[1]∂^ _#1 [1]μ^ _#1 [2]⟨#1 — #2 ⟩ [3]⟨#1 — #2 — #3 ⟩ [2]— #1 ⟩ ⟨#2 —

††thanks: These authors contributed equally.††thanks: These authors
contributed equally.

# Generalized time-reversal symmetry and effective theories for nonequilibrium
matter

Xiaoyang Huang<EMAIL_ADDRESS>Department of Physics and Center
for Theory of Quantum Matter, University of Colorado, Boulder CO 80309, USA
Jack H. Farrell<EMAIL_ADDRESS>Department of Physics and Center
for Theory of Quantum Matter, University of Colorado, Boulder CO 80309, USA
Aaron J. Friedman  Department of Physics and Center for Theory of Quantum
Matter, University of Colorado, Boulder CO 80309, USA  Isabella Zane
Department of Physics and Center for Theory of Quantum Matter, University of
Colorado, Boulder CO 80309, USA  Paolo Glorioso  Department of Physics,
Stanford University, Stanford CA 94305, USA  Department of Physics & Condensed
Matter Theory Center, University of Maryland, College Park, Maryland 20740,
USA  Andrew Lucas<EMAIL_ADDRESS>Department of Physics and
Center for Theory of Quantum Matter, University of Colorado, Boulder CO 80309,
USA

###### Abstract

The past decade has witnessed the development of systematic effective theories
for dissipative _thermal_ systems. Here, we describe an analogous effective
theory framework that applies to the classical stochastic dynamics of
_nonequilibrium_ systems. We illustrate this approach using a range of
examples, including nonreciprocal (predator-prey) dynamics, dissipative and
driven rigid-body motion, and active chiral fluids and solids. Many of these
systems exhibit a generalized time-reversal symmetry, which plays a crucial
role within our formalism, and in many cases can be implemented within the
Martin-Siggia-Rose path integral. This effective theory formalism yields
generalizations of the fluctuation-dissipation theorem and second law of
thermodynamics valid out of equilibrium. By stipulating a stationary
distribution and a set of symmetries—rather than postulating the stochastic
equations of motion directly—this formalism provides an alternative route to
building phenomenological models of driven and active matter. We hope that
this approach facilitates a systematic investigation of the universality
classes of active matter, and provides a common language for nonequilibrium
many-body physics from high energy to condensed matter.

###### Contents

  1. 1 Introduction
    1. 1.1 Review of Wilson’s paradigm
    2. 1.2 The challenge of dissipative, noisy dynamics
    3. 1.3 The purpose of this paper
    4. 1.4 Overdamped oscillator: a minimal example of our framework
    5. 1.5 Organization of the paper
  2. 2 Effective (field) theory for stochastic dynamics
    1. 2.1 Warm up: Discrete state spaces
    2. 2.2 From stochastic differential equations to the Fokker-Planck equation
    3. 2.3 Stationarity constraints on the Fokker-Planck equation
    4. 2.4 Construction of the Fokker-Planck generators
      1. 2.4.1 General form of $W$
      2. 2.4.2 Gaussian noise and the fluctuation-dissipation theorem
      3. 2.4.3 Non-Gaussian noise
      4. 2.4.4 Generating all possible T-breaking, noise-free terms
    5. 2.5 Entropy and the second law of thermodynamics
    6. 2.6 Symmetries and conservation laws
      1. 2.6.1 Strong and weak symmetries of the dynamics
      2. 2.6.2 Symmetries of the stationary distribution
    7. 2.7 Connection to the Martin-Siggia-Rose path integral
      1. 2.7.1 Details of the MSR formalism
      2. 2.7.2 Time-reversal symmetry
      3. 2.7.3 Time-reversal symmetry without integrating out noise
  3. 3 Dissipative generalizations of Hamiltonian dynamics
    1. 3.1 Hamiltonian dynamics and Poisson brackets
      1. 3.1.1 Hamiltonian mechanics with canonical phase space variables
      2. 3.1.2 Time-reversal symmetry
      3. 3.1.3 Symmetries and Noether’s theorem
      4. 3.1.4 Noncanonical symplectic forms
    2. 3.2 Dissipation towards a thermal state
    3. 3.3 Classical harmonic oscillator
      1. 3.3.1 Overdamped limit
      2. 3.3.2 Finite-mass limit
    4. 3.4 Vortex dynamics in superfluid thin films
    5. 3.5 Landau-Lifshitz-Gilbert spin dynamics
  4. 4 From thermal to active matter
    1. 4.1 An “odd” two-dimensional harmonic oscillator
    2. 4.2 Nonreciprocal Kuramoto model
      1. 4.2.1 The equilibrium (reciprocal) Kuramoto model
      2. 4.2.2 Nonreciprocity and generalized time-reversal symmetry
    3. 4.3 Dynamics of wealth
  5. 5 Spontaneous breaking of generalized time-reversal symmetry
    1. 5.1 General framework
      1. 5.1.1 Example: rigid-body rotation
      2. 5.1.2 Left and right symmetry groups
      3. 5.1.3 Dynamics along $G$
      4. 5.1.4 Spontaneously breaking generalized time-reversal symmetry
    2. 5.2 Generalizations of the nonreciprocal Kuramoto model
      1. 5.2.1 Kuramoto model with $N$ species
      2. 5.2.2 Motion on a two-dimensional sphere
  6. 6 Active and dissipative rigid-body dynamics
    1. 6.1 Hamiltonian formalism of rigid-body dynamics
    2. 6.2 Dissipative rigid-body dynamics
    3. 6.3 Active rigid-body dynamics
  7. 7 Field theory
    1. 7.1 Odd diffusitivity
    2. 7.2 Odd elasticity
      1. 7.2.1 Gyroscopic potential
      2. 7.2.2 Translationally pinned potential
    3. 7.3 Chiral fluid
      1. 7.3.1 Odd viscosity
      2. 7.3.2 Fluid with spinning particles
      3. 7.3.3 Odd stress and PT-odd charge
    4. 7.4 Fracton hydrodynamics
  8. 8 Lattice model for active phase separation
  9. 9 Outlook
    1. 9.1 Generalizations and comparisons to previous literature
    2. 9.2 Towards classifying active phases of matter
  10. A Topological exceptions to the form of dissipationless terms
  11. B Second law of thermodynamics
    1. B.1 Linear response
    2. B.2 A nonequilibrium system
  12. C Derivation of the MSR path integral
    1. C.1 Correlation functions
  13. D Convenient regularization of a nonlinear diffusion equation
  14. E Breaking gT in a four-species Kuramoto model on the 2-sphere

##  1 Introduction

A hallmark achievement of 20th-century physics was the development of the
Wilsonian renormalization group (RG), and with it, effective field theory
(EFT) Wilson (1971a, b); Hohenberg and Halperin (1977); Weinberg (1979);
Altland and Simons (2010). This framework provides a systematic approach to
predicting the emergent macroscopic behavior of _generic_ microscopic systems
based on coarse graining and symmetries. The EFT framework was initially
developed in the contexts of ground-state physics in quantum field theories
and the finite-temperature thermal physics of statistical field theories. In
this paper, we describe an EFT framework for general dissipative and
stochastic classical systems, both in and out of equilibrium.

###  1.1 Review of Wilson’s paradigm

In the context of classical systems, EFTs were first systematically developed
as statistical field theories describing thermal equilibrium. Starting from a
microscopic model, one can use the principles of RG Wilson (1971a, b);
Hohenberg and Halperin (1977); Weinberg (1979); Altland and Simons (2010) to
recover a simple effective theory (i.e., a free energy) that captures all
universal properties.

As a concrete example, consider the statistcal theory of the paramagnet-to-
ferromagnet phase transition. In 3D, an appropriate _microscopic_ model that
realizes this transition is the Ising model. Denoting by $S_{i}=\pm 1$ a
binary classical spin on site $i$ of a 3D lattice, the partition function for
the Ising model is

$Z_{\rm Ising}\,=\,\mathrm{e}^{-\beta
H}=\sum_{\\{S_{i}\\}}\exp\left[\beta\sum_{\expectationvalue{i,j}}S_{i}S_{j}\right]\,,~{}~{}$
(1)

where $\expectationvalue{i,j}$ denotes neighboring spins $i$ and $j$. The
model described by Eq. (1) hosts a phase transition at a critical temperature
corresponding $\beta=\beta_{c}=1/T_{c}$. At small $\beta<\beta_{c}$ (high
temperature), a typical configuration drawn with probability
$Z^{-1}\exp(-\beta H)$ is exponentially likely to have roughly half of each
$S=\pm 1$; at large $\beta>\beta_{c}$ (low temperature), the configurations
are exponentially more likely have a clear majority of $S$ aligned. While the
microscopic theory captured by Eq. (1) is useful for numerical simulations of
the transition (e.g., via Monte Carlo), it gives little insight into the long-
distance behavior one can expect in each phase (or at the critical point), nor
whether various _emergent_ macroscopic properties are _universal_ (i.e.,
insensitive to fine-tuned microscopic details).

The _philosophy_ of Wilsonian EFT differs substantially from the foregoing
approach. Instead of a microscopic lattice model, one instead starts by
considering the symmetries of the system and the natural IR modes that should
appear in the EFT. In the ferromagnetic phase of the Ising model, one expects
macroscopic ordering, corresponding to all spins aligning as either $S_{i}=+1$
or $S_{i}=-1$. We then coarse grain the lattice model into a continuum EFT by
defining, e.g.,

$\phi(x)\,=\,\sum_{i}S_{i}\frac{\mathrm{e}^{-(x-x_{i})^{2}/2\sigma^{2}}}{(2\pi\sigma^{2})^{3/2}}\,,~{}~{}$
(2)

so that when $\sigma\gg a$ is much larger than the lattice spacing, we expect
$\phi(x)$ to be reasonably smooth. We then postulate that the partition
function (1) is well approximated by

$Z_{\rm
Ising}\,\sim\,\exp\left[-\int\mathrm{d}^{3}x\,\left(A\,\absolutevalue{\nabla\phi}^{2}+B\,\phi^{2}+C\phi^{4}+D\phi^{6}+\cdots\right)\right]\,,~{}~{}$
(3)

up to normalization and higher-order terms in $\phi$, where the
($\beta$-dependent) phenomenological coefficients $A,B,C,D,\cdots$ are not
calculated from Eq. (1), but instead determined from experimental data. The
universal physics of the microscopic theory (1) is encoded in the effective
theory (3) via the parameter $B$, which like $\beta$ in the microscopic
theory, controls whether the system realizes a paramagnet or ferromagnet as
one tunes through the critical value $B_{c}$.

We comment that the _effective_ theory (3) follows from the microscopic theory
(1) under RG. Importantly, RGs organize degrees of freedom by their
“relevance” to the system’s long-distance and late-time behavior, and
systematically integrate out the short-distance degrees of freedom to recover
a “coarse-grained” theory. In the thermodynamic limit, the result is an
effective field theory (3) that is insensitive to the precise details of the
microscopic theory (1).

However, one can also write down an effective theory (3) _without_ coarse
graining a microscopic model. The crucial features of Eq. 3 are that the
effective free energy is (_i_) spatially local and (_ii_) has the same
$\mathbb{Z}_{2}$ symmetry as the partition function (1), where $S_{i}\to-
S_{i}$ becomes $\phi\to-\phi$. This symmetry reflects the fact that the
ferromagnetic phase is equally likely to have all spins up versus down. To
recover the effective theory, one need only write down all terms—organized by
their order in the field $\phi$ and number of derivatives $\partial$—that are
local and compatible with the anticipated symmetry of the model (in this case,
$\mathbb{Z}_{2}$). In the vicinity of the critical point $B=B_{c}$, the EFT
captured by Eq. (3) accurately predicts the same critical exponents and
universal properties as the discrete Ising model (1). Such an EFT describes
_any_ system with the same universal properties as the Ising model described
by Eq. (1), independent of microscopic details. In the remainder of this
paper, our perspective is that one may safely construct EFTs in the form of an
action (Lagrangian) or free energy by writing down all terms compatible with
locality and any desired symmetries in an organized fashion.

###  1.2 The challenge of dissipative, noisy dynamics

The EFT paradigm provides a systematic way to analyze and classify the
universal physics of equilibrium systems. We seek an analogue of this approach
for nonequilibrium systems and _active_ matter111While there does not seem to
be a formal definition for active matter, a heuristic definition is that
active matter corresponds to dynamics with “self-propelled” particles. In
general, such systems cannot be described by Hamiltonian mechanics—at least,
not straightforwardly—and thus, there is no notion of thermal equilibrium or
conventional thermodynamics.. For concreteness, we adopt the perspective
herein that “nonequilibrium” systems are those in which detailed balance—or a
more general “reversibility” (i.e., the stochastic notion of microscopic time-
reversal symmetry)—are violated.

Because nonequilibrium systems do not generally obey Hamiltonian dynamics—and
may seem to violate a second law of thermodynamics and a fluctuation-
dissipation theorem—it is unclear _a priori_ what aspects of the EFT paradigm
can be applied in such a setting. Some aspects of EFT—including the philosophy
of working with coarse-grained degrees of freedom and performing an RG
analysis—have been applied to nonequilibrium systems in the literature. Still,
we argue that the conventional approach to studying nonequilibrium matter has
some _philosophical_ differences compared to equilibrium EFTs. These
differences are summarized by the following fact: Effective theories for
dynamical systems typically start from equations of motion. For example,
Hohenberg and Halperin endeavored to classify all possible (stochastic)
equations of motion that could realize dissipative relaxation to thermal
equilibrium Hohenberg and Halperin (1977). The path-integral framework of
Martin, Siggia, and Rose (MSR) then assigns to such stochastic equations of
motion a corresponding _Lagrangian_ Martin  _et al._ (1973); one can then
apply RG to that Lagrangian, but _only after postulating the equations of
motion_ that went into the MSR path integral. The MSR framework also features
in Toner and Tu’s phenomenological model of _flocking_ Toner and Tu (1998),
which takes the form of a coarse-grained hydrodynamic equation of motion that,
roughly speaking, extends the Navier-Stokes equation to birds. While these
efforts have indeed identified new universality classes beyond the traditional
equilibrium setting, starting from equations of motion does make it more
challenging to recover _all_ of the benefits of the equilibrium EFTs: (_i_)
The symmetries that the EFT should obey are less clear than in equilibrium;
(_ii_) dynamical stability constraints are not manifest; and (_iii_) it is not
obvious whether the equations of motion might contain terms that, while
apparently allowed by symmetry, are in fact forbidden in any theory that arose
from coarse graining microscopic degrees of freedom.

This paper is organized around the goal of building an effective theory
framework for nonequilibrium dynamics that (_i_) takes the same philosophical
approach as the Wilsonian EFTs, (_ii_) avoids the potential pitfalls mentioned
above, and (_iii_) generalizes as much of equilibrium physics as possible to
nonequilibrium systems. Our motivation is twofold. On a _practical_ level, it
is easier to construct effective theories in terms of Lagrangians than
equations of motion, because the latter transform _covariantly_ under
symmetries, while the former are invariant, and thereby easier to classify. On
a _fundamental_ level, an EFT approach has the advantage that stability of the
resulting theory and invariance under appropriate symmetries become manifest.

We now give a brief overview of several examples of nonequilibrium systems for
which starting from microscopic equations of motion—even coarse-grained
ones—can fail to recognize constraints that a systematic “top-down” approach
(starting from a Lagrangian) would naturally incorporate.

  1. 1.

Consider a continuous active medium, such as an odd elastic solid Scheibner
_et al._ (2020); Fruchart  _et al._ (2022), which can be microscopically
modeled by an active mechanical system consisting of Hookean springs with
nonreciprocal couplings. The equations of motion for this microscopic medium
often lead to instabilities, which in turn drive the medium out of the regime
of validity of Hooke’s Law, which assumes that each spring is only weakly
perturbed out of equilibrium.

An EFT aims not to describe a system whose microscopic equations are known,
but rather, the coarse-grained dynamics of a system whose microscopic dynamics
are too complex to deduce _ab initio_. We would not observe in nature an odd
elastic solid, with a linear stress-strain relation, unless the phase was
stable. Hence, weseek to incorporate this stability criterion directly into
our EFT. This stability criterion is often related to a generalized notion of
time-reversal symmetry, which is challenging to see starting from equations of
motion. An EFT in which the time-reversal transformation is more transparent
thus makes it easier to understand the physical origin of any constraints on
coarse-grained stochastic equations of motion.

  2. 2.

In fluids that intrinsically break rotational invariance—i.e., in which
$\mathsf{O}\left(d\right)$ is broken to a discrete subgroup—there are numerous
terms that can be added to the hydrodynamic equations of motion compared to
the Navier-Stokes equations describing isotropic fluids Huang and Lucas
(2022); Friedman  _et al._ (2023). In _thermal equilibrium_ , many such terms
are forbidden by the requirement that the theory remains consistent when
coupled to curved spacetime; this is easiest to diagnose in a (Lagrangian)
effective field theory Huang and Lucas (2022). That EFT also elucidates the
connection between such anisotropic terms and _anomalies_ Delacrétaz and
Glorioso (2020); Qi  _et al._ (2023), which place further constraints on the
functional form of these anisotropies. Diagnosing whether these constraints
carry over to nonequilibrium and active systems—and if so, their fundamental
origin—likely requires “a Lagrangian EFT” for nonequilibrium systems as well.

  3. 3.

In kinetically constrained fluids Gromov  _et al._ (2020); Feldmeier  _et al._
(2020); Morningstar  _et al._ (2020); Hart  _et al._ (2022); Glorioso  _et
al._ (2022); Grosvenor  _et al._ (2021); Burchards  _et al._ (2022), there are
terms that are are forbidden from appearing in the stochastic equations of
motion despite being compatible with all known symmetries. Adding such terms
changes the universal dynamics (i.e., the critical exponents), realizing a
_different_ theory! Such terms cannot arise from the coarse-graining procedure
that recovers an EFT from a microscopic model, and this is known to hold even
out of equilibrium Guo  _et al._ (2022), as we discuss in Sec. 7.4. Hence,
knowledge of all symmetries is not sufficient to adequately constrain the
equations of motion.

Until recently, the phenomenology of dissipative and stochastic dynamics—even
those that relax to thermal equilibrium—was developed primarily through a
“bottom-up” approach starting from microscopic equations of motion. Over the
past decade, high-energy theorists have developed systematic “top-down”
_quantum_ EFTs starting from Lagrangians to describe the hydrodynamics of
dissipative thermal fluids Harder  _et al._ (2015); Crossley  _et al._ (2017);
Glorioso  _et al._ (2017); Haehl  _et al._ (2016); Jensen  _et al._ (2018).
Through careful consideration of the system’s symmetries, these efforts have
derived features—such as the usual fluctuation-dissipation theorem (FDT)—of
the corresponding hydrodynamics that previously had to be inserted by hand
(e.g., into the MSR path integral). Crucially, the FDT is guaranteed by the
Kubo-Martin-Schwinger (KMS) symmetry Kubo (1966); Martin and Schwinger (1959);
Araki (1978) that generalizes time-reversal symmetry to dissipative thermal
systems. A more careful analysis within EFT has also derived an _entropy_
current with nonnegative divergence realizing a second law of thermodynamics
in such dissipative systems Glorioso and Liu (2016). Hence, the celebrated
phenomenology of hydrodynamics in thermal equilibrium has been carefully
derived from dissipative effective field theories based on Lagrangians, rather
than equations of motion.

###  1.3 The purpose of this paper

This paper addresses the question: Can the EFT for dissipative thermal systems
can be extended to (certain) nonthermal systems? The goal of our approach is
to systematically obtain a stochastic (Langevin) equation for a set of
variables $\mathbf{q}=\\{q_{a}\\}$, such that the resulting equations avoid
all of the potential pitfalls above. Here $q_{a}$ represent slow degrees of
freedom that survive after coarse-graining out complicated microscopic
dynamics. For much of the paper $q_{a}$ will be a discrete set of variables,
but in Sec. 7 we discuss the natural extension of our methods to field
theories.

As we will see, it is easiest to systematically build stochastic equations by
constructing a _Fokker-Planck equation_ for the probability distribution
$P(\mathbf{q},t)$ of finding any configuration at time $t$. The Fokker-Planck
equation will be constructed using the philosophy of the Wilsonian EFT
paradigm, where it will be easier to impose the desired symmetries and
positivity constraints. More concretely, we proceed as follows:

  * •

The _starting point_ of our effective theory is to determine—or, more often,
to _postulate_ , à la Eq. (3)—the stationary probability distribution
$\exp(-\Phi)$ for the stochastic dynamics. In other words, we begin by
proposing the structure of the equal-time correlation functions of the
nonequilibrium fixed point222In contrast, $\exp(-\Phi)$ can at best be
calculated perturbatively in other approaches that instead postulate equations
of motion (in the absence of detailed balance). See, e.g., Ref. 31. On the
other hand, there could exist local equations of motion that lead to nonlocal
stationary distributions $\Phi$, and this is challenging to describe with our
framework..

  * •

Once the stationary distribution is known, we write down the most general
(Markovian) stochastic dynamics consistent with this stationary state. The
algorithmic approach to effective field theory also accounts for all
symmetries, conservation laws, and constraints present in the dynamics.

  * •

Stability criteria arise naturally from two places: the stability of the
proposed phase in $\Phi$, and the positivity of noise variance. A generalized
fluctuation-dissipation theorem (Tab. 1) and second law of thermodynamics (72)
can be formally stated. The Mermin-Wagner theorem Mermin and Wagner (1966) and
other constraints on equilibrium statistical physics extend neatly to our
nonequilibrium formalism.

  * •

With knowledge of $\Phi$, identifying terms that are “even” and “odd” under
detailed balance (50) (a stochastic notion of time-reversal symmetry) is
transparent. In particular, it is straightforward to impose generalized time-
reversal symmetries, labeled as T, and to understand generalized time-reversal
transformations within the Martin-Siggia-Rose (MSR) path integral (see Sec.
2.7).

Philosophically, our approach can be thought of as blending the MSR approach
to nonequilibrium matter with the KMS-invariant effective field theories for
dissipative thermal systems, which were organized systematically in terms of
symmetry-invariant building blocks, as in Wilsonian equilibrium
EFTs.333Another approach to try and extend the KMS-invariant EFT for a thermal
system to active matter was recently presented in Ref. Landry (2023). However,
while our approach is inspired by EFTs for thermal systems, we emphasize that
it is not simply recasting the same EFT used for thermal systems in a
“nonequilibrium setting”. In a thermal system, the Hamiltonian $H$ plays two
crucial roles: first, it generates time evolution, and second, the probability
of finding the system in any given state $q$ is $\mathrm{e}^{-\beta H(q)}$. In
our EFT for more general stochastic systems, we will see that the $\Phi$
introduced above need not directly relate to the dissipationless dynamics.
Indeed, we will see that many active systems are characterized by “equations
of motion” that are not directly related to the $\Phi$ which characterizes the
steady state of the stochastic dynamics. Indeed, part of this paper will
provide an algorithm to systematically study the most general possible active
or stochastic theory compatible with the desired stationary state.

A key advantage of taking this perspective is that the phase diagram of the
models that we build will always “decompose” along two orthogonal (sets of)
“axes”. Firstly, the phase diagram will contain all _thermodynamic_ phase
transitions captured by $\Phi$: for example, tuning the overall prefactor of
$\Phi$ could result in an ordering transition from a paramagnet to a
ferromagnet. Once we have found the phases captured by $\Phi$, we can then
systematically determine all of the possible dissipationless stochastic
dynamics: intuitively, these capture how the system can circulate around the
phase space without modifying $\Phi$. When such dissipationless terms are
deterministic, we can completely classify them (see e.g. Sec. 5). Additional
phase transitions can further then arise as a consequence of these
dissipationless dynamics. A simple example of this phenomenon is the
chirality-breaking phase transition found in nonreciprocal predator-prey
dynamics (Sec. 4.2). Putting these two ingredients together, we obtain a
schematic phase diagram illustrated in Fig. 1.

$v$
---
$\Phi$
---
0
---
Broken—Chiral R
---
Broken—Chiral L
---
Broken—Stationary
---
Unbroken—Stationary
---

Figure 1: A schematic “phase diagram” that summarizes our formalism, including
T-breaking phase transitions. Starting with $v=0$, we may tune the stationary
distribution $\Phi$ so that it has degenerate minima—this control over $\Phi$
is represented by the vertical axis of the figure. Phase transitions from the
disordered phase into the stationary broken phase are controlled by noise
strength and the minima of $\Phi$ according to statistical mechanics. But
separately, we may tune the dissipationless and T-breaking $v$ from positive
to negative, driving motion in one of two directions along the degenerate
minima of $\Phi$ and creating a T-broken phase. Control over $v$ is
represented by the horizontal axis, orthogonal to control over $\Phi$
(vertical axis).

Some phases of active matter, such as flocking Toner and Tu (1998) or
motility-induced phase separation Tailleur and Cates (2008), are not easy to
describe within our framework. As one example, one must sacrifice strict
spatial locality in $\Phi$ to realize the Toner-Tu fixed point, as one finds
nontrivial scaling exponents in spatial correlators. As incorporating this
directly into $\Phi$ may feel less motivated within the context of our
formalism, we have elected to focus on other recent theories of active matter
including active solids Scheibner  _et al._ (2020), fluids Han  _et al._
(2021), and nonreciprocal dynamics Fruchart  _et al._ (2021), where we (_i_)
reproduce many of the nontrivial active phases and phenomena observed
previously in the literature and (_ii_) provide generalizations of these
models.

###  1.4 Overdamped oscillator: a minimal example of our framework

Having outlined our approach to model building in words, we now provide an
example (with equations) describing the simplest possible stochastic dynamical
system: the overdamped harmonic oscillator. We gloss over a few technical
subtleties, which we instead be address more carefully in Sec. 2.

We first describe how one might approach this system by starting with
equations of motion directly, as is common in the literature. We start by
considering the dissipative system

$\dot{x}=-\gamma x\,,~{}~{}$ (4)

and, for simplicity, we assume that $x$ is the coordinate of a macroscopic
rigid body that is attached to a spring while suspended in a very viscous
fluid. The coefficient $\gamma$ denotes the damping of the body’s motion
towards the equilibrium position $x=0$, where the spring is unstretched. Note
that Eq. (4) explicitly breaks time-reversal symmetry: The arrow of time
corresponds to the pull of the particle towards its equilibrium position at
$x=0$. The condition $\gamma>0$ is required because the system flows towards,
rather than away, from equilibrium. Indeed, the solutions to Eq. (4) take the
form

$x(t)\,=\,A\,\mathrm{e}^{-\gamma t}\,,$ (5)

which is clearly not invariant under $t\to-t$.

Microscopically, collisions between the molecules of the fluid and the rigid
body impart small stochastic forces (thermal fluctuations) that modify Eq. (4)
according to (4) to

$\dot{x}=-\gamma x+\xi\,,$ (6)

where the noise term above satisfies

$\overline{\xi(t)\xi(t^{\prime})}=2D\delta(t-t^{\prime})\,,~{}~{}$ (7)

and we require that $D\geq 0$, as $\xi(t)$ is a real-valued random variable.
Since the trajectory $x(t)$ is _stochastic_ , we can calculate the probability
$P(x_{0},t)$ to find $x(t)=x_{0}$ given some initial condition, which obeys a
Fokker-Planck equation Fokker (1914); Planck (1917); Kolmogorov (1931);
Kramers (1940); Moyal (1949); Pawula (1967a, b); Kadanoff (2000):

$\partial_{t}P=\partial_{x}\left(\gamma xP\right)+D\partial_{x}^{2}P\,.$ (8)

We deduce that, at late times $t\to\infty$, the system relaxes to the
stationary distribution

$P(x)=\mathrm{e}^{-\Phi(x)}=\mathrm{e}^{-\gamma x^{2}/2D}\,.$ (9)

Importantly, the justification above for the noisy Fokker-Planck description
(8) of the overdamped oscillator began with an equation of motion,
subsequently incorporated noise, and ultimately deduced that, unsurprisingly,
the system is most likely to be found near $x=0$.

We now outline the implementation of our approach for the simple example
above. The starting point is to write down the general form of a master
equation that governs the evolution for the probability distribution $P(x,t)$
(8), without knowing anything about the microscopic details of the underlying
system. As is common in macroscopic approaches, the natural next step is to
write the master equation in terms of a derivative expansion:

$\partial_{t}P(x,t)=\partial_{x}\left[A(x)\left(\partial_{x}+B(x)\right)P(x,t)\right]+\cdots\,,$
(10)

where the dots denote higher-derivative terms that encode non-Gaussian noise.
Higher derivatives become progressively less important as far as the
macroscopic length scales of interest are much longer compared to lengths that
are intrinsic to the microscopic processes of the system. Keeping only second
derivatives in Eq. (10) implies that the noise is effectively Gaussian
because, in our example, the object on a spring is heavy compared to the
microscopic particles whose collisions with it lead to damping forces. Note
that all terms on the right-hand side of Eq. (10) are total derivatives as
required by conservation of probability. The next step is to postulate the
existence of a steady state $\exp(-\Phi(x))$. The choice of $\Phi(x)$ depends
on the of phenomenology of interest; in this example, we assume that $\Phi$
has a minimum at $x=0$, and we further restrict to the linear regime, so that
we can approximate

$\Phi(x)=\frac{1}{2}\beta kx^{2}+\cdots\,,~{}~{}$ (11)

where the dots stand for higher powers of $x$. In the above, we wrote the
prefactor as $\beta k$ for convenience, and we assume $\beta k\geq 0$ to
ensure stability at $x=0$. Requiring that $\Phi$ is a steady state for (10),
leads to the requirement that

$B(x)=\partial^{\vphantom{\prime}}_{x}\Phi(x)$ (12)

and hence Fokker-Planck equation

$\partial^{\vphantom{\prime}}_{t}P=\partial^{\vphantom{\prime}}_{x}\left[A(x)\left(\partial^{\vphantom{\prime}}_{x}+\beta
kx\right)P\right]\,,~{}~{}$ (13)

where we additionally require that $A(x)\geq 0$ to ensure that the system
relaxes _towards_ the steady state. Then, Eq. (13) is the most general Fokker-
Planck equation consistent with our requirements. Upon identifying

$A=D,\;\;\;\;\;\;\;\;A\beta k=\gamma\,,$ (14)

we precisely recovered Eq. (8). We thus uniquely determined the dynamics of
this stochastic system solely in terms of a minimal set of general
assumptions, without knowing any detail about the underlying dynamics.

When $A(x)=D$ is a constant, the approach above can also be elegantly
summarized using the MSR path integral Martin  _et al._ (1973). We do not
introduce the notation here, although expect it to be familiar enough for many
readers ( for details, see Sec. 2.7). Letting the noise field be $\pi$, the
MSR Lagrangian is

$L=\pi\dot{x}+\mathrm{i}\pi A\left(\pi-\mathrm{i}kx\right)\,,~{}~{}$ (15)

which can intuitively be read off from Fokker-Planck equation (10) upon
replacing $\partial_{x}\to-\mathrm{i}\pi$. Conventionally, if one started with
a stochastic equation (6), (15) can be written down using standard rules.
However, we will see that the form of (15) is completely fixed by demanding
stationarity with $\Phi$ given by (11); this constraint can be viewed as
implying that, upon transforming

$\pi\rightarrow-\pi+\mathrm{i}kx\,,~{}~{}$ (16)

the transformed $L$ is again a valid MSR Lagrangian (all terms have at least
one power of $\pi$). In fact, in this particular theory, $L$ is _invariant_
under this transformation.

The invariance under this transformation turns out to imply that this simple
one-dimensional model is compatible with microscopic _time-reversal symmetry_.
It is often desirable to demand this as a feature of a dissipative EFT: Since
$F=ma$ is time-reversal symmetric, this is a symmetry that must manifest in
the spring-fluid system, so long as the noise merely arises from “integrating
out” the bath degrees of freedom. In a dissipative system, this integrating-
out procedure implies that the stochastic system obeys the _detailed balance_
condition:

$\mathbb{P}\left[x(t_{1})=x_{1}\to
x(t_{2})=x_{2}\right]\mathrm{e}^{-\Phi(x_{1})}=\mathbb{P}\left[x(t_{1})=x_{2}\to
x(t_{2})=x_{1}\right]\mathrm{e}^{-\Phi(x_{2})}\,,~{}~{}$ (17)

and we refer to this property as _reversibility_ , which we denote by “T”
going forward.

As we show in Sec. 2, reversibility manifests in the requirement that the
generator of the Fokker-Planck equation is invariant under a simple
transformation, even as we generalize to systems with more degrees of freedom
and multiplicative and/or non-Gaussian noise. Therefore, reversibility can
prove to be a very powerful symmetry. And even if reversibility is not a
symmetry, we still find it extremely helpful to keep track of the
reversibility transformation: for example, it may often be a reasonable
physical postulate that the reversed theory is also a local theory. We will
see that this requirement is equivalent to the postulate that $\Phi$ is local.
So long as $\Phi$ is local, the formalism that we develop in this paper
provides non-trivial constraints on the stochastic effective theory.

Notice that the definition of reversibility given in Eq. (17) depends on
$\Phi$. One alternative definition of this symmetry that avoids introducing
$\Phi$ is

$\mathbb{P}[x(0)=a,x(t)=b,x(2t)=c,x(3t)=a]=\mathbb{P}[x(0)=a,x(t)=c,x(2t)=b,x(3t)=a].$
(18)

Unfortunately, it is not easy to implement such a symmetry manifestly within
effective theories, while – in contrast – (17) is. It is for this reason that,
in our framework, we first deduce $\Phi$ before writing down the equations of
motion.

###  1.5 Organization of the paper

The rest of this paper is organized as follows.

Sec. 2 contains a pedagogical mathematical tour through the Fokker-Planck
equations, and emphasizes how to properly implement reversibility and detailed
balance (and classify dissipative and dissipationless coefficients) for any
system (thermal or nonthermal). At the end of the section, we explain how in
certain limits, our approach relates to the MSR path integral and the thermal
EFTs Crossley  _et al._ (2017); Glorioso  _et al._ (2017) which were recently
developed.

Sec. 3 details the application of our formalism to thermal systems, whose
dissipationless dynamics are described by Hamiltonian mechanics. In doing so,
we provide elegant and algorithmic derivations of a few hallmark results, such
as the minimal Landau-Lifshitz-Gilbert damping for a classical spin.

Sec. 4 extends our formalism to active matter, beginning with a few simple
examples. In Sec. 5.1, we then describe a systematic classification of all
possible phases in which a generalized time-reversal symmetry is spontaneously
broken. This discussion gives a more systematic way of understanding the
possible phases of nonreciprocal matter, and includes a number of interesting
extensions of the predator-prey-like models of Fruchart  _et al._ (2021). The
example of active (or even simply damped) rigid body motion is presented in
Sec. 6; the nontrivial classical phase space of the rigid body renders our
formalism especially well-suited to the careful analysis of this system.

Sec. 7 then describes the generalization of our methods to continuum models
and field theories. We explain how to understand active solids and fluids in
our framework, and connect to some of the puzzles we raised in Sec. 1.2. A
general method for incorporating desired active terms into our effective
theories, in the simplifying limit of lattice models, is described briefly in
Sec. 8, in the context of active phase separation.

##  2 Effective (field) theory for stochastic dynamics

Here we develop the building blocks of the effective field theory (EFT)
framework that describes thermal, dissipative, and active classical matter. We
make precise the intuitive arguments of Sec. 1.4 on the role of reversibility
symmetry T—also known as _detailed balance_ in this context. As noted therein,
an effective theory for dissipative dynamics _must_ incorporate noise; this is
most naturally captured using an inherently _stochastic_ formalism. For
convenience, we represent the configurations of a system as elements of a
linear vector space, in analogy to quantum mechanics, and as prescribed by the
Doi-Peliti formulation Doi (1976a, b); Peliti, L. (1985); Peliti (1986). We
first consider the stochastic dynamics of classical systems with a discrete
configuration space, with dynamics generated by a continuous-time Markov
process. The continuum analogue corresponds to a stochastic differential
equation—e.g., the Langevin equation Langevin (1908); Lemons and Gythiel
(1997). However, we find it most transparent to consider the Fokker-Planck
equation (FPE) Fokker (1914); Planck (1917); Kolmogorov (1931); Kramers
(1940); Moyal (1949); Pawula (1967a, b); Kadanoff (2000) instead. Working in
the FPE language, we derive powerful constraints from the existence of a
stationary state (before connecting to T), a generalization of the
fluctuation-dissipation theorem Kubo (1966); Kwon  _et al._ (2005); Prost  _et
al._ (2009), and finally, the Martin-Siggia-Rose (MSR) path-integral approach
Martin  _et al._ (1973).

###  2.1 Warm up: Discrete state spaces

As a warm up to the more general Fokker-Planck dynamics of systems with
continuous state spaces, we first consider the simpler scenario of stochastic
classical systems with a _discrete_ configuration space. We consider a single
particle hopping between vertices $v\in V$ connected by edges $e\in E$ of a
graph $G=(V,E)$. The allowed configurations are associated with vertices $v\in
V$, and may realize physical positions, generic single-particle configurations
(so that $G$ is analogous to a “Hilbert space”), or even many-body
configurations (in which case $G$ is analogous to a “Fock space”), while the
allowed transitions are associated with edges $e\in E$ Doi (1976a, b); Peliti,
L. (1985); Peliti (1986).

Hence, the single-particle hopping problem captures generic classical systems
with discrete state spaces. The configurations (i.e., vertices $v\in V$)
correspond to positions, while the transitions (i.e., edges $e\in E$)
correspond to hops. We denote by $\ket{x}$ the state of the system in which
the particle is at the position (i.e., vertex) $x$; these kets form an
orthonormal basis for the space $\mathcal{H}$ of $N=\absolutevalue{V}$ total
allowed states given by

$\mathcal{H}\,=\,\mathbb{R}^{N}\,=\,\text{span}\,\left\\{\,\ket{v}\,\middle|\,v\in
V\,\right\\}~{}~{}~{}\text{with}~{}~{}~{}\left\langle
x\middle|y\right\rangle\,=\,\delta^{\,}_{x,y}~{}~{}\forall\,x,y\in V\,,~{}~{}$
(19)

where the state vectors $\ket{x}\in\mathcal{H}$ are real valued and the basis
states (or classical configurations) $\\{\ket{x}\\}$ are orthonormal and
complete. The utility of this basis is that it facilitates the construction of
a probability distribution

$\ket{P}\,=\,\sum\limits_{x\in
V}\,p_{x}\,\ket{x}~{}~{}~{}\text{with}~{}~{}~{}p_{x}\,=\,\left\langle
x\middle|P\right\rangle\,,~{}~{}$ (20)

where $p_{x}$ is the probability to find the particle at position $x\in V$
when sampled from the distribution $\ket{P}$. The relation $p_{x}=\left\langle
x\middle|P\right\rangle$ follows from completeness of the $\ket{x}$ basis,
with $p_{x}$ the overlap of the probability distribution $\ket{P}$ with the
configuration $\ket{x}$. Because $p_{x}$ is a probability, we must have
$\sum_{x}\,p_{x}=1$. We define

$\ket{u}\,=\,\sum\limits_{x\in
V}\,\ket{x}~{}~{}~{}\text{such~{}that}~{}~{}~{}\left\langle
u\middle|P\right\rangle\,=\,\sum\limits_{x\in V}\,p_{x}\equiv 1\,,~{}~{}$ (21)

where we refer to $\ket{u}$ as the _uniform distribution_. We also note that
the normalization of a state is unimportant, so long as the state is time
independent. Time evolution (i.e., dynamics) of $\ket{P}$ (20) follow from the
master equation

$\partial^{\vphantom{\prime}}_{t}\ket{P}\,=\,-W\ket{P}\,,~{}~{}$ (22)

where $W$ is the generator of infinitesimal translations in time (much like a
Hamiltonian; we use the symbol $W$ to denote the generator of stochastic
Markovian dynamics). Using the fact that probability is conserved (i.e.,
$\left\langle u\middle|P\right\rangle=1$ for any probability distribution
$\ket{P}$), taking the inner product of Eq. (22) with $\bra{u}$ gives

$\left\langle
u\middle|\partial^{\vphantom{\prime}}_{t}\middle|P\right\rangle\,=\,\partial^{\vphantom{\prime}}_{t}\left\langle
u\middle|P\right\rangle\,=\,\partial^{\vphantom{\prime}}_{t}1\,=\,0\,=\,-\left\langle*\middle|u\middle|W\right\rangle{P}~{}~{}\forall\,\ket{P}~{}~{}~{}\implies~{}~{}~{}\bra{u}W\,=\,0\,.~{}~{}$
(23)

If the initial probability distribution $p^{\,}_{x}(0)$ is known, the
probability distribution $p^{\,}_{x}(t)$ at any time $t$ recovers from

$\ket{P_{t}}\,=\,\mathbb{T}\,\mathrm{e}^{-\int_{0}^{t}\,\mathrm{d}t^{\prime}\,W(t^{\prime})}\ket{P_{0}}\,=\,\mathcal{U}(t,0)\ket{P_{0}}\,,~{}~{}$
(24)

where $\mathbb{T}$ is the “time-ordering operator” (which ensures that the
integral in the exponential is applied to the distribution $\ket{P_{0}}$ in
chronological order), and $\mathcal{U}(t,0)$ is the _propagator_ from time $0$
to time $t$. Relatedly, we define the probability that the system transitions
from the configuration $x$ at time $0$ to the configuration $y$ at time $t$
under $W$ via

$\mathbb{P}\left[x,0\to
y,t;W\right]=\left\langle*\middle|y\middle|\mathrm{e}^{-W\,t}\right\rangle{x}\,=\,\mathcal{U}^{\vphantom{\dagger}}_{yx}(t,0)\,,~{}~{}$
(25)

e.g., in the case where $W$ (22) is time independent (i.e., static); above,
$\mathcal{U}^{\,}_{yx}$ denotes the matrix elements of the propagator
$\mathcal{U}$ (24). A crucial ingredient in our framework is the existence of
a stationary distribution; we define

$S\equiv\sum\limits_{x\in
V}\,\mathrm{e}^{-\Phi_{x}}\,\left|x\middle\rangle\middle\langle
x\right|~{}~{}~{}\text{and}~{}~{}~{}\ket{P_{\rm ss}}\equiv
S\,\ket{u}=\sum\limits_{x\in V}\,\mathrm{e}^{-\Phi_{x}}\,\ket{x}\,,~{}~{}$
(26)

as the stationary operator and stationary state (distribution), respectively,
where $P^{\rm ss}_{x}=\left\langle x\middle|P^{\rm
ss}\right\rangle=\exp(-\Phi_{x})$.

Using the definitions above, the condition of stationarity is equivalent to

$\derivative{t}\ket{P_{\rm ss}}=-W\ket{P_{\rm ss}}=0\,,~{}~{}$ (27)

so that the stationary state is unchanged by the dynamics generated by $W$,
leading to $\mathcal{U}(t)\ket{P_{\rm ss}}=\ket{P_{\rm ss}}$. If $W$ generates
stochastic dynamics via Eq. (22) and has a stationary state (26) such that Eq.
(27) holds, then the operator

$\widetilde{W}\equiv S\,W^{T}\,S^{-1}\,,~{}~{}$ (28)

generates stochastic dynamics and satisfies Eqs. (23) and (27) for the _same_
$P^{\,}_{\rm ss}$ (26) as $W$. In this sense, $\widetilde{W}$ is the “time-
reversed” partner to $W$, in that it generates the same sequence of
configurations as $W$, albeit in reverse chronological order. Accordingly, we
refer to the $\mathbb{Z}^{\,}_{2}$ transformation (28) as a _reversibility_
operation T. Numerous properties of the time-reversed generator (28) are
discussed in Sec. 2.3, in the context of continuous state spaces; here, we
briefly state several key results for discrete state spaces.

For a particle hopping on the graph $G=(V,E)$ with vertex set $V$ and
(directed) edge set $E$, the most general form of the generator $W$ (22) of
time evolution compatible with a stationary operator $S$ (26) takes the form

$W=-\sum\limits_{e\in E}\,Q_{e}\,\left|e\middle\rangle\middle\langle
e\right|\,S^{-1}+\sum\limits_{C=\text{cyc}(e_{1},\dots,e_{n})}\alpha_{C}~{}\sum\limits_{i=1}^{n}\left(\left|e_{i}\middle\rangle\middle\langle
e_{i-1}\right|-\left|e_{i-1}\middle\rangle\middle\langle
e_{i}\right|\right)\,S^{-1}\,,~{}~{}$ (29)

where we define a directed edge configuration
$\ket{e_{xy}}\equiv\ket{x}-\ket{y}$ (ignoring normalization) if the edge
$e_{xy}\in E$ corresponds to a hop from $x$ to $y$ (for vertices $x,y\in V$),
the ordered set of $n$ directed edges $\\{e_{i}\\}$ forms a _closed_ loop in
$V$, and $C=\mathrm{cyc}(e_{1},\dots,e_{n})$ denotes a cyclic permutation
around that closed loop (with weight $\alpha_{C}$). The statement that
$e_{1},\dots e_{n}$ forms a closed loop on the graph $G$ is equivalent to the
condition

$\sum\limits_{i=1}^{n}\,\ket{e_{i}}=0\,,~{}~{}$ (30)

and the sum over $C$ in Eq. (29) corresponds to all possible cycles in the
directed graph.

Because $W$ (29) is a Markov “rate” matrix, it must satisfy the conditions
$\left\langle i\middle|W\middle|j\right\rangle\leq 0$ for all $i\neq j$,
$\left\langle i\middle|W\middle|i\right\rangle\geq 0$ for all $i$, and each
row must sum to zero (where, e.g., $i$ and $j$ correspond to vertices). Under
time reversal T (28), the $Q^{\,}_{e}$ terms remain invariant, while
$\alpha_{C}\to-\alpha_{C}$ switches sign. Hence, the two terms in Eq. (29)
correspond to the most general possible T-even and T-odd terms allowed,
respectively.

To see this, first consider the case in which the stationary distribution
$P_{\rm ss}$ (26) is _uniform_. Since all configurations are equiprobable
(i.e., $\Phi=\log N$ is a constant), the operator $W$ (29) must be _doubly_
stochastic, which means that both the rows and columns of $W$ sum to zero.
This implies that the uniform distribution $\ket{u}$ (21) is both a left and
right null vector of $W$. Crucially, we note that every configuration in the
“edge basis” $\\{\ket{e}\\}$ is orthogonal to the uniform distribution,
meaning that diagonal operators $\left|e\middle\rangle\middle\langle e\right|$
are compatible with $W$ being doubly stochastic, and realize the T-even terms
in Eq. (29). Having constrained the symmetric terms in $W$, it is easy to
check that nonsymmetric terms must act on at least three vertices to be
compatible with a uniform stationary state. This is satisfied by considering
off-diagonal terms in the edge basis, so long as the edges are nearest
neighbors _and_ form a closed loop, as in Eq. (29). This is most transparent
if one considers a system with three configurations, such that $W$ (29)
realizes “flow” around a triangle; the loop condition simply extends this
example to arbitrary graphs. Finally, to capture generic, nonuniform
stationary distributions $\ket{S}=S\ket{u}$, we simply take $W=W_{u}\,S^{-1}$,
where $W^{\,}_{u}$ has a uniform stationary distribution proportional to
$\ket{u}$), which ensures that $W$ obeys $W\,\ket{P_{\rm
ss}}=W_{u}\,S^{-1}\,S\,\ket{u}=W_{u}\,\ket{u}=0$ as required by Eq. (27).

###  2.2 From stochastic differential equations to the Fokker-Planck equation

We now extend the discussion of Sec. 2.1 to the more general context in which
there is a continuum of allowed configurations. Consider a set of $M$
continuous variables $\mathbf{q}=(q_{1},\dots,q_{M})$, which may describe,
e.g., the positions (and momenta) of $M$ particles, conserved hydrodynamic
modes, the phase-space coordinates of a dynamical system, or other continuous
classical degrees of freedom. In general, each component $q_{a}$ is a
continuous, real-valued variable, and a function of time $t$. While we do not
consider this possibility here, in principle, one may also consider systems
with _both_ discrete variables $q_{b}$ and continuous variables $q_{a}$. We
also remark that one can write down a stochastic _field theory_ using
analogous tools to the discrete particle mechanics we discuss here. By taking
a standard continuum limit, in Sec. 2.7 we recover the equivalent Martin-
Siggia-Rose (MSR) Martin  _et al._ (1973) path-integral representation.
However, there are a number of subtleties in the MSR formalism: While the
formalism is useful in a number of field theoretic (i.e., continuum) settings,
our perspective is that the operator approach developed herein is much more
conceptually transparent, and especially with respect to reversibility (which
plays a crucial role in this work).

A stochastic differential equation (SDE) for $\mathbf{q}$ is realized by a
Langevin equation of the form

$\partial^{\vphantom{\prime}}_{t}q_{a}\,=\,f_{a}(\mathbf{q})+\xi_{a}(t)\,,~{}~{}$
(31)

where $\mathbf{f}$ is an $M$-component force that depends on $\mathbf{q}$, and
${\bf\it\xi}$ is a time-dependent noise source (specified below).
Unfortunately, Eq. (31) is not well defined without stipulating a particular
discretization convention for derivatives. Multiple prescriptions (e.g., the
Stratonovich and Itô formulations of stochastic calculus) have been proposed
Lau and Lubensky (2007). For this reason, we argue that an algorithmic
effective theory approach to Eq. (31) requires a completely unambiguous
framework. Moreover, it can be conceptually advantageous to consider the
evolution of _all_ possible trajectories in a statistical sense, analogous to
studying the evolution of _distributions_ of random variables, rather than the
stochastic evolution of individual random variables themselves. These
_probability distributions_ $P(\mathbf{q})$ obey the Fokker-Planck equation
(FPE), realized by the standard partial differential equation of Eq. (34),
which does _not_ suffer from the complications of stochastic calculus, and is
thus unambiguous.

As is common in the literature on stochastic dynamics Fokker (1914); Planck
(1917); Doi (1976a, b); Peliti, L. (1985); Peliti (1986)—and reviewed below—it
is convenient to associate the allowed configurations of a classical system
with a vector space, analogous to Hilbert spaces from quantum mechanics. One
can then interpret the probability distribution $P(\mathbf{q})$ as a
mathematical _analogy_ to the wavefunction $\psi(x)$ in quantum mechanics, and
apply the intuition and methods from quantum dynamics to the underlying
stochastic dynamics of Eq. (31) via $P(\mathbf{q})$. Formally, distinct
configurations of the system are labelled by “kets” of the form
$\ket{\mathbf{q}}=\ket{q_{1},\dots,q_{M}}$. We define the inner product
between two configurations $\ket{\mathbf{q}}$ and $\ket{\mathbf{q^{\prime}}}$
as

$\left\langle\mathbf{q}\middle|\mathbf{q^{\prime}}\right\rangle\,=\,\delta^{M}\left(\mathbf{q}-\mathbf{q^{\prime}}\right)\,=\,\prod_{n=1}^{M}\,\delta\left(q^{\vphantom{\prime}}_{n}-q^{\prime}_{n}\right)\,,~{}~{}$
(32)

in analogy to the inner product(s) for finite- and infinite-dimensional
Hilbert spaces.

The utility of this bra-ket formulation is that one can identify a probability
distribution $P(\mathbf{q})$ for a stochastic classical system analogous to
the wavefunction $\psi(x)$ of a deterministic quantum system. The probability
density function $P(\mathbf{q})$ can be expressed as an overlap with the
distribution vector $\ket{P}$, given by

$P(\mathbf{q})\,=\,\left\langle\mathbf{q}\middle|P\right\rangle\,,~{}~{}$ (33)

where the effective theory we develop captures the dynamics of
$P(\mathbf{q})$.

In particular, the dynamics associated with the SDE (31) is naturally and
unambiguously described by the Fokker-Planck equation (FPE), given in the Itô
formulation (where all derivatives are placed on the _left_) by

$\partial^{\vphantom{\prime}}_{t}P\,=\,-\partial^{\vphantom{\prime}}_{a}\left(f_{a}\,P-\frac{1}{2}\partial^{\vphantom{\prime}}_{b}(Q^{\vphantom{\prime}}_{ab}\,P)+\dots\right)\,,~{}~{}$
(34)

where $\partial^{\vphantom{\prime}}_{*}{a}$ is a shorthand for
$\partialderivative*{q_{a}}$, and the matrix $Q$ relates to the variance of
noise term ${\bf\it\xi}$ in Eq. (31) via

$Q^{\vphantom{\prime}}_{ab}\delta(t-t^{\prime})\,\equiv\,\expectationvalue{\xi_{a}(t)\xi_{b}(t^{\prime})}\,\to\,\Delta\,\delta^{\,}_{a,b}\,\delta\left(t-t^{\prime}\right)\,,~{}~{}$
(35)

where the latter relation applies to additive Gaussian noise—for which
${\bf\it\xi}$ is independent of $\mathbf{q}$ and only the second moment (35)
is nonzero—with strength (i.e., variance) $\Delta$, which we take to be
independent of $a$ for simplicity.

For more general noise sources ${\bf\it\xi}(\mathbf{q})$ in Eq. (31), we
always require that $\expectationvalue{\xi_{a}(t)}=0$, while nonvanishing
higher-order cumulants of ${\bf\it\xi}$ contribute to the $\dots$ terms in Eq.
(34). For clarity of presentation, many of the examples we consider in later
sections involve additive, uncorrelated Gaussian noise, which satisfies Eq.
(35); however, our formalism extends to more generic, non-Gaussian and/or
multiplicative noise sources, as we describe in Sec. 2.4.

Importantly, Eq. (34) can be recast in the form of a _master equation_ ,

$\frac{\mathrm{d}}{\mathrm{d}t}\ket{P(t)}\,=\,-W\ket{P(t)}\,,~{}~{}$ (36)

also called the “quantum Hamiltonian” description Schütz (2001) due to its
resemblance of the Schrödinger equation (up to being real valued; see also Doi
(1976a, b); Peliti, L. (1985); Peliti (1986)). The _differential_ operator $W$
generates time evolution, and is known as the “rate matrix” in the context of
continuous-time Markov chains. We infer its form from Eq. (34),

$W\,\equiv\,\partial^{\vphantom{\prime}}_{a}f_{a}-\frac{1}{2}\,\partial^{\vphantom{\prime}}_{a}\partial^{\vphantom{\prime}}_{b}Q^{\vphantom{\prime}}_{ab}\,,~{}~{}$
(37)

where, because $W$ is an operator, the partial derivatives above act not only
on $F$ and $Q$, but also on any object to which $W$ is applied (as in quantum
mechanics). Regarding Eq. (36), the probability distribution $P(\mathbf{q})$
(33) at time $t$ can be expressed in terms of initial conditions using the
_propagator_ ${\cal U}$, which is implicitly defined via

$\ket{P(t)}\,=\,\mathbb{T}\,\mathrm{e}^{-\int_{0}^{t}\,\mathrm{d}t^{\prime}\,W(t^{\prime})}\,\ket{P(0)}\,=\,\mathcal{U}(t,0)\,\ket{P(0)}\,,~{}~{}$
(38)

where the operator $\mathbb{T}$ enforces time ordering of the integral in the
exponential, and $\mathcal{U}(t,0)$ is the propagator from time 0 to time $t$
(we simply write ${\cal U}(t)$ when $W$ is time independent). Additionally,
the probability that the initial configuration $\mathbf{q}_{i}$ at time $0$
evolves under $W$ into the configuration $\mathbf{q}_{f}$ at time $t$ is given
by the transition amplitude

$\displaystyle\mathbb{P}[\mathbf{q}_{i},0\to\mathbf{q}_{f},t;W]\,=\,\left\langle\mathbf{q}_{f}\middle|\exp(-\int_{0}^{t}\mathrm{d}t^{\prime}\,W(t^{\prime}))\middle|\mathbf{q}_{i}\right\rangle\,\equiv\,\mathcal{U}^{\vphantom{\dagger}}_{fi}(t,0)\,,~{}~{}$
(39)

where $\mathcal{U}^{\,}_{fi}(t,0)$ are matrix elements of the propagator (38),
which simplifies when $W$ is time independent.

Note that Eq. (39) resembles a quantum _amplitude_ for some “path” $i\to f$,
suggesting a formulation à la the Feynman path-integral representation of the
evolution of quantum states. Indeed, such a prescription is well established
by the Martin-Siggia-Rose (MSR) path integral Martin  _et al._ (1973), which
we consider in Sec. 2.7. However, for reasons that will become apparent in the
path-integral context, it is actually more elegant (and straightforward) to
implement reversibility T in terms of the operator $W$ (37). This represents a
rare instance in which the effective theory is _more difficult_ to implement
systematically (to all orders in nonlinearities) in a Lagrangian (path-
integral) formalism.

Finally, our interest lies in Fokker-Planck dynamics that admit a _stationary_
state $\ket{P_{\rm ss}}$ satisfying

$W\,\ket{P_{\rm ss}}=-\frac{\mathrm{d}}{\mathrm{d}t}\,\ket{P_{\rm
ss}}=0\,,~{}~{}$ (40)

or equivalently, $\mathcal{U}\,\ket{P_{\rm ss}}=\ket{P_{\rm ss}}$, so that
$\ket{P_{\rm ss}}$ is invariant under the dynamics generated by $W$. Note that
if $W$ is time independent, at least one choice of $\ket{P_{\rm ss}}$ is
guaranteed to exist, but it may not be unique. For simplicity, throughout this
work, we assume that $\ket{P^{\,}_{\rm ss}}$ exists and is nonsingular, so
that a stationary operator $S$ (26) exists and is _invertible_ ; we also
explain how to incorporate conservation laws in our EFT framework (a
physically relevant scenario in which $\ket{P_{\rm ss}}$ may fail to be
unique) in Sec. 2.6. Under the foregoing assumptions—and without loss of
generality—we define the stationary state $\ket{P_{\rm ss}}$ according to

$P_{\rm ss}(\mathbf{q})\,\equiv\,\left\langle\mathbf{q}\middle|P_{\rm
ss}\right\rangle\,=\,\mathrm{e}^{-\Phi(\mathbf{q})}\,\Longleftrightarrow\,\ket{P_{\rm
ss}}=\int\mathrm{d}\mathbf{q}\,\mathrm{e}^{-\Phi(\mathbf{q})}\,\ket{\mathbf{q}}\,,~{}~{}$
(41)

which is a simple Ansatz given Eq. (40) and the requirement that $P_{\rm
ss}(\mathbf{q})=\left\langle\mathbf{q}\middle|P_{\rm ss}\right\rangle$ be
positive. We need not worry about overall normalization, since we only
consider linear operations on $\ket{P_{\rm ss}}$.

###  2.3 Stationarity constraints on the Fokker-Planck equation

We now discuss several features of the dynamics generated by the FPE (34) that
follow from stationarity, including the important relations to reversibility
T. First, it will prove useful to define several quantities. For example,
related to the stationary _state_ $\ket{P_{\rm ss}}$ (40) with distribution
$\Phi$ (41) is the stationary _operator_ given by

$S\,=\,\int\mathrm{d}\mathbf{q}\,\mathrm{e}^{-\Phi(\mathbf{q})}\,\left|\mathbf{q}\middle\rangle\middle\langle\mathbf{q}\right|~{}~{}~{}\implies~{}~{}~{}\ket{P_{\rm
ss}}\,=\,S\,\ket{u}\,,~{}~{}$ (42)

in analogy to Eq. (26) in Sec. 2.1, where $\ket{u}$ is the _uniform_
distribution

$\ket{u}\,\equiv\,\int\mathrm{d}\mathbf{q}\,\ket{\mathbf{q}}~{}~{}~{}\implies~{}~{}~{}u(\mathbf{q})\,=\,1\,,~{}~{}$
(43)

up to normalization, in analogy to Eq. (21) for discrete spaces in Sec. 2.1.

As before, numerous useful properties follow from $\ket{u}$ (43). For example,
conservation of probability mandates that $\left\langle
u\middle|P(t)\right\rangle=\int\mathrm{d}\mathbf{q}\,P(\mathbf{q})=1$. Noting
that $u$ (43) is independent of time, applying $\bra{u}$ to Eq. (36) leads to

$\bra{u}\frac{\mathrm{d}}{\mathrm{d}t}\ket{P}\,=\,\frac{\mathrm{d}}{\mathrm{d}t}\,\left\langle
u\middle|P\right\rangle\,=\,\frac{\mathrm{d}}{\mathrm{d}t}1\,=\,0~{}~{}~{}\implies~{}~{}~{}\bra{u}\,W\,=\,0\,,~{}~{}$
(44)

for any $\ket{P}$, where $0$ is technically the null vector. In other words,
$W^{T}$ annihilates the uniform distribution $\ket{u}$ (43).

The assumption of stationarity implies that the probability distribution
$\ket{P_{\rm ss}}\equiv S\ket{u}$ (41) is a time-independent (i.e.,
stationary) solution to the FPE (34), and thus

$W\,\ket{P_{\rm ss}}=W\,S\ket{u}=0\,,~{}~{}$ (45)

as in Eq. (40), while $\mathcal{U}\,\ket{P_{\rm ss}}=\ket{P_{\rm ss}}$. The
two preceding expressions constrain the action of $W$ to the left and right.

We now consider the crucial transformation $W\to\widetilde{W}$, corresponding
to a $\mathbb{Z}^{\,}_{2}$ _reversibility_ operation T. In analogy to Eq. (28)
in the case of discrete state spaces, the “time-reversed” evolution generator
$\widetilde{W}$ is defined in terms of the original generator $W$ (37) and the
stationary operator $S$ (42) via

$\widetilde{W}\,\equiv\,S\,W^{T}\,S^{-1}\,,~{}~{}$ (46)

which exists whenever Kolmogorov’s criterion is satisfied, and which has the
_same_ stationary distribution as $W$ by Kelly’s Lemma Kolmogorov (1931);
Kelly (1979). The above can be rewritten component wise as

$\displaystyle(\,\widetilde{W}^{\,T}\,)^{\vphantom{\prime}}_{xy}=\frac{P^{\vphantom{\prime}}_{\rm
ss}(x)\,W^{\vphantom{\prime}}_{xy}}{P^{\vphantom{\prime}}_{\rm
ss}(y)}\,,~{}~{}$ (47)

where there is no sum over $y$ or $x$. The relation in Eq. (47) is well known
in the theory of Markov chains Kelly (1979).

We now motivate this definition and the interpretation of $\widetilde{W}$ (46)
as the time-reversed counterpart of $W$ (37). Consider the probability that
the system starts in the state $\ket*{\mathbf{q}_{i}}$ and ends in the state
$\ket*{\mathbf{q}_{f}}$. Note that

$\left\langle\mathbf{q}_{f}\middle|\exp(-W\,t)\middle|\mathbf{q}_{i}\right\rangle\,=\,\left\langle\mathbf{q}_{i}\middle|\exp(-W^{T}\,t)\middle|\mathbf{q}_{f}\right\rangle\,,~{}~{}$
(48)

yet $W^{T}$ does not, in general, generate valid Fokker-Planck dynamics
captured by Eq. (36), since it obeys neither Eq. (44) nor Eq. (45). However,
as in the case of discrete state spaces considered in Sec. 2.1, the operator
$\widetilde{W}$ (46) _is_ a proper generator of Fokker-Planck dynamics with
the _same stationary distribution_ $\ket{S}$ (41) as $W$ Kelly (1979). To see
this, it is sufficient to check that Eqs. (44) and (45) hold for
$\widetilde{W}$, i.e.,

$\displaystyle\bra{u}\widetilde{W}\,$
$\displaystyle=\,\bra{u}\left(S^{-1}\,W\,S\right)^{T}\,=\,\bra{u}S^{T}W^{T}(S^{-1})^{T}\,=\,0$
(49a) $\displaystyle\widetilde{W}S\ket{u}\,$
$\displaystyle=\,\left(S^{-1}WS\right)^{T}S\ket{u}\,=\,S\,W^{T}S^{-1}S\ket{u}\,=\,S\left(\bra{u}W\right)^{T}\,=\,S\,(0)=0\,,~{}~{}$
(49b)

so that $\widetilde{W}$ is also a stochastic matrix with stationary
distribution $P_{\rm ss}=\exp(-\Phi)$ (41).

Using the results above, we identify the _global balance condition_

$\mathbb{P}[\mathbf{q}_{i}(0)\to\mathbf{q}_{f}(t)]\,=\,\left\langle*\middle|\mathbf{q}_{f}\middle|\mathrm{e}^{-W\,t}\right\rangle{\mathbf{q}_{i}}\,=\,\left\langle*\middle|\mathbf{q}_{i}\middle|S^{-1}\mathrm{e}^{-\widetilde{W}\,t}S\right\rangle{\mathbf{q}_{f}}\,=\,\mathrm{e}^{\Phi_{i}-\Phi_{f}}\,\widetilde{\mathbb{P}}[\mathbf{q}_{f}(0)\to\mathbf{q}_{i}(t)]\,,~{}~{}$
(50)

given the stationary distribution $P^{\rm ss}=\mathrm{e}^{-\Phi}$ (41).
Lastly, the reversibility T is an involution:

$\widetilde{\widetilde{W}}\,=\,S\,\widetilde{W}^{T}\,S^{-1}\,=\,S\,\left(S\,W^{T}\,S^{-1}\right)^{T}\,S^{-1}\,=S\,S^{-1}\,\left(W^{T}\right)^{T}\,S\,S^{-1}\,=W\,,~{}~{}$
(51)

and is therefore $\mathbb{Z}^{\,}_{2}$, as required. Applying T twice returns
the same operator $W\to\widetilde{W}\to W$.

If the Fokker-Planck generator $W$ (37) obeys $W=\widetilde{W}$, the system is
said to obey _detailed balance_ , while all $W$s with a stationary state
$P_{\rm ss}$ (41) obey _global balance_ (47). In the Markov-chain literature,
$\widetilde{W}$ (46) is known as the “backward”—or _time-reversed_ —Markov
generator. In the setting of stochastic effective theories, we adopt the
perspective that the transformation $W\to\widetilde{W}$ (46) corresponds to a
_reversibility_ transformation; accordingly, we associate detailed balance
with reversibility T. Theories with $W=\widetilde{W}$ obey detailed balance
and are “T even”; theories with $W\neq\widetilde{W}$ break detailed balance,
and thus T; in particular, T-breaking theories with $W=-\widetilde{W}$ are “T
odd.” The connection between the transformation of Eq. (46) and time reversal
is more transparent when accompanied by the familiar, _microscopic_ notion of
time reversal (e.g., $t\to-t$, $p\to-p$, etc.); we discuss such details in
later sections, where we consider how T (46) acts on quantities other than the
generator $W$ (37). The formalism we develop herein is a powerful theoretical
tool, especially for systems in which reversibility T plays an important role
and is a symmetry of $W$, either on its own or in the generalized sense of a
gT symmetry (i.e., in combination with another $\mathbb{Z}^{\,}_{2}$
transformation). Even if time reversal T (or gT) is not preserved, there is
still great value in separating the T-even versus T-odd contributions to $W$.
As we explain in Sec. 2.4, the T-even and T-odd contributions are related to
dissipative versus dissipationless dynamics, and identifying the corresponding
terms leads to generalizations of the second law of thermodynamics in Eq. (78)
and a fluctuation-dissipation theorem in Eq. (60).

###  2.4 Construction of the Fokker-Planck generators

Inspired by Wilsonian effective theory Wilson (1971a, b); Hohenberg and
Halperin (1977); Weinberg (1979); Altland and Simons (2010), we now provide a
general construction for the differential operator $W$ (37), which must be
compatible with (_i_) the existence of a stationary distribution $P_{\rm
ss}\propto\exp(-\Phi)$ (41), (_ii_) normalization of the probability
distribution $P(\mathbf{q},t)$ (33), and (_iii_) the existence of a time-
reversed evolution operator $\widetilde{W}$ (46) that fulfils the same
conditions as $W$ itself. While massaging this FPE-based formalism into a more
conventional path-integral EFT involves a few subtleties (which we discuss in
Sec. 2.7), the FPE approach nonetheless captures the philosophical
underpinnings of Wilson’s EFT formalism in the context of stochastic dynamics
(31).

For simplicity, we assume herein that the reversibility operation T (46) that
maps $W\to\widetilde{W}$ also maps the trajectory $q_{a}(t)\to q_{a}(-t)$.
Later, in Sec. 3.1, we provide an examples in the context of Hamiltonian
mechanics that involve momentum variables $q_{a}=p_{a}$, for which
$q_{a}(t)\to q_{a}(-t)=-q_{a}(t)$ under T. We comment further on the treatment
of T—and modifications to the results recovered herein—in the presence of
T-odd variables $q_{a}$.

####  2.4.1 General form of $W$

We begin by defining the generalized chemical potentials—from which various
aspects of time-reversal symmetry T and the EFT framework follow—in terms of
the stationary distribution $\Phi(\mathbf{q})$ (41) via

$\mu^{\vphantom{\prime}}_{a}\,\equiv\,\frac{\partial\Phi}{\partial
q_{a}}\,,~{}~{}$ (52)

and we now introduce a generic—and particularly convenient—expression for the
evolution operator $W$ (37). The conditions of Eqs. (44) and (45) suggest that
the evolution operator $W$ (37) may be written

$W\,=-\frac{1}{2}\partial^{\vphantom{\prime}}_{a}\mathcal{Q}_{ab}(\partial^{\vphantom{\prime}}_{b}+\mu^{\vphantom{\prime}}_{b})\,,~{}~{}$
(53)

where the factor of $\partial^{\vphantom{\prime}}_{*}{a}$ on the left ensures
satisfaction of Eq. (44) by annihilating the uniform state $\bra{u}$ (43)
acting to the left, the factor of
$(\partial^{\vphantom{\prime}}_{*}{b}+\mu^{\vphantom{\prime}}_{*}{b})$ on the
right ensures satisfaction of Eq. (45) by annihilating the stationary operator
$S$ (42) acting to the right, and $\mathcal{Q}$ is a matrix that we expect to
be related to $Q$ (35). However, compared to $Q$, the object $\mathcal{Q}$ in
Eq. (53) may be nonlinear in the coordinates $\mathbf{q}$, include further
derivatives of arbitrary order, etc.

It can be helpful to introduce the notation

$\pi^{\vphantom{\prime}}_{a}\,=-\mathrm{i}\,\partial^{\vphantom{\prime}}_{a}~{}~{}~{}\text{and}~{}~{}~{}\mathcal{H}\,=-\mathrm{i}\,W\,,~{}~{}$
(54)

so that, in analogy with quantum mechanics, $\pi_{a}$ is the “conjugate
operator” to $q_{a}$ and $\mathrm{i}\,\mathcal{H}$ is the infinitesimal
generator of time translations. Note that $\cal H$ resembles a quantum
Hamiltonian Doi (1976a, b); Peliti, L. (1985); Peliti (1986); we reserve $H$
for Hamiltonians mechanics, where $H$ relates to both $W$ and $\Phi$ (41).
This notation provides a useful shorthand in certain scenarios, especially
when we discuss genuine stochastic PDEs and field theories. In fact, the field
$\pi_{a}$ naturally arises in the path integral construction discussed in Sec.
2.7. Of course, the notation of Eq. (54) awkwardly introduces complex numbers
into a formalism that is otherwise real valued, and so we generally utilize
the FPE notation (i.e., $\partial^{\vphantom{\prime}}_{*}{a}$ and $W$), until
our discussion of continuum field theories (where one requires functional
Fokker-Planck equations) in Sec. 7.

It will also prove useful to derive an expression for the time-reversed
operator $\widetilde{W}$ (46) analogous to Eq. (53). We note that the
derivative operator $\partial^{\vphantom{\prime}}_{*}{a}$ is antisymmetric
under T, and that

$S\partial^{\vphantom{\prime}}_{a}S^{-1}\,=\,\partial^{\vphantom{\prime}}_{a}+[\partial^{\vphantom{\prime}}_{a}\Phi]\,=\,\partial^{\vphantom{\prime}}_{a}+\mu^{\vphantom{\prime}}_{a}\,,~{}~{}$
(55)

where the derivative in $[\partial^{\vphantom{\prime}}_{*}{a}\Phi]$ acts
_only_ on $\Phi$, and not to the right (in general, derivatives inside square
braces act only within those braces). We use the relation above to recover the
expression

$\displaystyle\widetilde{W}\,$
$\displaystyle=\,-\frac{1}{2}\,S\,W^{T}S^{-1}\,=\,-\frac{1}{2}\,S\,\left(\partial^{\vphantom{\prime}}_{a}\,\mathcal{Q}^{\vphantom{\dagger}}_{ab}\,\left(\partial^{\vphantom{\prime}}_{b}+\mu^{\vphantom{\prime}}_{b}\right)\right)^{T}\,S^{-1}\,=\,-\frac{1}{2}\,S(-\partial^{\vphantom{\prime}}_{b}+\mu^{\vphantom{\prime}}_{b})\,\mathcal{Q}^{T}_{ab}(-\partial^{\vphantom{\prime}}_{a})S^{-1}$
$\displaystyle=\,-\frac{1}{2}\,S(\partial^{\vphantom{\prime}}_{b}-\mu^{\vphantom{\prime}}_{b})S^{-1}S\mathcal{Q}^{T}_{ab}S^{-1}S(\partial^{\vphantom{\prime}}_{a})S^{-1}\,=\,-\frac{1}{2}\,\partial^{\vphantom{\prime}}_{b}\left(S\mathcal{Q}^{T}_{ab}S^{-1}\right)(\partial^{\vphantom{\prime}}_{a}+\mu^{\vphantom{\prime}}_{a})\,,~{}~{}$
(56)

which resembles Eq. (53) for $W$, up to a transpose on the matrix
$\mathcal{Q}$.

We now consider $W$ (53) in more detail. We first note that Eq. (53) is
generally complicated, and sensitive to the various details contained in
$\mathcal{Q}$. The matrix $\mathcal{Q}$ (57) may be nonlinear in $\mathbf{q}$
and capture non-Gaussian and/or multiplicative noise, indicated by the $\dots$
in Eq. (34). As is standard practice both in EFTs and in the treatment of
stochastic (FPE) dynamics, we consider a _derivative expansion_ of
$\mathcal{Q}$ in powers of $\partial^{\vphantom{\prime}}_{*}{a}$, organizing
terms by their derivative order; in all cases of interest in this paper, the
low-order terms are sufficient to capture the relevant physics.

####  2.4.2 Gaussian noise and the fluctuation-dissipation theorem

For concreteness, we begin by considering the derivative-free terms in
$\mathcal{Q}$. These capture the _only_ nontrivial contribution in the
presence of purely Gaussian noise, for which moments higher than the variance
vanish. Also note that derivative-free matrices $\mathcal{Q}$ commute with the
stationary operator $S$ (42). In general, Gaussian noise still allows for
$\mathcal{Q}$ to be a functional $\mathcal{Q}(\mathbf{q})$ of the coordinates
$\mathbf{q}$ (i.e., multiplicative noise). We decompose
$\mathcal{Q}(\mathbf{q})$ according to

$\mathcal{Q}_{ab}\,=\,Q_{ab}-V_{ab}\,,~{}~{}$ (57)

where, on the right-hand side, $Q$ and $V$ represent undetermined symmetric
and antisymmetric matrices, respectively.

This notation is intentional: The matrix $Q$ in Eq. (57) is the (symmetric)
noise-variance matrix $Q$ (35)—i.e., the _same_ object $Q$ that appears in Eq.
(37). Subtleties relating to the ordering of derivatives may arise at the
level of the SDE (31), but are unimportant at the level of the FPE (34). For
higher-order (non-Gaussian) noise (65), this matching between the SDE and FPE
is unambiguous at the highest derivative order. Then, applying the time-
reversal operation T (46) to the derivative-free part of $\mathcal{Q}$ (57)
gives

$\displaystyle
S\,\Big{(}Q-V\Big{)}^{T}\,S^{-1}\,=\,S\,\Big{(}Q^{T}-V^{T}\Big{)}\,S^{-1}\,=\,S\,\big{(}Q+V\big{)}\,S^{-1}\,=\,Q+V\,,~{}~{}$
(58)

meaning that the symmetric term $Q$ in $\mathcal{Q}$ (57) is invariant under
time reversal T (46), while the antisymmetric term $V$ flips sign. So when
$\mathcal{Q}$ contains no derivatives, we will see that the symmetric part $Q$
of $\mathcal{Q}$ with _dissipative_ (T-even) dynamics and the antisymmetric
part $V$ with dissipation _less_ (T-odd) dynamics.

In the simplest scenario—in which $\mathcal{Q}$ (57) contains no
derivatives—we expand $W$ (53) using Eq. (57) to find

$\displaystyle W\,$
$\displaystyle=\,-\frac{1}{2}\,\partial^{\vphantom{\prime}}_{a}\,\mathcal{Q}_{ab}\,(\partial^{\vphantom{\prime}}_{b}+\mu^{\vphantom{\prime}}_{b})\,=\,-\frac{1}{2}\,\partial^{\vphantom{\prime}}_{a}\,(Q_{ab}-V_{ab}\,)\,(\partial^{\vphantom{\prime}}_{b}+\mu^{\vphantom{\prime}}_{b})$
$\displaystyle=\,\frac{1}{2}\,\partial^{\vphantom{\prime}}_{a}\big{(}\mu^{\vphantom{\prime}}_{b}+\partial^{\vphantom{\prime}}_{b}\big{)}\big{(}V_{ab}-Q_{ab}\big{)}+\frac{1}{2}\,\partial^{\vphantom{\prime}}_{a}\Big{(}\big{[}\partial^{\vphantom{\prime}}_{b}Q_{ab}\big{]}-\big{[}\partial^{\vphantom{\prime}}_{b}V_{ab}\big{]}\Big{)}\,,~{}~{}$
(59)

where the divergences in square braces
$[\partial^{\vphantom{\prime}}_{*}{b}Q_{ab}]$ are _functions_ (the derivative
does not continue acting to the right), and in the lower line, we place
$\partial^{\vphantom{\prime}}_{*}{b}$ to the left of the constant matrix
$Q_{ab}$ to match the Itô formulation of Eqs. (31), (35), and (37). Comparing
Eq. (2.4.2) to Eq. (37), the final term on the right-hand side of in Eq.
(2.4.2) corresponds to the dissipative contribution from the _noise_ via $Q$
(35), while the other terms correspond to the force

$2\,f_{a}=-Q_{ab}\,\mu^{\vphantom{\prime}}_{b}+V_{ab}\,\mu^{\vphantom{\prime}}_{b}+\big{[}\partial^{\vphantom{\prime}}_{b}Q_{ab}\big{]}-\big{[}\partial^{\vphantom{\prime}}_{b}V_{ab}\big{]}\,,~{}~{}$
(60)

in Eq. (37), which contains _both_ dissipative and dissipationless terms. In
particular, if the symmetric part $Q$ of $\mathcal{Q}$ is independent of
$\mathbf{q}$ and free of derivatives—as in Eq. (2.4.2) above—then $Q$ is
simply the variance of white noise (35).

The constrained form of the dissipative part of the force (60) is a
manifestation of a _fluctuation-dissipation theorem_ (FDT) Kubo (1966); Kwon
_et al._ (2005); Prost  _et al._ (2009). The fact that the dissipative force
is $-Q_{ab}\,\mu^{\vphantom{\prime}}_{*}{b}/2$—where $Q_{ab}$ is the noise
variance (35)—is essential to preserving stationarity. Note that $\Phi$ (41)
shows up in the FDT (60) via $\mu^{\vphantom{\prime}}_{*}{}$ (52).

We now recast the FDT in a more familiar form Prost  _et al._ (2009). Consider
the correlation function $\expectationvalue{q_{a}(t)q_{b}(0)}$, where
$\expectationvalue{\cdot}$ denotes an average with respect to the stationary
distribution $S$ (41). This correlation function can be rewritten as

$\expectationvalue{q_{a}(t)\,q_{b}(0)}\,=\,\frac{\mathrm{d}}{\mathrm{d}\epsilon}\,\left.\int\mathrm{d}\mathbf{q}\,\mathrm{e}^{-\Phi+\epsilon
q_{b}}\,q_{a}(t)\right|^{\vphantom{\prime}}_{\epsilon=0}\,,~{}~{}$ (61)

where the right-hand side essentially captures a “shift” of the chemical
potential
$\mu^{\vphantom{\prime}}_{*}{b}\to\mu^{\vphantom{\prime}}_{*}{b}+\epsilon$
(52) for $t<0$. Defining

$\chi^{\vphantom{\prime}}_{ab}(t)\,\equiv\,\frac{\delta\expectationvalue{q_{a}(t)}}{\delta\mu^{\vphantom{\prime}}_{b}(0)}\,,~{}~{}$
(62)

we can rewrite the correlation function of Eq. (61) in the familiar form

$\expectationvalue{q_{a}(t)\,q_{b}(0)}\,=\,\int\limits_{t}^{\infty}\mathrm{d}s\,\chi^{\vphantom{\prime}}_{ab}(s)\,~{}~{}$
(63)

where $\chi$ is a generalized susceptibility (the type depends on $q_{a}$).

In the context of active matter, the lore is that there is no FDT Kubo (1966);
Kwon  _et al._ (2005); Prost  _et al._ (2009). However, if a stationary
distribution $P_{\rm ss}=\exp(-\Phi)$ (41) is known to exist (for a given
$W$)—or, if $W$ (53) is constructed to be compatible with some $\Phi$—then the
FDT (63) is a direct mathematical consequence of $P_{\rm ss}$ (41), which
emerges naturally as a constraint on the force $f_{a}$ (60). The difference
between our perspective and that of the active-matter literature is more
precisely stated as follows: If a specific active system has local equations
of motion, but a nonlocal $\Phi$, the FDT (63) relates physics in a local
theory to physics in a nonlocal theory. The emergence (or lack thereof) of an
FDT can then be traced to whether or not a “simple” stationary state of the
form $P_{\rm ss}\propto\exp(-\Phi)$ (41) can be identified.

####  2.4.3 Non-Gaussian noise

We now extend this construction from the derivative-free example above to more
general operators $\mathcal{Q}$. To this end, we would like to organize the
derivative expansion of $\mathcal{Q}_{ab}$ such that the symmetry of $W$ under
T (46) is related to the symmetry of $\mathcal{Q}_{ab}$ under
$a\leftrightarrow b$. To handle non-Gaussian noise (corresponding to more
derivatives in $\mathcal{Q}$), we make use of the following useful (albeit
intimidating) derivative expansion à la Kramers-Moyal Kramers (1940); Moyal
(1949); Pawula (1967a, b). We define

$\partial^{-}_{a}\,\equiv\,\partial^{\vphantom{\prime}}_{a}\,=\,\partialderivative{q_{a}}~{}~{}~{}\text{and}~{}~{}~{}\partial^{+}_{a}\,\equiv\,\partial^{\vphantom{\prime}}_{a}+\mu^{\vphantom{\prime}}_{a}\,,~{}~{}$
(64)

using which we expand the matrix $\mathcal{Q}$ (53) order by order according
to

$\displaystyle\mathcal{Q}^{\vphantom{\dagger}}_{ab}\,$
$\displaystyle=\,\mathcal{Q}_{ab}^{(0)}+(\partial^{-}_{c}\mathcal{Q}^{(1)}_{ab,c}+\mathcal{Q}^{(1)}_{ab,c}\partial^{+}_{c})+\cdots\,=\,\sum\limits_{n=0}^{\infty}\left(\partial^{-}_{c_{1}}\cdots\partial^{-}_{c_{n}}\mathcal{Q}^{(n)}_{ab,c_{1}\dots
c_{n}}\,+\mathcal{Q}^{(n)}_{ab,c_{n}\ldots
c_{1}}\partial^{+}_{c_{n}}\cdots\partial^{+}_{c_{1}}\right)\,,~{}~{}$ (65)

where $Q^{(n)}_{abc_{1}\ldots c_{n}}$ are $\mathbf{q}$-dependent coefficients
that are related to cumulants of the noise, and summation over repeated
$c_{k}$ indices is implicit. In Eq. (65), the rightmost sum contains all
orderings: Pulling some factors of $\partial^{\pm}$ through a
$\mathcal{Q}^{(n)}$ only leads to a multiplicative factor, which can simply be
absorbed into another term $\mathcal{Q}^{(n)}$ at lower order $n$. Changing
any $\partial^{+}\to\partial^{-}$ (or vice versa) similarly leads to a
multiplicative factor, allowing us to arrange for all terms to contain _only_
$\partial^{+}$ or $\partial^{-}$ at each order, acting on the right or left of
$\mathcal{Q}$, respectively, without loss of generality.

The motivation for the Kramers-Moyal-esque expansion Kramers (1940); Moyal
(1949); Pawula (1967a, b) of $\mathcal{Q}$ (65) is that each term in the sum
transforms “nicely” under the time-reversal transformation T (46). Whether a
term in the sum in Eq. (65) corresponds to dissipative or dissipationless
dynamics depends only on the symmetry (or antisymmetry) of the corresponding
matrix $\mathcal{Q}^{(n)}$ in the $a,b$ indices! In particular, for even $n$,
the sign of $W$ under time reversal is equivalent to the sign of
$\mathcal{Q}_{ab}$ under interchanging $a\leftrightarrow b$, while for odd
$n$, the sign of $W$ under time reversal is opposite the sign of
$\mathcal{Q}_{ab}$ under interchanging $a\leftrightarrow b$. In this way, the
expansion in Eq. (65) provides for the identification of dissipative and
dissipationless terms that appear in the presence of higher-derivative (i.e.,
non-Gaussian) noise.

Formally, the derivative expansion in Eq. (65) provides a convenient means by
which to organize the various terms that may contribute to $\mathcal{Q}$ (57).
More physically, this derivative expansion can be viewed on a similar footing
as, e.g., those of hydrodynamic theories Harder  _et al._ (2015); Crossley
_et al._ (2017); Glorioso  _et al._ (2017); Haehl  _et al._ (2016); Jensen
_et al._ (2018); Glorioso and Liu (2016), where
$\partial^{\vphantom{\prime}}_{*}{a}$ is associated with a small length scale
$\ell\ll 1$. In the FPE context, the intuition is that the length scale of
variation of a given probability distribution $P(t,\mathbf{q})$ is much larger
than any microscopic length scales (e.g., the mean free path of the elementary
constituents) that appear in the coefficients of higher-derivative terms. This
implies that higher derivatives in Eq. (65)—which correspond to higher-order,
non-Gaussian correlations of the noise $\xi(t)$ in Eq. (35)—become
progressively less important.

####  2.4.4 Generating all possible T-breaking, noise-free terms

Having established the general form of $W$ (53) compatible with the existence
of a stationary state (40) and global—but not necessarily _detailed_ —balance
(46), we now discuss the generalized T-breaking (i.e., dissipationless)
contributions to the right-hand side of Eq. (53). For simplicity, we ignore
the effects of noise for the time being. Up to a subtlety described below,
this simply amounts to choosing the most general from of the antisymmetric
term $V_{ab}$ in Eq. (57). Up to second order in derivatives, taking
$\mathcal{Q}_{ab}=-V_{ab}=V_{ba}$ we find, from Eq. (2.4.2),

$W\,=\,\frac{1}{2}\,\partial^{\vphantom{\prime}}_{a}V^{\vphantom{\dagger}}_{ab}(\partial^{\vphantom{\prime}}_{b}+\mu^{\vphantom{\prime}}_{b})\,=\,\frac{1}{2}\,\partial^{\vphantom{\prime}}_{a}\left(\partial^{\vphantom{\prime}}_{b}V^{\vphantom{\dagger}}_{ab}+\mu^{\vphantom{\prime}}_{b}V^{\vphantom{\dagger}}_{ab}-[\partial^{\vphantom{\prime}}_{b}V^{\vphantom{\dagger}}_{ab}]\right)\,,~{}~{}$
(66)

where all derivatives are on the left, as in Eq. (2.4.2), so that the
rightmost expression above realizes the standard Itô formulation. Here and
below, any derivative terms in square brackets indicate that the derivative
operators act only within those brackets. Hence, the term (66) is a purely
deterministic contribution to the FPE (34). Accordingly, the dissipationless
part $v_{a}$ of the Itô force $f_{a}$ introduced in Eq. (31) has components
given by

$\displaystyle
v_{a}\,=-\frac{1}{2}\,[\partial^{\vphantom{\prime}}_{b}V_{ab}]+\,\frac{1}{2}\mu^{\vphantom{\prime}}_{b}\,V_{ab}\,,~{}~{}$
(67)

so that $v_{a}$ leads to a nonvanishing probability current $J_{a}$, even when
evaluate on a steady state. To see this, we rewrite the Fokker-Planck equation
(34) as a _continuity equation_ ,

$\displaystyle\partial^{\vphantom{\prime}}_{t}P\,=-\partial^{\vphantom{\prime}}_{a}J^{\vphantom{\prime}}_{a}\,,~{}~{}$
(68)

where $J_{a}$ is the probability current; for the stationary state $P^{\rm
ss}=\exp(-\Phi)$, the corresponding probability current is

$\displaystyle J_{a}^{\,\rm ss}\,=\,v_{a}\,\mathrm{e}^{-\Phi}\,,~{}~{}$ (69)

and since $P^{\rm ss}$ is _stationary_ , the probability current must satisfy
$\partial^{\vphantom{\prime}}_{*}{a}\,J_{a}^{\,\mathrm{ss}}=0$—i.e.,
$J_{a}^{\,\mathrm{ss}}$ is divergenceless. Taking the divergence of Eq. (69)
leads to the following partial differential equation for $v_{a}$,

$\displaystyle\partial^{\vphantom{\prime}}_{a}v_{a}-\mu^{\vphantom{\prime}}_{a}\,v_{a}\,=\,0\,,~{}~{}$
(70)

which is compatible with Eq. (67) Kwon and Ao (2011). Thus, Eq. (67) is most
the general solution to Eq. (70), except when $q_{a}$ lives on a topologically
nontrivial manifold, such as the torus. This is explained in App. A. Restoring
the noise, we have

$W\,=\,-\frac{1}{2}\partial^{\vphantom{\prime}}_{a}Q_{ab}\left(\partial^{\vphantom{\prime}}_{b}-\mu^{\vphantom{\prime}}_{b}\right)+\frac{1}{2}\partial^{\vphantom{\prime}}_{a}v_{a}\,,~{}~{}$
(71)

where it is instructive to compare to Eq. (37).

| Dissipationless terms | Dissipative terms
---|---|---
Fokker-Planck operator ($W$): |  $\partial_{a}v_{a}$ | $-\frac{1}{2}\partial_{a}Q_{ab}(\partial_{b}+\mu_{b})$
Constraint on Itô “force” ($f_{a}$): |  $(\partial^{\vphantom{\prime}}_{a}v_{a})-\mu^{\vphantom{\prime}}_{a}v_{a}=0$ | $g_{a}=-\frac{1}{2}Q_{ab}\mu^{\vphantom{\prime}}_{b}+\frac{1}{2}\partial^{\vphantom{\prime}}_{b}Q_{ab}$
Itô Langevin equation: |  $\partial^{\vphantom{\prime}}_{t}q_{a}=v_{a}+\xi_{a}(t)$ | $\partial^{\vphantom{\prime}}_{t}q_{a}=g_{a}+\xi_{a}(t)$
Relation to noise ($\xi_{a}$): | N/A | $\expectationvalue{\xi_{a}(t)\xi_{b}(t^{\prime})}=Q_{ab}\delta(t-t^{\prime})$
Table 1: Summary of constraints from stationarity of $\Phi$ on the
dissipationless and dissipative parts of the Fokker-Planck generator $W$ (53)
and the corresponding Langevin equation (31) in the Itô formulation.

###  2.5 Entropy and the second law of thermodynamics

In dissipative theories of _thermal_ systems, the existence of well-defined
entropy currents—and an associated _second law of thermodynamics_ —lead to
powerful constraints on the resulting EFTs Glorioso and Liu (2016). Thus, it
is natural to ask whether the EFT formalism we have developed for _generic_
stochastic dynamics—which may relax to _nonequilibrium_ stationary
distributions $P^{\rm ss}$ (41)—admits a notion of entropy $S(\mathbf{q},t)$,
and a second law of thermodynamics,

$\displaystyle\derivative{S(\mathbf{q},t)}{t}\,\geq\,0\,.~{}~{}$ (72)

In fact, there is a significant body of literature (see, e.g., Schnakenberg
(1976); Tomé and de Oliveira (2012)) concerning the generalization of entropy
(currents)—and the second law (72)—to generic, nonequilibrium stochastic
systems (e.g., with nonthermal stationary states). It is also natural to
expect that, if such notions exist, they should lead to useful constraints on
nonequilibrium EFTs, including those describing _active_ matter. The present
discussion is further motivated by the fact that, in the literature, entropy
production is often used as a measure of the extent to which a system is
“active” Tomé and de Oliveira (2012).

There is a formal notion of entropy density $s[P(\mathbf{q},t)]$ which leads
to a second law of thermodynamics (72) in our EFT framework. Moreover, when
$P(\mathbf{q},t)$ is close to the stationary distribution $P^{\rm
ss}\propto\exp(-\Phi(\mathbf{q}))$ (41), these notions of entropy and the
second law (72) closely resemble those of dissipative EFTs for thermal fluids
Glorioso and Liu (2016). For this reason—and others discussed below—we believe
this definition of entropy to be a logical starting point for generic,
stochastic EFTs. However, we caution that the definition of entropy we recover
may differ from that typically associated with entropy production in active
and nonequilibrium dynamics Schnakenberg (1976); Tomé and de Oliveira (2012).
Additionally, we restrict the derivation below to Gaussian noise (either
additive or multiplicative) for simplicity. We conclude this discussion by
commenting on the extension of this notion of entropy for nonequilibrium EFTs
beyond Gaussian noise.

We can work out the “correct” choice of the entropy $D[P]$ for a probability
distribution $P(\mathbf{q},t)$ (33) in a pedagogical way. The most important
property that $D[P]$ must satisfy is that $D_{\rm max}=D[P^{\rm ss}]$—i.e.,
that the stationary distribution $P^{\rm ss}$ (41) _uniquely_ maximizes the
entropy $D$. A natural guess for the entropy density $s(\mathbf{q},t)$ for the
probability distribution $P(\mathbf{q},t)$ (33) is the standard one
$s(\mathbf{q},t)=-P(\mathbf{q},t)\,\log P(\mathbf{q},t)$, such that the global
entropy

$D_{\rm Shannon}(t)\,=\,-\int\mathrm{d}\mathbf{q}\,P(\mathbf{q},t)\,\log
P(\mathbf{q},t)\,,~{}~{}$ (73)

is the _Shannon entropy_ for $P(\mathbf{q},t)$. However, it is straightforward
to see that this is not the correct choice. Importantly, the Shannon entropy
(73) for the _uniform distribution_ $P^{\rm uni}={\cal
V}^{-1}\int\mathrm{d}\mathbf{q}$ (43) is $D_{\rm uni}=\log{\cal V}$, where
${\cal V}=\int\mathrm{d}\mathbf{q}$. Meanwhile, the Shannon entropy (73) for
the stationary distribution $P^{\rm ss}={\cal
Z}^{-1}\,\int\mathrm{d}\mathbf{q}\,\exp(-\Phi(\mathbf{q}))$ (41)—where ${\cal
Z}=\int\mathrm{d}\mathbf{q}\,\exp(-\Phi(\mathbf{q}))$ is analogous to a
partition function for $P^{\rm ss}$—is given by

$\displaystyle D_{\rm Shannon}\left[P^{\rm ss}\right]\,$
$\displaystyle=-\int\mathrm{d}\mathbf{q}\frac{e^{-\Phi(\mathbf{q})}}{\cal
Z}\,\log\frac{e^{-\Phi(\mathbf{q})}}{\cal Z}\,=\,\log{\cal Z}+\frac{1}{\cal
Z}\int\mathrm{d}\mathbf{q}\,\Phi(\mathbf{q})\,e^{-\Phi(\mathbf{q})}\,\leq\,D_{\rm
Shannon}\left[P^{\rm uni}\right]\,,~{}~{}$ (74)

where the final relation follows from Gibbs’ inequality, which says that
$-\int\mathrm{d}\mathbf{q}\,P^{\rm ss}(\mathbf{q})\log P^{\rm
ss}(\mathbf{q})\leq-\int\mathrm{d}\mathbf{q}\,P^{\rm ss}(\mathbf{q})\log
P^{\rm uni}=\log{\cal V}$. In other words, Gibbs’ inequality guarantees that
the Shannon entropy (73) for the stationary distribution $P^{\rm ss}$ (41)
saturates $\log{\cal V}$ (i.e., is maximal) if and only if $P^{\rm ss}$ is
uniform. Hence, the Shannon entropy (73) is only a valid definition of the
entropy for stochastic EFTs in which the stationary distribution $P^{\rm ss}$
(41) is uniform (43), which is not generally the case for the systems of
interest (e.g., active matter).

For generic stationary distributions $P^{\rm ss}$ (41), we require a
functional of $D[P(\mathbf{q},t)]$ that is maximal when
$P(\mathbf{q},t)=P^{\rm ss}$ . This suggests that the correct entropy $D[P]$
should be of the form of a _relative entropy_ of the true probability
distribution $P(\mathbf{q},t)$ and the stationary distribution $P^{\rm ss}$
(41). A natural choice—which follows from Gibbs’ inequality—is the Kullback-
Leibler (KL) divergence (or relative entropy), $D_{\rm KL}\left[P^{\rm
ss}(\mathbf{q})\middle\|P(\mathbf{q},t)\right]=\int\mathrm{d}\mathbf{q}\,P^{\rm
ss}(\mathbf{q})\,\log\left[\,P^{\rm
ss}(\mathbf{q})/P(\mathbf{q},t)\,\right]\leq 0$. However, this shows that the
KL divergence is _minimized_ when $P(\mathbf{q},t)=P^{\rm ss}$, suggesting
that the correct notion of entropy in generic stochastic EFTs is given by
_minus_ the KL divergence,

$D\left[P(\mathbf{q},t)\right]\,\equiv\,\int\mathrm{d}\mathbf{q}\,\mathrm{e}^{-\Phi(\mathbf{q})}\,\log\frac{P(\mathbf{q},t)}{\mathrm{e}^{-\Phi(\mathbf{q})}}\,,~{}~{}$
(75)

where we have ignored the normalization of $P^{\rm ss}$ and $P(\mathbf{q},t)$,
since this merely amounts to an unimportant rescaling and overall constant
shift of $D$ (75). We now explore this quantity in more detail.

First, we prove that a second law of thermodynamics (72) holds for $D(t)$
(75). Recall that, by Gibbs’ inequality, $D(t)$ is maximal only when
$P\propto\exp(-\Phi)$. For convenience, we write $P(\mathbf{q},t)$ in the form

$P(\mathbf{q},t)\,=\,\mathrm{e}^{-\Phi(\mathbf{q})-\Psi(\mathbf{q},t)}\,,~{}~{}$
(76)

where $\Psi$ is small when $P$ is close to the stationary distribution $P^{\rm
ss}$ (41). We then find that

$\displaystyle\derivative{D}{t}\,$
$\displaystyle=\,\int\mathrm{d}\mathbf{q}\,\mathrm{e}^{-\Phi}\,\derivative{\,}{t}\log
P\,=\,\int\mathrm{d}\mathbf{q}\,\mathrm{e}^{-\Phi}\,\frac{1}{P}\,\derivative{P}{t}\,=\,\int\mathrm{d}\mathbf{q}\,\mathrm{e}^{-\Phi}\,\mathrm{e}^{\Phi+\Psi}\,\left(-W\,P\right)$
$\displaystyle=\,\frac{1}{2}\int\mathrm{d}\mathbf{q}\,\mathrm{e}^{\Psi}\,\partial^{\vphantom{\prime}}_{a}\,\mathcal{Q}_{ab}\,\left(\partial^{\vphantom{\prime}}_{b}+\mu^{\vphantom{\prime}}_{b}\right)\mathrm{e}^{-\Phi-\Psi}\,=\,-\frac{1}{2}\int\mathrm{d}\mathbf{q}\,\mathrm{e}^{\Psi}\,\partial^{\vphantom{\prime}}_{a}\,\mathcal{Q}_{ab}\,\mathrm{e}^{-\Phi-\Psi}\,\left(\partial^{\vphantom{\prime}}_{b}\Psi\right)$
$\displaystyle=\,\frac{1}{2}\int\mathrm{d}\mathbf{q}\,\left(\partial^{\vphantom{\prime}}_{a}\Psi\right)\,\mathrm{e}^{\Psi}\,\mathcal{Q}_{ab}\,\mathrm{e}^{-\Phi-\Psi}\,\left(\partial^{\vphantom{\prime}}_{b}\Psi\right)\,,~{}~{}$
(77)

where, in the last line, we used integration by parts and the fact that
$\mathrm{e}^{\Psi}\mathcal{Q}_{ab}\,\mathrm{e}^{-\Phi-\Psi}\left(\partial^{\vphantom{\prime}}_{*}{b}\Psi\right)$
vanishes at the boundary of phase space. For simple noise distributions—in
which $\mathcal{Q}_{ab}$ depends only on $\mathbf{q}$ and not
${\bf\it\partial}$—we observe that

$\derivative{D}{t}\,\to\,\frac{1}{2}\int\mathrm{d}\mathbf{q}\,(\partial^{\vphantom{\prime}}_{a}\psi)\,\mathcal{Q}_{ab}\,\mathrm{e}^{-\Phi}\,(\partial^{\vphantom{\prime}}_{b}\psi)\,\geq\,0\,,~{}~{}$
(78)

by the fact that $\mathcal{Q}_{ab}\sim Q_{ab}$ (57) is positive semidefinite.
Thus, if we define $D$ (75) to be the entropy of the probability distribution
$P(\mathbf{q},t)$ for a system that relaxes to the stationary distribution
$P^{\rm ss}(\mathbf{q})=\exp(-\Phi(\mathbf{q}))$ (41), and for noise
distribution in which $\mathcal{Q}$ (57) contains no derivatives, we obtain a
second law of thermodynamics, captured by Eq. (78). This fact is well
understood in, e.g., the statistics literature.

For non-Gaussian noise, which contributes higher-derivative corrections to the
Fokker-Planck generator $W$ (53) via Eq. (65), it is no longer obvious that
$D(t)$ (75) must be a nondecreasing function of $t$. However, $D(t)$ should
increase due to the data-processing inequality Cover and Thomas (2012). This
suggests that our generalized notion of a second law of thermodynamics
provides nontrivial constraints on any effective theory of non-Gaussian
stochastic dynamics, which cannot be deduced by simply writing down all
invariant building blocks in a derivative expansion in $W$. It would be
interesting to more systematically analyze these constraints, but we expect
that they are related to the fact that in the presence of non-Gaussian noise,
the moments of the noise distribution are not independent and must be
compatible with the Cauchy-Schwarz inequality. It would be interesting to
explicitly check this by using the relations between microscopic noise and
higher-derivative terms in the FPE, following Ref. 60.

The formal notion of entropy that we have defined depends _nonlinearly_ on the
solution to the Fokker-Planck equation (34). Thus, it is not defined for
_single_ stochastic trajectories—in fact, it is not possible for such an
entropy to be well defined. For example, in the overdamped oscillator
discussed in Sec. 1, such an entropy would have to be maximal when $x=0$, but
a system initialized with $x=0$ can reach a state with $x\neq 0$, meaning that
such an entropy function would decrease on this trajectory. To define a
suitable entropy function $S(q_{a})$ on phase space, the best one can hope for
is that this function is nondecreasing _on average_. For linear stochastic
processes, such an entropy function always exists (see App. B.1 for details).
For nonlinear processes, we have not systematically found such a function, but
we anticipate that combining the construction above with the arguments of Ref.
61 could imply the existence of such an entropy function in certain models
(e.g., in hydrodynamic systems).

###  2.6 Symmetries and conservation laws

Having developed the basic EFT formalism for stochastic dynamics (34) in terms
of the stationary distribution $\Phi$ (41), we now discuss the role of
symmetries and conservation laws. As we will see, even though symmetries may
lead to nonunique stationary distributions (e.g., depending on the “symmetry
sector” to which the conserved charge belongs), they are elegantly imposed
within our framework, using an analogue of Noether’s theorem for stochastic
dynamics in Eq. (80). Note that there are two subtly different ways by which
symmetry can enter our EFT formalism, corresponding to symmetries of the
dynamics versus symmetries of the stationary distribution $\Phi$ (41). The
former case admits two types of conserved quantities, corresponding to
“strong” and “weak” symmetries: A _strong_ symmetry holds for each individual,
random (stochastic) trajectory, while a _weak_ symmetry only holds upon
averaging over trajectories. However, there also exist situations where a
symmetry of the stationary distribution $P^{\rm ss}\propto\exp(\Phi)$ is not a
symmetry of the dynamics (even on average). We address each case in turn, and
detail how such symmetries constrain the evolution operator $W$ (53).

####  2.6.1 Strong and weak symmetries of the dynamics

We first consider the case of _strong_ symmetries, in which a quantity
$F(\mathbf{q})$ is conserved (i.e., $\derivative*{F}{t}=0$) on _all_
trajectories, and derive the consequences of such a conservation law on $W$
(53). Since $F$ is constant in time by construction, multiplying and dividing
$W$ by (any function of) thereof should not change $W$. In particular,

$\displaystyle\mathrm{e}^{-F}\,W\,\mathrm{e}^{F}\,=\,W\,,~{}~{}$ (79)

must hold for any such $F$. Since $W$ is generically a differential operator,
we must account for the action of $W$ on the left-hand side of Eq. (79) on the
function $\exp(F)$ to its right. We then find that, for $F$ to be conserved,
$W$ must be invariant under the following shift of the differential operator
$\partial^{\vphantom{\prime}}_{*}{a}$,

$\displaystyle
W(\partial^{\vphantom{\prime}}_{a},q_{a})\,=\,W(\partial^{\vphantom{\prime}}_{a}+[\partialderivative*{F}{q_{a}}],q_{a})\,,~{}~{}$
(80)

where the quantity $[\partialderivative*{F}{q_{a}}]$ in square brackets is a
function—not a differential operator. Thus, the form of $W$ (53) is further
constrained by _each_ conserved quantity $F$; this also extends to the case of
vector conserved quantities, in which each component of $F$ is conserved.
Examples of strong symmetries appear in Secs. 3.5 and 6.

We now consider the case of _weak_ symmetries, in which case, instead of
$\derivative*{F}{t}=0$, we instead have

$\displaystyle\derivative{t}\expectationvalue{F(t)}\,=\,\derivative{t}\int\differential{\mathbf{q}}F(\mathbf{q})P(\mathbf{q},t)\,=\,0\,,~{}~{}$
(81)

where $P(\mathbf{q},t)$ is the probability distribution at time $t$, and
$\expectationvalue{F}$ denotes the average of $F$ over random trajectories
(since $F$ is only a conserved quantity on average, and not along individual
trajectories). Note that $F(\mathbf{q})$ has no explicit time dependence (it
depends on time only implicitly via its dependence on $\mathbf{q}$); as a
result,

$\displaystyle
0\,=\,\int\differential{\mathbf{q}}\,F(\mathbf{q})\,\partial^{\vphantom{\prime}}_{t}P(\mathbf{q})\,=\,\int\differential{\mathbf{q}}\,F(\mathbf{q})\,[-W\,P(\mathbf{q})]\,=\,\int\differential{\mathbf{q}}\,P(\mathbf{q})[-W^{T}F(\mathbf{q})]\,,~{}~{}$
(82)

and, since $W$ (53) always has a derivative operator on the left (to preserve
normalization in the FPE), we find

$\displaystyle
W^{T}F(\mathbf{q})\,=\,\frac{1}{2}\,(\partial^{\vphantom{\prime}}_{a}-\mu^{\vphantom{\prime}}_{a})\mathcal{Q}_{ba}\,\partial^{\vphantom{\prime}}_{b}F\,=\,0\,,~{}~{}$
(83)

so that, in many cases, we may simply enforce the condition. Thus, both strong
and weak symmetries in stochastic dynamics may be enforced via constraints on
$W$ (53).

####  2.6.2 Symmetries of the stationary distribution

Alternatively, there may also exist “symmetries” of the stationary
distribution function $\Phi$ (41) that do _not_ correspond to quantities $F$
that are conserved by the dynamics, even on average. Nonetheless, there
remains a sense in which a particular invariance of $\Phi$ corresponds to a
genuine symmetry; moreover, these nondynamical symmetries are relevant to our
EFT formalism and have implications for the time-evolution operator $W$ (53).

In particular, such symmetries directly facilitate the construction of certain
dissipationless terms in $W$. For instance, suppose that a given stationary
distribution $\Phi[\mathbf{q}]$ (41) is invariant under coordinate shifts

$q_{a}\,\to\,q_{a}+\epsilon F_{a}(\mathbf{q})\,,~{}~{}$ (84)

which implies that the generator $\mathbf{F}$ and chemical potential
${\bf\it\mu}$ are orthogonal, i.e.,

$\mathbf{F}\cdot{\bf\it\mu}\,=\,F_{a}\,\mu^{\vphantom{\prime}}_{a}\,=\,\partialderivative{q_{a}}{\epsilon}\,\partialderivative{\Phi}{q_{a}}\,=\,\derivative{\Phi}{\epsilon}\,=\,0\,,~{}~{}$
(85)

and we also take $F$ to be divergenceless (i.e.,
$\partial^{\vphantom{\prime}}_{*}{a}F_{a}=0$). In that case, a dissipationless
term

$\displaystyle v_{a}\,\sim\,F_{a}\,,~{}~{}$ (86)

as defined in Eq. (67) automatically satisfies the constraint of Eq. (70) on
all dissipationless contributions to the Fokker-Planck force $\mathbf{f}$
(60), and is therefore allowed. More generally, dissipationless terms
$v^{\,}_{a}$ are always allowed to move the system along _level sets_ (84) of
the stationary distribution function $\Phi$ (41).

A particularly important application of this idea arises when the distribution
$\Phi$ is invariant under some group $G$. This provides for the inclusion of
nontrivial (and quite exotic) dissipationless terms, even though there may be
no corresponding dynamically conserved quantity. We treat this more general
case in Sec. 5.1, where we also discuss the resulting possibility of
spontaneous breaking of generalized time-reversal symmetry gT.

We note that symmetries of the distribution function $\Phi$ (41) also offer a
means by which to generate additional terms in $W$ (53), starting from some
“minimal” choice of $W$ compatible with $\Phi$. To see this, note that a
continuous symmetry of $\Phi$ generated via Eq. (84) acts on a generic
probability distribution $P$ (33) via

$\displaystyle P\,\to\,P^{\prime}\,=\,\mathrm{e}^{\epsilon
F_{a}\partial^{\vphantom{\prime}}_{a}}P\,,~{}~{}$ (87)

where invariance of $\Phi$ (41) under the transformation of Eq. (84) generated
by $\mathbf{F}$ requires that $\Phi^{\prime}=\Phi$ (which recovers from taking
$P=P^{\vphantom{\prime}}_{\rm ss}$ and $P^{\prime}=P_{\rm ss}^{\prime}$).
Applying $\exp(\epsilon F_{a}\partial^{\vphantom{\prime}}_{*}{a})$ to both
sides of $\partial^{\vphantom{\prime}}_{*}{t}P=-WP$ (36) leads to

$\displaystyle\partial^{\vphantom{\prime}}_{t}P\,=\,-WP\implies\mathrm{e}^{-\epsilon
F_{a}\partial^{\vphantom{\prime}}_{*}{a}}\partial^{\vphantom{\prime}}_{t}P^{\prime}\,=\,-W\mathrm{e}^{-\epsilon
F_{a}\partial^{\vphantom{\prime}}_{*}{a}}P^{\prime}~{}~{}\implies~{}~{}\partial^{\vphantom{\prime}}_{t}P^{\prime}\,=\,-\mathrm{e}^{+\epsilon
F_{a}\partial^{\vphantom{\prime}}_{*}{a}}W\mathrm{e}^{-\epsilon
F_{a}\partial^{\vphantom{\prime}}_{*}{a}}P^{\prime}\,\equiv\,-W^{\prime}P^{\prime}\,,~{}~{}$
(88)

and, since $P_{\rm ss}^{\vphantom{\prime}}=P_{\rm ss}^{\prime}$ (41), we have
found that $W^{\prime}\equiv\mathrm{e}^{\epsilon
F_{a}\partial^{\vphantom{\prime}}_{*}{a}}W\mathrm{e}^{-\epsilon
F_{a}\partial^{\vphantom{\prime}}_{*}{a}}$ is a valid Fokker-Planck generator
with the _same_ stationary distribution $P_{\rm ss}=\exp(\Phi)$ (41) as $W$!
In particular, for infinitesimal transformations, we find

$\displaystyle
W^{\prime}\,=\,W+\epsilon\left[F_{a}\partial^{\vphantom{\prime}}_{a},W\right]\,,~{}~{}$
(89)

where $[A,B]=AB-BA$ is the commutator. The shift captured by Eq. (89) shows
how additional terms (consistent with the same stationary distribution) may be
added to a given Fokker-Planck generator $W$.

###  2.7 Connection to the Martin-Siggia-Rose path integral

We now connect the foregoing operator formalism to the path-integral
description of stochastic Langevin dynamics (31) and FPEs (34) due to Martin,
Siggia, and Rose (MSR) Martin  _et al._ (1973). The latter affords a
Lagrangian description of dissipative systems, so that familiar path-integral
methods can be used to study active and dissipative dynamics not only of
point-particle systems, but continuum systems (i.e., field theories). As with
the Fokker-Planck operator formalism developed thus far, the mathematical
implementation and physical role of the time-reversal operation T are deeply
important to the MSR description of the stochastic EFT. The MSR implementation
of T appears—at least superficially—quite similar to that of the Kubo-Martin-
Schwinger (KMS) invariance of systems both in thermal equilibrium Araki
(1978); Martin and Schwinger (1959) and beyond Guo  _et al._ (2022); Qi  _et
al._ (2023). However, we note that there are several subtleties; accordingly,
we view the corresponding $\mathbb{Z}_{2}$ operation as the path-integral
implementation of T.

We also note that T is often cumbersome to implement in the MSR framework—at
least in full generality—due to subtleties related to the regularization
(i.e., Itô versus Stratonovich) of SDEs such as Eq. (31). We relegate the
appertaining technical details to App. C; other subtleties related to the
regularization are discussed in Ref. 53. In most of this work, we instead
formulate EFTs in terms of the FPE operator $W$ (53), in which T is
straightforward. However, we note that a direct connection between the FPE and
MSR implementations of T exists in certain simple limits. Accordingly—and
owing to the utility of the MSR formalism in describing continuum models, and
its relevance to recent developments concerning active and dissipative
systems—we now review the MSR formalism, the implementation of T therein, and
its relation to the operator-based Fokker-Planck approach discussed
previously.

####  2.7.1 Details of the MSR formalism

The MSR path integral reproduces the transition probability (39) according to

$\displaystyle\mathbb{P}[\mathbf{q}_{i}(0)\,\to\,\mathbf{q}_{f}(t);\mathcal{H}]\,=\,\left\langle*\middle|\mathbf{q}_{f}(t)\middle|\mathrm{e}^{-\int_{0}^{t}\mathrm{d}t^{\prime}\,W(\mathbf{q}(t^{\prime}))}\right\rangle{\mathbf{q}_{i}(0)}\,=\,\int\limits_{\mathbf{q}(0)=\mathbf{q}_{i}}^{\mathbf{q}(t)=\mathbf{q}_{f}}\mathrm{D}\mathbf{q}\,\mathrm{D}{\bf\it\pi}~{}\exp\left[\mathrm{i}\,\int\limits^{t}_{0}\mathrm{d}t^{\prime}\,(\pi_{a}\partial^{\vphantom{\prime}}_{t}q_{a}-\mathcal{H}(\mathbf{q},{\bf\it\pi}))\right]\,,~{}~{}$
(90)

where the conjugate momenta
$\pi_{a}\sim-\mathrm{i}\partial^{\vphantom{\prime}}_{*}{a}$ and Hamiltonian
$\mathcal{H}\sim-\mathrm{i}\,W$ are defined in Eq. (54) for FPE dynamics. As a
reminder, $\cal H$ is generally _not_ the microscopic Hamiltonian (which
appears in Sec. 3), and the coordinates $\mathbf{q}$ include all phase-space
variables describing the dynamical system—for $N$ point particles in $d$
spatial dimensions, $\mathbf{q}$ includes all $dN$ canonical positions $x_{a}$
_and_ momenta $p_{b}$. The fields $\pi_{a}$ are the conjugate momenta to
$q_{a}$ in the sense of the EFT formalism, and are often easier to work with
than $\partial^{\vphantom{\prime}}_{*}{a}$ in the context of continuum
dynamics.

Before explaining the details of the MSR formalism and its connection to the
FPE operator language, we first introduce a slightly modified notation to
incorporate _multiplicative_ noise. A stochastic process with random noise
source $\xi_{a}(t)$—where $\expectationvalue{\xi_{a}(t)}=0$ and
$\expectationvalue{\,\xi_{a}(t)\,\xi_{b}(t^{\prime})\,}=\Delta\,\delta^{\,}_{a,b}\,\delta(t-t^{\prime})$—can
be extended to account for multiplicative noise via
$\eta_{a}(\mathbf{q},t)=M_{ab}(\mathbf{q})\,\xi_{b}(t)$. Essentially, the bare
noise $\xi_{b}$ is modulated by $M_{ab}(\mathbf{q})$, so that

$Q_{ab}(\mathbf{q})\,=\,\int\mathrm{d}t^{\prime}\,\expectationvalue{\eta_{a}(\mathbf{q},t)\eta_{b}(\mathbf{q},t^{\prime})}\,=\,M_{ac}(\mathbf{q})\,M_{bd}(\mathbf{q})\int\mathrm{d}t^{\prime}\,\expectationvalue{\xi_{c}(t)\xi_{d}(t^{\prime})}\,=\,M_{ac}(\mathbf{q})\,M_{bc}(\mathbf{q})\,,~{}~{}$
(91)

is the symmetric, positive-semidefinite noise matrix (35). In the case
$Q_{ab}(\mathbf{q})=\Delta\,\delta^{\,}_{a,b}$, the multiplicative noise
reduces to Gaussian additive noise, described by the noise matrix (35) with
variance $\Delta$.

Now, consider a stochastic process captured by a Langevin SDE (31) and subject
to multiplicative noise $\eta_{a}$ with noise matrix $Q$ (91). The
corresponding MSR path integral (with boundary conditions suppressed) is given
by

$\displaystyle
Z[{\bf\it\xi}]\,=\,\int\mathrm{D}\mathbf{q}\,\mathcal{J}[\mathbf{q}]~{}\delta\left(\partial^{\vphantom{\prime}}_{t}q_{a}-f_{a}(\mathbf{q})-M_{ab}(\mathbf{q})\,\xi_{b}(t)\right)\,=\,1\,,~{}~{}$
(92)

where $\mathcal{J}[\mathbf{q}]$ is the Jacobian of the operators inside the
delta function (see App. C for technical details), and ${\bf\it\xi}$ appears
in Eq. (92) because $Z$ is not yet averaged over noise. Note that Eq. (92) is
equal to one because the integral is indefinite, and realizes the transition
amplitude of Eq. (90) when appropriate boundary conditions are imposed. For
convenience, we consider the Itô regularization, for which
$\mathcal{J}[\mathbf{q}]=1$. The noise-averaged partition function (92) is

$\displaystyle Z\,$
$\displaystyle=\,\int\mathrm{D}{\bf\it\xi}~{}Z[{\bf\it\xi}]\,\exp\left(-\frac{1}{2}\int\mathrm{d}t\,\xi_{a}^{2}\right)$
$\displaystyle=\,\int\mathrm{D}{\bf\it\xi}\,\mathrm{D}\mathbf{q}\,\mathrm{D}{\bf\it\pi}~{}\exp\left(\mathrm{i}\int\mathrm{d}t\,\pi_{a}(\partial^{\vphantom{\prime}}_{t}q_{a}-f_{a}(\mathbf{q})-M_{ab}(\mathbf{q})\,\xi_{b})\right)~{}\exp\left(-\frac{1}{2}\int\mathrm{d}t\,\xi_{a}^{2}\right)\,,~{}~{}$
(93) and integrating out the noise (i.e., evaluating the integral over
${\bf\it\xi}$) leads to $\displaystyle Z\,$
$\displaystyle=\,\int\mathrm{D}\mathbf{q}\,\mathrm{D}{\bf\it\pi}~{}\exp\left(\mathrm{i}\int\mathrm{d}t\,(\pi_{a}\,\partial^{\vphantom{\prime}}_{t}q_{a}-\pi_{a}f_{a}(\mathbf{q})+\mathrm{i}\frac{1}{2}\pi_{a}Q_{ab}\pi_{b})\right)$
$\displaystyle=\,\int\mathrm{D}\mathbf{q}\,\mathrm{D}{\bf\it\pi}~{}\exp\left(\mathrm{i}\int\mathrm{d}t\,L(\mathbf{q},{\bf\it\pi})\right)\equiv\int\mathrm{D}\mathbf{q}\,\mathrm{D}{\bf\it\pi}~{}\exp\left(\mathrm{i}\int\mathrm{d}t\,(\pi_{a}\partial^{\vphantom{\prime}}_{t}q_{a}-\mathcal{H}(\mathbf{q},{\bf\it\pi}))\right)\,,~{}~{}$
(94)

which implicitly defines the MSR _Lagrangian_

$L(\mathbf{q},{\bf\it\pi})\,=\,\pi_{a}\partial^{\vphantom{\prime}}_{t}q_{a}-\mathcal{H}(\mathbf{q},{\bf\it\pi})\,=\,\pi_{a}\partial^{\vphantom{\prime}}_{t}q_{a}-\pi_{a}f_{a}(\mathbf{q})+\frac{\mathrm{i}}{2}\pi_{a}\,Q_{ab}(\mathbf{q})\,\pi_{b}\,,~{}~{}$
(95)

where we have written both Eqs. (94) and (95) such that replacing
$\pi_{a}\to-\mathrm{i}\,\partial^{\vphantom{\prime}}_{*}{a}$ (54) recovers
$W=\mathrm{i}\,\mathcal{H}$ (53).

####  2.7.2 Time-reversal symmetry

In developing the operator-based Wilsonian EFT for stochastic dynamics, we
have emphasized the simple nature of the “time-reversal” transformation T
acting on the Fokker-Planck generator $W$ (46). There is also a natural
definition of T in the MSR formulation of the same stochastic effective
theories. Importantly, the latter implementation of T is (_i_) generally more
convenient in the context of continuum systems described by field theories and
(_ii_) closely connected to the KMS transformation Araki (1978); Martin and
Schwinger (1959), around which the thermal EFTs of, e.g., hydrodynamic systems
are organized Glorioso and Liu (2018); Guo  _et al._ (2022); Qi  _et al._
(2023); Mullins  _et al._ (2023). We first present the path-integral
implementation of T (97) in a simple (yet physically relevant) limit in which
it is directly related to its FPE-operator analogue (46). We then justify in
the path-integral framework that the corresponding transformation (97) is the
correct MSR implementation of T. Finally, we comment on the connection between
these two implementations of T beyond the simple limit above.

Consider a T-invariant stochastic model (i.e., with ${\cal
W}=\widetilde{W}\implies{\cal H}=\widetilde{\cal H}$) subject to a Gaussian
noise source with constant variance matrix $Q_{ab}$ (35). Ignoring for the
moment the dissipationless term $v_{a}$ (67) in $\mathbf{f}$ (60), the MSR
Lagrangian for such a stochastic system is given by

$L\,=\,\pi_{a}\partial^{\vphantom{\prime}}_{t}q_{a}+\frac{\mathrm{i}}{2}\pi_{a}\,\Delta_{ab}\,\left(\pi_{b}-\mathrm{i}\mu^{\vphantom{\prime}}_{b}\right)\,,~{}~{}$
(96)

which follows from Eqs. (35), (60), and (95), where the elements
$Q_{ab}=\Delta_{ab}$ (35) are independent of both $\mathbf{q}$ and
${\bf\it\partial}$. We consider the inclusion of dissipationless terms—along
with multiplicative and higher-derivative noise—momentarily.

The corresponding _MSR implementation of T_ in the limit of Gaussian noise and
T-even $W$ (53) is given by

$\displaystyle t\,\to-t~{},~{}~{}q_{a}\to\pm
q_{a}~{},~{}~{}\text{and}~{}~{}\pi_{a}\to\mp\left(\pi_{a}-\mathrm{i}\mu^{\vphantom{\prime}}_{a}\right)\,,~{}~{}$
(97)

where $\pm$ denotes whether $q_{a}$ is even ($+$) or odd ($-$) under T (i.e.,
$x\to x$ and $p\to-p$), and we find that
$\mu^{\vphantom{\prime}}_{*}{a}\to\pm\mu^{\vphantom{\prime}}_{*}{a}$ under T
(97). We note that the transformation in Eq. (97) is mathematically equivalent
to the implementation of KMS in the path-integral (EFT) description of thermal
dissipative systems Araki (1978); Martin and Schwinger (1959); Glorioso and
Liu (2018); Qi  _et al._ (2023); Guo  _et al._ (2022).

By applying the transformation T (97) to
$\mathbb{P}[\mathbf{q}_{i}(0)\to\mathbf{q}_{f}(t)]$ (90), we can compare the
probability to go from configuration $\mathbf{q}_{i}$ to $\mathbf{q}_{f}$ in
time $t$ to the probability to go from $\mathbf{q}_{f}$ to $\mathbf{q}_{i}$ in
time $t$ under the time-reversed process (46),

$\displaystyle\widetilde{\mathbb{P}}\,[\mathbf{q}_{i}(0)\to\mathbf{q}_{f}(t);\mathcal{H}]\,$
$\displaystyle=\,\int_{\mathbf{q}(0)=\mathbf{q}_{f}}^{\mathbf{q}(t)=\mathbf{q}_{i}}\mathrm{D}\mathbf{q}\,\mathrm{D}{\bf\it\pi}~{}\exp\left(\mathrm{i}\int_{0}^{t}\mathrm{d}t^{\prime}\,\left[(\pi_{a}-\mathrm{i}\mu^{\vphantom{\prime}}_{a})\partial^{\vphantom{\prime}}_{t}q_{a}-\widetilde{\mathcal{H}}(\mathbf{q},{\bf\it\pi})\right]\right)\,,~{}~{}$
(98) where $\widetilde{\cal H}=-\mathrm{i}\widetilde{W}$ (46) is the time-
reversed MSR Hamiltonian. Using the standard chain rule, we note that
$\mu^{\vphantom{\prime}}_{*}{a}\partial^{\vphantom{\prime}}_{*}{t}q^{\,}_{a}=(\partialderivative*{\Phi}{q_{a}})\,\partial^{\vphantom{\prime}}_{*}{t}q_{a}=\derivative*{\Phi}{t}$,
so that the probability amplitude above becomes
$\displaystyle\widetilde{\mathbb{P}}\,[\mathbf{q}_{i}(0)\to\mathbf{q}_{f}(t);\mathcal{H}]\,$
$\displaystyle=\,\mathrm{e}^{\Phi(\mathbf{q}_{i})-\Phi(\mathbf{q}_{f})}\,\int_{\mathbf{q}(0)=\mathbf{q}_{f}}^{\mathbf{q}(t)=\mathbf{q}_{i}}\mathrm{D}\mathbf{q}\,\mathrm{D}{\bf\it\pi}~{}\exp\left(\mathrm{i}\int_{0}^{t}\mathrm{d}t^{\prime}\,\left[\pi_{a}\partial^{\vphantom{\prime}}_{t}q_{a}-\widetilde{\mathcal{H}}(q,\pi)\right]\right)$
$\displaystyle=\,\mathrm{e}^{\Phi(\mathbf{q}_{i})-\Phi(\mathbf{q}_{f})}~{}\mathbb{P}[\mathbf{q}_{f}(0)\to\mathbf{q}_{i}(t);\widetilde{\mathcal{H}}]\,,~{}~{}$
(99)

which is the probability to transition from $\mathbf{q}_{f}$ to
$\mathbf{q}_{i}$ in time $t$ under $\widetilde{\cal H}$ (46)—the time-reversed
partner to ${\cal H}=-\mathrm{i}W$—in analogy to Eq. (50). Note that we have
restricted to models in which ${\cal H}$ is invariant under T, and thus obeys
_detailed balance_ (17). When $Q_{ab}$ (35) corresponds to Gaussian noise
(i.e., contains no derivatives $\partial^{\vphantom{\prime}}_{*}{a}$), we find
that

$\displaystyle{\cal
H}\,=\,-\frac{\mathrm{i}}{2}\pi_{a}\,Q_{ab}\,\left(\pi_{a}-\mathrm{i}\,\mu^{\vphantom{\prime}}_{a}\right)\,$
$\displaystyle\to\,\widetilde{\cal
H}\,=-\frac{\mathrm{i}}{2}\left(\pi_{a}-\mathrm{i}\,\mu^{\vphantom{\prime}}_{a}\right)\,Q_{ab}\,\pi_{a}\,=\,\,{\cal
H},~{}~{}$ (100)

meaning that ${\cal H}=\widetilde{\cal H}$, as required. Note that the
transformation T (97)—and the connection to its FPE counterpart (46)—are
complicated in the presence of non-Gaussian noise, where the derivatives
$\partial^{\vphantom{\prime}}_{*}{a}$ in the expansion for ${\cal Q}_{ab}$
(65) spoil the derivations above. For such noise sources, the connection
between the two implementations of T is more elusive. However for Gaussian
noise, invariance under the transformation of Eq. (97) implies that

$W_{\rm MSR-
even}\,=-\frac{1}{2}\partial^{\vphantom{\prime}}_{a}\left(\partial^{\vphantom{\prime}}_{b}+\mu^{\vphantom{\prime}}_{b}\right)Q_{ab}\,=\,\mathrm{i}\,{\cal
H}_{\rm MSR-even}\,=\,\mathrm{i}\,\widetilde{\cal
H}\,=\,\widetilde{W}\,,~{}~{}$ (101)

in the FPE operator language, since the Fokker-Planck generator $W$ (53) is
straightforwardly related to the MSR Hamiltonian $\cal{H}$ (95) via Eq. (54)
when written in the Itô formalism (with all derivatives to the left).

In the presence of _multiplicative_ noise, Eq. (101) fails, as $Q_{ab}$
depends on $\mathbf{q}$. Moreover, even in the case of dissipation toward
equilibrium, the connection to thermal KMS Glorioso and Liu (2018) is
complicated in the presence of multiplicative noise. In App. C, we show that
Eq. (97) does _not_ preserve the Itô regularization444A similar issue is also
known to arise in standard quantum mechanical path integrals for particles for
which $H\neq p^{2}/2m+V(q)$ Kleinert (2009).; thus, the correct implementation
of T involves supplementing Eq. (97) with an appropriate change in the
regularization itself555In previous work Gao  _et al._ (2020), the KMS
transformation was implemented using a regularization stated in the
_frequency_ domain, in which case it is not straightforward to cast the
dynamics in terms of a master equation.. In App. C, in the case of
multiplicative Gaussian noise, we explain how to implement the path integral
(90) with a suitable regularization such that T (97) is manifestly a symmetry.
We relegate a study of more general, non-Gaussian noise terms to future work.

However, we also note that, from a field-theoretic _renormalization-group_
perspective, multiplicative noise is typically _irrelevant_. Thus, any
potential discrepancy between the MSR implementation of T (97) and both the
FPE analogue (46) and the KMS transformation for thermal systems Araki (1978);
Martin and Schwinger (1959) may ultimately be unimportant to the universal
properties and classification of active and dissipative systems.

We now consider more general models, which may include _dissipationless_
contributions $v_{a}$ (67) to the Fokker-Planck force $\mathbf{f}$ (60). In
this case, we adjust the (T-even) MSR Lagrangian (96) as

$L\,=\,L_{\rm MSR-
even}+\pi_{a}\,v_{a}\,=\,\pi_{a}\partial^{\vphantom{\prime}}_{t}q_{a}+\frac{\mathrm{i}}{2}\pi_{a}\,Q_{ab}(\pi_{b}-\mathrm{i}\mu^{\vphantom{\prime}}_{b})+\pi_{a}v_{a}\,,~{}~{}$
(102)

where $v_{a}$ (67) is generically a function of $\mathbf{q}$. Suppose that
$v_{a}\to\widetilde{v}_{a}$ under T, then for each label $a$, the
$\pi_{a}v_{a}$ term in the MSR Lagrangian (102) transforms as

$\pi_{a}v_{a}\,\to\,\mp(\pi_{a}-\mathrm{i}\mu^{\vphantom{\prime}}_{a})\widetilde{v}_{a}\,,~{}~{}$
(103)

and if $\widetilde{v}_{a}=v_{a}$, the $V$ term in Eq. (57)—corresponding to
$\pi_{a}v_{a}$ in $L$ (102)—flips sign under T, as expected. On the other
hand, if $\widetilde{v}_{a}=-v_{a}$, the $\pi_{a}v_{a}$ in $L$ (102) is
invariant under T (such dissipationless terms obey detailed balance, and are
compatible with Hamiltonian dynamics). We now observe an important
complication: The extra $\mathrm{i}\mu^{\vphantom{\prime}}_{*}{a}v_{a}$ term
in Eq. (103) that is “spawned” when we apply T (97) to $L$ (102). For the
Lagrangian $L$ (102) to transform under T into another valid MSR Lagrangian in
the Itô regularization, we must have $\mu^{\vphantom{\prime}}_{*}{a}v_{a}=0$
since all terms must have at least one power of $\pi_{a}$. However, this is
_not_ the constraint on T-odd terms derived in Eq. (70) unless
$\partial^{\vphantom{\prime}}_{*}{b}V_{ab}=0$. This is guaranteed so long as
$V_{ab}$ is constant, but may not hold in general. The origin of this
complication is the subtlety surrounding regularizations noted below Eq.
(101). The takeaway is that there are extra constraints on the dissipationless
terms $\pi_{a}v_{a}$ in the MSR Lagrangian due to stationarity: Dynamics
compatible with a stationary distribution $\Phi$ (41) have a time-reversed
evolution operator $\widetilde{W}$ (46) that is also a valid FPE generator;
however, if $\mu^{\vphantom{\prime}}_{*}{a}v_{a}\neq 0$, then the time-
reversed MSR Lagrangian does not generate valid Fokker-Planck dynamics, as all
terms in the MSR Lagrangian (95) must contain at least one power of $\pi_{a}$.

The subtleties discussed above concerning multiplicative noise and
dissipationless contributions arise when $\partial_{b}Q_{ab}$ or
$\partial_{b}V_{ab}$ are nonzero. However, for continuum field theories, such
as hydrodynamics, we show explicitly in App. D that it is possible to choose a
spatial discretization of $Q_{ab}$ and $V_{ab}$ such that their divergences
vanish. This discretization, which can be thought of as a regularization of
the model, leads to an MSR realization of time reversal (97) that is _exactly_
equivalent to the Fokker-Planck realization of T (46). The crucial assumption
is that the expansion in spatial gradients is treated perturbatively. This
observation further suggests that precision tests Delacretaz and Mishra (2023)
of nonlinear fluctuating hydrodynamics would _not_ detect any discrepancy
between reversibility (46) and KMS invariance Araki (1978); Martin and
Schwinger (1959); meaning that such discrepancies can only arise in theories
where higher-gradients are not perturbative.

####  2.7.3 Time-reversal symmetry without integrating out noise

The MSR framework also makes transparent how a noisy SDE (31) is
_microscopically_ T invariant, and thus obeys detailed balance. For
simplicity, consider additive Gaussian noise with
$Q_{ab}=\Delta\,\delta^{\,}_{a,b}$ (35), so that T (97) acts as

$t\,\to-t~{},~{}~{}q_{a}\,\to\pm
q_{a}~{},~{}~{}\pi_{a}\,\to\mp\left(\pi_{a}-\mathrm{i}\mu^{\vphantom{\prime}}_{a}\right)~{},~{}~{}\xi_{a}\,\to\mp\left(\xi_{a}-\Delta\mu^{\vphantom{\prime}}_{a}\right)\,,~{}~{}$
(104)

where $\pm$ refers to whether $q_{a}$ is T even ($+$) or T odd ($-$). It is
straightforward to verify that the MSR Lagrangian $L$ (93) prior to
integrating out the noise $\xi_{a}$ is invariant under the transformation T
(104).

In other words, Eq. (104) is the transformation in the path-integral EFT
corresponding to T (i.e., reversibility). Considering the damped harmonic
oscillator described by Eq. (4) of Sec. 1, from our perspective, it makes
sense to say that $\dot{x}=-\gamma x+\xi$, with $\gamma=\Delta\mu/2x$ as in
Eq. (6), is time-reversal symmetric (or T even). Under time reversal T, the
noise variable $\xi$ shifts so as to undo precisely the naïve sign change to
the dissipative term. This does not happen by chance, but is a direct
consequence of the existence of a stationary state.

##  3 Dissipative generalizations of Hamiltonian dynamics

Before we consider the application of the effective field theory (EFT)
framework of Sec. 2 to nonequilibrium _active_ matter, we first consider EFTs
for dissipative thermal systems. The formalism of Sec. 2 provides a simple
theoretical description of Hamiltonian systems in thermal equilibrium, and a
natural prescription for adding dissipation to any such model. Such systems
evolve toward a particular stationary distribution, corresponding to

$\Phi\,=\,\beta\,H\,,~{}~{}$ (105)

meaning that stationary distribution $\Phi$ is simply the Boltzmann
distribution for the underlying Hamiltonian $H$ at inverse temperature
$\beta$. Such a distribution naturally describes thermal equilibrium of a
system with its environment, where the Hamiltonian $H$ is also responsible for
time evolution. We now describe several simplifying aspects of Hamiltonian
dynamics, along with a natural procedure for coupling Hamiltonian systems to
dissipative baths. While the treatment of equilibrium systems herein largely
reproduces well-known results, it does so in a more systematic manner using a
simple algorithm compared to existing literature, and helps to build intuition
for the EFT formalism in anticipation of its later application to active
matter in the rest of the paper.

We first review of Hamiltonian mechanics in Sec. 3.1, using the EFT formalism
of Sec. 2. In particular, given a stationary state $\Phi=\beta H$ (105), we
relate the Hamiltonian $H$ to a Fokker-Planck generator $W$ compatible with
$\Phi$; we note that this does not require the use of the canonical phase-
space variables, and can incorporate symmetries of the dynamics and stationary
distribution $\Phi$. In Sec. 3.2, we prescribe how to add dissipation to
generic Hamiltonian systems (including those with particular symmetries) in
equilibrium with a thermal bath; this is remarkably straightforward in the EFT
framework of Sec. 2. Finally, in the remaining subsections, we survey a
variety dissipative generalizations of paradigmatic Hamiltonian theories,
including an example featuring nontrivial Poisson brackets (in Sec. 3.5).

###  3.1 Hamiltonian dynamics and Poisson brackets

Here we briefly review the important features of Hamiltonian dynamics in the
absence of dissipation (i.e., noise), and explain the application of the
effective theory formalism developed in Sec. 2 to this setting.

####  3.1.1 Hamiltonian mechanics with canonical phase space variables

Numerous important aspects of Hamiltonian mechanics are captured by the
_Poisson bracket_ $\\{\cdot,\cdot\\}$, which is defined for arbitrary
functions of the phase-space variables. We frequently refer to the simple
example of $N$ particles in $d$ spatial dimensions, described by $Nd$
positions $x_{a}$ and $Nd$ momenta $p_{a}$, which satisfy the canonical
Poisson brackets

$\displaystyle\quantity{x_{a},x_{b}}\,$
$\displaystyle=\,\quantity{p_{a},p_{b}}\,=\,0,$ (106a)
$\displaystyle\quantity{x_{a},p_{b}}\,$
$\displaystyle=-\quantity{p_{b},x_{a}}\,=\,\delta^{\,}_{a,b}\,,~{}~{}$ (106b)

where the label $a$ combines the particle and spatial-axis indices (with a
total of $Nd$ values). The Poisson bracket $\\{\cdot,\cdot\\}$ is defined for
_all_ classical dynamical systems, including those with noncanonical phase-
space variables; it also extends to generic functions of the phase-space
variables via the chain rule of calculus.

As a reminder, in the EFT formalism of Sec. 2, the coordinate vector
$\mathbf{q}$ contains _all_ phase-space variables. In the example above,
$\mathbf{q}$ is a length-$2Nd$ vector of all positions $x_{a}$ _and_ momenta
$p_{b}$, such that Hamilton’s equations may be written simply as
$\partial^{\vphantom{\prime}}_{*}{t}q_{a}=\quantity{q_{a},H}$, with $H$ the
Hamiltonian. For dissipationless Hamiltonian dynamics, we have

$W\,=\,T\,\partial^{\vphantom{\prime}}_{a}\quantity{q_{a},q_{b}}\mu^{\vphantom{\prime}}_{b}\,,~{}~{}$
(107)

where $T\equiv\beta^{-1}$ is the temperature (with $k_{B}=1$). Because the
stationary distribution $\Phi$ (105) is proportional to the Hamiltonian $H$,
the “chemical potentials” $\mu^{\vphantom{\prime}}_{*}{b}$ in Eq. (107) are
the canonical ones—the Legendre transforms (or thermodynamic conjugates) of
the corresponding coordinate $q_{b}$. For example, for a single
nonrelativistic particle in one dimension subject to the force
$F(x)=V^{\prime}(x)$ (i.e., $H=p^{2}/2m+V(x)$), we have

$\mu^{\vphantom{\prime}}_{a}\,=\,\partialderivative{[\beta
H]}{q_{a}}~{}~{}\implies~{}~{}\mu^{\vphantom{\prime}}_{x}\,=\,\frac{1}{T}\partialderivative{V}{x}\,=-\beta
F(x)~{},~{}~{}\mu^{\vphantom{\prime}}_{p}\,=\,\frac{p}{mT}\,=\,\beta\dot{x}\,,~{}~{}$
(108)

and thus, for dissipationless Hamiltonian mechanics, knowledge of the Poisson
bracket (106) and the Hamiltonian $H$, directly implies the form of the
Fokker-Planck generator $W$ (111) of time evolution.

It is also straightforward to recover the corresponding MSR Lagrangian
$L(\mathbf{q},\dot{\mathbf{q}},{\bf\it\pi})$ (95),

$\displaystyle
L\,=\,\pi_{a}\partial^{\vphantom{\prime}}_{t}q_{a}-\pi_{a}\quantity{q_{a},H}\,,~{}~{}$
(109)