the conditional probability for the system to start from
$\psi(t_{\text{i}})=\eta_{\Theta}\uppsi_{\text{f}}$ and traversing the time-
reversed path $\Theta\psi(t)=\eta_{\Theta}\psi(\Theta t)$, in the presence of
time-reversed background sources
$\eta_{\Theta}\underline{\mathsf{s}}_{r,a}(\Theta t)$. Relegating details to
section A.5, we find

$\displaystyle\frac{\mathbb{P}_{\Theta}(\eta_{\Theta}\psi|\eta_{\Theta}\uppsi_{\text{f}},t_{\text{i}})}{\mathbb{P}(\psi|\uppsi_{\text{i}},t_{\text{i}})}={\rm
e}^{\beta_{0}\Delta\Omega-\beta_{0}W_{\psi}}\leavevmode\nobreak\ ,$ (17)

where
$\Delta\Omega=\Omega(\uppsi_{\text{f}},t_{\text{f}})-\Omega(\uppsi_{\text{i}},t_{\text{i}})$
is the free energy differential between the end states, and

$\displaystyle\exp(-\beta_{0}W_{\psi})$
$\displaystyle=\left\langle\exp\\!\left(-\beta_{0}\int\mathrm{d}t\,\mathrm{d}^{d}x\,\mathcal{W}\right)\right\rangle_{\psi}\leavevmode\nobreak\
,$ (18)

denotes the dissipative work done on the system during forward path $\psi(t)$,
averaged over the thermal noise, i.e.

$\displaystyle\mathcal{W}$
$\displaystyle=-\mathcal{O}\partial_{t}\mathsf{s}_{r}+\ell
r_{{\mathsf{F}}}\Delta E+\ell
r_{{\mathsf{E}}}{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}\Delta T$
$\displaystyle=-\Big{(}\mathcal{O}\partial_{t}\mathsf{s}_{r}+\ell
r_{{\mathsf{F}}}\partial_{t}\Phi^{{\mathsf{F}}}_{r}+\ell
r_{{\mathsf{E}}}\partial_{t}\Phi^{{\mathsf{E}}}_{a}\Big{)}+\ell
r_{{\mathsf{E}}}{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T_{0}\leavevmode\nobreak\ .$ (19)

defined as the total energy supplied by the sources, fuel, and environment in
eq. 11, minus the heat lost to the environment in eq. 14. Eq. 17 is the field-
theoretic realisation of _microscopic reversibility_ from non-equilibrium
statistical mechanics [67, 68]. It states that, in the presence of activity,
the likelihood of a process vs. its time-reverse is no longer governed by just
the free energy differential $\Delta\Omega$ between the end states, but also
the dissipative work $W_{\psi}$ performed on the system [56, 60].

An equivalent statement for the transition probabilities can be obtained by
summing over all paths, i.e.

$\displaystyle\frac{\mathbb{P}_{\Theta}(\eta_{\Theta}\uppsi_{\text{i}},t_{\text{f}}|\eta_{\Theta}\uppsi_{\text{f}},t_{\text{i}})}{\mathbb{P}(\uppsi_{\text{f}},t_{\text{f}}|\uppsi_{\text{i}},t_{\text{i}})}={\rm
e}^{\beta_{0}\Delta\Omega-\beta_{0}W}\leavevmode\nobreak\ ,$ (20)

where $W$ is now averaged over all paths, defined as

$\displaystyle\exp(-\beta_{0}W)=\frac{\displaystyle\int_{\uppsi_{\text{i}}}^{\uppsi_{\text{f}}}\mathcal{D}\psi\exp(-\beta_{0}W_{\psi})\,\mathbb{P}(\psi|\uppsi_{\text{f}},t_{\text{i}})}{\displaystyle\int_{\uppsi_{\text{i}}}^{\uppsi_{\text{f}}}\mathcal{D}\psi\,\mathbb{P}(\psi|\uppsi_{\text{f}},t_{\text{i}})}\leavevmode\nobreak\
.$ (21)

Eq. 20 is the generalisation of the principle of _detailed balance_ in the
presence of external work $W$. When all the background fields are time-
independent and the fuel source is turned off, i.e.
$\partial_{t}\mathsf{s}_{r},\Delta E=0$, and the system temperature has
equilibrated with the environment temperature, i.e. $T=T_{{\mathsf{E}}}$, we
recover the original statement of detailed balance [69, 70].

A consequence of detailed balance is that, in the absence of external driving
or activity, a transition is more likely to occur than its time-reversal if it
decreases the free energy. Since this statement applies for arbitrary states,
a passive system left to its own devices will ultimately settle into the state
with least free energy, known as the principle of _thermodynamic stability_.
However, comparing free energies is not sufficient to determine the preferred
state of an active system, and one must also account for the heat lost to the
environment. In principle, one may use eq. 20 to determine the strength of
activity required to induce an active phase transition to a state with higher
free energy that is forbidden in passive systems. We leave such explorations
for future work.

##  3 Active superfluids

To draw a qualitative picture of our framework, let us consider a toy model
featuring a conserved number density $n$, together with the energy density
$\epsilon$. The energy balance and charge conservation equations can be
summarised as

$\displaystyle\partial_{t}{\epsilon}+\partial_{i}\epsilon^{i}$
$\displaystyle=E_{i}j^{i}{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\,-\,\ell
r_{{\mathsf{F}}}\partial_{t}\Phi^{{\mathsf{F}}}-\ell
r_{{\mathsf{E}}}\partial_{t}\Phi^{{\mathsf{E}}}}\leavevmode\nobreak\ ,$
$\displaystyle\partial_{t}n+\partial_{i}j^{i}$
$\displaystyle=0\leavevmode\nobreak\ ,$ (22)

where $j^{i}$ denotes the number flux. We have denoted the active
contributions arising from the fuel and environment fields in blue for
emphasis. We have also introduced the background U(1) gauge field sources
$A_{t},A_{i}$ coupled to $n,j^{i}$ that contribute to the energy balance
equation via the associated electric field
$E_{i}=\partial_{i}A_{t}-\partial_{t}A_{i}$. Charge conservation may be
understood as a consequence of an underlying U(1) symmetry that acts on the
background fields as $A_{t}\to A_{t}+\partial_{t}\Lambda,A_{i}\to
A_{i}+\partial_{i}\Lambda$. We may also introduce background “clock fields”
$n_{t},n_{i}$ coupled to $\epsilon,\epsilon^{i}$, however the resultant
equations are quite involved due to non-linearities and we have relegated a
full treatment to appendix A. Note that the energy balance equation in section
3 differs from our previous expression in eq. 11 by a redefinition
$\epsilon\to\epsilon-A_{t}n$, $\epsilon^{i}\to\epsilon^{i}-A_{t}j^{i}$ to make
the equations gauge invariant.

Furthermore, our toy model consists of a complex scalar field $\Psi$ charged
under the U(1) symmetry as $\Psi\to\Psi{\rm e}^{-i\Lambda}$. This can be used
to study the superfluid phase where $\Psi$ attains a nonzero expectation value
$\langle\Psi\rangle=\Psi_{0}$ and the U(1) symmetry is spontaneously broken.
The massless fluctuations of $\Psi$ around the ground state are parametrised
as $\Psi=\Psi_{0}{\rm e}^{i\phi}$, where $\phi$ is the superfluid Goldstone
field. Its dynamics is governed by the Josephson equation that will be
determined in our formalism by varying the SK effective action, given in eq.
38.

###  3.1 Fields and symmetries

The primary dynamical ingredients in a SK-EFT featuring energy balance and
charge conservation are: temperature $T$, chemical potential $\mu$, and the
partner stochastic noise fields $X^{t}_{a}$, $\varphi_{a}$. To accommodate the
possibility of spontaneous symmetry breaking, we also introduce the complex
scalar field $\Psi$ and its stochastic partner $\Psi_{a}$. The background U(1)
gauge field is doubled in the SK framework to $A_{r,a\,t},A_{r,a\,i}$. We may
identify $A_{rt},A_{ri}$ as the classical gauge field $A_{t},A_{i}$ and shall
use the two notations interchangeably. The same holds for the doubled clock
fields $n_{r,a\,t},n_{r,a\,i}$.

The effective theory is invariant under doubled U(1) global symmetries as well
as doubled time-translations discussed around eq. 10. The action of $r$-type
time-translations is given by time-diffeomorphisms on all the fields as usual;
see appendix A. The remaining symmetries act on various dynamical and
background fields as

$\displaystyle X^{t}_{a}$ $\displaystyle\to-\chi^{t}_{a}\leavevmode\nobreak\
,$ $\displaystyle\varphi_{a}$
$\displaystyle\to\varphi_{a}-\Lambda_{a}-X_{a}^{t}\partial_{t}\Lambda_{r}\leavevmode\nobreak\
,$ $\displaystyle\Psi_{r,a}$ $\displaystyle\to{\rm
e}^{-i\Lambda_{r}}\Psi_{r,a}\leavevmode\nobreak\ ,$ $\displaystyle A_{t}$
$\displaystyle\to A_{t}+\partial_{t}\Lambda_{r}\leavevmode\nobreak\ ,$
$\displaystyle A_{i}$ $\displaystyle\to
A_{i}+\partial_{i}\Lambda_{r}\leavevmode\nobreak\ ,$ $\displaystyle A_{at}$
$\displaystyle\to
A_{at}+\partial_{t}\Lambda_{a}+\chi^{t}_{a}\partial_{t}A_{t}-A_{t}\partial_{t}\chi^{t}_{a}\leavevmode\nobreak\
,$ $\displaystyle A_{ai}$ $\displaystyle\to
A_{ai}+\partial_{i}\Lambda_{a}+\chi^{t}_{a}\partial_{t}A_{i}-A_{t}\partial_{i}\chi^{t}_{a}\leavevmode\nobreak\
,$ $\displaystyle n_{at}$ $\displaystyle\to
n_{at}+\chi^{t}_{a}\partial_{t}n_{t}-n_{t}\partial_{t}\chi^{t}_{a}\leavevmode\nobreak\
,$ $\displaystyle n_{ai}$ $\displaystyle\to
n_{ai}+\chi^{t}_{a}\partial_{t}n_{i}-n_{t}\partial_{i}\chi^{t}_{a}\leavevmode\nobreak\
,$ (23)

while $T$, $\mu$, $n_{t}$, $n_{i}$ are invariant. In the following, it will be
useful to define the quantities

$\displaystyle N_{at}$
$\displaystyle=n_{t}\partial_{t}X^{t}_{a}+X^{t}_{a}\partial_{t}n_{t}+n_{at}\leavevmode\nobreak\
,$ $\displaystyle N_{ai}$
$\displaystyle=n_{t}\partial_{i}X^{t}_{a}+X^{t}_{a}\partial_{t}n_{i}+n_{ai}\leavevmode\nobreak\
,$ $\displaystyle B_{at}$
$\displaystyle=\partial_{t}\varphi_{a}+A_{t}\partial_{t}X^{t}_{a}+X^{t}_{a}\partial_{t}A_{t}+A_{at}\leavevmode\nobreak\
,$ $\displaystyle B_{ai}$
$\displaystyle=\partial_{i}\varphi_{a}+A_{t}\partial_{i}X^{t}_{a}+X^{t}_{a}\partial_{t}A_{i}+A_{ai}\leavevmode\nobreak\
,$ (24)

which are invariant under section 3.1. We also define the gauge-covariant
derivatives of the complex scalar fields as $\mathrm{D}_{t}=\partial_{i}\pm
iA_{rt}$, $\mathrm{D}_{i}=\partial_{i}\pm iA_{ri}$, with positive sign for
$\Psi$,$\Psi_{a}$ and negative sign for $\Psi^{*}$, $\Psi^{*}_{a}$.

T-even | T-odd
---|---
$x^{i}$, $\partial_{i}$ |  $t$, $\partial_{t}$
$n$, $\epsilon$, $\tau^{ij}$ |  $j^{i}$, $\epsilon^{i}$, $\pi_{i}$
$A_{t}$, $A_{at}$, $n_{t}$, $n_{at}$, $h_{ij}$, $h_{aij}$ |  $A_{i}$, $A_{ai}$, $n_{i}$, $n_{ai}$, $v^{i}$, $v^{i}_{a}$
|  $\Phi^{{\mathsf{E}}}$, $\Phi^{{\mathsf{E}}}_{a}$, $\Phi^{{\mathsf{F}}}$,
$\Phi^{{\mathsf{F}}}_{a}$
$T$, $\mu$, $X^{i}_{a}$ |  $X^{t}_{a}$, $\varphi_{a}$, $u^{i}$
$\operatorname{Re}\Psi$, $\operatorname{Re}\Psi_{a}$ |  $\operatorname{Im}\Psi$, $\operatorname{Im}\Psi_{a}$, $\phi$, $\phi_{a}$
$Q_{ij}$, $\mathcal{Q}_{ij}$ |
Table 1: Time-reversal even and odd quantities used in this work. The $r$-type
background fields are identified with the unlabelled ones. The director field
$q_{i}$ and its noise partner ${\mathpzc q}_{ai}$ may be time-reversal even or
odd based on the system under consideration.

The KMS transformation of the background fields is given in section 2.2,
whereas it acts as just as a time-reversal transformation on the physical
dynamical fields, with the time-reversal eigenvalues summarised in table 1.
Note that the time-reversal acts oppositely on the real and imaginary parts of
$\Psi$, so $\Psi\xrightarrow{\mathrm{{KMS}}}\Psi^{*}$. The $a$-type noise
dynamical fields are taken to transform under KMS as

$\displaystyle X^{t}_{a}$
$\displaystyle\xrightarrow{\mathrm{{KMS}}}-\hat{X}^{t}_{a}\equiv-X^{t}_{a}-i\left(\beta-\beta_{0}\right)\leavevmode\nobreak\
,$ $\displaystyle\varphi_{a}$
$\displaystyle\xrightarrow{\mathrm{{KMS}}}-\hat{\varphi}_{a}\equiv-\varphi_{a}-i\Big{(}\beta(\mu-
A_{t})-\beta_{0}\mu_{0}\Big{)}\leavevmode\nobreak\ ,$ $\displaystyle\Psi_{a}$
$\displaystyle\xrightarrow{\mathrm{{KMS}}}-\hat{\Psi}_{a}^{\dagger}\equiv\Psi_{a}^{*}+i\beta\mathrm{D}_{t}\Psi^{*}-\beta\mu\Psi^{*}\leavevmode\nobreak\
,$ $\displaystyle\Psi_{a}^{\dagger}$
$\displaystyle\xrightarrow{\mathrm{{KMS}}}-\hat{\Psi}_{a}\equiv-\Psi_{a}-i\beta\mathrm{D}_{t}\Psi-\beta\mu\Psi\leavevmode\nobreak\
,$ (25)

where
$\beta=1/({\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T)$. Here
$\beta_{0}=1/({\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T_{0})$ and $\mu_{0}$ are the inverse global temperature chemical
potential. Note that these transformations do not respect complex-conjugation,
but do respect the SK-unitarity operation (13). We can define the hatted-
versions of the quantities in section 3.1 by replacing the $a$-type field with
their hatted versions, such that
$f_{a}\xrightarrow{\mathrm{{KMS}}}\eta_{\Theta}\hat{f}_{a}$.

Finally, when activity is present, the $a$-type time-translation symmetry
requires that the fuel and environment background fields from section 2.2
appear in specific combinations

$\displaystyle\Delta
E=-\partial_{t}\Phi^{{\mathsf{F}}}_{r},\qquad{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T_{{\mathsf{E}}}=\partial_{t}\Phi^{{\mathsf{E}}}_{r}\leavevmode\nobreak\
,$
$\displaystyle\Pi_{a}^{{{\mathsf{F}}},{{\mathsf{E}}}}=\ell\Phi_{a}^{{{\mathsf{F}}},{{\mathsf{E}}}}+\ell
X_{a}^{t}\partial_{t}\Phi^{{{\mathsf{F}}},{{\mathsf{E}}}}_{r}\leavevmode\nobreak\
,$ (26)

with the hatted versions

$\displaystyle\hat{\Pi}_{a}^{{\mathsf{F}}}=\Pi_{a}^{{\mathsf{F}}}-i\ell\beta\Delta
E\leavevmode\nobreak\
,\quad\hat{\Pi}_{a}^{{\mathsf{E}}}=\Pi_{a}^{{\mathsf{E}}}-i\ell\beta{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}\Delta T\leavevmode\nobreak\ ,$ (27)

so that
$\Pi^{{{\mathsf{F}}},{{\mathsf{E}}}}_{a}\xrightarrow{\mathrm{{KMS}}}-\hat{\Pi}^{{{\mathsf{F}}},{{\mathsf{E}}}}_{a}$.

###  3.2 Schwinger-Keldysh effective action

We are now ready to construct the SK effective action. As the simplest first
step, we invoke a physical assumption that the active system under
consideration only has control over how the heat is dumped into the
environment and not on how it is drawn from the fuel source. This entails,
e.g., that the rate $r_{{\mathsf{F}}}$ of fuel consumption does not depend on
the thermodynamic variables $T$, $\mu$, or their derivatives. In terms of the
SK formalism, this means that the “fuel part” of the Lagrangian is fixed to
the simple form

$\displaystyle\mathcal{L}_{{\mathsf{F}}}=i{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T\gamma_{{\mathsf{F}}}\Pi_{a}^{{\mathsf{F}}}\hat{\Pi}_{a}^{{\mathsf{F}}}\leavevmode\nobreak\
,$ (28)

which respects all the SK symmetries in section 3.1 as well as the unitarity
constraints in eq. 12. The coefficient $\gamma_{{\mathsf{F}}}$ appearing here
is the same as we saw in eq. 2. Noting the composition of
$\Pi^{{\mathsf{F}}}_{a}$ in eq. 26, we see that the fuel fields only talk to
the fluid via $T$ and $X^{t}_{a}$. This simple fuel Lagrangian is sufficient
to obtain all the universal features of active hydrodynamics agnostic of the
details of the fuel burning process. One may easily generalise eq. 28 if one
wishes to simultaneously describe the fuel sector. By contrast, we shall allow
the environment fields to non-trivially couple with all the hydrodynamic
degrees of freedom.

Let us start with the “fluid part” of the SK Lagrangian depending only on the
dynamical fields $T$, $\mu$, and their noise partners. For example, we may
write down

$\displaystyle\mathcal{L}_{\text{f}}$ $\displaystyle=-\epsilon
N_{at}+nB_{at}+i{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T^{2}\kappa
N_{ai}\hat{N}_{a}^{i}+i{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T\sigma B_{ai}\hat{B}_{a}^{i}$
$\displaystyle\quad{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}+\,i{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T\gamma_{{\mathsf{E}}}\Pi_{a}^{{\mathsf{E}}}\hat{\Pi}_{a}^{{\mathsf{E}}}}\leavevmode\nobreak\
.$ (29)

The terms in the first line are comprised of the SK model for energy diffusion
[71] and charge diffusion [72]. In addition to the densities $n$ and
$\epsilon$, we have introduced the thermal conductivity $\kappa$, charge
conductivity $\sigma$, and heat relaxation coefficient
$\gamma_{{\mathsf{E}}}$. All the coefficients are functions of the
thermodynamic variables $T$ and $\mu$. Generically, we may also introduce a
thermo-electric conductivity $\sigma_{\times}$ coupling the energy and charge
sectors that has been considered in section B.1.

Next, we consider the “superfluid part” that contains the order parameter
$\Psi$ and its noise partner in addition to the hydrodynamic fields. Consider

$\displaystyle\mathcal{L}_{\Psi}$
$\displaystyle=f_{\Psi}\mathrm{D}^{i}\Psi\Big{(}\mathrm{D}_{t}\Psi^{*}N_{ai}+i\Psi^{*}B_{ai}-\mathrm{D}_{i}\Psi_{a}^{*}\Big{)}-\frac{\partial
V}{\partial\Psi^{*}}\Psi^{*}_{a}$
$\displaystyle\quad+i{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T\sigma_{\Psi}\Big{(}\Psi_{a}^{*}{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\,-\,\frac{i\mu\lambda_{\phi{{\mathsf{E}}}}}{{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T_{{\mathsf{E}}}}\Psi^{*}\Pi_{a}^{{\mathsf{E}}}}\Big{)}\Big{(}\hat{\Psi}_{a}{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\,+\,\frac{i\mu\lambda_{\phi{{\mathsf{E}}}}}{{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T_{{\mathsf{E}}}}\Psi\hat{\Pi}_{a}^{{\mathsf{E}}}}\Big{)}$
$\displaystyle\quad{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\,+\,i\frac{T}{T_{{\mathsf{E}}}}f_{\Psi}\lambda_{n{{\mathsf{E}}}}\Psi^{*}i\mathrm{D}^{i}\Psi\Big{(}B_{ai}\hat{\Pi}_{a}^{{\mathsf{E}}}-\hat{B}_{ai}\Pi_{a}^{{\mathsf{E}}}\Big{)}}$
$\displaystyle\quad{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\,+\,i\frac{T}{T_{{\mathsf{E}}}}a_{{\mathsf{E}}}\left(\Psi^{*}\hat{\Psi}_{a}\Pi_{a}^{{\mathsf{E}}}-\Psi\Psi^{*}_{a}\hat{\Pi}_{a}^{{\mathsf{E}}}\right)}-(\ldots)^{\dagger}\leavevmode\nobreak\
.$ (30)

The last term denotes the SK-unitarity-conjugate defined using eq. 13. We have
also introduced the superfluid density parameter $f_{\Psi}$, diffusion
parameter $\sigma_{\Psi}$, and a few active parameters
$\lambda_{\phi{{\mathsf{E}}}}$, $\lambda_{n{{\mathsf{E}}}}$, and
$a_{{\mathsf{E}}}$ whose significance will be clear momentarily. $\Psi$ is
subjected to a potential $V$, which is a function of $|\Psi|^{2}=\Psi^{*}\Psi$
and may take the representative form

$\displaystyle V=a|\Psi|^{2}+\frac{1}{2}a_{4}|\Psi|^{4}\leavevmode\nobreak\ ,$
(31)

where $a$, $a_{4}$ are phenomenological parameters that determine the shape of
the potential. All these coefficients may be functions of $T$ and $\mu$. A
more exhaustive analysis of the allowed terms in the superfluid Lagrangian
appears in section B.2.

One may check that the theory is invariant under the global symmetries in
section 3.1. The first two conditions in eq. 12 are satisfied by construction,
while the third one requires

$\displaystyle\kappa\geq 0\leavevmode\nobreak\ ,\quad\sigma\geq
0\leavevmode\nobreak\ ,\quad\sigma_{\Psi}\geq 0\leavevmode\nobreak\
,\quad\gamma_{{{\mathsf{E}}},{{\mathsf{F}}}}\geq 0\leavevmode\nobreak\ ,$ (32)

which guarantees the positivity of $\operatorname{Im}\mathcal{L}$. The KMS
symmetry requires that $n$, $\epsilon$, and $f_{\Psi}$ are derived from the
free energy density $\mathcal{F}$ via the thermodynamic relations

$\displaystyle\mathrm{d}\mathcal{F}$
$\displaystyle=-s\,\mathrm{d}T-n\,\mathrm{d}\mu+f_{\Psi}\mathrm{d}(\mathrm{D}^{i}\Psi^{*}\mathrm{D}_{i}\Psi)+\frac{\partial
V}{\partial|\Psi|^{2}}\mathrm{d}|\Psi|^{2}\leavevmode\nobreak\ ,$
$\displaystyle\epsilon$ $\displaystyle=Ts+\mu
n+\mathcal{F}\leavevmode\nobreak\ ,$ (33)

where $s$ is the entropy density. Noting these relations, one may verify that
the SK Lagrangian is KMS-invariant up to a total derivative term
$-i\partial_{t}(\beta\mathcal{F})$ that drops out from the effective action.
Comparing with our discussion in section 2.4, we have
$\beta_{0}\Omega=\int\mathrm{d}^{d}x\,\beta\mathcal{F}$. Assuming $f_{\Psi}$
to be independent of $\Psi$, the free energy density takes the familiar
Landau-Ginzburg form

$\displaystyle\mathcal{F}=-p+f_{\Psi}\mathrm{D}^{i}\Psi^{*}\mathrm{D}_{i}\Psi+V\leavevmode\nobreak\
,$ (34)

where $p$ is the thermodynamic pressure of the fluid.

###  3.3 Spontaneous symmetry breaking

Extremising the SK effective action with respect to $\Psi_{a}$ yields the
equation of motion for the order parameter

$\displaystyle\mathrm{D}_{t}\Psi$
$\displaystyle=\frac{1}{\sigma_{\Psi}}\mathrm{D}_{i}(f_{\Psi}\mathrm{D}^{i}\Psi)-\frac{1}{\sigma_{\Psi}}\left(\frac{\partial
V}{\partial|\Psi|^{2}}+{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color<EMAIL_ADDRESS>a_{{\mathsf{E}}}}\right)\Psi$
$\displaystyle\qquad+i\mu\left(1+{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\aleph\lambda_{\phi{{\mathsf{E}}}}}\right)\Psi\leavevmode\nobreak\
,$ (35)

where $\aleph=\ell\Delta T/T_{{\mathsf{E}}}$ denotes the strength of activity.
In general, the equation of motion also contains an auxiliary noise part,
which is classically set to zero by extremising with respect to $T$, $\mu$,
and $\Psi$, provided that the $a$-type background noise fields are turned off.
This part is important in stochastic hydrodynamics when computing correlation
functions of hydrodynamic operators. In the SK framework, however, correlation
functions are computed directly using the effective action, so we do not need
to concern ourselves with the explicit form of such stochastic corrections.

The qualitative behaviour of $\Psi$ depends on the form of the potential $V$
in eq. 31 and the coefficient $a_{{\mathsf{E}}}$. If the combination $a+\aleph
a_{{\mathsf{E}}}>0$, the order parameter $\Psi$ is gapped with the gap-scale
$(a+{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color<EMAIL_ADDRESS>a_{{\mathsf{E}}}})/\sigma_{\Psi}$ and we are in the fluid phase. On the other
hand, if $a+\aleph a_{{\mathsf{E}}}<0$, the potential is minimised at

$\displaystyle\langle|\Psi|^{2}\rangle=\Psi_{0}^{2}=-\frac{a+{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color<EMAIL_ADDRESS>a_{{\mathsf{E}}}}}{a_{4}}\leavevmode\nobreak\ ,$ (36)

thereby spontaneously breaking the U(1) symmetry and leading us to the
superfluid phase.

Depending on the signs of the parameters $a$ and $a_{{\mathsf{E}}}$, the
activity may induce or destroy the superfluidity order. In particular, if
$a>0$ and $a_{{\mathsf{E}}}<0$, the system exists in the fluid phase in the
famine state and activity induces superfluidity beyond the critical scale

$T_{0}=T_{{\mathsf{E}}}\left(1-\frac{a}{\ell
a_{{\mathsf{E}}}}\right)\leavevmode\nobreak\ ,\quad\Delta
E={\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T_{{\mathsf{E}}}\sqrt{-\frac{\gamma_{{\mathsf{E}}}}{\gamma_{{\mathsf{F}}}}\frac{a}{\ell
a_{{\mathsf{E}}}}}\leavevmode\nobreak\ .$ (37)

In contrast, if $a<0$ and $a_{{\mathsf{E}}}>0$, the system already exists in
the superfluid phase in the famine state and activity beyond the critical
scale (37) destroys superfluidity. In the remaining two cases, activity does
not alter the phase of the system. We should emphasise that our framework of
active hydrodynamics is only really reliable for “small activity”, controlled
by $\ell$, so that we do not stray too far from thermal equilibrium. In this
sense, strictly speaking, our model can describe active phase transitions when
the famine state is already close to criticality, i.e the dimensionless ratio
$|a/a_{{\mathsf{E}}}|$ is sufficiently small.

The U(1) phase of $\Psi$ in the superfluid phase becomes a massless Goldstone
mode. Parametrising the phase fluctuations as $\Psi=\Psi_{0}{\rm e}^{i\phi}$
in section 3.3, we can obtain the Josephson equation for $\phi$, i.e.

$\displaystyle\xi_{t}$
$\displaystyle=\lambda_{\phi}\mu+\frac{1}{\sigma_{\phi}}\partial_{i}\big{(}f_{s}\xi^{i}\big{)}\leavevmode\nobreak\
,$ (38)

where $\xi_{t}=\partial_{t}\phi+A_{t}$, $\xi_{i}=\partial_{i}\phi+A_{i}$ are
the superfluid potential and velocity respectively, and we have identified the
superfluid density $f_{s}$ and diffusion parameter $\sigma_{\phi}$ as

$\displaystyle f_{s}=2\Psi_{0}^{2}f_{\Psi}\leavevmode\nobreak\
,\qquad\sigma_{\phi}=2\Psi_{0}^{2}\sigma_{\Psi}\leavevmode\nobreak\ .$ (39)

The coefficient $\lambda$ in front of the chemical potential term in eq. 38 is
1 for passive superfluids, but the presence of activity improves it to

$\displaystyle\lambda_{\phi}=1+{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color<EMAIL_ADDRESS>,$ (40)

controlled by the active coefficient $\lambda_{\phi{{\mathsf{E}}}}$. In other
words, the superfluid Goldstone experiences a screened chemical potential
$\lambda_{\phi}\mu$ in the presence of activity instead of the true
thermodynamic chemical potential $\mu$. This behaviour is reminiscent of the
pseudo-spontaneous symmetry breaking pattern found in superfluids in the
presence of approximate U(1) symmetry [45].

The Lagrangian (3.2) is useful for describing the dynamics of the full U(1)
order parameter $\Psi$, together with the transition between the fluid and
active superfluid phases. However, once we have settled into the superfluid
phase sufficiently far from the phase transition point, we may integrate out
$|\Psi|$ and its noise partner to arrive at the low-energy description
exclusively for the massless Goldstone $\phi$ and the hydrodynamic degrees of
freedom. In practice, this integrating-out procedure is quite technical, so a
better strategy is to directly build the SK effective theory for $\phi$ from
scratch. Since our construction is rooted in symmetries, the final effective
description would be the same as that obtained after the integrating-out
procedure, albeit up to renormalisation of coefficients and higher-derivative
corrections. To this end, we introduce the noise field $\phi_{a}$ partner to
the superfluid Goldstone, with the KMS transformation

$\displaystyle\phi_{a}\xrightarrow{\mathrm{{KMS}}}-\hat{\phi}_{a}\equiv-\phi_{a}-i\beta(\xi_{t}-\mu)\leavevmode\nobreak\
.$ (41)

In terms of this, we have

$\displaystyle\mathcal{L}_{\text{sf}}$
$\displaystyle=f_{s}\xi^{i}\Big{(}\xi_{t}N_{ai}-B_{ai}-\partial_{i}\phi_{a}\Big{)}$
$\displaystyle\quad+i{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T\sigma_{\phi}\left(\phi_{a}+{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\frac{\mu\lambda_{\phi{{\mathsf{E}}}}}{{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T_{{\mathsf{E}}}}\Pi^{{\mathsf{E}}}_{a}}\right)\left(\hat{\phi}_{a}+{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\frac{\mu\lambda_{\phi{{\mathsf{E}}}}}{{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T_{{\mathsf{E}}}}\hat{\Pi}^{{\mathsf{E}}}_{a}}\right)$
$\displaystyle\quad{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}-\,i{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}Tf_{s}\lambda_{n{{\mathsf{E}}}}\xi^{i}\left(B_{ai}\hat{\Pi}_{a}^{{\mathsf{E}}}-\hat{B}_{ai}\Pi_{a}^{{\mathsf{E}}}\right)}\leavevmode\nobreak\
,$ (42)

that together with section 3.2 describes the superfluid phase. All
coefficients appearing here may be functions of $T$, $\mu$, and
$\xi^{2}=\xi_{i}\xi^{i}$. One may check that this yields the correct Josephson
equation in eq. 38. The low-energy thermodynamic relations are given as

$\displaystyle\mathrm{d}\mathcal{F}$
$\displaystyle=-s\,\mathrm{d}T-n\,\mathrm{d}\mu+\frac{1}{2}f_{s}\mathrm{d}\xi^{2}\leavevmode\nobreak\
,$ $\displaystyle\epsilon$ $\displaystyle=Ts+\mu
n+\mathcal{F}\leavevmode\nobreak\ ,$ (43)

which ensure the KMS symmetry. Assuming constant $\Psi_{0}$, it is
straightforward to obtain section 3.3 from section 3.2 by setting
$\Psi=\Psi_{0}{\rm e}^{i\phi}$ and $\Psi_{a}=i\Psi_{0}\phi_{a}{\rm
e}^{i\phi}$. More generally, $\Psi_{0}$ may be a function of $T$, $\mu$, and
$\xi^{2}$, in which case a more careful derivation is warranted as given in
section B.2.

###  3.4 Constitutive relations and the second law

The energy balance and charge conservation equations in section 3 are obtained
by extremising the SK effective action with respect to $X_{a}^{t}$ and
$\varphi_{a}$. The corresponding constitutive relations are given by

$\displaystyle\epsilon^{i}$
$\displaystyle=-f_{s}\xi_{t}\xi^{i}-\kappa\,\partial^{i}T\leavevmode\nobreak\
,$ $\displaystyle j^{i}$
$\displaystyle=-\lambda_{n}f_{s}\xi^{i}-\sigma\left(T\partial^{i}\frac{\mu}{T}-E^{i}\right)\leavevmode\nobreak\
,$ (44)

where we have identified

$\displaystyle\lambda_{n}=1{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\,+\,\aleph\lambda_{n{{\mathsf{E}}}}}\leavevmode\nobreak\
.$ (45)

In the presence of activity, the superfluid density $f_{s}$ as observed by
$\phi$ in eq. 38 is different from the superfluid density $\lambda_{n}f_{s}$
as appearing in $j^{i}$ in section 3.4. The source of activity in eqs. 38 and
3.4 is attributed to the heat exchange rates

$\displaystyle r_{{\mathsf{E}}}$
$\displaystyle=\ell\gamma_{{\mathsf{E}}}{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}\Delta
T-f_{s}\lambda_{n{{\mathsf{E}}}}\xi^{i}\left(T\partial_{i}\frac{\mu}{T}-E_{i}\right)-\mu\lambda_{\phi{{\mathsf{E}}}}\partial_{i}(f_{s}\xi^{i})\leavevmode\nobreak\
,$ $\displaystyle r_{{\mathsf{F}}}$
$\displaystyle=\ell\gamma_{{\mathsf{F}}}\Delta E\leavevmode\nobreak\ ,$ (46)

obtained by varying the SK action with respect to $\Phi^{{\mathsf{F}}}_{a}$
and $\Phi^{{\mathsf{E}}}_{a}$ respectively. We have recovered the rates in eq.
2 for our model, together with the higher-derivative corrections. Due to the
simplifications in our toy model, we do not see any active corrections to the
energy flux $\epsilon^{i}$, but in general they also receive corrections
similar to those in the charge flux $j^{i}$. Furthermore, in writing sections
3.4 and 3.4, we have already integrated out the magnitude $|\Psi|$ of the
order parameter. This is why the active parameter $a_{{\mathsf{E}}}$ from
section 3.3 does not appear in section 3.4 explicitly and is hidden within the
renormalisations of other active coefficients. A more comprehensive discussion
of these considerations is presented in section B.2.

The SK structure outlined above conspires to give rise to the active
modification to the second law of thermodynamics given in eq. 14. Using
section 3.3, we find that

$\displaystyle\partial_{t}s+\partial_{i}s^{i}{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\,+\,\ell{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}r_{{\mathsf{E}}}}$
$\displaystyle=\kappa\left(\frac{1}{T}\partial_{i}T\right)^{2}+\frac{\sigma}{T}\left(T\partial_{i}\frac{\mu}{T}-E_{i}\right)^{2}+\frac{1}{T\sigma_{\phi}}\big{(}\partial_{i}(f_{s}\xi^{i})\big{)}^{2}$
$\displaystyle\qquad{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}+\,\frac{\gamma_{{\mathsf{E}}}}{T}\ell^{2}{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}^{2}\Delta T^{2}+\frac{\gamma_{{\mathsf{F}}}}{T}\ell^{2}\Delta
E^{2}}\geq 0\leavevmode\nobreak\ ,$ (47)

where the heat flux is given as

$\displaystyle Ts^{i}$ $\displaystyle=\epsilon^{i}-\mu
j^{i}+f_{s}\xi^{i}(\xi_{t}-\mu)$
$\displaystyle=-\kappa\,\partial^{i}T+\mu\sigma\left(T\partial^{i}\frac{\mu}{T}-E^{i}\right){\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\,+\,\aleph\mu\lambda_{n{{\mathsf{E}}}}f_{s}\xi^{i}}\leavevmode\nobreak\
.$ (48)

The positivity of the right-hand side of section 3.4 is guaranteed by the
positivity constraints on the dissipative coefficients in eq. 32.

###  3.5 Linearised mode spectrum and violations of fluctuation-dissipation
theorem

Let us use the SK-EFT for active superfluids developed in the previous
subsections to compute the linearised mode spectrum and symmetric and retarded
two-point correlation functions. We shall also see how activity gives rise to
systematic violations of FDT in section 2.2. Consider the steady states of the
hydrodynamic model

$\displaystyle T=T_{0}\leavevmode\nobreak\
,\qquad\mu=\mu_{0}\leavevmode\nobreak\
,\qquad\phi=\lambda_{\phi}\mu_{0}t\leavevmode\nobreak\ ,$ $\displaystyle
X^{t}_{a}=\varphi_{a}=\phi_{a}=0\leavevmode\nobreak\ ,$ (49)

with $T_{0}$ given in eq. 3. In particular, note that $\phi\neq\mu_{0}t$ as it
would be in the thermal equilibrium state in the absence of activity. For
illustrative purposes, let us turn off the cross-susceptibility,
$\partial\epsilon/\partial\mu=0$, and assume that all the transport
coefficients are constants. We also focus on states with a particular value of
the chemical potential satisfying $\lambda_{\phi}\mu_{0}=0$. Under these
assumptions, the energy and charge fluctuations decouple from each other and
give rise to a relaxed energy diffusion mode and a superfluid sound mode

$\displaystyle\omega$
$\displaystyle=-i\Gamma_{\epsilon}-iD_{\epsilon}k^{2}+\ldots\leavevmode\nobreak\
,$ $\displaystyle\omega$ $\displaystyle=\pm
v_{s}k-\frac{i}{2}(D_{n}+D_{\phi})k^{2}+\ldots\leavevmode\nobreak\ ,$ (50)

where we have identified the parameters

$\displaystyle D_{\epsilon}=\frac{\kappa}{c_{v}}\leavevmode\nobreak\ ,\qquad
D_{n}=\frac{\sigma}{\chi}\leavevmode\nobreak\ ,$ $\displaystyle
v_{s}=\sqrt{\frac{\lambda_{n}f_{s}}{\chi}},\qquad
D_{\phi}=\frac{f_{s}}{\sigma_{\phi}}\leavevmode\nobreak\ ,$ (51)

where $\chi=\partial n/\partial\mu$ denotes the charge susceptibility and
$c_{v}=\partial\epsilon/\partial T$ the heat capacity. The energy relaxation
rate $\Gamma_{\epsilon}$ has been given in eq. 4.

The retarded and symmetric correlation functions can be obtained by varying
the SK generating functional with respect to the doubled background sources;
see eq. 6. For example, for the charge density correlators we find

$\displaystyle G^{\mathrm{R}}_{nn}$
$\displaystyle=\frac{\lambda_{n}f_{s}k^{2}-\left(i\omega-
D_{\phi}k^{2}\right)\sigma k^{2}}{\left(i\omega-
D_{n}k^{2}\right)\left(i\omega-
D_{\phi}k^{2}\right)+v_{s}^{2}k^{2}}\leavevmode\nobreak\ ,$ (52a)
$\displaystyle G^{\mathrm{S}}_{nn}$
$\displaystyle=2{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T_{0}\frac{|i\omega-
D_{\phi}k^{2}|^{2}\sigma\,k^{2}+\lambda_{n}^{2}f_{s}^{2}/\sigma_{\phi}\,k^{4}}{\left|\left(i\omega-
D_{n}k^{2}\right)\left(i\omega-
D_{\phi}k^{2}\right)+v_{s}^{2}k^{2}\right|^{2}}\leavevmode\nobreak\ ,$
$\displaystyle=\frac{2{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T_{0}}{\omega}{\rm Im}\,G^{\mathrm{R}}_{nn}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
+\frac{2{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T_{0}\chi
D_{\phi}\lambda_{n{{\mathsf{E}}}}v_{s}^{2}k^{4}}{\left|\left(i\omega-
D_{n}k^{2}\right)\left(i\omega-
D_{\phi}k^{2}\right)+v_{s}^{2}k^{2}\right|^{2}}\aleph\leavevmode\nobreak\ .$
(52b)

Note also that the symmetric correlator is strictly non-negative because of
the constraint on $\sigma$ in eq. 32. However, these correlators violate FDT
when activity is present, i.e. $\aleph\neq 0$. We can similarly work out the
other correlators involving $j^{i}$, $\epsilon$, or $\epsilon^{i}$.

##  4 Active nematics

Having gained some insights using our simple toy model, we now proceed to
apply our formalism of active hydrodynamics to a physically richer model of
active matter, namely active nematics, with applications such as bacterial
populations [73, 74, 75, 76], microtubule-motor protein mixtures [77, 78],
epithelial cells [79, 80, 81] and swarming sperm cells [82]. A nematic liquid
crystal is characterised by the presence of long-range orientational order in
a physical system, thereby spontaneously breaking the global rotational
symmetry. The order parameter for nematicity is a symmetric traceless tensor
$Q_{ij}=\langle a_{i}a_{j}-1/d\,\delta_{ij}\rangle_{\text{micro}}$,
constructed by averaging over the orientations $a_{i}$ of all individual
constituents with $a^{i}a_{i}=1$. Note that $Q_{ij}$ is invariant under the
flip of individual orientations $a_{i}\to-a_{i}$, hence the nematic phase
describes elongated rod-like constituents without a defined head or tail. We
will mostly be interested in a uniaxial nematic, where the individual
constituents align themselves along a single macroscopic director field
$p_{i}$, with $p^{i}p_{i}=1$, i.e.

$\displaystyle
Q_{ij}=Q_{0}\left(p_{i}p_{j}-\frac{1}{d}\delta_{ij}\right)\leavevmode\nobreak\
,$ (53)

with $Q_{0}$ representing the strength of alignment [8].

The theory of active nematic hydrodynamics is characterised by the associated
energy, momentum, and mass conservation equations taking the form

$\displaystyle\partial_{t}\epsilon+\partial_{i}\epsilon^{i}$
$\displaystyle={\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\,-\,\ell
r_{{\mathsf{F}}}\partial_{t}\Phi^{{\mathsf{F}}}-\ell
r_{{\mathsf{E}}}\partial_{t}\Phi^{{\mathsf{E}}}}\leavevmode\nobreak\ ,$
$\displaystyle\partial_{t}\pi^{i}+\partial_{j}\tau^{ij}$
$\displaystyle={\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color<EMAIL_ADDRESS>r_{{\mathsf{F}}}\partial^{i}\Phi^{{\mathsf{F}}}+\ell
r_{{\mathsf{E}}}\partial^{i}\Phi^{{\mathsf{E}}}}\leavevmode\nobreak\ ,$
$\displaystyle\partial_{t}\rho+\partial_{i}\pi^{i}$
$\displaystyle=0\leavevmode\nobreak\ ,$ (54)

where we have introduced the mass density $\rho$, momentum density
$\pi^{i}=\rho u^{i}$, stress tensor $\tau^{ij}$, with $u^{i}$ being the fluid
velocity. Note that, generically, the spatially inhomogeneous profiles of the
fuel and environment fields $\Phi^{{{\mathsf{F}}},{{\mathsf{E}}}}$ also impart
momentum to the system in section 4, but these terms drop out for the
homogeneous configuration in eq. 8. The conservation equations together can be
seen as determining the dynamics of the fluid temperature $T$, fluid velocity
$u^{i}$, and mass chemical potential $\mu$. We still need an equation of
motion for the nematic order parameter $Q_{ij}$, which will be obtained by
extremising the SK effective action for the theory.

###  4.1 Fields and symmetries

Since we have introduced conserved momentum into our setup, we need to add new
degrees of freedom in the SK framework, i.e. fluid velocity $u^{i}$ and the
associated noise field $X^{i}_{a}$, in addition to $T$, $\mu$, $X^{t}_{a}$,
and $\varphi_{a}$ already introduced in section 3.1. Furthermore, to describe
the nematic phase we need to introduce the order parameter $Q_{ij}$ and its
noise partner $\mathcal{Q}_{aij}$. The dynamical fields realise doubled U(1)
symmetry, and doubled space- and time-translation symmetries. The action of
$r$-type space- and time-translations is given as usual diffeomorphisms on all
the fields. The remaining symmetries act as

$\displaystyle X^{t}_{a}$ $\displaystyle\to-\chi^{t}_{a}\leavevmode\nobreak\
,$ $\displaystyle X^{i}_{a}$
$\displaystyle\to-\chi^{i}_{a}\leavevmode\nobreak\ ,$
$\displaystyle\varphi_{a}$
$\displaystyle\to\varphi_{a}-\Lambda_{a}-X_{a}^{t}\partial_{t}\Lambda_{r}-X_{a}^{i}\partial_{i}\Lambda_{r}\leavevmode\nobreak\
,$ (55)

while leaving $T$, $\mu$, $u^{i}$, $Q_{ij}$, and $\mathcal{Q}_{aij}$
invariant. We avoid introducing the associated background fields in the main
text for simplicity, but a detailed treatment can be found in the appendices.
We also restrict our attention to systems that feature Galilean boost
symmetry. $N_{at}$, $N_{ai}$ defined in section 3.1 are Galilean-covariant,
but $B_{at}$, $B_{ai}$ are not and we instead define

$\displaystyle\mathcal{B}_{at}$
$\displaystyle=B_{at}+\frac{1}{2}N_{at}\vec{u}^{2}\leavevmode\nobreak\ ,$
$\displaystyle\mathcal{B}_{ai}$
$\displaystyle=B_{ai}+\partial_{t}X_{ai}-u_{i}N_{at}+\frac{1}{2}N_{ai}\vec{u}^{2}\leavevmode\nobreak\
,$ $\displaystyle\mathcal{H}_{aij}$
$\displaystyle=2\partial_{(i}X_{aj)}-2N_{a(i}u_{j)}\leavevmode\nobreak\ .$
(56)

The chemical potential $\mu$ also needs to be improved to the Galilean-
invariant mass chemical potential $\varpi=\mu+\frac{1}{2}\vec{u}^{2}$. Due to
new spacetime symmetries, the definitions in eq. 26 also need to be modified
to

$\displaystyle\Delta
E=-\left(\partial_{t}+u^{i}\partial_{i}\right)\\!\Phi^{{\mathsf{F}}}_{r},\qquad{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T_{{\mathsf{E}}}=\left(\partial_{t}+u^{i}\partial_{i}\right)\\!\Phi^{{\mathsf{E}}}_{r}\leavevmode\nobreak\
,$
$\displaystyle\Pi_{a}^{{{\mathsf{F}}},{{\mathsf{E}}}}=\ell\Phi_{a}^{{{\mathsf{F}}},{{\mathsf{E}}}}+\ell\left(X_{a}^{t}\partial_{t}+X_{a}^{i}\partial_{i}\right)\\!\Phi^{{{\mathsf{F}}},{{\mathsf{E}}}}_{r}\leavevmode\nobreak\
.$ (57)

More details can be found in the appendices.

As with the superfluid model, the KMS transformation acts on the new physical
dynamical fields $u^{i}$, $Q_{ij}$ as merely a time-reversal transformation,
with the time-reversal eigenvalues given in table 1, while the noise fields
are taken to transform as

$\displaystyle X^{i}_{a}$
$\displaystyle\xrightarrow{\mathrm{{KMS}}}\hat{X}^{i}_{a}\equiv
X^{i}_{a}+i\beta u^{i},$ $\displaystyle\mathcal{Q}_{aij}$
$\displaystyle\xrightarrow{\mathrm{{KMS}}}\hat{\mathcal{Q}}_{aij}\equiv\mathcal{Q}_{aij}+i\beta\frac{\mathrm{d}}{\mathrm{d}t}Q_{ij}$
$\displaystyle\qquad\qquad\qquad+i\beta\left(\partial_{[i}u_{k]}Q_{j}{}^{k}+\partial_{[j}u_{k]}Q_{i}{}^{k}\right)\leavevmode\nobreak\
,$ (58)

where $\mathrm{d}/\mathrm{d}t\equiv\partial_{t}+u^{i}\partial_{i}$ denotes the
time-derivative long the fluid flow. The derivation of these transformations
can be found in sections A.2 and B.4.

To model a nematic liquid crystal, $Q_{ij}$ and $\mathcal{Q}_{aij}$ need to be
traceless. We can achieve this by including the following Lagrange multiplier
terms in the SK Lagrangian

$\displaystyle\mathcal{L}_{\operatorname{tr}}$
$\displaystyle=\operatorname{tr}(Q)\operatorname{tr}(\mathcal{Q}_{a})$
$\displaystyle\qquad+\frac{1}{2}\operatorname{tr}(Q)^{2}\left(N_{at}+u^{i}N_{ai}+\frac{1}{2}\mathcal{H}_{ai}{}^{i}\right)\leavevmode\nobreak\
,$ (59)

where the terms in the second line are necessitated by KMS. Therefore,
$\operatorname{tr}(Q)$ and $\operatorname{tr}(\mathcal{Q}_{a})$ are set to
zero onshell.

###  4.2 Schwinger-Keldysh effective action

We can use the ingredients outlined above to write down the SK effective
action for an active nematic liquid crystal. The “fuel part” of the Lagrangian
is still given by eq. 28, but with the modified definitions in section 4.1.
The “fluid part” is given as

$\displaystyle\mathcal{L}_{\text{f}}$
$\displaystyle=-\varepsilon\left(N_{at}+u^{i}N_{ai}\right)+\rho\left(\mathcal{B}_{at}+u^{i}\mathcal{B}_{ai}\right)$
$\displaystyle\\!\\!+\frac{1}{2}\left(\rho\,u^{i}u^{j}-\mathcal{F}\,\delta^{ij}\right)\mathcal{H}_{aij}+i{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T^{2}\kappa N_{ai}\hat{N}_{a}^{i}$
$\displaystyle\\!\\!+\frac{i}{2}{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T\eta\,\mathcal{H}_{aij}\hat{\mathcal{H}}^{ij}_{a}+\frac{i}{4}{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T\\!\left(\zeta-{\textstyle\frac{2}{d}}\eta\right)\\!\mathcal{H}_{ai}^{\leavevmode\nobreak\
\leavevmode\nobreak\ i}\hat{\mathcal{H}}_{aj}{}^{j}$
$\displaystyle\\!\\!{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\,+\,i{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T\gamma_{{\mathsf{E}}}\Pi_{a}^{{\mathsf{E}}}\hat{\Pi}_{a}^{{\mathsf{E}}}+\frac{i}{2}\frac{T}{T_{{\mathsf{E}}}}p_{{\mathsf{E}}}\delta^{ij}\Big{(}\mathcal{H}_{aij}\hat{\Pi}^{{\mathsf{E}}}_{a}-\hat{\mathcal{H}}_{aij}\Pi^{{\mathsf{E}}}_{a}\Big{)}}\leavevmode\nobreak\
.$ (60)

The passive contributions here are taken from [83], where we have introduced
the internal energy density $\varepsilon$, mass density $\rho$, free energy
density $\mathcal{F}$, thermal conductivity $\kappa$, shear viscosity $\eta$,
and bulk viscosity $\zeta$. Among the active contributions, the
$\gamma_{{\mathsf{E}}}$ term is analogous to the one in section 3.2, while
$p_{{\mathsf{E}}}$ is a new term that would give rise to an active correction
to pressure.

Moving onto the “nematic part”, the simplest SK Lagrangian for the order
parameter $Q_{ij}$ is written analogous to the superfluid model in section
3.2, but slightly more involved due to the tensorial structure of $Q_{ij}$. To
wit

$\displaystyle\mathcal{L}_{Q}$
$\displaystyle=K_{Q}\partial^{k}Q^{ij}\left(\frac{\mathrm{d}}{\mathrm{d}t}Q_{ij}N_{ak}+\frac{1}{2}\partial^{l}Q_{ij}\mathcal{H}_{akl}\right)$
$\displaystyle+2K_{Q}\partial^{i}Q^{l[k}Q^{j]}_{\leavevmode\nobreak\
l}\Big{(}\partial_{k}\mathcal{H}_{aij}+\partial_{k}u_{j}N_{ai}+2\partial_{(i}u_{k)}N_{aj}\Big{)}$
$\displaystyle-\left(\frac{\partial V}{\partial
Q_{ij}}+K_{Q}\partial^{k}Q^{ij}\partial_{k}\right)\mathcal{Q}_{aij}$
$\displaystyle+i{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T\sigma_{Q}\left(\mathcal{Q}_{a}^{ij}-\frac{1}{2}\gamma^{ijkl}\mathcal{H}_{akl}\right)\\!\left(\hat{\mathcal{Q}}_{aij}-\frac{1}{2}\gamma_{ijkl}\hat{\mathcal{H}}^{kl}_{a}\right)$
$\displaystyle{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\,+\,\frac{i}{2}{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T\left(\lambda_{{\mathsf{E}}}Q^{ij}+\frac{a_{{\mathsf{E}}}}{{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T_{{\mathsf{E}}}}Q_{kl}\gamma^{klij}\right)\Big{(}\mathcal{H}_{aij}\hat{\Pi}^{{\mathsf{E}}}_{a}-\hat{\mathcal{H}}_{aij}\Pi^{{\mathsf{E}}}_{a}\Big{)}}$
$\displaystyle{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\,-\,i\frac{T}{T_{{\mathsf{E}}}}a_{{\mathsf{E}}}Q^{ij}\Big{(}\mathcal{Q}_{aij}\hat{\Pi}^{{\mathsf{E}}}_{a}-\hat{\mathcal{Q}}_{aij}\Pi^{{\mathsf{E}}}_{a}\Big{)}}\leavevmode\nobreak\
,$ (61)

where we have introduced a single elastic constant $K_{Q}$, nematic
conductivity $\sigma_{Q}$, and the nematic shear coupling tensor
$\gamma_{ijkl}$ that is symmetric-traceless in the first two indices and
symmetric in the last two. The potential $V$ may depend arbitrarily on
$Q_{ij}$. For instance, we may take a simple form [8]

$\displaystyle
V=\frac{1}{2}a\operatorname{tr}(Q^{2})-\frac{1}{3}a_{3}\operatorname{tr}(Q^{3})+\frac{1}{4}a_{4}\operatorname{tr}(Q^{2})^{2}\leavevmode\nobreak\
,$ (62)

where $a$, $a_{3}$, and $a_{4}$ are phenomenological parameters similar to eq.
31. The form of the potential is sufficient to describe the nematic phase in
$d\leq 3$ spatial dimensions, because higher-traces of $Q_{ij}$, i.e.
$\operatorname{tr}(Q^{n})$ for $n\geq 4$, are not independent. We will need to
account for these traces in higher-dimensions. The active coefficients
$a_{{\mathsf{E}}}$ and $\lambda_{{\mathsf{E}}}$ play a similar role to their
namesakes from the active superfluid model.

The SK Lagrangian outlined above is invariant under all the global symmetries
of the SK framework. The Lagrangian also satisfies the conditions in eq. 12,
provided that we demand

$\displaystyle\kappa\geq 0\leavevmode\nobreak\ ,\quad\eta\geq
0\leavevmode\nobreak\ ,\quad\zeta\geq 0\leavevmode\nobreak\
,\quad\sigma_{Q}\geq 0\leavevmode\nobreak\
,\quad\gamma_{{{\mathsf{E}}},{{\mathsf{F}}}}\geq 0\leavevmode\nobreak\ .$ (63)

Finally, the SK Lagrangian is KMS-invariant, provided that $\varepsilon$,
$\rho$, and $K_{Q}$ are related to $\mathcal{F}$ via the thermodynamic
relations

$\displaystyle\varepsilon$
$\displaystyle=Ts+\rho\varpi+\mathcal{F}\leavevmode\nobreak\ ,$
$\displaystyle\mathrm{d}\mathcal{F}$
$\displaystyle=-s\,\mathrm{d}T-\rho\,\mathrm{d}\varpi$
$\displaystyle\qquad+\frac{1}{2}K_{Q}\mathrm{d}\Big{(}\partial^{k}Q_{ij}\partial_{k}Q^{ij}\Big{)}+\frac{\partial
V}{\partial Q_{ij}}\mathrm{d}Q_{ij}\leavevmode\nobreak\ ,$
$\displaystyle\mathrm{d}\varepsilon$
$\displaystyle=T\,\mathrm{d}s+\varpi\,\mathrm{d}\rho$
$\displaystyle\qquad+\frac{1}{2}K_{Q}\mathrm{d}\Big{(}\partial^{k}Q_{ij}\partial_{k}Q^{ij}\Big{)}+\frac{\partial
V}{\partial Q_{ij}}\mathrm{d}Q_{ij}\leavevmode\nobreak\ .$ (64)

With these in place, the SK Lagrangian is KMS-invariant up to a boundary term
$-i\partial_{t}(\beta\mathcal{F})-i\partial_{i}(\beta\mathcal{F}\,u^{i})$.
Assuming $K_{Q}$ to be constant, we arrive at the Landau-de Gennes free energy
[8, 84, 85]

$\displaystyle\mathcal{F}=-p+\frac{1}{2}K_{Q}\partial^{k}Q_{ij}\partial_{k}Q^{ij}+V\leavevmode\nobreak\
.$ (65)

The generalisation to multiple elastic constants is straightforward and has
been discussed in the appendix.

###  4.3 Spontaneous symmetry breaking

The equation of motion for the nematic order parameter $Q_{ij}$ can be
obtained by extremising the SK effective action with respect to
$\mathcal{Q}_{aij}$. We find

$\frac{\mathrm{d}}{\mathrm{d}t}Q_{ij}=\frac{1}{\sigma_{Q}}\Big{(}\mathsf{H}_{ij}{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\,-\,\aleph
a_{{\mathsf{E}}}Q_{ij}}\Big{)}+\mathsf{S}_{ijkl}\partial^{k}u^{l}\leavevmode\nobreak\
,$ (66)

where $\mathsf{H}_{ij}$ is the thermodynamic conjugate to $Q_{ij}$ and
$\mathsf{S}_{ijkl}$ is the generalised advection tensor, defined as

$\displaystyle\mathsf{H}_{ij}$
$\displaystyle=\partial_{k}\big{(}K_{Q}\partial^{k}Q_{ij}\big{)}-\frac{\partial
V}{\partial Q_{ij}}-\text{(trace)}\leavevmode\nobreak\ ,$
$\displaystyle\mathsf{S}_{ijkl}$
$\displaystyle=Q_{k(i}\delta_{j)l}-Q_{l(i}\delta_{j)k}+\gamma_{ijkl}\leavevmode\nobreak\
.$ (67)

Whether the system ends up in the nematic phase or the fluid phase depends on
the potential $V$ in eq. 62 and the activity parameter $a_{{\mathsf{E}}}$. The
system admits a fluid phase for $a+\aleph a_{{\mathsf{E}}}>0$ with $\langle
Q_{ij}\rangle=0$. Whereas, when $a+\aleph
a_{{\mathsf{E}}}<\frac{d-2}{8d}a_{3}^{2}/a_{4}$, the system admits a nematic
phase with $\langle
Q_{ij}\rangle=Q_{0}\left(p^{0}_{i}p^{0}_{j}-1/d\,\delta_{ij}\right)$, where
$p^{0}_{i}$ is a fixed unit vector and $Q_{0}$ is given by [85]

$\displaystyle Q_{0}$
$\displaystyle=\frac{a_{3}}{|a_{3}|}\sqrt{\frac{d}{(d-1)a_{4}}}$
$\displaystyle\times\left(\sqrt{\frac{d-2}{8d}\frac{a_{3}^{2}}{a_{4}}}+\sqrt{\frac{d-2}{8d}\frac{a_{3}^{2}}{a_{4}}-a{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\,-\,\aleph
a_{{\mathsf{E}}}}}\right)\leavevmode\nobreak\ .$ (68)

Both the fluid and nematic phases are admitted in the overlapping regime
$0<a+\aleph a_{{\mathsf{E}}}<\frac{d-2}{8d}a_{3}^{2}/a_{4}$. In thermal
equilibrium, the system prefers the phase with lower free energy, with a
first-order phase transition between them at the coexistence point
$a_{*}=\frac{d-2}{9d}a_{3}^{2}/a_{4}$. We do not have the luxury to compare
the free energies to determine thermodynamic stability of a non-equilibrium
steady state in the presence of activity; see our discussion at the end of
section 2.4. However, for small activity, we can expect the first-order phase
transition to happen somewhere in the vicinity of
$a_{*}\approx\frac{d-2}{9d}a_{3}^{2}/a_{4}$.

Similar to our discussion for active superfluids around eq. 37, activity may
induce or destroy the nematic phase transition when $a$ and $a_{{\mathsf{E}}}$
have opposite signs.

The fluctuations of $Q_{ij}$ in the fluid phase are gapped and the low-energy
description is just given by the Lagrangian (4.2). In the nematic phase,
however, we need to account for the Goldstone modes associated with the
spontaneously broken rotation symmetry generators, parametrised in terms of
the nematic director $p_{i}$ given in eq. 53. The associated equation of
motion can be read off from eq. 66 as

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}p_{i}$
$\displaystyle=\frac{\bar{p}_{i}^{j}}{\sigma_{p}}\partial_{k}\Big{(}K\partial^{k}p_{j}\Big{)}+\left(p^{[k}\bar{p}^{l]}_{i}+\gamma
p^{(k}\bar{p}^{l)}_{i}\right)\partial_{k}u_{l}\leavevmode\nobreak\ ,$ (69)

where $\bar{p}_{ij}=\delta_{ij}-p_{i}p_{j}$ is the projector transverse
$p_{i}$, and we have identified the Frank constant $K$, relaxation coefficient
$\sigma_{p}$, and the shear coupling coefficient $\gamma$ as

$\displaystyle K=2Q_{0}^{2}K_{Q}\leavevmode\nobreak\
,\qquad\sigma_{p}=2Q_{0}^{2}\sigma_{Q}\leavevmode\nobreak\ ,$
$\displaystyle\gamma=\frac{2/Q_{0}}{d-1}p_{i}p_{k}\bar{p}_{jl}\gamma^{ijkl}\leavevmode\nobreak\
.$ (70)

The effects of activity are hidden within $Q_{0}$.

We can write down a low-energy effective theory for the director $p_{i}$
directly, akin to the superfluid model in section 3.3. To this end, we need to
define the KMS noise partner ${\mathpzc p}_{ai}$ to the director, satisfying
$p^{i}{\mathpzc p}_{ai}=0$, with the KMS transformation

$\displaystyle{\mathpzc
p}_{ai}\xrightarrow{\mathrm{{KMS}}}\eta_{\mathrm{T}}\,\hat{\\!{\mathpzc
p}}_{ai}\equiv\eta_{\mathrm{T}}\\!\left({\mathpzc
p}_{ai}+i\beta\frac{\mathrm{d}}{\mathrm{d}t}p_{i}+i\beta\partial_{[i}u_{j]}p^{j}\right)\leavevmode\nobreak\
.$ (71)

Note that since the original nematic order parameter $Q_{ij}$ has a $p_{i}\to-
p_{i}$ symmetry, we are free to choose either time-reversal eigenvalue for
$p_{i}$, ${\mathpzc p}_{ai}$. In terms of these, the active nematic Lagrangian
takes the simple form

$\displaystyle\mathcal{L}_{\text{nem}}$
$\displaystyle=-\left(\frac{\partial\mathcal{F}}{\partial
p_{i}}+\frac{\partial\mathcal{F}}{\partial(\partial_{k}p_{i})}\partial_{k}\right){\mathpzc
p}_{ai}$
$\displaystyle+\frac{\partial\mathcal{F}}{\partial(\partial_{k}p_{i})}\\!\left(\frac{\mathrm{d}}{\mathrm{d}t}p_{i}N_{ak}+\frac{1}{2}\partial^{l}p_{i}\mathcal{H}_{akl}\right)$
$\displaystyle+\frac{\partial\mathcal{F}}{\partial(\partial_{i}p_{[k})}p^{j]}\left(\partial_{k}\mathcal{H}_{aij}+\partial_{k}u_{j}N_{ai}+2\partial_{(i}u_{k)}N_{aj}\right)$
$\displaystyle+i{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T\sigma_{p}\\!\left({\mathpzc
p}_{a}^{i}-\frac{\gamma}{2}\bar{p}^{ij}p^{k}\mathcal{H}_{akj}\right)\\!\left(\,\hat{\\!{\mathpzc
p}}_{ai}-\frac{\gamma}{2}\bar{p}_{ij}p_{k}\hat{\mathcal{H}}^{kj}_{a}\right)$
$\displaystyle{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\,+\,\frac{i}{2}{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}TQ_{0}\lambda_{{\mathsf{E}}}p^{i}p^{j}\Big{(}\mathcal{H}_{a\langle
ij\rangle}\hat{\Pi}^{{\mathsf{E}}}_{a}-\hat{\mathcal{H}}_{a\langle
ij\rangle}\Pi^{{\mathsf{E}}}_{a}\Big{)}}\leavevmode\nobreak\ ,$ (72)

with the normalisation conditions imposed by the Lagrange multiplier terms

$\displaystyle\mathcal{L}_{\text{norm}}$
$\displaystyle=(p^{i}p_{i}-1)(p^{j}{\mathpzc p}_{aj})$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
+\frac{1}{4}(q^{k}q_{k}-1)^{2}\left(N_{at}+u^{i}N_{ai}+\frac{1}{2}\mathcal{H}_{ai}{}^{i}\right)\leavevmode\nobreak\
.$ (73)

The low-energy thermodynamic relations are given as

$\displaystyle\varepsilon$
$\displaystyle=Ts+\rho\varpi+\mathcal{F}\leavevmode\nobreak\ ,$
$\displaystyle\mathrm{d}\mathcal{F}$
$\displaystyle=-s\,\mathrm{d}T-\rho\,\mathrm{d}\varpi+\frac{1}{2}K\mathrm{d}\Big{(}\partial^{i}p^{j}\partial_{i}p_{j}\Big{)}\leavevmode\nobreak\
,$ (74)

which ensure the KMS-invariance of the theory. As with the superfluid case,
section 4.3 can be obtained from section 4.2 quite simply under the assumption
that $Q_{0}$ is constant, by identifying $Q_{ij}$ as in eq. 53 and
$\mathcal{Q}_{aij}=2Q_{0}p_{(i}{\mathpzc p}_{aj)}$. For a non-constant
$Q_{0}$, a more careful calculation needs to be performed to obtain the
renormalisation of various coefficients appearing in section 4.3, which we
leave for future work.

###  4.4 Constitutive relations and the second law

Extremising the SK effective action with respect to $X^{t}_{a}$, $X^{i}_{a}$,
and $\varphi_{a}$, we recover the energy, momentum, and mass conservation
equations, with the energy density
$\epsilon=\varepsilon+\frac{1}{2}\rho\vec{u}^{2}$, momentum density
$\pi^{i}=\rho u^{i}$, and mass density $\rho$. The associated energy flux and
stress tensor

$\displaystyle\epsilon^{i}$
$\displaystyle=\left(\varepsilon+\frac{1}{2}\rho\vec{u}^{2}-\mathcal{F}{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\,+\,p_{{\mathsf{E}}}\ell{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}\Delta
T}\right)u^{i}{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\,+\,\lambda_{{\mathsf{E}}}Q^{ij}u_{j}\ell{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}\Delta T}$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
-\partial^{i}Q^{kl}\partial_{t}Q_{kl}-u_{j}\left(\gamma^{klij}-2\delta^{k[i}Q^{j]l}\right)\frac{\delta\mathcal{F}}{\delta
Q_{kl}}$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ -2\eta
u_{j}\partial^{\langle i}u^{j\rangle}-\zeta
u^{i}\partial_{k}u^{k}-\kappa\,\partial^{i}T{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}\pgfsys@color@cmyk@stroke{0}{0}{0}{0.50}\pgfsys@color@cmyk@fill{0}{0}{0}{0.50}\,-\,\partial_{k}\big{(}\mathcal{X}^{[ik]j}u_{j}\big{)}}\leavevmode\nobreak\
,$ $\displaystyle\tau^{ij}$
$\displaystyle=\rho\,u^{i}u^{j}-\left(\mathcal{F}{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\,-\,p_{{\mathsf{E}}}\ell{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}\Delta
T}\right)\delta^{ij}{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\,+\,\lambda_{{\mathsf{E}}}Q^{ij}\ell{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}\Delta T}$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
+\partial^{i}Q^{kl}\partial^{j}Q_{kl}-\left(\gamma^{klij}-2\delta^{k[i}Q^{j]l}\right)\frac{\delta\mathcal{F}}{\delta
Q_{kl}}$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
-\eta\partial^{\langle
i}u^{j\rangle}-\zeta\delta^{ij}\partial_{k}u^{k}{\color[rgb]{0.5,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0.5}\pgfsys@color@cmyk@stroke{0}{0}{0}{0.50}\pgfsys@color@cmyk@fill{0}{0}{0}{0.50}\,-\,2\partial_{k}\mathcal{X}^{[ik]j}}\leavevmode\nobreak\
.$ (75)

The grayed out terms are total-derivatives and drop out of the conservation
equations whose form is discussed in section B.4; the remaining contributions
are sometimes referred to as the “canonical” constitutive relations. Note that
while the full stress tensor is symmetric, the canonical part is not. The
distinctive active feature in these constitutive relations
$\sim\lambda_{{\mathsf{E}}}Q^{ij}$ term in the stress tensor that is absent in
passive nematics [85, 86]. The active contributions in blue may be attributed
to the rates

$\displaystyle r_{{\mathsf{E}}}$
$\displaystyle=\ell\gamma_{{\mathsf{E}}}{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}\Delta
T+p_{{\mathsf{E}}}\partial_{i}u^{i}+iTa_{{\mathsf{E}}}Q^{ij}\hat{\mathcal{Q}}_{aij}$
$\displaystyle\qquad+\left(\lambda_{{\mathsf{E}}}Q^{ij}+a_{{\mathsf{E}}}Q_{kl}\gamma^{klij}\right)\partial_{i}u_{j}\leavevmode\nobreak\
,$ $\displaystyle r_{{\mathsf{F}}}$
$\displaystyle=\ell\gamma_{{\mathsf{F}}}\Delta E\leavevmode\nobreak\ .$ (76)

We can also reduce these constitutive relations in terms of the director
$p_{i}$ instead of the full $Q_{ij}$, but we do not perform this exercise
here.

Using the thermodynamic relations in section 4.2, we can verify that the
constitutive relations satisfy the second law of thermodynamics

$\displaystyle\partial_{t}s+\partial_{i}s^{i}{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\,+\,\ell{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}r_{{\mathsf{E}}}}$
$\displaystyle=\kappa\left(\frac{1}{T}\partial_{i}T\right)^{2}+\frac{2\eta}{T}(\partial_{\langle
i}u_{j\rangle})^{2}+\frac{\zeta}{T}(\partial_{i}u^{i})^{2}+\frac{1}{T\sigma_{Q}}(\mathsf{H}_{ij})^{2}$
$\displaystyle\qquad{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\,+\,\frac{\gamma_{{\mathsf{E}}}}{T}\ell^{2}{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}^{2}\Delta T^{2}+\frac{\gamma_{{\mathsf{F}}}}{T}\ell^{2}\Delta
E^{2}}\geq 0\leavevmode\nobreak\ ,$ (77)

where the heat flux is given by

$\displaystyle Ts^{i}$
$\displaystyle=\epsilon^{i}-\mathcal{F}u^{i}-\left(\varpi-{\textstyle\frac{1}{2}}\vec{u}^{2}\right)\rho
u^{i}-u_{j}\tau^{ji}$
$\displaystyle\qquad+K_{Q}\partial^{i}Q^{kl}\frac{\mathrm{d}}{\mathrm{d}t}Q_{kl}$
$\displaystyle=Tsu^{i}+\kappa\,\partial^{i}T\leavevmode\nobreak\ ,$ (78)

The positivity of the right-hand side of section 4.4 follows from the
constraints in eq. 63.

Having formulated the SK effective action and equations of motion for our
thermal theory of active nematics, one may proceed as section 3.5 and perform
a linearised analysis of the mode spectrum and correlation functions. This
would also enable us to compute the violations of FDT analogous to eq. 52.
Interestingly, active nematics suffer from a linear instability to splay or
bend deformations arising from the $\tau^{ij}\sim\lambda_{{\mathsf{E}}}Q^{ij}$
term in the constitutive relations [13, 12]. This drives the system to a state
of nematic turbulence [9, 10, 11], invalidating any linearised results.
Therefore, we conclude our formulation of thermal active nematic hydrodynamics
here and hope to return with insights on nematic turbulence in future work.

##  5 A buzz of activity

In this work we have developed a new hydrodynamic framework for active matter
with local temperature fluctuations. The primary ingredient in this framework
is the first law of thermodynamics, i.e. _energy balance_ : while the internal
energy of an active system is not necessarily conserved, it must be balanced
by the work done by the fuel source that drives activity and the heat lost to
the environment in the process. This means that to appropriately model the
thermal fluctuations of active matter, we need to add both a fuel source and a
distinct energy sink into the hydrodynamic framework, modelled in this work
using the background scalar fields $\Phi^{{\mathsf{F}}}$ and
$\Phi^{{\mathsf{E}}}$ respectively. In field-theoretic terms, these are seen
as background sources coupled to the operators $r_{{\mathsf{F}}}$ and
$r_{{\mathsf{E}}}$, measuring the rates of fuel consumption and heat loss
respectively. Another important ingredient that was used in our framework is
the existence of a _famine state_ : when an active system runs out of fuel, it
must behave passively, i.e. it must obey the FDTs.

The SK-EFT formalism is perfectly suited for our purposes. It is a symmetry-
based effective field theory approach to stochastic systems that, in
principle, applies arbitrary far from equilibrium [49, 50, 51, 52, 53, 54,
55]. In particular, the energy balance and (non-linear) FDT requirements are
built into the SK-EFT formalism through time-translation symmetry and discrete
KMS symmetry. While an active system by itself is not invariant under time-
translations, the symmetry does apply when acting simultaneously on the
fuel/environment background fields; see [45, 46, 47] for a similar procedure
for other symmetries. Furthermore, by requiring that the active system
respects the original KMS symmetry in the famine state, we derived a new
_active KMS symmetry_ , which makes the SK-EFT formalism suitable for
modelling active matter in contact with a thermal bath (see section 2.2).

The time-translation symmetry, active KMS symmetry, and the unitarity
constraints built into the SK-EFT framework, together yield the second law of
thermodynamics appropriate for active systems. The active second law of
thermodynamics dictates that only the total entropy, i.e. the sum of the
entropy of the system and the entropy lost to the environment, is increasing.
This is unlike passive systems, where entropy increases locally (see section
2.3). We also explored how the active KMS symmetry is related to microscopic
reversibility and gives rise to an active correction to the principle of
detailed balance in terms of the work performed by the external fields, as
well as the fuel/environment sources (see section 2.4).

We have applied our hydrodynamic framework for active matter to two
interesting examples, namely active superfluids in section 3 and active
nematics in section 4. In both these examples, we discussed how coupling to
fuel/environment sources may generate entirely new active transport parameters
in the hydrodynamic equations that are forbidden for passive systems. These
active parameters are entirely fixed in terms of the fuel consumption and heat
loss rate operators, $r_{{\mathsf{F}}}$ and $r_{{\mathsf{E}}}$. These
coefficients characterise the explicit structure of FDT violation in
frequency- and wavevector-dependent retarded and symmetric correlation
functions.

Our hydrodynamic framework is particularly useful when the active system under
consideration is operating sufficiently close to thermal equilibrium, meaning
that the strength of activity
$\aleph=\ell(T-T_{{\mathsf{E}}})/T_{{\mathsf{E}}}$ is small, where $\ell$ is a
small bookkeeping parameter. Assuming $\ell$ to scale on par with the spatial
derivatives $\mathcal{O}(\partial)$, which are treated as small in
hydrodynamics, this allows us to systematically organise the active and
passive corrections to the hydrodynamic constitutive relations order-by-order
in derivatives. In sections 3 and 4, we have focused on leading-order active
corrections, while we will comment on a few potentially interesting
subleading-order corrections later in this section.

While we have focused our attention to simple models of active superfluids and
active nematics for concreteness, the active hydrodynamic framework developed
in this work opens up possibilities for symmetry-based systematic modelling of
active phenomena far beyond these specific examples. We outline a few of these
potential avenues in the following.

#### Dissipation vs response vs fluctuation:

Ordinarily, as a result of FDT in a passive hydrodynamics, the transport
coefficients controlling dissipation in the constitutive relations are the
same coefficients controlling the strength of the retarded and symmetric
correlators. However, this may no longer be the case in active hydrodynamics.
Consider, for example, the fluid part of the superfluid effective action
written in section 3.2. We may add to this new active terms of the kind

$\displaystyle-i\frac{T}{T_{{\mathsf{E}}}}\left(\sigma_{\rm
t}^{{\mathsf{E}}}T\partial^{i}\frac{\mu}{T}-\sigma_{\rm
r}^{{\mathsf{E}}}E^{i}\right)\left(B_{ai}\hat{\Pi}^{{\mathsf{E}}}_{a}+\hat{B}_{ai}\Pi^{{\mathsf{E}}}_{a}\right)\leavevmode\nobreak\
,$ (79)

which modify the conductivity contribution in the flux to
$j^{i}\sim-\sigma_{\rm t}T\partial^{i}(\mu/T)+\sigma_{\rm r}E^{i}$, where
$\sigma_{\rm t,\rm r}=\sigma+\aleph\sigma_{\rm t,\rm r}^{{\mathsf{E}}}$.
Switching off the superfluid part, the three kinds of conductivities,
$\sigma$, $\sigma_{\rm t}$, and $\sigma_{\rm r}$ show up differently in the
retarded and symmetric correlators of density as

$\displaystyle G^{\mathrm{R}}_{nn}$ $\displaystyle=\frac{-\sigma_{\rm
r}\,k^{2}}{i\omega-\sigma_{\rm t}/\chi\,k^{2}}\leavevmode\nobreak\ ,\quad
G^{\mathrm{S}}_{nn}=\frac{2{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T_{0}\sigma\,k^{2}}{\left|i\omega-\sigma_{\rm
t}/\chi\,k^{2}\right|^{2}}\leavevmode\nobreak\ .$ (80)

The “transport conductivity” $\sigma_{\rm t}$ controls the poles of both the
retarded and symmetric correlators, and thus the linearised mode spectrum. On
the other hand, the “response conductivity” $\sigma_{\rm r}$ and the
“fluctuation conductivity” $\sigma$ control the strength of retarded and
symmetric correlators respectively. All three notions of conductivity may
generically be different in an active system. We should note that for weak
activity, $\ell\sim\mathcal{O}(\partial)$, the terms in eq. 79 formally appear
at $\mathcal{O}(\partial^{2})$ in the constitutive relations, which makes
sense because these are active corrections on top of the already derivative
suppressed dissipative corrections. Therefore, at least for weakly active
systems, they are generically less important than the other active corrections
considered in section 3.

A similar strategy may be employed for adding active corrections to
viscosities in a hydrodynamic model with conserved momentum, i.e.

$-i\frac{T}{T_{{\mathsf{E}}}}\left(\eta^{{\mathsf{E}}}_{\rm
t}\partial^{\langle i}u^{j\rangle}+\frac{1}{2}\zeta^{{\mathsf{E}}}_{\rm
t}\delta^{ij}\partial_{k}u^{k}\right)\left(\mathcal{H}_{aij}\hat{\Pi}_{a}^{{\mathsf{E}}}+\hat{\mathcal{H}}_{aij}\Pi_{a}^{{\mathsf{E}}}\right)\leavevmode\nobreak\
,$ (81)

that corrects the viscosity terms in the stress tensor as
$\tau^{ij}\sim-2\eta_{\rm t}\partial^{\langle i}u^{j\rangle}-\zeta_{\rm
t}\delta^{ij}\partial_{k}u^{k}$, where $\eta_{\rm
t}=\eta+\aleph\eta^{{\mathsf{E}}}_{\rm t}$ and $\zeta_{\rm
t}=\eta+\aleph\zeta^{{\mathsf{E}}}_{\rm t}$ are the “transport viscosities”.
Unlike conductivities, however, spacetime symmetries generically force the
“response viscosities” to be the same as “transport viscosities”, while these
may generically differ from the “fluctuation viscosities” $\eta$ and $\zeta$
in an active system.

Staring at eqs. 79 and 81, we may notice that these are similar in form to the
$\lambda_{n{{\mathsf{E}}}}$ term in section 3.2 or the
$\lambda_{{\mathsf{E}}}$ term in section 4.2, except that the signs in the
parenthesis are different due to different time-reversal eigenvalues. This is
indeed the common theme for generating active corrections in the constitutive
relations, by writing down cross terms with the activity sources; see [27, 5,
25].

#### Sign-indefinite dissipation:

A consequence of the active corrections to the dissipative transport
coefficients is that they may be sign-indefinite. Going back to the
conductivity corrections in eq. 79, we note that neither of $\sigma_{\rm
t}^{{\mathsf{E}}}$ or $\sigma_{\rm r}^{{\mathsf{E}}}$ are required to be non-
negative by the SK unitarity constraints in eq. 12; only $\sigma$ needs to be
non-negative. This is essentially requiring that the symmetric correlator,
which computes the variance of a stochastic random variable, must be non-
negative, while no such requirement exists for the retarded correlator in the
absence of FDT. This is to say that $\sigma_{\rm t}$ or $\sigma_{\rm r}$ may
turn negative for sufficiently strong activity. Interestingly, since
$\sigma_{\rm t}$ controls the pole of the correlation functions in eq. 80,
this also means that such active systems will become unstable for sufficiently
strong activity. While a purely diffusive system becomes unstable as soon as
$\sigma_{\rm t}$ turns negative, this feature is sensitive to the details of
the mode spectrum. For example, the superfluid mode spectrum in section 3.5
remains stable until $\sigma_{\rm t}$ becomes sufficiently negative to
overcome the Goldstone diffusion parameter $D_{\phi}=f_{s}/\sigma_{\phi}$.
Similar considerations also apply for the transport viscosities $\eta_{\rm t}$
and $\zeta_{\rm t}$.

#### Odd viscosity and odd elasticity:

Another class of active phenomena that has gained recent traction are odd
viscosity and odd elasticity, which can arise in two-dimensional parity-
violating systems. For the former, let us consider the case where parity is
broken but time-reversal symmetry is present. In that case, odd viscosity is
prohibited for passive systems, however for active systems one can add a term
to the SK-EFT Lagrangian

$\displaystyle-i\frac{T}{T_{{\mathsf{E}}}}\tilde{\eta}\,\delta^{ik}\epsilon^{jl}\partial_{(k}u_{l)}\left(\mathcal{H}_{aij}\hat{\Pi}_{a}^{{\mathsf{E}}}+\hat{\mathcal{H}}_{aij}\Pi_{a}^{{\mathsf{E}}}\right)\leavevmode\nobreak\
,$ (82)

where $\epsilon^{ij}$ is the 2d anti-symmetric Levi-Civita symbol. This
results in an odd-elasticity term in the stress tensor of the kind
$\tau^{ij}\sim-2\aleph\tilde{\eta}\,\delta^{k(i}\epsilon^{j)l}\partial_{(k}u_{l)}$,
which is forbidden by the Onsager’s relations in passive time-reversal
symmetric hydrodynamics. Note that for weak activity,
$\ell\sim\mathcal{O}(\partial)$, the contribution in eq. 82 appears at
$\mathcal{O}(\partial^{2})$. In contrast, odd-elasticity appears already at
$\mathcal{O}(\partial)$ at weak activity. Directly importing the form in eq.
82, we may write

$\displaystyle
i\frac{T}{T_{{\mathsf{E}}}}K^{o}\,\delta^{ik}\epsilon^{jl}u_{kl}\left(\mathcal{H}_{aij}\hat{\Pi}_{a}^{{\mathsf{E}}}-\hat{\mathcal{H}}_{aij}\Pi_{a}^{{\mathsf{E}}}\right)\leavevmode\nobreak\
,$ (83)

where $u_{ij}$ is the strain tensor and the sign inside the parenthesis is $+$
because $u_{ij}$ is even under time-reversal symmetry. This results in the
odd-elasticity contribution to the stress tensor that take the form
$\tau^{ij}\sim 2\aleph K^{o}\,\delta^{k(i}\epsilon^{j)l}u_{kl}$, introduced in
[14].

#### Explicitly broken symmetries:

Just like the explicitly broken time-translation symmetry, many active systems
of interest also explicitly break other internal and/or spacetime symmetries.
Examples include: broken number/charge conservation in Malthusian active
matter [87, 88, 89] where entities may replicate or die over time; broken
spatial-translations and/or boosts when the system is subjected to physical
barriers or friction [3]; broken rotations when subjected to electromagnetic
fields or acceleration [16]; and also broken topological symmetries [90, 91,
47]. Explicitly broken or approximate symmetries can be modelled by
introducing the appropriate sink background fields in the hydrodynamic
framework [45, 47], just like $\Phi^{{\mathsf{E}}}$ plays the role of an
energy sink in our construction.

For instance, we introduce a momentum sink for broken spatial-translations in
the form of a background scalar field $\Phi^{I}$, one for each spatial
direction, with the noise partner $\Phi^{I}_{a}$ in the SK framework. They
feature the regular KMS transformation (2.2) with time-reversal eigenvalue
$+1$, and take values $\delta^{I}_{i}x^{i}$ and $0$ respectively for a
homogeneous momentum source. The SK-EFT Lagrangian now contains new terms such
as

$\displaystyle
i{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T\sigma_{\Phi}\Pi^{I}_{a}\hat{\Pi}_{Ia}-i\frac{T}{T_{{\mathsf{E}}}}\sigma^{{\mathsf{E}}}_{\Phi{\rm
t}}\frac{\mathrm{d}\Phi_{I}}{\mathrm{d}t}\left(\Pi_{Ia}\hat{\Pi}^{{\mathsf{E}}}_{a}+\hat{\Pi}_{Ia}\Pi^{{\mathsf{E}}}_{a}\right)\leavevmode\nobreak\
,$ (84)

where
$\Pi^{I}_{a}=\ell\Phi_{a}^{I}+\ell(X_{a}^{t}\partial_{t}+X_{a}^{i}\partial_{i})\Phi^{I}$,
and $\ell$ is taken to collectively control all forms of explicit symmetry
breaking. Together, they give rise to a contribution on the right-hand side of
the momentum conservation equation in section 4, which for a homogeneous
background configuration takes the form
$\partial_{t}\pi^{i}\sim-\ell^{2}\sigma_{\Phi{\rm t}}u^{i}$, where
$\ell\sigma_{\Phi{\rm
t}}=\ell\sigma_{\Phi}+\aleph\sigma^{{\mathsf{E}}}_{\Phi{\rm t}}$. Since
$\sigma_{\Phi\rm t}$ must be strictly non-negative for passive systems,
provided that none of the other momentum-imparting background fields are
turned on, the only allowed solution is $u^{i}=0$. However, when activity is
turned on, $\sigma_{\Phi\rm t}$ is no longer sign-definite and the momentum
equation may admit other favourable solutions where $\sigma_{\Phi\rm t}$ goes
to 0 instead.

This effect is most evident for fluids without boost symmetry, in which case
the coefficients may take the form $\sigma_{\Phi}=A+B\vec{u}^{2}$ and
$\sigma^{{\mathsf{E}}}_{\Phi{\rm t}}=A_{{\mathsf{E}}}$, with $A,B>0$ and
$A_{{\mathsf{E}}}<0$. When $\ell A+\aleph A_{{\mathsf{E}}}>0$, the system
prefers the solution with $u^{i}=0$. Whereas, when $\ell A+\aleph
A_{{\mathsf{E}}}<0$, the system spontaneously picks a state with
$\vec{u}^{2}=-(A+\aleph/\ell\,A_{{\mathsf{E}}})/B$. This is precisely the
structure underlying the Toner-Tu model of flocking [3]. Using the combination
of broken translations, broken boosts, and activity, one may also generate all
other terms in the Toner-Tu model; we leave an in-depth analysis for future
work.

#### Active polar:

In section 4, we studied active nematic liquid crystals described by the order
parameter $Q_{ij}$, whose low-energy dynamics reduces to the director $p_{i}$,
together with the constraint $\vec{p}^{2}=1$ and $p_{i}\to-p_{i}$ symmetry. If
we were to relax these, we may describe an active polar liquid crystal where
the microscopic constituents feature distinct heads and tails [3, 4, 5, 6, 7].
The free energy density $\mathcal{F}$ now contains a potential
$V(\vec{p}^{2})=\frac{1}{2}a\,\vec{p}^{2}+\frac{1}{4}a_{4}\vec{p}^{4}$ that
controls the transition between the ordered and disordered phases similar to
our discussion in section 4.3. The explicit structure of the hydrodynamic
description depends on the time-reversal eigenvalue of $p_{i}$, which the
nematic phase is agnostic to on account of the $p_{i}\to-p_{i}$ symmetry. For
example, $p_{i}$ is time-reversal-odd if it models the individual spins of
microscopic constituents or time-reversal-even if it models the individual
dipole moments.

For time-reversal-even $p_{i}$, the free energy density $\mathcal{F}$ may
contain new terms at one-derivative order, e.g.
$f_{1}\,p^{i}\partial_{i}\rho$,
$\frac{1}{2}f_{2}\,p^{i}\partial_{i}\vec{p}^{2}$, which contribute to SK-EFT
Lagrangian in section 4.3 accordingly. They generate a number of new terms in
the equation for $\partial_{t}p_{i}$ in eq. 69, i.e.
$\lambda_{2}p_{i}\partial_{k}p^{k}$, $\lambda_{3}\partial_{i}\vec{p}^{2}$,
$\lambda_{4}p_{i}p^{k}\partial_{k}\vec{p}^{2}$, $\lambda_{5}\partial_{i}\rho$,
$\lambda_{6}p_{i}p^{k}\partial_{k}\rho$, with the coefficients
$\lambda_{2,\ldots,6}$ fixed in terms of $f_{1,2}$. Importantly, the advective
term like $\lambda_{1}p^{k}\partial_{k}p_{i}$ is forbidden in purely passive
polar liquid crystals. By contrast, all these terms are forbidden in a passive
polar liquid crystal for time-reversal-odd $p_{i}$. These constraints may be
overcome by introducing activity into the model, via new contributions in the
SK-EFT Lagrangian

$\displaystyle-i\frac{T}{T_{{\mathsf{E}}}}\left(\lambda_{1}p^{k}\partial_{k}p^{i}+\ldots\right)\left({\mathpzc
p}_{ai}\hat{\Pi}^{{\mathsf{E}}}_{a}\mp\hat{\mathpzc
p}_{ai}\Pi^{{\mathsf{E}}}_{a}\right)\leavevmode\nobreak\ ,$ (85)

where the upper/lower sign is applicable for even/odd $p_{i}$ under time-
reversal. Note that even though activity can be used to generate every
possible term in the equation for $\partial_{t}p_{i}$, casting both the even
and odd cases on the same footing, the two scenarios remain qualitatively
distinct. In particular, for weak activity, the terms allowed by passivity in
either of these cases dominate over those only allowed by activity.

Along the same lines, we noted that passive polar dynamics sets the advective
coefficient $\lambda_{1}=0$ for both time-reversal even/odd $p_{i}$, and this
must be generated purely from activity. This should be contrasted with the
Toner-Tu model of flocking, where the role of the vector order parameter
$p_{i}$ is played by the velocity $u^{i}$ of the individual constituents, as
opposed to the fixed properties like spin or dipole moment. In this case, the
passive dynamics is actually governed by the Navier-Stokes equations arising
from momentum conservation and does allow for the advective term with
$\lambda_{1}=1$ as seen from section 4.4. We need activity to make this term
different from 1.

#### Electrically-driven fluids:

Instead of a fuel source $\Phi^{{\mathsf{F}}}$, we may consider driving
activity through one of the other background fields in the description, such
as electric fields $E_{i}$ in a charged fluid [62]. Since electric fields
impart both energy $E_{i}j^{i}$ and momentum $E_{i}n$ to the fluid, we are
forced to include both the energy sink $\Phi^{{\mathsf{E}}}_{r,a}$ and the
momentum sink $\Phi^{I}_{r,a}$ introduced around eq. 84. Taking
$j^{i}=nu^{i}$, energy and momentum balance leads to the steady states

$\displaystyle
T=T_{{\mathsf{E}}}\left(1+\frac{n^{2}E_{i}E^{i}}{\ell^{4}{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}^{2}T_{{\mathsf{E}}}^{2}\gamma_{{\mathsf{E}}}\sigma_{\Phi\rm
t}}\right)\leavevmode\nobreak\ ,\qquad
u^{i}=\frac{nE^{i}}{\ell^{2}\sigma_{\Phi\rm t}}\leavevmode\nobreak\ .$ (86)

Starting from this steady state, it is possible to retrace the discussion in
this paper and formulate a hydrodynamic theory which includes electric field
driving-induced contributions that are not bound by the local second law of
thermodynamics.

#### Active phase transitions:

In sections 3.3 and 4.3, we discussed how the strength of activity may be used
as a control parameter to induce phase transitions between the ordered and
disordered states in our active hydrodynamic models that may be forbidden in
passive systems. Active phase transitions are theoretically challenging
because they do not have a notion of free energy minimisation and thus our
usual theory of phase transitions based on Euclidean statistical field theory
does not apply. The SK-EFT framework developed in this work perfectly sets the
stage for a Wilsonian renormalisation group approach to study activity-induced
phase transitions. While we have not pursued this line of inquiry in this
work, we anticipate that the models proposed in this work will help further
our understanding of active phase transitions and we plan to return to these
considerations in future work.

#### Kinetic theory and holography:

In this work, we have developed a systematic framework for constructing
hydrodynamic models for active matter based on symmetries. However, such
modelling typically features a number of undetermined transport coefficients
that need to be fixed either through experiments or through an explicit
microscopic calculation. At weak coupling, one can use the techniques of
kinetic theory to derive the transport coefficients [20, 92]. It will be
interesting to revisit this approach for thermal active matter in the presence
of fuel source and energy sink, as done previously for hydrodynamics with
momentum sinks [93].

However, our analytical tools are quite limited in the strong coupling regime
where the hydrodynamic models are most reliable. Holography or the AdS/CFT
correspondence provides an alternative route to derive the qualitative
features of transport coefficients in a strongly coupled fluid, using a
higher-dimensional gravitational theory [22, 23, 24]. However, the standard
paradigm of holography only applies to relativistic systems, which has
hindered its utilisation for studying active matter models that typically do
not feature energy conservation. By contrast, the formalism of active
hydrodynamics developed in this work systematically accounts for energy
conservation and dynamical temperature, and while we have only applied this to
non-relativistic systems, we do not anticipate any conceptual obstructions to
extending these to relativistic systems. If so, it will be interesting to
return to the prospect of active holography in another work.

###### Acknowledgements.

We would like to thank Dominik Hahn for helpful discussions. The authors are
partly supported by the Dutch Institute for Emergent Phenomena (DIEP) cluster
at the University of Amsterdam and JA via the DIEP programme Foundations and
Applications of Emergence (FAEME). The work of AJ was partly funded by the
European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska-Curie grant agreement NonEqbSK No. 101027527. Part of this
project was carried out during the “Hydrodynamics at All Scales” workshop at
the Nordic Institute for Theoretical Physics (NORDITA), Stockholm.

## References

  * Schrodinger [1946] E. Schrodinger,  _What is life?: the physical aspect of the living cell_ (The Macmillan company, 1946).
  * Marchetti  _et al._ [2013] M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrodynamics of soft active matter, Rev. Mod. Phys. 85, 1143 (2013).
  * Toner and Tu [1998] J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E 58, 4828 (1998), arXiv:cond-mat/9804180 [cond-mat.stat-mech] .
  * Toner  _et al._ [2005] J. Toner, Y. Tu, and S. Ramaswamy, Hydrodynamics and phases of flocks, Annals of Physics 318, 170 (2005).
  * Jülicher  _et al._ [2007] F. Jülicher, K. Kruse, J. Prost, and J.-F. Joanny, Active behavior of the cytoskeleton, Physics Reports 449, 3 (2007), nonequilibrium physics: From complex fluids to biological systems III. Living systems.
  * Joanny and Prost [2009] J. Joanny and J. Prost, Active gels as a description of the actin-myosin cytoskeleton, HFSP Journal 3, 94 (2009), pMID: 19794818, https://doi.org/10.2976/1.3054712 .
  * Romanczuk  _et al._ [2012] P. Romanczuk, M. Bär, W. Ebeling, B. Lindner, and L. Schimansky-Geier, Active Brownian Particles. From Individual to Collective Stochastic Dynamics, European Physical Journal Special Topics 202, 1 (2012), arXiv:1202.2442 [cond-mat.other] .
  * Doostmohammadi  _et al._ [2018] A. Doostmohammadi, J. Ignés-Mullol, J. M. Yeomans, and F. Sagués, Active nematics, Nature Communications 9, 3246 (2018).
  * Alert  _et al._ [2020] R. Alert, J.-F. Joanny, and J. Casademunt, Universal scaling of active nematic turbulence, Nature Physics 16, 682–688 (2020).
  * Alert  _et al._ [2022] R. Alert, J. Casademunt, and J.-F. Joanny, Active turbulence, Annual Review of Condensed Matter Physics 13, 143 (2022).
  * Assante  _et al._ [2023] R. Assante, D. Corbett, D. Marenduzzo, and A. Morozov, Active turbulence and spontaneous phase separation in inhomogeneous extensile active gels, Soft Matter 19, 189 (2023).
  * Aditi Simha and Ramaswamy [2002] R. Aditi Simha and S. Ramaswamy, Hydrodynamic fluctuations and instabilities in ordered suspensions of self-propelled particles, Phys. Rev. Lett. 89, 058101 (2002).
  * Ramaswamy and Rao [2007] S. Ramaswamy and M. Rao, Active-filament hydrodynamics: instabilities, boundary conditions and rheology, New Journal of Physics 9, 423 (2007).
  * Scheibner  _et al._ [2019] C. Scheibner, A. Souslov, D. Banerjee, P. Surowka, W. T. M. Irvine, and V. Vitelli, Odd elasticity, arXiv e-prints , arXiv:1902.07760 (2019), arXiv:1902.07760 [cond-mat.soft] .
  * Jülicher  _et al._ [2022] F. Jülicher, J. Prost, and J. Toner, Broken living layers: Dislocations in active smectic liquid crystals, Phys. Rev. E 106, 054607 (2022), arXiv:2207.04562 [cond-mat.soft] .
  * Yamada  _et al._ [2003] D. Yamada, T. Hondou, and M. Sano, Coherent dynamics of an asymmetric particle in a vertically vibrating bed, Phys. Rev. E 67, 040301 (2003).
  * Xiao  _et al._ [2020] S. Xiao, T. Wang, T. Liu, C. Zhou, X. Jiang, and J. Zhang, Active metamaterials and metadevices: a review, Journal of Physics D Applied Physics 53, 503002 (2020).
  * Paxton  _et al._ [2004] W. F. Paxton, K. C. Kistler, C. C. Olmeda, A. Sen, S. K. St. Angelo, Y. Cao, T. E. Mallouk, P. E. Lammert, and V. H. Crespi, Catalytic nanomotors: Autonomous movement of striped nanorods, Journal of the American Chemical Society 126, 13424 (2004), pMID: 15479099, https://doi.org/10.1021/ja047697z .
  * Palacci  _et al._ [2013] J. Palacci, S. Sacanna, A. P. Steinberg, D. J. Pine, and P. M. Chaikin, Living crystals of light-activated colloidal surfers, Science 339, 936 (2013), https://www.science.org/doi/pdf/10.1126/science.1230020 .
  * Pitaevskii and Lifshitz [2012] L. Pitaevskii and E. Lifshitz,  _Physical Kinetics: Volume 10_, v. 10 (Elsevier Science, 2012).
  * Vicsek  _et al._ [1995] T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, Novel Type of Phase Transition in a System of Self-Driven Particles, Phys. Rev. Lett. 75, 1226 (1995), arXiv:cond-mat/0611743 [cond-mat.stat-mech] .
  * Maldacena [1999] J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38, 1113 (1999), arXiv:hep-th/9711200 .
  * Policastro  _et al._ [2001] G. Policastro, D. T. Son, and A. O. Starinets, The Shear viscosity of strongly coupled N=4 supersymmetric Yang-Mills plasma, Phys. Rev. Lett. 87, 081601 (2001), arXiv:hep-th/0104066 .
  * Bhattacharyya  _et al._ [2008] S. Bhattacharyya, V. E. Hubeny, S. Minwalla, and M. Rangamani, Nonlinear Fluid Dynamics from Gravity, JHEP 02 (02), 045, arXiv:0712.2456 [hep-th] .
  * Julicher  _et al._ [2018] F. Julicher, S. W. Grill, and G. Salbreux, Hydrodynamic theory of active matter, Reports on Progress in Physics 81, 076601 (2018).