Time-reversal symmetries and equilibrium-like Langevin equations Lokrshi Prawar Dadhichi and Klaus Kroy Institute for Theoretical Physics Leipzig University 04103 Leipzig Germany

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Time-reversal symmetries and equilibrium-like Langevin equations

Lokrshi Prawar Dadhichi and Klaus Kroy∗

Institute for Theoretical Physics, Leipzig University, 04103 Leipzig, Germany

Graham has shown in Z. Physik B 26, 397-405 (1977) that a ﬂuctuation-dissipation relation can

be imposed on a class of non-equilibrium Markovian Langevin equations that admit a stationary so-

lution of the corresponding Fokker-Planck equation. The resulting equilibrium form of the Langevin

equation is associated with a nonequilibrium Hamiltonian. Here we provide some explicit insight

into how this Hamiltonian may loose its time reversal invariance and how the “reactive” and “dis-

sipative” ﬂuxes loose their distinct time reversal symmetries. The antisymmetric coupling matrix

between forces and ﬂuxes no longer originates from Poisson brackets and the “reactive” ﬂuxes con-

tribute to the (“housekeeping”) entropy production, in the steady state. The time-reversal even

and odd parts of the nonequilibrium Hamiltonian contribute in qualitatively diﬀerent but physically

instructive ways to the entropy. We ﬁnd instances where ﬂuctuations due to noise are solely respon-

sible for the dissipation. Finally, this structure gives rise to a new, physically pertinent instance of

frenesy.

I. INTRODUCTION

A Langevin equation is a stochastic diﬀerential equa-

tion describing generic mesoscopic dynamics driven by

both systematic and stochastically ﬂuctuating forces [1].

The concept is suitable for slow degrees of freedom cou-

pled to a large number of fast degrees of freedom that

can be subsumed into a “noisy” stochastic force. There

are multiple ways to derive Langevin equations from mi-

croscopic descriptions [1,2]. In equilibrium, the deter-

ministic part of the dynamics is governed by an eﬀective

(i.e., typically coarse grained) Hamiltonian. It there-

fore relies on a crucial feature of equilibrium dynam-

ics, namely that the coarse-graining, by which one ex-

ploits the scale separation between the slow systematic

and fast stochastic degrees of freedom, leads one to a

free energy governing the slow variables that itself obeys

Hamiltonian symmetries. In this case, the stochastic

noise strength can moreover be fully speciﬁed mesoscop-

ically, by a ﬂuctuation-dissipation relation (FDR) [3–5]

that obliges the fast degrees of freedom to act as an eﬀec-

tive thermostat for the slow variables, so that the solu-

tions obtained for the latter from the Langevin equation

coincide with those from the classical Gibbs ensembles,

at late times. In this framework, dissipative and reactive

(reversible/conservative) contributions can be clearly dis-

tinguished.

Traditionally, the FDR is thus intimately associated

with thermal equilibrium [3–5] and its failure with a loss

of equilibrium. Indeed, a set of Langevin equations de-

scribing a generic nonequilibrium system is not obliged to

obey Hamiltonian dynamics nor any FDR. However, the

FDR has time and again been generalized to nonequilib-

rium conditions —pars pro toto we here refer to Refs. [6–

15], and references therein. Interestingly Graham [16]

and later, inspired by his work, Eyink et al. [17] gave a

formal procedure to extend the FDR to generic nonequi-

∗lpdadhichi@gmail.com;klaus.kroy@uni-leipzig.de

librium mesoscopic Markov systems, whenever the exis-

tence of a stationary solution of the associated Fokker–

Planck equation (FPE) can be taken for granted. In

principle, it provides a formal eﬀective Hamiltonian-like

structure reminiscent of a potential of mean force [18],

given by the logarithm of said stationary solution of the

FPE, to which we want to refer as the nonequilibrium

Hamiltonian (NH). As we recall and explicitly lay out

in the following, the nonequilibrium Langevin equations

rephrased in terms of this NH have the same structure

as in equilibrium. The NH is a Lyponov function for

the Langevin dynamics around the steady state, so that

the latter is unique and stable [16]. The resemblance

of the equations with the equilibrium structure, includ-

ing a formal FDR, naturally raises the question, where

the condition of nonequilibrium got hidden? As pointed

out in Refs. [16,17], it is hidden in the symmetry under

time reversal of the dynamical equation. Here, we dwell

deeper into this question and, through an exactly solvable

model (studied widely in active matter), demonstrate

how “dissipative” and “reactive” parts of the nonequilib-

rium Langevin equations violate the familiar equilibrium

time-reversal signatures. Additionally we show that the

time-reversal even and odd part of the NH contribute

to the entropy production in qualitatively diﬀerent ways.

Apart from being a Lyponov function for a given steady

state, the (negative) NH is also related to the entropy

and the excess heat produced in a quasistatic operation

turning it into another steady state [19]. This provides

multiple reasons to study the eﬀect of perturbing the

NH. Unlike the usual practice for nonequilibrium sys-

tems, where perturbing forces are directly added to the

equations of motion (which can be understood as a force

balance), here the perturbing force appears in the NH

with a special coupling [16]. In this context, we establish

the link to frenesy [20,21], which is a measure of the

“undirected” currents in a system.

More precisely, we show that, in such equilibrium-like

Langevin equations, the nonequilibrium condition mani-

fests itself in the following ways:

arXiv:2210.14255v2 [cond-mat.stat-mech] 21 Mar 2023

•The NH need not be time reversal invariant.

•The antisymmetric couplings do not arise from

Poisson brackets.

•The “reactive” currents also produce entropy.

Similarly, ﬂuctuations from the steady state,

due to the noise, can produce entropy (“ac-

tive/dissipative” ﬂuctuations).

We also make the following important observations:

•The parts of the NH with diﬀerent time-reversal

signatures contribute in qualitatively diﬀerent ways

to the entropy production.

•The NH opens a new meaningful way to perturb the

system and hence provides a second interesting in-

stance of excess frenesy, beyond the usual one. We

also point out that this perturbation can be used to

derive a variant of, the Harada-Sasa relation [22].

The paper is organised as follows: in Sec. II we recall

the structure of equilibrium Langevin equations, Sec. III

summarizes Graham’s work [16] and establishes the

structural similarity between Langevin equations with

nonequilibrium steady states and equilibrium Langevin

equations. In Sec. IV, we analytically solve the FPE

corresponding to a linear nonequilibrium Langevin equa-

tion to explicitly reveal this equilibrium-like structure.

We discuss its interesting features, and, in Sec. V, apply

this formalism to a much studied model in the physics

of soft active matter, namely so-called active Ornstein–

Uhlenbeck particles (AOUPs) [23]. In Sec. VI, the role of

diﬀerent parts of NH in the entropy production is stud-

ied. Finally, Sec. VII provides the link to frenesy in this

context and a comparison with previous studies [21].

II. EQUILIBRIUM STRUCTURE

We construct equations of motion for dynamical vari-

ables Cwith position-like and momentum-like compo-

nents Qand P, respectively, even and odd under time-

reversal (hereafter denoted by T). We allow Cto be

ﬁnite or inﬁnite-dimensional, depending on whether we

are dealing with a system parameterized in terms of par-

ticle degrees of freedom or with a spatially extended sys-

tem described by a stochastic ﬁeld theory. In particle

systems in thermal equilibrium, Qand Pnormally refer

to canonically conjugate variables, but in more strongly

coarse-grained formulations [24], and in particular in gen-

eralized non-equilibrium Langevin systems, they do not

have to, nor do they need to have the same number of

components. The stochastic equations of motion describ-

ing the thermal equilibrium dynamics of Care [1,24–26]

∂tC=−(Γ+W)·∂CH+T ∂C·W+ξ(1)

where H(C) is the eﬀective Hamiltonian, Γis a symmetric

matrix of dissipative couplings between the variables that

governs the FDR

hξ(t)ξ(t0)i= 2TΓδ(t−t0),(2)

and Wis an antisymmetric matrix of reactive couplings.

For brevity, we are using the notation for discrete de-

grees of freedom, which can however straightforwardly

be upgraded for the case that Cis supposed to be a ﬁeld

variable; e.g., the term W·∂CHwould read

Zdx0Wµν (x,x0)δH

δCν(x0)(3)

(summation over νimplied).

Terms involving Wmust have, component by compo-

nent, the same signature under Tas ∂tC, and those in-

volving Γmust have the opposite T-signature. Thus the

QQ and PP components of the matrix Γmust them-

selves be even under T, while its QP and PQ compo-

nents must be odd. In equilibrium, Wis identiﬁed with

the Poisson bracket between the dynamical variables as

discussed later in this section [24,25].

Standard derivations of generalized Langevin equa-

tions [2,24,25,27] require the additional term T∇C·W

in Eq. (1) for the steady state to be ∝e−H/T . While it

vanishes in familiar equilibrium Langevin equations [28],

there are natural instances in active matter where it is

nonzero [29]. Moreover, when Γdepends on C, the noise

term in (1) is multiplicative. It then produces a spurious

drift. For the steady-state distribution to remain e−H/T ,

we must then include, as a counter term, the additional

drift T(∇C·Γ−αg· ∇Cg), in Eq. (1), where g·g=2Γ

[26], and the continuous parameter α∈[0,1] parameter-

izes diﬀerent physical interpretations of the noise. Simi-

larly, WPP should be odd under Tand therefore suitably

P-dependent.

For an equilibrium system, the reactive (reversible)

term emerges from the Poisson bracket of the variable

with the Hamiltonian [24,25].

∂tCµ={H, Cµ} ≡ −Wµν ∂CνH(4)

The antisymmetric coupling matrix Wµν =−Wνµ =

{Cµ,Cν}has the structure of a Poisson bracket of the

dynamical (ﬁeld) variables. And, again, an extra term

∂µWµν is required in the reactive part to attain a Boltz-

mann equilibrium distribution. Hydrodynamic Poisson

brackets are usually calculated directly from a micro-

scopic model [30] or indirectly inferred from symmetries

[31].

In general, the above eﬀective Hamiltonian structure,

and hence the clear identiﬁcation of reactive and dissipa-

tive currents, breaks down for nonequilibrium systems,

which naturally raises the question how much of it can

be rescued for the special subclass of Markov systems

that admit non-equilibrium steady states (NESS).

III. GRAHAM’S EQUILIBRIUM-LIKE

STRUCTURE

Graham [16] and later, inspired by his work, Eyink et

al. [17] gave a formal procedure how to write nonequilib-

rium Markovian equations in an equilibrium-like form, in

which Eqs. (1), (2) still pertain. In particular, Eyink et

al. pointed out that the “dissipative” (symmetric) cou-

pling governing the strength of the noise correlation actu-

ally establishes a FDR of the ﬁrst kind, as discussed fur-

ther below. Here we summarise the results that are use-

ful for our present purpose. Namely, a general Langevin

equation (for discrete degrees of freedom)

Cµ=Jµ(C) + gi

µ(C)ξi(5)

where

hξi(t)ξj(t0)i= 2δij δ(t−t0)Qµν (C) = gi

µ(C)gi

ν(C) (6)

can be written as

Cµ=−(Qµν +La

µν )∂Cνφ+∂CνLa

µν +gi

µ(C)ξi(7)

with antisymmetric couplings La

µν ≡Fµν e−φ, where Fµν

itself is antisymmetric and deﬁned by

rµP0(C)≡∂CνFµν .(8)

We now clarify this notation. First, φ≡ln P0(C) is the

logarithm of the steady-state solution P0(C) of the asso-

ciated Fokker–Planck equation corresponding to Eq. (5),

namely

∂tP(C, t) = −∂Cµ[Jµ(C)P(C, t)−Qµν (C)∂CνP(C, t)]

(9)

Following the equilibrium paradigm [24], the determinis-

tic ﬂux Jµ(C) is broken into “dissipative” and “reactive”

contributions, dµ(C) and rµ(C)≡Jµ(C)−dµ(C), respec-

tively, where the former has the explicit form

dµ(C) = Qµν (C)∂Cνφ(10)

Using Eq. (8) and the deﬁnition La

µν ≡Fµν e−φ, the re-

active current can be cast into the explicit form

rµ(C) = La

µν ∂Cνφ+∂CνLa

µν (11)

The point we want to make here is that the structure

of the dissipative and reactive terms in the equilibrium

and equilibrium-like description is exactly the same, i.e.

Eqs. (7) and (6) are structurally identical to Eqs. (1)

and (2). The identiﬁcation of the symmetric coupling

with the noise strength is also present in both cases, es-

tablishing an FDR of the ﬁrst kind, as noted in Ref. [17].

One very important diﬀerence is that, unlike in equilib-

rium, the antisymmetric coupling is not obliged to orig-

inate from a Poisson bracket, in the generic case. It can

however easily be seen that the generic antisymmetric

coupling, Laboils down to a Poisson bracket W, in the

equilibrium limit, where φ=−H. Then, the antisym-

metric coupling is Weφe−φ, and equating this with La

gives

Fµν =Wµν e−H(12)

Taking its derivative and using (8), we then ﬁnd

rν=Wµν ∂CµH−∂CµWµν ,(13)

consistent with the equilibrium formalism [24,25].

In summary, Eqs. (1) and (7) have an identical form,

and the symmetry of the coupling coeﬃcient with respect

to an interchange of its indices is also the same. However,

when the dynamics is governed by Eq.(1), the system is

in equilibrium, whereas Eq. (7) can describe both equilib-

rium as well as nonequilibrium dynamics. This prompts

the question where is the nonequilibrium condition hid-

den in Eq. (7)? It is clear from the above discussion

that an explicit description of a nonequilibrium dynam-

ics by Eq. (7) requires the knowledge of its steady-state

distribution. In general, the latter will be very diﬃcult

to ﬁnd. Therefore, we study in the following an exactly

solvable linear system to provide explicit answers to the

theoretical questions raised and to illustrate the general

statements promised in the Introduction I.

IV. A SOLVABLE MODEL

As described above, for a given (eﬀective) Hamiltonian

and noise correlation matrix in an equilibrium system,

the reactive and dissipative terms come out naturally

with the correct time reversal signatures. But things get

more complicated once the system is out of equilibrium.

Here we discuss the linear case, which can be solved ex-

actly, to see how the classiﬁcation of the terms as reactive

and dissipative looses meaning. Our starting point is the

following set of coupled linear equations.

˙α=−1

Γ∂αH+ ˜υ∂βH+ξα(t) (14)

β=−˜υ∂αH−1

γ∂βH+ξβ(t) (15)

associated with the quadratic Hamiltonian

H=1

2Kα2+1

2kβ2(16)

where Kand kare positive stiﬀness constants; ˜υis a pos-

itive constant. The Markovian noise correlation matrix

shall be given by

D= 2 1

Γ0

γδ(t−t0) (17)

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Time-reversalsymmetriesandequilibrium-likeLangevinequationsLokrshiPrawarDadhichiandKlausKroyInstituteforTheoreticalPhysics,LeipzigUniversity,04103Leipzig,GermanyGrahamhasshowninZ.PhysikB26,397-405(1977)thatauctuation-dissipationrelationcanbeimposedonaclassofnon-equilibriumMarkovianLangevinequations...

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Time-reversal symmetries and equilibrium-like Langevin equations Lokrshi Prawar Dadhichi and Klaus Kroy Institute for Theoretical Physics Leipzig University 04103 Leipzig Germany

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