Properties of the dissipation functions for passive and active systems

Harsh Soni School of Physical Sciences, IIT Mandi, Kamand, Mandi, HP 175005 India

Abstract

The dissipation function for a system is defined as the natural logarithm of the ratio between probabilities of a trajectory and its time-reversed trajectory, and its probability distribution follows a well-known relation called the fluctuation theorem. Using the generic Langevin equations, we derive the expressions of the dissipation function for passive and active systems. For passive systems, the dissipation function depends only on the initial and the final values of the dynamical variables of the system, not on the trajectory of the system. Furthermore, it does not depend explicitly on the reactive or dissipative coupling coefficients of the generic Langevin equations. In addition, we study a 1D case numerically to verify the fluctuation theorem with the form of the dissipation function we obtained. For active systems, we define the work done by active forces along a trajectory. If the probability distribution of the dynamical variables is symmetric under time reversal, in both cases, the average rate of change of the dissipation function with trajectory duration is nothing but the average entropy production rate of the system and reservoir.

I Introduction

Irreversibility of a system can be quantified by using the dissipation function which is defined as the natural logarithm of the ratio of the probability density of a trajectory to that of its time-reversed trajectory. The probability distribution function of the dissipation function exhibits an interesting symmetry relation known as the fluctuation theorem [1, 2, 3]. The fluctuation theorem has been substantially explored using theory [4, 5, 6, 7, 8, 9, 10, 11, 12] and experiment [13, 14, 15, 16, 17, 18, 19]. For stochastic processes, it has been investigated mainly for the single-particle or single-variable case [5, 6, 20]. Moreover, little attention has been paid to the systems described by the Langevin equations with multiplicative noise, except for a few studies [20, 12].

This paper discusses the fluctuation relations for a wide class of systems described by the generic Langevin equations [21, 22]. Assuming that the slow variables of a system vary much slower than its microscopic degrees of freedom, one can consider that the system is always in local thermodynamic equilibrium at temperature $T$ . The dynamics of such systems is well-explained by the generic Langevin equations. We consider the active as well as passive systems. We use the path integral approach to calculate the probability density of a trajectory of the system with $\alpha$ -discritization [23, 20, 24].

Our main results are as follows. We first show that the generic Langevin equations describe a passive system, irrespective of the value of $\alpha$ . We then derive the expression of the dissipation function for passive systems relaxing towards thermodynamic equilibrium. Interestingly, the dissipation function is independent of the trajectory followed by the system; it only depends on the initial and the final values of the dynamical variables of the system. Moreover, it is not an explicit function of the coefficients appearing in the generic Langevin equation. Using Brownian dynamics simulation, we also verify the fluctuation theorem for a 1D single-particle problem with state-dependent diffusion. Finally, we construct an expression of the dissipation function for the active systems, and we define the work done by the active forces. For both active and passive systems, the average rate of change of the dissipation function with the time duration is the same as the rate of change of the entropy of the system and reservoir, assuming that the probability distribution of the dynamical variables is invariant under time reversal.

In the next section, we will discuss passive systems, and in section III, we will explore active systems.

II Passive systems

This section is arranged as follows. In the next subsection, we summarize the generic Langevin equations. We then calculate the ratio of the probabilities of a trajectory and its time-reversed trajectory (see subsection II.2). In subsection II.3, fluctuation relations and the dissipation function for the passive systems are presented. In subsection II.4, we talk about the quenched systems, along with an example of a system of a single colloidal particle.

II.1 Generic Langevin equations

Here we consider a passive system relaxing towards equilibrium. Its macroscopic dynamics is described by a set of $n$ number of slow variables $\bm{A}\equiv\{A_{1},A_{2},....A_{n}\}$ . Let $A_{i}\to s_{i}A_{i}$ under time reversal, where $s_{i}=1$ and $s_{i}=-1$ if $A_{i}$ is even and odd under time reversal, respectively; e.g., $s_{i}=1$ for position and $s_{i}=-1$ for momentum. The generic Langevin equations for the system at temperature $T$ can be written in the following form [21, 22]:

\dfrac{dA_{i}}{dt}=-\Gamma_{ij}\dfrac{\partial\mathcal{H}}{\partial A_{j}}+k_{\text{B}}T\dfrac{\partial\Gamma_{ij}}{\partial A_{j}}+\eta_{i}(t).

(1)

where $\mathcal{H}$ is the coarse-grained or effective Hamiltonian of the system and the coefficients $\Gamma_{ij}$ satisfy the following property:

\Gamma_{ij}=s_{i}s_{j}\Gamma_{ji}.

(2)

In Eq. (1), the terms with $s_{i}s_{j}=-1$ are the Poisson bracket or reactive terms, whereas the terms with $s_{i}s_{j}=1$ are the dissipative terms [25]. The last term $\eta_{i}(t)$ represents the rapid fluctuations due to the dynamics of the microscopic degrees of freedom of the system. We assume that $\eta_{i}(t)$ is white Gaussian noise, and its autocorrelation function is given by

\left\langle\eta_{i}(t)\eta_{j}(t^{\prime})\right\rangle=2k_{\text{B}}T\Gamma^{\text{s}}_{ij}\delta(t-t^{\prime}),

(3)

where $\Gamma^{\text{s}}_{ij}\equiv(\Gamma_{ij}+\Gamma_{ji})/2$ is the symmetric part of $\Gamma_{ij}$ . From Eq.(2), $\Gamma^{\text{s}}_{ij}$ shows the following symmetry property:

\Gamma^{\text{s}}_{ij}=s_{i}s_{j}\Gamma^{\text{s}}_{ij}.

(4)

Further, it is assumed to be invertible. Here in Eq. (1), we use Einstein notation, which will be carried through the rest of the paper, unless otherwise stated. Writing $\eta_{i}(t)$ as the linear combination of time series of the white Gaussian noise $\xi_{j}(t)$ having no correlation with each other i.e. $\left\langle\xi_{i}(t)\xi_{j}(t)\right\rangle=\delta_{ij}\delta(t-t^{\prime})$ :

\eta_{i}(t)=N_{ij}\xi_{j}(t),

(5)

where, from Eq. (3), $N_{ij}$ is given by the solution of the equations:

N_{ik}N_{jk}=2k_{\text{B}}T\Gamma^{\text{s}}_{ij}.

(6)

Since $N_{ik}$ must be real, $\Gamma^{\text{s}}_{ij}$ must have positive eigenvalues [26]. As $\Gamma^{\text{s}}_{ij}$ is considered to be invertible, $N_{ik}$ is invertible as well. It should be noted that $N_{jk}$ is not uniquely defined by the above equation. However, $N_{jk}$ is just a dummy matrix which does not appear anywhere in the final results. Substituting (5) into Eq. (1):

\dfrac{dA_{i}}{dt}=-\Gamma_{ij}\dfrac{\partial\mathcal{H}}{\partial A_{j}}+k_{\text{B}}T\dfrac{\partial\Gamma_{ij}}{\partial A_{j}}+N_{ij}\xi_{j}(t).

(7)

The above stochastic equations have no ambiguity when $N_{ij}$ does not depend explicitly on $\bm{A}$ . However, $N_{ij}$ is the function of $\bm{A}$ for many systems; in such cases, the above equations are not well-defined unless their discrete scheme is specified. We here use $\alpha$ -discretization method to discretize the above equations [20, 23, 24], which leads to a drift of $\alpha N_{lj}\partial N_{ij}/\partial A_{l}$ to the value of $A_{i}$ due to the noise term [27, 24]. On the contrary, the noise terms in the generic Langevin equations represent the thermal fluctuations and do not contribute to the average dynamics of the slow variables. One can eliminate the noise-induced drift by adding a correction term $-\alpha N_{lj}\partial N_{ij}/\partial A_{l}$ to Eq. (7). Therefore, the generic Langevin equations can be completely described as follows:

	$\displaystyle\dfrac{dA_{i}}{dt}$	$\displaystyle=$	$\displaystyle-\Gamma_{ij}\dfrac{\partial\mathcal{H}}{\partial A_{j}}+k_{\text{B}}T\dfrac{\partial\Gamma_{ij}}{\partial A_{j}}-\alpha N_{lj}\dfrac{\partial N_{ij}}{\partial A_{l}}+N_{ij}\xi_{j}$		(8)
		$\displaystyle\equiv$	$\displaystyle\mathcal{F}_{i}+N_{ij}\xi_{j}$		(9)

with their discrete form

dA_{i}(l)=\epsilon\mathcal{F}_{i}(\bar{\bm{A}}^{\text{f}}_{l})+\sqrt{\epsilon}N_{ij}(\bar{\bm{A}}^{\text{f}}_{l})\xi_{j}^{l},

(10)

where $\epsilon$ is the time step, $dA_{i}(l)\equiv A_{i}(\epsilon l)-A_{i}(\epsilon(l-1))$ , $\bar{\bm{A}}^{\text{f}}_{l}\equiv\alpha\bm{A}(\epsilon l)+(1-\alpha)\bm{A}(\epsilon(l-1))$ ,

\mathcal{F}_{i}\equiv-\Gamma_{ij}\dfrac{\partial\mathcal{H}}{\partial A_{j}}+k_{\text{B}}T\dfrac{\partial\Gamma_{ij}}{\partial A_{j}}-\alpha N_{lj}\dfrac{\partial N_{ij}}{\partial A_{l}},

(11)

and,

\xi_{j}^{l}\equiv\dfrac{1}{\sqrt{\epsilon}}\int^{l\epsilon}_{(l-1)\epsilon}\xi_{j}(t)dt.

(12)

are the series of random numbers having normal distribution with standard deviation 1 and mean 0. The parameter $\alpha$ can take any “absolute constant” between 0 and 1; $\alpha=0$ and $\alpha=1/2$ cases are referred to as Itô and Stratonovich methods, respectively. As we have another parameter $\alpha$ in the problem now, one of the questions we ask here is, do different values of $\alpha$ correspond to different systems? If yes, do all the values of $\alpha$ belong to passive systems?

Based on the behavior under time reversal, dividing $\mathcal{F}_{i}$ into the following three parts $\mathcal{F}^{\text{s}}_{i}$ , $\mathcal{F}^{\text{a}}_{i}$ and $\mathcal{F}^{\text{N}}_{i}$ :

	$\displaystyle\mathcal{F}^{\text{s}}_{i}(\bm{A})=-\Gamma^{\text{s}}_{ij}\dfrac{\partial\mathcal{H}}{\partial A_{j}}+k_{\text{B}}T\dfrac{\partial\Gamma^{\text{s}}_{ij}}{\partial A_{j}}$		(13)
	$\displaystyle\mathcal{F}^{a}_{i}(\bm{A})=-\Gamma^{a}_{ij}\dfrac{\partial\mathcal{H}}{\partial A_{j}}+k_{\text{B}}T\dfrac{\partial\Gamma^{\text{a}}_{ij}}{\partial A_{j}}$		(14)
	$\displaystyle\mathcal{F}^{\text{N}}_{i}=-\alpha N_{kj}\dfrac{\partial N_{ij}}{\partial A_{k}}$		(15)

where $\Gamma^{\text{a}}_{ij}\equiv(\Gamma_{ij}-\Gamma_{ji})/2$ is the antisymmetric part of $\Gamma_{ij}$ . From Eq.(2), $\Gamma^{\text{a}}_{ij}$ exhibits the following symmetry property (not in Einstein notation):

\Gamma^{\text{a}}_{ij}=-s_{i}s_{j}\Gamma^{\text{a}}_{ij}.

(16)

Since $\mathcal{H}(\bm{s}\circ\bm{A})=\mathcal{H}(\bm{A})$ and $\Gamma^{\text{s}}_{ij}(\bm{s}\circ\bm{A})=\Gamma^{\text{s}}_{ij}(\bm{A})$ [21], from Eqs. (4) & (16), under time reversal,

	$\displaystyle\bm{\mathcal{F}}^{\text{s}}(\bm{A})\to\bm{\mathcal{F}}^{\text{s}}(\bm{s}\circ\bm{A})=\bm{s}\circ\bm{\mathcal{F}}^{\text{s}}(\bm{A}),$		(17)
	$\displaystyle\bm{\mathcal{F}}^{\text{a}}(\bm{A})\to\bm{\mathcal{F}}^{\text{a}}(\bm{s}\circ\bm{A})=-\bm{s}\circ\bm{\mathcal{F}}^{\text{a}}(\bm{A}).$		(18)

where $\circ$ stands for Hadamard product i.e. $\bm{s}\circ\bm{A}\equiv\{s_{1}A_{1},s_{2}A_{2},....s_{n}A_{n}\}$ . For given $\Gamma^{\text{s}}_{ij}$ , $\mathcal{F}^{\text{s}}_{i}(\bm{A})$ and $\mathcal{F}^{\text{a}}_{i}(\bm{A})$ do not depend on $N_{ij}$ . In general, $\mathcal{F}^{\text{N}}_{i}$ does not follow any of the above time reversal symmetries.

II.2 The ratio between the probability densities of a trajectory and its time-reversed trajectory

Let $p_{0}(\bm{A})$ be the probability distribution function of $\bm{A}$ at $t=0$ . In $\epsilon\to 0$ limit, the probability density of a trajectory of the system $(\bm{A}_{0},\bm{A}_{1},\bm{A}_{2},.....\bm{A}_{N})$ (here $\bm{A}_{l}\equiv\bm{A}(l\epsilon)$ ) between $t=0$ and $t=\tau\equiv N\epsilon$ is given by [12] (see Appendix A)

$\displaystyle P$	$\displaystyle\simeq$	$\displaystyle p_{0}(\bm{A}_{0})\prod^{N}_{l=1}\left\{\dfrac{(2k_{\text{B}}T)^{-1/2}}{(2\pi\epsilon)^{n/2}}\exp\Biggl{[}-\dfrac{1}{4\epsilon k_{\text{B}}T}\left[dA_{i}(l)-\epsilon\mathcal{F}^{\text{s}}_{i}(\bar{\bm{A}}^{\text{f}}_{l})-\epsilon\mathcal{F}^{\text{a}}_{i}(\bar{\bm{A}}^{\text{f}}_{l})\right]{{{\color[rgb]{0,0,0}(\Gamma^{s}}}^{-1})_{ij}}(\bar{\bm{A}}^{\text{f}}_{l})\left[dA_{j}(l)-\epsilon\mathcal{F}^{\text{s}}_{j}(\bar{\bm{A}}^{\text{f}}_{l})-\epsilon\mathcal{F}^{\text{a}}_{j}(\bar{\bm{A}}^{\text{f}}_{l})\right]\Biggr{.}\right.$	(19)
		$\displaystyle\left.-\alpha\left[\left(dA_{i}(l)-\epsilon\mathcal{F}^{\text{s}}_{i}(\bar{\bm{A}}^{\text{f}}_{l})-\epsilon\mathcal{F}^{\text{a}}_{i}(\bar{\bm{A}}^{\text{f}}_{l})\right)\left[({\Gamma^{s}}^{-1})_{ij}\dfrac{\partial\Gamma^{s}_{jk}}{\partial A_{k}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}+\epsilon\left[\dfrac{\partial\mathcal{F}^{\text{s}}_{i}}{\partial A_{i}}+\dfrac{\partial\mathcal{F}^{\text{a}}_{i}}{\partial A_{i}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}\right]\right]\text{det}\left(\bm{\Gamma^{\text{s}}}(\bar{\bm{A}}^{\text{f}}_{l})\right)^{-1/2}$
		$\displaystyle\times\left.\exp\left[\alpha^{2}\epsilon k_{\text{B}}T\left[\dfrac{\partial^{2}\Gamma^{s}_{ij}}{\partial A_{i}\partial A_{j}}-\dfrac{\partial\Gamma^{s}_{ik}}{\partial A_{k}}({\Gamma^{s}}^{-1})_{ij}\dfrac{\partial\Gamma^{s}_{jp}}{\partial A_{p}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}\right]\right\}.$

The $\epsilon^{3/2}$ - and higher-order terms have been neglected here. It is apparent from the above expression that, for given $\Gamma^{\text{s}}_{ij}$ , $P$ is independent of $N_{ij}$ . So no statistical property of the system has a dependence upon the choice of $N_{ik}$ . Therefore, as mentioned earlier, $N_{ik}$ is merely a dummy matrix. The probability density $P$ depends on $\alpha$ ; thus, the different values of $\alpha$ correspond to different systems. Later in this subsection, we will see that Eq. (8) provides the dynamics of a passive system for any $\alpha$ . The time-reversed trajectory of the trajectory $(\bm{A}_{0},\bm{A}_{1},\bm{A}_{2},.....\bm{A}_{N})$ would be $(\bm{s}\circ\bm{A}_{N},\bm{s}\circ\bm{A}_{N-1},.....\bm{s}\circ\bm{A}_{1})$ , so its probability density can be calculated by replacing $\bm{A}_{i}$ by $\bm{s}\circ\bm{A}_{N-i}$ in the above equation; that is, (see Appendix B)

$\displaystyle P_{r}$	$\displaystyle\simeq$	$\displaystyle\prod^{N}_{l=1}\left\{\dfrac{(2k_{\text{B}}T)^{-1/2}}{(2\pi\epsilon)^{n/2}}\exp\Biggl{[}-\dfrac{1}{4\epsilon k_{\text{B}}T}\left[-dA_{i}(l)-\epsilon\mathcal{F}^{\text{s}}_{i}(\bar{\bm{A}}^{\text{r}}_{l})+\epsilon\mathcal{F}^{\text{a}}_{i}(\bar{\bm{A}}^{\text{r}}_{l})\right]{{{\color[rgb]{0,0,0}(\Gamma^{s}}}^{-1})_{ij}}(\bar{\bm{A}}^{\text{r}}_{l})\left[-dA_{j}(l)-\epsilon\mathcal{F}^{\text{s}}_{j}(\bar{\bm{A}}^{\text{r}}_{l})+\epsilon\mathcal{F}^{\text{a}}_{j}(\bar{\bm{A}}^{\text{r}}_{l})\right]\Biggr{.}\right.$	(20)
		$\displaystyle\left.-\alpha\left[\left(-dA_{i}(l)-\epsilon\mathcal{F}^{\text{s}}_{i}(\bar{\bm{A}}^{\text{r}}_{l})+\epsilon\mathcal{F}^{\text{a}}_{i}(\bar{\bm{A}}^{\text{r}}_{l})\right)\left[({\Gamma^{s}}^{-1})_{ij}\dfrac{\partial\Gamma^{s}_{jk}}{\partial A_{k}}\right]_{\bar{\bm{A}}^{\text{r}}_{l}}+\epsilon\left[\dfrac{\partial\mathcal{F}^{\text{s}}_{i}}{\partial A_{i}}-\dfrac{\partial\mathcal{F}^{\text{a}}_{i}}{\partial A_{i}}\right]_{\bar{\bm{A}}^{\text{r}}_{l}}\right]\right]\text{det}\left(\bm{\Gamma^{\text{s}}}(\bar{\bm{A}}^{\text{r}}_{l})\right)^{-1/2}$
		$\displaystyle\times\left.\exp\left[\alpha^{2}\epsilon k_{\text{B}}T\left[\dfrac{\partial^{2}\Gamma^{s}_{ij}}{\partial A_{i}\partial A_{j}}-\dfrac{\partial\Gamma^{s}_{ik}}{\partial A_{k}}({\Gamma^{s}}^{-1})_{ij}\dfrac{\partial\Gamma^{s}_{jp}}{\partial A_{p}}\right]_{\bar{\bm{A}}^{\text{r}}_{l}}\right]\right\}p_{0}(\bm{s}\circ\bm{A}_{N}).$

In $\epsilon\to 0$ limit, the ratio $P/P_{r}$ takes the following form (see Appendix 84):

\dfrac{P}{P_{r}}=\dfrac{p_{0}(\bm{A}(0))}{p_{0}(\bm{s}\circ\bm{A}(\tau))}\exp\left[-\dfrac{1}{k_{\text{B}}T}\left(\mathcal{H}(\bm{A}(\tau))-\mathcal{H}(\bm{A}(0))\right)\right]

(21)

In the stationary state, $p_{0}(\bm{A})=\exp(-\mathcal{H}(\bm{A})/k_{\text{B}}T)/\mathcal{Z}$ (see Appendix H), the above equation then yields

P=P_{r}.

(22)

It implies that any system whose dynamics is given by Eq. (8) has the time reversal symmetry in its stationary state, regardless of the value of $\alpha$ . Hence, Eq. (8) describes a passive system for any value of $\alpha$ between 0 and 1.

II.3 Fluctuation relations and the dissipation function for the passive systems

One can readily show that [12] the dissipation function $\mathcal{R}_{\tau}$ for the trajectory $(\bm{A}_{0},\bm{A}_{1},\bm{A}_{2},.....\bm{A}_{N})$ defined as

\mathcal{R_{\tau}}=\ln\left[\dfrac{P}{P_{r}}\right]

(23)

satisfies the relation

{{\color[rgb]{0,0,0}\dfrac{\mathcal{P}(\mathcal{R}_{\tau}=X)}{\mathcal{P}(\mathcal{R}_{\tau}=-X)}=\exp(X),}}

(24)

where $\mathcal{P}$ is the probability distribution of $\mathcal{R}_{\tau}$ . The above relation is known as the fluctuation theorem [2, 5]. From Eq. (21),

\mathcal{R}_{\tau}=\ln\dfrac{p_{0}(\bm{A}(0))}{p_{0}(\bm{s}\circ\bm{A}(\tau))}-\dfrac{1}{k_{\text{B}}T}\left[\mathcal{H}(\bm{A}(\tau))-\mathcal{H}(\bm{A}(0))\right].

(25)

Intriguingly, $\mathcal{R}_{\tau}$ does not depend explicitly on $\Gamma_{ij}$ . Moreover, it depends only the initial and final values of $\bm{A}$ , not on the trajectory followed by $\bm{A}$ . Note that the ratio $P/P_{r}$ in Eq. (21) also has the same functional properties.

One can also define the dissipation function for the system as follows [1]:

\mathcal{R}^{\prime}_{\tau}=\ln\dfrac{p(\bm{A}(0),\bm{A}(\tau);\tau)}{p(\bm{s}\circ\bm{A}(\tau),\bm{s}\circ\bm{A}(0);\tau)},

(26)

where $p(\bm{A}(0),\bm{A}(\tau);\tau)$ is the net probability that the system goes from $\bm{A}(0)$ to $\bm{A}(\tau)$ in time $\tau$ :

p(\bm{A}(0),\bm{A}(\tau);\tau)=\sum P,

(27)

where the summation is performed over all the trajectories between $\bm{A}(0)$ and $\bm{A}(\tau)$ . Since the ratio $P/P_{r}$ is independent of the trajectory between $\bm{A}(0)$ and $\bm{A}(\tau)$ , $\mathcal{R^{\prime}}_{\tau}=\mathcal{R}_{\tau}$ .

The integrated form of the relation (24) is

\left\langle\exp(-\mathcal{R}_{\tau})\right\rangle=1,

(28)

where angular bracket stands for the ensmeble average. Using this relation, one can show that [28, 5]

\left\langle\mathcal{R}_{\tau}\right\rangle\geq 0.

(29)

In equilibrium, $P=P_{r}$ , thus $\left\langle\mathcal{R}_{\tau}\right\rangle=\mathcal{R}_{\tau}=0$ . So $\left\langle\mathcal{R}_{\tau}\right\rangle$ behaves like the change in the entropy of system and it can be used to evaluate that how far the system is from equilibrium. Since the solution of the Fokker-Planck equation corresponding to Eq. (8) does not depend on $\alpha$ , $\left\langle\mathcal{R}_{\tau}\right\rangle$ is constant in $\alpha$ (see Appendix I).

Generalizing the expression of the dissipation function given in Eq. (25) for the trajectories starting at arbitrary time $t$ :

	$\displaystyle\mathcal{R}_{\tau}(t)$	$\displaystyle=$	$\displaystyle\ln\dfrac{p_{t}(\bm{A}(t))}{p_{t}(\bm{s}\circ\bm{A}(t+\tau))}$		(30)
			$\displaystyle-\dfrac{1}{k_{\text{B}}T}\left[\mathcal{H}(\bm{A}(t+\tau))-\mathcal{H}(\bm{A}(t))\right].$		(30)

The instantaneous irreversibility can be evaluated by calculating $\mathcal{R}_{\tau}(t)$ in $\tau\to 0$ limit, that is,

{{\color[rgb]{0,0,0}\mathcal{R}_{\tau}(t)\simeq\dot{{s}}(t)\tau+\ln\dfrac{p_{t}(\bm{A}(t+\tau))}{p_{t}(\bm{s}\circ\bm{A}(t+\tau))},}}

(31)

where

\dot{{s}}(t)=-\left.\dfrac{d}{dt^{\prime}}\left(\ln p_{t}(\bm{A}(t^{\prime}))+\dfrac{1}{k_{\text{B}}T}\mathcal{H}(\bm{A}(t^{\prime}))\right)\right|_{t^{\prime}=t}.

(32)

As discussed in Appendix E, $k_{\text{B}}\left\langle\dot{{s}}(t)\right\rangle$ is nothing but the rate of change of total entropy of the system and the reservoir (see Eq. (107)). According to Eq. (31), the irreversible behavior of the system results from entropy production and from the asymmetric behavior of $p_{t}(\bm{A})$ under time reversal. If $p_{t}(\bm{s}\circ\bm{A})\neq p_{t}(\bm{A})$ , the system is instantaneously irreversible since

\mathcal{R}_{0}(t)=\ln\dfrac{p_{t}(\bm{A}(t))}{p_{t}(\bm{s}\circ\bm{A}(t))}\neq 0.

(33)

Since $\mathcal{H}(\bm{s}\circ\bm{A})=\mathcal{H}(\bm{A})$ , the equilibrium probability distribution $p_{\text{eq}}(\bm{A})\equiv\exp(-\mathcal{H}(\bm{A})/k_{\text{B}}T)/\mathcal{Z}$ always follows the symmetry property $p_{\text{eq}}(\bm{s}\circ\bm{A})=p_{\text{eq}}(\bm{A})$ ; it is a fundamental property of $p_{\text{eq}}(\bm{A})$ . The nonzero value of $\left\langle\mathcal{R}_{0}(t)\right\rangle$ for an out-of-equilibrium system signifies that the system violates this symmetry. Note that $\left\langle\mathcal{R}_{0}(t)\right\rangle\geq 0$ . An example of such systems is as follows: consider a colloidal particle moving with a nonzero average velocity $\bm{v}_{0}$ and having the probability distribution $p_{0}(\bm{v})=C\exp(-(\bm{v}-\bm{v}_{0})^{2}/2)$ , at $t=0$ . Under time reversal $\bm{v}\to-\bm{v}$ , so $\bm{s}=\{-1,-1,-1\}$ . Hence $p_{0}(\bm{s}\circ\bm{v})\neq p_{0}(\bm{v})$ .

For $p_{t}(\bm{s}\circ\bm{A})=p_{t}(\bm{A})$ case, $\mathcal{R}_{0}(t)=0$ , so from Eq. (31),

\dot{{s}}(t)=\lim_{\tau\to 0}\dfrac{\mathcal{R}_{\tau}(t)}{\tau}.

(34)

Thus, the average rate of change of $\mathcal{R}_{\tau}(t)$ with $\tau$ is the same as the rate of the total entropy production of the system and the reservoir; from Eq. (29), the second law of thermodynamics is evident, $\left\langle\dot{{s}}(t)\right\rangle>0$ . In the next subsection, we discuss a broad class of passive systems with $p_{t}(\bm{s}\circ\bm{A})=p_{t}(\bm{A})$ .

The form of the dissipation function used by Seifert et al. [5] is briefly discussed in Appendix F.

II.4 The dissipation function for quenched systems

Here we consider that the system is initially in a thermodynamic equilibrium state and the state variables of the system $\bm{\beta}\equiv\{\beta_{1},\beta_{2},...,\beta_{n}\}$ are abruptly changed at $t=0$ . Then the system will start evolving towards the equilibrium state corresponding to the modified values of $\bm{\beta}$ . Writing the coarse-grained Hamiltonian of the system as the function of $\bm{\beta}$ : $\mathcal{H}\equiv\mathcal{H}(\bm{A};\bm{\beta})$ . Let $\bm{\beta}=\bm{\beta}_{\text{I}}$ at $t=0$ then

p_{0}(\bm{A})=\dfrac{1}{\mathcal{Z}(\bm{\beta}_{\text{I}})}\exp\left[-\dfrac{\mathcal{H}(\bm{A};\bm{\beta}_{\text{I}})}{k_{\text{B}}T}\right]

(35)

From Eq.(25), for a quench from $\bm{\beta}=\bm{\beta}_{\text{I}}$ to $\bm{\beta}=\bm{\beta}_{\text{F}}$ at $t=0$ , the dissipation function for the system takes the following form

	$\displaystyle\mathcal{R}_{\tau}=\dfrac{1}{k_{\text{B}}T}\left[\mathcal{H}(\bm{A}(0);\bm{\beta}_{\text{F}})-\mathcal{H}(\bm{A}(0);\bm{\beta}_{\text{I}})\right.$
	$\displaystyle-\left.(\mathcal{H}(\bm{A}(\tau);\bm{\beta}_{\text{F}})-\mathcal{H}(\bm{A}(\tau);\bm{\beta}_{\text{I}}))\right].$		(36)

We will now discuss an example of quenched systems.

II.4.1 Colloidal particle in a harmonic potential well

Consider a colloidal particle trapped in a harmonic potential $U=k\mathbf{r}^{2}/2$ , where $k$ is the stiffness of the potential. Imagine that initially the particle is in thermodynamic equilibrium with $k=k_{0}$ and the value of $k$ is instantaneously changed from $k_{0}$ to $k_{1}$ at $t=0$ [15]. Ignoring the kinetic energy, the coarse-grained Hamiltonian for this system would be simply $\mathcal{H}=U$ . Then, from Eq. (II.4), the dissipation function for a trajectory between $\mathbf{r}=\mathbf{r}_{0}$ and $\mathbf{r}=\mathbf{r}_{\tau}$ in time $\tau$ is given by

R_{\tau}=\dfrac{1}{2}(k_{0}-k_{1})(\mathbf{r}_{\tau}^{2}-\mathbf{r}^{2}_{0})

(37)

The above expression was derived by Carberry et al. [29, 15] for spatially uniform diffusion constant. As we have considered the dependence of $\Gamma_{ij}$ on $\bm{A}$ in the derivation of $\mathcal{R}_{\tau}$ , the above expression of the dissipation function is more general; it is valid for the systems having state dependent diffusion as well. In order to verify our prediction, we numerically solve the Langevin equation for a colloidal particle with the diffusion coefficient varying with position. For simplicity, we consider the 1D case. From Eq. (8), the overdamped Langevin equation for the colloidal particle reads

\dfrac{dx}{dt}=-\dfrac{1}{k_{\text{B}}T}D(x)kx+(1-\alpha)\dfrac{dD(x)}{dx}+\sqrt{2D(x)}\eta(t),

(38)

with its discreate form (see Eq. (10))

	$\displaystyle x(t+dt)=x(t)-\dfrac{1}{k_{\text{B}}T}D(\bar{x}^{\text{f}})k\bar{x}^{\text{f}}$	$\displaystyle+$	$\displaystyle(1-\alpha)\left[\dfrac{dD(x)}{dx}\right]_{\bar{x}^{\text{f}}}$		(39)
		$\displaystyle+$	$\displaystyle\sqrt{2D(\bar{x}^{\text{f}})dt}\eta_{t},$		(39)

where $\bar{x}^{\text{f}}=\alpha x(t+dt)+(1-\alpha)x(t)$ and $D(x)$ is the state-dependent diffusion. The above equation is a self-consistent equation of $x(t+dt)$ for given $x(t)$ . There are many examples of the systems having state-dependent diffusion; e.g., a colloidal particle near a wall [23]. We here consider a hypothetical system having Gaussian profile of the diffusion coefficient:

D(x)=D_{0}\exp\left(-\dfrac{x^{2}}{L^{2}}\right).

(40)

Refer to caption — Figure 1: $\ln[\mathcal{P}(X)/\mathcal{P}(-X)]$ vs $X$ for a 1D colloidal particle in a potential well $U=kx^{2}/2$ with the diffusion coefficient $D(x)=D_{0}\exp(-x^{2}/L^{2})$ , where $\mathcal{P}$ is the probability distribution function for the dissipation function given by Eq. (37). The value of $k$ is suddenly changed from $k=k_{0}$ to $k=k_{1}$ at $t=0$ . Here $L/\sqrt{k_{1}/k_{\text{B}}T}=1$ , $k_{1}/k_{0}=4$ , the time duration of the trajectory $\tau=3k_{\text{B}}T/D_{0}k_{1}$ , and 8,00,000 no. of trajectories are used for the statistics.

To obtain the trajectory of the particle, at each time step, we solve the Eq. (39) for $x(t+dt)$ with fixed point iteration method with the accuracy of $10^{-4}$ . In Fig. 1, we show $\ln[\mathcal{P}(X)/\mathcal{P}(-X)]$ vs $X$ : clearly, the dissipation function given by Eq. (37) obeys the fluctuation relation, for all the values of $\alpha$ .

III The dissipation function for active systems

In this section, we consider the active systems [30] whose dynamics is governed by the equations of motion having the following form:

\dfrac{dA_{i}}{dt}=\mathcal{F}_{i}+\mathcal{X}_{i}+N_{ij}\xi_{j}(t),

(41)

where the addition term $\mathcal{X}_{i}$ represents the active driving forces. Due to the presence of the active forces, the active systems are always away from equilibrium. However, they can achieve a nonequilibrium steady state. Writing ${\mathcal{X}}_{i}$ as the sum of two terms ${\mathcal{X}}^{\text{s}}_{i}$ and ${\mathcal{X}}^{\text{a}}_{i}$ such that $\bm{\mathcal{X}}^{\text{s}}(\bm{s}\circ\bm{A})=\bm{s}\circ\bm{\mathcal{X}}^{\text{s}}(\bm{A})$ and $\bm{\mathcal{X}}^{\text{a}}(\bm{s}\circ\bm{A})=-\bm{s}\circ\bm{\mathcal{X}}^{\text{a}}(\bm{A})$ (see Eqs. (115) & (116), Appendix G). It should be noted that $N_{ij}$ serves as a dummy matrix here as well because the form of $P$ will be the same as that in Eq. (19), except that $\mathcal{F}^{\text{s}}_{i}$ and $\mathcal{F}^{\text{a}}_{i}$ will have additional active components ${\mathcal{X}}^{\text{s}}_{i}$ and ${\mathcal{X}}^{\text{a}}_{i}$ , respectively. Following the approach used in subsection II.2, we obtain the following expression of the dissipation function:

	$\displaystyle\mathcal{R}_{\tau}(t)=\ln\dfrac{p_{t}(\bm{A}(t))}{p_{t}(\bm{s}\circ\bm{A}(t+\tau))}-\dfrac{1}{k_{\text{B}}T}\left[\mathcal{H}(\bm{A}(t+\tau))\right.$
	$\displaystyle\left.-\mathcal{H}(\bm{A}(t))\right]+\dfrac{1}{k_{\text{B}}T}\int^{t+\tau}_{t}w(t^{\prime})dt^{\prime},$		(42)

where

	$\displaystyle w(t)$	$\displaystyle=$	$\displaystyle\dfrac{\partial\mathcal{H}(\bm{A}(t))}{\partial A_{i}}\mathcal{X}^{\text{a}}_{i}(\bm{A}(t))-k_{\text{B}}T\dfrac{\partial\mathcal{X}^{\text{a}}_{i}(\bm{A}(t))}{\partial A_{i}}$		(43)
		$\displaystyle+$	$\displaystyle{{\color[rgb]{0,0,0}({\Gamma^{\text{s}}}^{-1})_{ij}}}(\bm{A}(t))\mathcal{X}^{\text{s}}_{j}(\bm{A}(t))\left[\dfrac{dA_{i}}{dt}-\mathcal{Y}^{\text{a}}_{i}(\bm{A}(t))\right]$		(43)

and

\mathcal{Y}^{\text{a}}_{i}=\mathcal{X}^{\text{a}}_{i}+\dfrac{\partial\mathcal{H}}{\partial A_{k}}\Gamma^{\text{a}}_{ki}-k_{\text{B}}T\dfrac{\partial\Gamma^{\text{a}}_{ki}}{\partial A_{k}}.

(44)

The integration in Eq. (43) is performed using midpoint rule. In contrast to passive systems, $\mathcal{R}_{\tau}(t)$ here depends on $\Gamma_{ij}$ , though not on $\alpha$ . Moreover, $\mathcal{R}_{\tau}(t)$ is trajectory-dependent, so $\left\langle\mathcal{R}_{\tau}(t)\right\rangle$ is a function of $\alpha$ because the probability density $P$ of a trajectory depends on $\alpha$ (as in passive systems, see Eq. (19)). The ensemble average of $w(t)$ is given by (with an assumption, see Appendix G)

	$\displaystyle\left\langle w(t)\right\rangle$	$\displaystyle=$	$\displaystyle\left\langle\left(\dfrac{dA_{i}}{dt}-\dfrac{\partial\mathcal{H}}{\partial A_{k}}\Gamma^{\text{a}}_{ki}+k_{\text{B}}T\dfrac{\partial\Gamma^{\text{a}}_{ki}}{\partial A_{k}}\right){{\color[rgb]{0,0,0}({\Gamma^{\text{s}}}^{-1})_{ij}}}\mathcal{X}^{\text{s}}_{j}\right\rangle$		(45)
			$\displaystyle-\left\langle\left(\dfrac{dA_{i}}{dt}-\mathcal{Y}^{\text{a}}_{i}\right){{\color[rgb]{0,0,0}({\Gamma^{\text{s}}}^{-1})_{ij}}}\mathcal{X}^{\text{a}}_{j}\right\rangle,$		(45)

where the first term is the average rate of work done by active force ${{\color[rgb]{0,0,0}({\Gamma^{\text{s}}}^{-1})_{ij}}}\mathcal{X}^{\text{s}}_{j}$ and the second term is by the active force ${{\color[rgb]{0,0,0}({\Gamma^{\text{s}}}^{-1})_{ij}}}\mathcal{X}^{\text{a}}_{j}$ . So $w(t)$ can be interpreted as the rate of work done by active forces along the trajectory at time $t$ . For $p_{t}(\bm{s}\circ\bm{A})=p_{t}(\bm{A})$ , the rate of change of dissipation function with $\tau$ (see Eqs. (34) and (III)) is given by

	$\displaystyle\dot{{s}}(t)=\dfrac{1}{k_{\text{B}}T}w(t)-\left.\dfrac{d}{dt^{\prime}}\left(\ln p_{t}(\bm{A}(t^{\prime}))+\dfrac{1}{k_{\text{B}}T}\mathcal{H}(\bm{A}(t^{\prime}))\right)\right\|_{t^{\prime}=t}.$
			(46)

and its average reads (see Appendix G)

	$\displaystyle\left\langle\dot{{s}}(t)\right\rangle=\dfrac{1}{k_{\text{B}}T}\left\langle\left(\dfrac{J_{i}(\bm{A},t)}{p_{t}(\bm{A})}-\mathcal{Y}^{\text{a}}_{i}(\bm{A})\right){{\color[rgb]{0,0,0}({\Gamma^{\text{s}}}^{-1})_{ij}}}(\bm{A})\right.$
	$\displaystyle\times\left.\left(\dfrac{J_{j}(\bm{A},t)}{p_{t}(\bm{A})}-\mathcal{Y}^{\text{a}}_{j}(\bm{A})\right)\right\rangle,$		(47)

where $J_{i}(\bm{A},t)$ is the probability current for $\bm{A}$ [23]:

	$\displaystyle J_{i}(\bm{A},t)$	$\displaystyle=$	$\displaystyle\left(-{\Gamma^{\text{s}}}_{ij}(\bm{A})\dfrac{\partial\mathcal{H}(\bm{A})}{\partial A_{j}}+\mathcal{Y}^{\text{a}}_{i}(\bm{A})+\mathcal{X}^{\text{s}}_{j}(\bm{A})\right)p_{t}(\bm{A})$		(48)
			$\displaystyle\qquad\qquad\qquad\qquad-k_{\text{B}}T\Gamma^{\text{s}}_{ij}(\bm{A})\dfrac{\partial p_{t}(\bm{A})}{dA_{j}}.$		(48)

Using Eq (6), it is easy to show that $\left\langle\dot{{s}}(t)\right\rangle>0$ , as expected from the integrated fluctuation theorem (28). Again, $k_{\text{B}}\left\langle\dot{{s}}(t)\right\rangle$ is nothing but the rate of the total entropy production of the system and the reservoir (see Appendix E).

As discussed for the passive systems in subsection II.3, for $p_{t}(\bm{s}\circ\bm{A})\neq p_{t}(\bm{A})$ case, time reversal asymmetry in $p_{t}(\bm{A})$ also contributes to irreversibility. This contribution can also be observed in the stationary states of many active systems. Active systems with polar alignment [31, 32, 33, 34] are examples of this type of system; as the velocities of the particles are globally aligned, the velocity distribution is not an even function for these systems. The passive systems, being in equilibrium in their stationary states, cannot demonstrate this irreversibility.

A few special cases for the active systems are as follows: (a) if initially the system is in equilibrium (that is, $\mathcal{X}_{i}=0$ ) and the active term $\mathcal{X}_{i}$ is switched on at $t=0$ then $p_{0}(\bm{A})=\exp(-\mathcal{H}(\bm{A})/k_{\text{B}}T)/\mathcal{Z}$ . In this case, $\mathcal{R}_{\tau}$ is just the net work done by the active forces during the trajectory (see Eq. (III)):

\mathcal{R}_{\tau}={{\color[rgb]{0,0,0}\dfrac{1}{k_{\text{B}}T}\int^{\tau}_{0}}}w(t^{\prime})dt^{\prime}.

(49)

(b) In the stationary state (i.e. in $t\to\infty$ limit), the time averaged work done by the active forces,

w_{\text{av}}=\lim_{\tau\to\infty}\dfrac{1}{\tau}\int^{t+\tau}_{t}dt^{\prime}w(t^{\prime}),

(50)

is independent of $t$ . So, in $\tau\to\infty$ limit, in Eq. (III), the last term is proportional to $\tau$ and we can ignore the first two terms. Then,

	$\displaystyle\mathcal{R}_{\tau}$	$\displaystyle\simeq$	$\displaystyle\dfrac{1}{k_{\text{B}}T}\int^{t+\tau}_{t}w(t^{\prime})dt^{\prime}.$		(51)
		$\displaystyle\simeq$	$\displaystyle\dfrac{1}{k_{\text{B}}T}w_{\text{av}}\tau.$		(52)

From Eq. (24), the probability distribution $\mathcal{P}_{\text{w}}(w_{\text{av}})$ of $w_{\text{av}}$ satisfies the relation

X=\lim_{\tau\to\infty}k_{\text{B}}T\dfrac{1}{\tau}\ln\dfrac{\mathcal{P}_{\text{w}}(w_{\text{av}}=X)}{\mathcal{P}_{\text{w}}(w_{\text{av}}=-X)},

(53)

This is called the steady-state fluctuation theorem [28].

IV Conclusion

Starting with the generic Langevin equations, using path integral approach, we first calculated the ratio of the probability densities of a trajectory and its time-reversed trajectory for passive systems using $\alpha$ -discritization: it is independent of the value of $\alpha$ . Irrespective of the value of $\alpha$ , the stationary solutions of generic Langevin equations have time reversal symmetry, so the generic Langevin equations with any value of $\alpha$ describes a passive system. Next we calculated the dissipation function for the passive systems which is found to be independent of the trajectory of the system, it depends only on the intial and the final values of the dynamical variables of the system. Furthermore, it is not an explicit function of coefficients of the generic Langevin equations. We also verify the fluctuation theorem for a 1D particle trapped in a potential well whose stiffness is suddenly changed, with the state-dependent diffusion. Finally, we obtained the expression of the dissipation function for active systems and defined the work done by the active forces. For both passive and active systems, the average of the rate of change of dissipation function with the duration of the trajectory is just the entropy production rate of the system and the reservoir.

Appendix A The probability density of a trajectory for passive systems

The generic Langevin equations for passive systems in discrete form (see Eq. (10)):

dA_{i}(l)=\epsilon\mathcal{F}_{i}(\bar{\bm{A}}^{\text{f}}_{l})+\sqrt{\epsilon}N_{ij}(\bar{\bm{A}}^{\text{f}}_{l})\xi_{j}^{l},

(54)

where $dA_{i}(l)\equiv A_{i}(\epsilon l)-A_{i}(\epsilon(l-1))$ , $\bar{\bm{A}}^{\text{f}}_{l}\equiv\alpha\bm{A}(\epsilon l)+(1-\alpha)\bm{A}(\epsilon(l-1))$ , and

\mathcal{F}_{i}\equiv-\Gamma_{ij}\dfrac{\partial\mathcal{H}}{\partial A_{j}}+k_{\text{B}}T\dfrac{\partial\Gamma_{ij}}{\partial A_{j}}-\alpha N_{lj}\dfrac{\partial N_{ij}}{\partial A_{l}}.

(55)

Solving the above equations for $\xi_{i}^{l}$ , we obtain

\xi_{i}^{l}=\dfrac{1}{\sqrt{\epsilon}}{{\color[rgb]{0,0,0}(N^{-1})_{ij}}}(\bar{\bm{A}}^{\text{f}}_{l})(dA_{j}(l)-\epsilon\mathcal{F}_{j}(\bar{\bm{A}}^{\text{f}}_{l})).

(56)

Since $\xi_{i}^{l}$ are the uncorrelated series of random numbers having normal distribution with zero mean and variance one, the probability density function of a trajectory of the system $(\bm{A}_{0},\bm{A}_{1},\bm{A}_{2},.....\bm{A}_{N})$ (here $\bm{A}_{l}\equiv\bm{A}(l\epsilon)$ ) between $t=0$ and $t=\tau\equiv N\epsilon$ is given by [12]

P=p_{0}(\bm{A}_{0})\left|\mathcal{J}\right|\prod^{N}_{l=1}\dfrac{1}{(2\pi)^{n/2}}\exp\left[-\dfrac{1}{2\epsilon}\left[dA_{i}(l)-\epsilon\mathcal{F}_{i}(\bar{\bm{A}}^{\text{f}}_{l})\right]{{\color[rgb]{0,0,0}(N^{-1})_{ki}}}(\bar{\bm{A}}^{\text{f}}_{l}){{\color[rgb]{0,0,0}(N^{-1})_{kj}}}(\bar{\bm{A}}^{\text{f}}_{l})\left[dA_{j}(l)-\epsilon\mathcal{F}_{j}(\bar{\bm{A}}^{\text{f}}_{l})\right]\right],

(57)

where $p_{0}(\bm{A})$ is the probability distribution of $\bm{A}$ at $t=0$ and $\mathcal{J}$ is the Jacobean determinant for the transformation of the variables of the probability density function from the $\xi_{i}^{l}$ to $A_{j}(\epsilon m)$ . From Eq. (56), the $Nn\times Nn$ Jacobean matrix for the variable transformation is given by

$\displaystyle\mathcal{J}^{il}_{jm}$	$\displaystyle=$	$\displaystyle\dfrac{\partial\xi_{i}^{l}}{\partial A_{j}(\epsilon m)}$	(58)
	$\displaystyle=$	$\displaystyle\dfrac{1}{\epsilon^{1/2}}\left[\left[\left[\dfrac{\partial{{\color[rgb]{0,0,0}(N^{-1})_{ik}}}}{\partial A_{j}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}dA_{k}(l)-\epsilon\left[\dfrac{\partial{{\color[rgb]{0,0,0}(N^{-1})_{ik}}}}{\partial A_{j}}\mathcal{F}_{k}+\dfrac{\partial\mathcal{F}_{k}}{\partial A_{j}}{{\color[rgb]{0,0,0}(N^{-1})_{ik}}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}\right](\alpha\delta_{lm}+(1-\alpha)\delta_{(l-1)m})\right.$
		$\displaystyle\Bigg{.}+{{\color[rgb]{0,0,0}(N^{-1})_{ij}}}(\bar{\bm{A}}^{\text{f}}_{l})(\delta_{lm}-\delta_{(l-1)m})\Bigg{]}$
	$\displaystyle=$	$\displaystyle\dfrac{1}{\epsilon^{1/2}}\left[\left[\left[\dfrac{\partial{{\color[rgb]{0,0,0}(N^{-1})_{ik}}}}{\partial A_{j}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}\left(dA_{k}(l)-\epsilon\mathcal{F}_{k}(\bar{\bm{A}}^{\text{f}}_{l})\right)-\epsilon\left[\dfrac{\partial\mathcal{F}_{k}}{\partial A_{j}}{{\color[rgb]{0,0,0}(N^{-1})_{ik}}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}\right](\alpha\delta_{lm}+(1-\alpha)\delta_{(l-1)m})\right.$
		$\displaystyle\Bigg{.}+{{\color[rgb]{0,0,0}(N^{-1})_{ij}}}(\bar{\bm{A}}^{\text{f}}_{l})(\delta_{lm}-\delta_{(l-1)m})\Bigg{]}.$

The above matrix is a block triangular matrix of $n\times n$ submatrices with fixed ( $l,m$ ), so its determinant will be the multiplication of all the diagonal submatrices (i.e. with $l=m$ ):

\mathcal{J}=\prod^{N}_{l=1}\dfrac{1}{\epsilon^{n/2}}{{\color[rgb]{0,0,0}\text{det}\left(\bm{M}(l)\right)}},

(59)

where

	$\displaystyle M_{ij}(l)$	$\displaystyle=$	$\displaystyle{{\color[rgb]{0,0,0}(N^{-1})_{ij}}}(\bar{\bm{A}}^{\text{f}}_{l})+\alpha\left[\dfrac{\partial{{\color[rgb]{0,0,0}(N^{-1})_{ik}}}}{\partial A_{j}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}\left(dA_{k}(l)-\epsilon\mathcal{F}_{k}(\bar{\bm{A}}^{\text{f}}_{l})\right)-\epsilon\alpha\left[\dfrac{\partial\mathcal{F}_{k}}{\partial A_{j}}{{\color[rgb]{0,0,0}(N^{-1})_{ik}}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}$		(60)
		$\displaystyle=$	$\displaystyle{{\color[rgb]{0,0,0}(N^{-1})_{ip}}}(\bar{\bm{A}}^{\text{f}}_{l})\left[\delta_{pj}+\alpha\left[N_{pq}\dfrac{\partial{{\color[rgb]{0,0,0}(N^{-1})_{qk}}}}{\partial A_{j}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}\left(dA_{k}(l)-\epsilon\mathcal{F}_{k}(\bar{\bm{A}}^{\text{f}}_{l})\right)-\alpha\epsilon\left[\dfrac{\partial\mathcal{F}_{p}}{\partial A_{j}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}\right].$		(60)

Using the power series expression of $\ln({{\color[rgb]{0,0,0}\text{det}\left(\bm{I}+\delta\bm{\mathcal{B}}\right)}})$ for any matrix $\bm{\mathcal{B}}$ in $\delta\to 0$ limit (such that $||\delta\bm{\mathcal{B}}||<1$ ), that is,

\ln({{\color[rgb]{0,0,0}\text{det}\left(\bm{I}+\delta\bm{\mathcal{B}}\right)}})=\left[\text{Tr}[\bm{\mathcal{B}}]\delta-\dfrac{1}{2}\text{Tr}[\bm{\mathcal{B}}\cdot\bm{\mathcal{B}}]\delta^{2}+\mathcal{O}(\delta^{3})\right],

(61)

the determinant of $\bm{M}(l)$ can be written as

$\displaystyle{{\color[rgb]{0,0,0}\text{det}\left(\bm{M}(l)\right)}}$	$\displaystyle=$	$\displaystyle{{\color[rgb]{0,0,0}\text{det}\left(\bm{N}^{-1}(\bar{\bm{A}}^{\text{f}}_{l})\right)}}\exp\left[\alpha\left[\left[N_{jq}\dfrac{\partial{{\color[rgb]{0,0,0}(N^{-1})_{qk}}}}{\partial A_{j}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}\left(dA_{k}(l)-\epsilon\mathcal{F}_{k}(\bar{\bm{A}}^{\text{f}}_{l})\right)-\epsilon\left[\dfrac{\partial\mathcal{F}_{j}}{\partial A_{j}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}\right]\right.$	(62)
		$\displaystyle\left.{{\color[rgb]{0,0,0}-\dfrac{1}{2}\alpha^{2}\left[N_{pq}\dfrac{\partial(N^{-1})_{qk}}{\partial A_{j}}N_{jr}\dfrac{\partial(N^{-1})_{ri}}{\partial A_{p}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}dA_{k}(l)dA_{i}(l)+\mathcal{O}(\epsilon^{3/2})}}\right],$
	$\displaystyle=$	$\displaystyle{{\color[rgb]{0,0,0}\text{det}\left(\bm{N}^{-1}(\bar{\bm{A}}^{\text{f}}_{l})\right)}}\exp\left[-\alpha\left[\left[{{\color[rgb]{0,0,0}(N^{-1})_{qk}}}\dfrac{\partial N_{jq}}{\partial A_{j}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}\left(dA_{k}(l)-\epsilon\mathcal{F}_{k}(\bar{\bm{A}}^{\text{f}}_{l})\right)+\epsilon\left[\dfrac{\partial\mathcal{F}_{j}}{\partial A_{j}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}\right]\right.$
		$\displaystyle\left.{{\color[rgb]{0,0,0}-\dfrac{1}{2}\alpha^{2}\epsilon\left[\dfrac{\partial N_{pm}}{\partial A_{j}}\dfrac{\partial N_{jm}}{\partial A_{p}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}+\mathcal{O}(\epsilon^{3/2})}}\right].$

Note that $dA_{k}(l)$ has a $\epsilon^{1/2}$ -term, so $dA_{i}(l)dA_{k}(l)$ is of the order of $\epsilon$ . Relations (85) & (90) (see Appendix D) have been used to get the last term of the above equation. Eq. (60) then reads

	$\displaystyle\mathcal{J}$	$\displaystyle=$	$\displaystyle\prod^{N}_{l=1}\dfrac{1}{\epsilon^{n/2}}{{\color[rgb]{0,0,0}\text{det}\left(\bm{N}^{-1}(\bar{\bm{A}}^{\text{f}}_{l})\right)}}\exp\left[-\alpha\left[\left[{{\color[rgb]{0,0,0}(N^{-1})_{qk}}}\dfrac{\partial N_{jq}}{\partial A_{j}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}\left(dA_{k}(l)-\epsilon\mathcal{F}_{k}(\bar{\bm{A}}^{\text{f}}_{l})\right)+\epsilon\left[\dfrac{\partial\mathcal{F}_{j}}{\partial A_{j}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}\right]\right.$		(63)
			$\displaystyle\left.{{\color[rgb]{0,0,0}-\dfrac{1}{2}\alpha^{2}\epsilon\left[\dfrac{\partial N_{pm}}{\partial A_{j}}\dfrac{\partial N_{jm}}{\partial A_{p}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}+\mathcal{O}(\epsilon^{3/2})}}\right].$		(63)

Substituting the above expression of $\mathcal{J}$ into Eq.(57):

	$\displaystyle P$	$\displaystyle=$	$\displaystyle p_{0}(\bm{A}_{0})\prod^{N}_{l=1}\left\{\dfrac{1}{(2\pi\epsilon)^{n/2}}{{\color[rgb]{0,0,0}\left\|\text{det}\left(\bm{N}^{-1}(\bar{\bm{A}}^{\text{f}}_{l})\right)\right\|}}\exp\left[-\dfrac{1}{2\epsilon}\left[dA_{i}(l)-\epsilon\mathcal{F}_{i}(\bar{\bm{A}}^{\text{f}}_{l})\right]{{\color[rgb]{0,0,0}(N^{-1})_{ki}}}(\bar{\bm{A}}^{\text{f}}_{l}){{\color[rgb]{0,0,0}(N^{-1})_{kj}}}(\bar{\bm{A}}^{\text{f}}_{l})\left[dA_{j}(l)-\epsilon\mathcal{F}_{j}(\bar{\bm{A}}^{\text{f}}_{l})\right]\right]\right.$		(64)
			$\displaystyle\left.\times\exp\left[-\alpha\left[\left(dA_{i}(l)-\epsilon\mathcal{F}_{i}(\bar{\bm{A}}^{\text{f}}_{l})\right)\left[{{\color[rgb]{0,0,0}(N^{-1})_{ji}}}\dfrac{\partial N_{kj}}{\partial A_{k}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}+\epsilon\left[\dfrac{\partial\mathcal{F}_{i}}{\partial A_{i}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}\right]{{\color[rgb]{0,0,0}-\dfrac{1}{2}\alpha^{2}\epsilon\left[\dfrac{\partial N_{pm}}{\partial A_{j}}\dfrac{\partial N_{jm}}{\partial A_{p}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}+\mathcal{O}(\epsilon^{3/2})}}\right]\right\}.$		(64)

From Eq. (2), ${{\color[rgb]{0,0,0}(N^{-1})_{ki}}}{{\color[rgb]{0,0,0}(N^{-1})_{kj}}}={{\color[rgb]{0,0,0}(\Gamma^{s}}}^{-1})_{ij}/2k_{\text{B}}T$ , so

	$\displaystyle P$	$\displaystyle=$	$\displaystyle p_{0}(\bm{A}_{0})\prod^{N}_{l=1}\left\{\dfrac{{{\color[rgb]{0,0,0}(2k_{\text{B}}T)^{-1/2}}}}{(2\pi\epsilon)^{n/2}}{{\color[rgb]{0,0,0}\text{det}\left(\bm{\Gamma^{\text{s}}}(\bar{\bm{A}}^{\text{f}}_{l})\right)^{-1/2}}}\exp\left[-\dfrac{1}{4\epsilon k_{\text{B}}T}\left[dA_{i}(l)-\epsilon\mathcal{F}_{i}(\bar{\bm{A}}^{\text{f}}_{l})\right]{{{\color[rgb]{0,0,0}(\Gamma^{s}}}^{-1})_{ij}}(\bar{\bm{A}}^{\text{f}}_{l})\left[dA_{j}(l)-\epsilon\mathcal{F}_{j}(\bar{\bm{A}}^{\text{f}}_{l})\right]\right]\right.$
			$\displaystyle\left.\times\exp\left[-\alpha\left[\left(dA_{i}(l)-\epsilon\mathcal{F}_{i}(\bar{\bm{A}}^{\text{f}}_{l})\right)\left[{{\color[rgb]{0,0,0}(N^{-1})_{ji}}}\dfrac{\partial N_{kj}}{\partial A_{k}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}+\epsilon\left[\dfrac{\partial\mathcal{F}_{i}}{\partial A_{i}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}\right]{{\color[rgb]{0,0,0}-\dfrac{1}{2}\alpha^{2}\epsilon\left[\dfrac{\partial N_{pm}}{\partial A_{j}}\dfrac{\partial N_{jm}}{\partial A_{p}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}+\mathcal{O}(\epsilon^{3/2})}}\right]\right\}.$

Now let us break $\mathcal{F}_{i}$ into two terms,

\mathcal{F}^{0}_{i}=-\Gamma_{ij}\dfrac{\partial\mathcal{H}}{\partial A_{j}}+k_{\text{B}}T\dfrac{\partial\Gamma_{ij}}{\partial A_{j}}

(66)

and

\mathcal{F}^{\text{N}}_{i}=-\alpha N_{lj}\dfrac{\partial N_{ij}}{\partial A_{l}}.

(67)

Replacing $\mathcal{F}_{i}$ by $\mathcal{F}^{0}_{i}+\mathcal{F}^{\text{N}}_{i}$ in Eq. (A):

$\displaystyle P$	$\displaystyle=$	$\displaystyle p_{0}(\bm{A}_{0})\prod^{N}_{l=1}\left\{\dfrac{(2k_{\text{B}}T)^{-1/2}}{(2\pi\epsilon)^{n/2}}\text{det}\left(\bm{\Gamma^{\text{s}}}(\bar{\bm{A}}^{\text{f}}_{l})\right)^{-1/2}\exp\left[-\dfrac{1}{4\epsilon k_{\text{B}}T}\left[dA_{i}(l)-\epsilon\mathcal{F}^{0}_{i}(\bar{\bm{A}}^{\text{f}}_{l})\right]{{(\Gamma^{s}}^{-1})_{ij}}(\bar{\bm{A}}^{\text{f}}_{l})\left[dA_{j}(l)-\epsilon\mathcal{F}^{0}_{j}(\bar{\bm{A}}^{\text{f}}_{l})\right]\right]\right.$	(68)
		$\displaystyle\times\exp\left[-\alpha\left[\left(dA_{i}(l)-\epsilon\mathcal{F}^{0}_{i}(\bar{\bm{A}}^{\text{f}}_{l})\right)\left[(N^{-1})_{ji}\dfrac{\partial N_{kj}}{\partial A_{k}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}+\epsilon\left[\dfrac{\partial\mathcal{F}^{0}_{i}}{\partial A_{i}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}\right]\right]$
		$\displaystyle\times\exp\left[\dfrac{1}{2k_{\text{B}}T}\mathcal{F}^{\text{N}}_{i}(\bar{\bm{A}}^{\text{f}}_{l}){{(\Gamma^{s}}^{-1})_{ij}}(\bar{\bm{A}}^{\text{f}}_{l})\left[dA_{j}(l)-\epsilon\mathcal{F}^{0}_{j}(\bar{\bm{A}}^{\text{f}}_{l})\right]-\dfrac{\epsilon}{4k_{\text{B}}T}\left[\mathcal{F}^{\text{N}}_{i}{{(\Gamma^{s}}^{-1})_{ij}}\mathcal{F}^{\text{N}}_{j}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}\right]$
		$\displaystyle\left.\times\exp\left[\alpha\epsilon\left[\mathcal{F}^{\text{N}}_{i}(N^{-1})_{ji}\dfrac{\partial N_{kj}}{\partial A_{k}}-\dfrac{\partial\mathcal{F}^{\text{N}}_{i}}{\partial A_{i}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}-\dfrac{1}{2}\alpha^{2}\epsilon\left[\dfrac{\partial N_{pm}}{\partial A_{j}}\dfrac{\partial N_{jm}}{\partial A_{p}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}+\mathcal{O}(\epsilon^{3/2})\right]\right\}.$

Using Eq. (6) (that is, $N_{ik}N_{jk}=2k_{\text{B}}T\Gamma^{\text{s}}_{ij}$ ), one can write

$\displaystyle(N^{-1})_{ji}\dfrac{\partial N_{kj}}{\partial A_{k}}$	$\displaystyle=$	$\displaystyle\delta_{im}(N^{-1})_{jm}\dfrac{\partial N_{kj}}{\partial A_{k}}$	(69)
	$\displaystyle=$	$\displaystyle({\Gamma^{s}}^{-1})_{ip}{\Gamma^{s}}_{pm}(N^{-1})_{jm}\dfrac{\partial N_{kj}}{\partial A_{k}}$
	$\displaystyle=$	$\displaystyle\dfrac{1}{2k_{\text{B}}T}({\Gamma^{s}}^{-1})_{ip}N_{pj}\dfrac{\partial N_{kj}}{\partial A_{k}}$
	$\displaystyle=$	$\displaystyle\dfrac{1}{2k_{\text{B}}T}({\Gamma^{s}}^{-1})_{ip}\left(\dfrac{\partial(N_{kj}N_{pj})}{\partial A_{k}}-N_{kj}\dfrac{\partial N_{pj}}{\partial A_{k}}\right)$
	$\displaystyle=$	$\displaystyle({\Gamma^{s}}^{-1})_{ip}\dfrac{\partial{\Gamma^{s}}_{pk}}{\partial A_{k}}+\dfrac{1}{2k_{\text{B}}T\alpha}({\Gamma^{s}}^{-1})_{ip}\mathcal{F}^{\text{N}}_{p}.$

With the above expression, Eq. (68) reduces to

$\displaystyle P$	$\displaystyle=$	$\displaystyle p_{0}(\bm{A}_{0})\prod^{N}_{l=1}\left\{\dfrac{(2k_{\text{B}}T)^{-1/2}}{(2\pi\epsilon)^{n/2}}\text{det}\left(\bm{\Gamma^{\text{s}}}(\bar{\bm{A}}^{\text{f}}_{l})\right)^{-1/2}\exp\Biggl{[}-\dfrac{1}{4\epsilon k_{\text{B}}T}\left[dA_{i}(l)-\epsilon\mathcal{F}^{0}_{i}(\bar{\bm{A}}^{\text{f}}_{l})\right]{{{\color[rgb]{0,0,0}(\Gamma^{s}}}^{-1})_{ij}}(\bar{\bm{A}}^{\text{f}}_{l})\left[dA_{j}(l)-\epsilon\mathcal{F}^{0}_{j}(\bar{\bm{A}}^{\text{f}}_{l})\right]\Biggr{.}\right.$
		$\displaystyle\left.-\alpha\left[\left(dA_{i}(l)-\epsilon\mathcal{F}^{0}_{i}(\bar{\bm{A}}^{\text{f}}_{l})\right)\left[({\Gamma^{s}}^{-1})_{ij}\dfrac{\partial\Gamma^{s}_{jk}}{\partial A_{k}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}+\epsilon\left[\dfrac{\partial\mathcal{F}^{0}_{i}}{\partial A_{i}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}\right]\right]$
		$\displaystyle\times\left.\exp\left[\dfrac{\epsilon}{4k_{\text{B}}T}\left[\mathcal{F}^{\text{N}}_{i}({\Gamma^{s}}^{-1})_{ij}\left(\mathcal{F}^{\text{N}}_{j}+4\alpha k_{\text{B}}T\dfrac{\partial\Gamma^{s}_{jk}}{\partial A_{k}}\right)-4\alpha k_{\text{B}}T\dfrac{\partial\mathcal{F}^{\text{N}}_{i}}{\partial A_{i}}-2k_{\text{B}}T\alpha^{2}\dfrac{\partial N_{pm}}{\partial A_{j}}\dfrac{\partial N_{jm}}{\partial A_{p}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}+\mathcal{O}(\epsilon^{3/2})\right]\right\}.$

Using the relation $N_{ik}N_{jk}=2k_{\text{B}}T\Gamma^{\text{s}}_{ij}$ , further simplifying the last term of the above equation yields

$\displaystyle P$	$\displaystyle=$	$\displaystyle p_{0}(\bm{A}_{0})\prod^{N}_{l=1}\left\{\dfrac{(2k_{\text{B}}T)^{-1/2}}{(2\pi\epsilon)^{n/2}}\text{det}\left(\bm{\Gamma^{\text{s}}}(\bar{\bm{A}}^{\text{f}}_{l})\right)^{-1/2}\exp\Biggl{[}-\dfrac{1}{4\epsilon k_{\text{B}}T}\left[dA_{i}(l)-\epsilon\mathcal{F}^{0}_{i}(\bar{\bm{A}}^{\text{f}}_{l})\right]{{{\color[rgb]{0,0,0}(\Gamma^{s}}}^{-1})_{ij}}(\bar{\bm{A}}^{\text{f}}_{l})\left[dA_{j}(l)-\epsilon\mathcal{F}^{0}_{j}(\bar{\bm{A}}^{\text{f}}_{l})\right]\Biggr{.}\right.$	(71)
		$\displaystyle\left.-\alpha\left[\left(dA_{i}(l)-\epsilon\mathcal{F}^{0}_{i}(\bar{\bm{A}}^{\text{f}}_{l})\right)\left[({\Gamma^{s}}^{-1})_{ij}\dfrac{\partial\Gamma^{s}_{jk}}{\partial A_{k}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}+\epsilon\left[\dfrac{\partial\mathcal{F}^{0}_{i}}{\partial A_{i}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}\right]\right]$
		$\displaystyle\times\left.\exp\left[\alpha^{2}\epsilon k_{\text{B}}T\left[\dfrac{\partial^{2}\Gamma^{s}_{ij}}{\partial A_{i}\partial A_{j}}-\dfrac{\partial\Gamma^{s}_{ik}}{\partial A_{k}}({\Gamma^{s}}^{-1})_{ij}\dfrac{\partial\Gamma^{s}_{jp}}{\partial A_{p}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}+\mathcal{O}(\epsilon^{3/2})\right]\right\}.$

We further split $\mathcal{F}^{0}_{i}$ into the two terms,

\mathcal{F}^{\text{s}}_{i}(\bm{A})=-\Gamma^{\text{s}}_{ij}\dfrac{\partial\mathcal{H}}{\partial A_{j}}+k_{\text{B}}T\dfrac{\partial\Gamma^{\text{s}}_{ij}}{\partial A_{j}}

(72)

and

\mathcal{F}^{a}_{i}(\bm{A})=-\Gamma^{a}_{ij}\dfrac{\partial\mathcal{H}}{\partial A_{j}}+k_{\text{B}}T\dfrac{\partial\Gamma^{\text{a}}_{ij}}{\partial A_{j}},

(73)

such that, under time reversal (see (17) and (18)),

	$\displaystyle\bm{\mathcal{F}}^{\text{s}}(\bm{A})\to\bm{\mathcal{F}}^{\text{s}}(\bm{s}\circ\bm{A})=\bm{s}\circ\bm{\mathcal{F}}^{\text{s}}(\bm{A}),$		(74)
	$\displaystyle\bm{\mathcal{F}}^{\text{a}}(\bm{A})\to\bm{\mathcal{F}}^{\text{a}}(\bm{s}\circ\bm{A})=-\bm{s}\circ\bm{\mathcal{F}}^{\text{a}}(\bm{A}).$		(75)

Eq. (A) then becomes

$\displaystyle P$	$\displaystyle=$	$\displaystyle p_{0}(\bm{A}_{0})\prod^{N}_{l=1}\left\{\dfrac{(2k_{\text{B}}T)^{-1/2}}{(2\pi\epsilon)^{n/2}}\exp\Biggl{[}-\dfrac{1}{4\epsilon k_{\text{B}}T}\left[dA_{i}(l)-\epsilon\mathcal{F}^{\text{s}}_{i}(\bar{\bm{A}}^{\text{f}}_{l})-\epsilon\mathcal{F}^{\text{a}}_{i}(\bar{\bm{A}}^{\text{f}}_{l})\right]{{{\color[rgb]{0,0,0}(\Gamma^{s}}}^{-1})_{ij}}(\bar{\bm{A}}^{\text{f}}_{l})\left[dA_{j}(l)-\epsilon\mathcal{F}^{\text{s}}_{j}(\bar{\bm{A}}^{\text{f}}_{l})-\epsilon\mathcal{F}^{\text{a}}_{j}(\bar{\bm{A}}^{\text{f}}_{l})\right]\Biggr{.}\right.$	(76)
		$\displaystyle\left.-\alpha\left[\left(dA_{i}(l)-\epsilon\mathcal{F}^{\text{s}}_{i}(\bar{\bm{A}}^{\text{f}}_{l})-\epsilon\mathcal{F}^{\text{a}}_{i}(\bar{\bm{A}}^{\text{f}}_{l})\right)\left[({\Gamma^{s}}^{-1})_{ij}\dfrac{\partial\Gamma^{s}_{jk}}{\partial A_{k}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}+\epsilon\left[\dfrac{\partial\mathcal{F}^{\text{s}}_{i}}{\partial A_{i}}+\dfrac{\partial\mathcal{F}^{\text{a}}_{i}}{\partial A_{i}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}\right]\right]\text{det}\left(\bm{\Gamma^{\text{s}}}(\bar{\bm{A}}^{\text{f}}_{l})\right)^{-1/2}$
		$\displaystyle\times\left.\exp\left[\alpha^{2}\epsilon k_{\text{B}}T\left[\dfrac{\partial^{2}\Gamma^{s}_{ij}}{\partial A_{i}\partial A_{j}}-\dfrac{\partial\Gamma^{s}_{ik}}{\partial A_{k}}({\Gamma^{s}}^{-1})_{ij}\dfrac{\partial\Gamma^{s}_{jp}}{\partial A_{p}}\right]_{\bar{\bm{A}}^{\text{f}}_{l}}+\mathcal{O}(\epsilon^{3/2})\right]\right\}.$

Clearly, for given $\Gamma^{\text{s}}_{ij}$ , $P$ is independent of the choice of $N_{ij}$ .

Appendix B The probability density for the time-reversed trajectory

As the time-reversed trajectory of the trajectory $(\bm{A}_{0},\bm{A}_{1},\bm{A}_{2},.....\bm{A}_{N})$ is $(\bm{s}\circ\bm{A}_{N},\bm{s}\circ\bm{A}_{N-1},.....\bm{s}\circ\bm{A}_{1})$ , under time reversal, $\bm{A}_{l}\to\bm{s}\circ\bm{A}_{N-l}$ and therefore,

$\displaystyle dA_{i}(l)$	$\displaystyle=$	$\displaystyle A_{i}(\epsilon l)-A_{i}(\epsilon(l-1))$	(77)
	$\displaystyle\to$	$\displaystyle s_{i}(A_{i}(\epsilon(N-l))-A_{i}(\epsilon(N-l+1))$
	$\displaystyle\to$	$\displaystyle-s_{i}dA_{i}(N-l+1),$

(Einstein’s convention is not used here) and

$\displaystyle\bar{\bm{A}}^{\text{f}}_{l}$	$\displaystyle=$	$\displaystyle\alpha\bm{A}_{l}+(1-\alpha)\bm{A}_{l-1}$	(78)
	$\displaystyle\to$	$\displaystyle(\alpha\bm{s}\circ\bm{A}_{N-l}+(1-\alpha)\bm{s}\circ\bm{A}_{N-l+1})$
	$\displaystyle\to$	$\displaystyle\bm{s}\circ\bar{\bm{A}}^{\text{r}}_{N-l+1},$

where $\bar{\bm{A}}^{\text{r}}_{l}\equiv(1-\alpha)\bm{A}(l)+\alpha\bm{A}(l-1)$ . With the above transformations, using the relation $\Gamma^{\text{s}}_{ij}=s_{i}s_{j}\Gamma^{\text{s}}_{ij}$ and Eqs. (74), (75) & (76), we obtain the following expression of the probability density of the time-reversed trajectory:

$\displaystyle P_{r}$	$\displaystyle=$	$\displaystyle\prod^{N}_{l=1}\left\{\dfrac{(2k_{\text{B}}T)^{-1/2}}{(2\pi\epsilon)^{n/2}}\exp\Biggl{[}-\dfrac{1}{4\epsilon k_{\text{B}}T}\left[-dA_{i}(l^{\prime})-\epsilon\mathcal{F}^{\text{s}}_{i}(\bar{\bm{A}}^{\text{r}}_{l^{\prime}})+\epsilon\mathcal{F}^{\text{a}}_{i}(\bar{\bm{A}}^{\text{r}}_{l^{\prime}})\right]{{{\color[rgb]{0,0,0}(\Gamma^{s}}}^{-1})_{ij}}(\bar{\bm{A}}^{\text{r}}_{l^{\prime}})\left[-dA_{j}(l^{\prime})-\epsilon\mathcal{F}^{\text{s}}_{j}(\bar{\bm{A}}^{\text{r}}_{l^{\prime}})+\epsilon\mathcal{F}^{\text{a}}_{j}(\bar{\bm{A}}^{\text{r}}_{l^{\prime}})\right]\Biggr{.}\right.$	(79)
		$\displaystyle\left.-\alpha\left[\left(-dA_{i}(l^{\prime})-\epsilon\mathcal{F}^{\text{s}}_{i}(\bar{\bm{A}}^{\text{r}}_{l^{\prime}})+\epsilon\mathcal{F}^{\text{a}}_{i}(\bar{\bm{A}}^{\text{r}}_{l^{\prime}})\right)\left[({\Gamma^{s}}^{-1})_{ij}\dfrac{\partial\Gamma^{s}_{jk}}{\partial A_{k}}\right]_{\bar{\bm{A}}^{\text{r}}_{l^{\prime}}}+\epsilon\left[\dfrac{\partial\mathcal{F}^{\text{s}}_{i}}{\partial A_{i}}-\dfrac{\partial\mathcal{F}^{\text{a}}_{i}}{\partial A_{i}}\right]_{\bar{\bm{A}}^{\text{r}}_{l^{\prime}}}\right]\right]\text{det}\left(\bm{\Gamma^{\text{s}}}(\bar{\bm{A}}^{\text{r}}_{l^{\prime}})\right)^{-1/2}$
		$\displaystyle\times\left.\exp\left[\alpha^{2}\epsilon k_{\text{B}}T\left[\dfrac{\partial^{2}\Gamma^{s}_{ij}}{\partial A_{i}\partial A_{j}}-\dfrac{\partial\Gamma^{s}_{ik}}{\partial A_{k}}({\Gamma^{s}}^{-1})_{ij}\dfrac{\partial\Gamma^{s}_{jp}}{\partial A_{p}}\right]_{\bar{\bm{A}}^{\text{r}}_{l^{\prime}}}+\mathcal{O}(\epsilon^{3/2})\right]\right\}p_{0}(\bm{s}\circ\bm{A}_{N}),$

where $l^{\prime}=N-l+1$ . In the above equation, the index $l^{\prime}$ runs from $N$ to 1 so we can replace $\prod^{N}_{l=1}$ by $\prod^{1}_{l^{\prime}=N}\equiv\prod^{N}_{l^{\prime}=1}$ . Hence

$\displaystyle P_{r}$	$\displaystyle=$	$\displaystyle\prod^{N}_{l=1}\left\{\dfrac{(2k_{\text{B}}T)^{-1/2}}{(2\pi\epsilon)^{n/2}}\exp\Biggl{[}-\dfrac{1}{4\epsilon k_{\text{B}}T}\left[-dA_{i}(l)-\epsilon\mathcal{F}^{\text{s}}_{i}(\bar{\bm{A}}^{\text{r}}_{l})+\epsilon\mathcal{F}^{\text{a}}_{i}(\bar{\bm{A}}^{\text{r}}_{l})\right]{{{\color[rgb]{0,0,0}(\Gamma^{s}}}^{-1})_{ij}}(\bar{\bm{A}}^{\text{r}}_{l})\left[-dA_{j}(l)-\epsilon\mathcal{F}^{\text{s}}_{j}(\bar{\bm{A}}^{\text{r}}_{l})+\epsilon\mathcal{F}^{\text{a}}_{j}(\bar{\bm{A}}^{\text{r}}_{l})\right]\Biggr{.}\right.$	(80)
		$\displaystyle\left.-\alpha\left[\left(-dA_{i}(l)-\epsilon\mathcal{F}^{\text{s}}_{i}(\bar{\bm{A}}^{\text{r}}_{l})+\epsilon\mathcal{F}^{\text{a}}_{i}(\bar{\bm{A}}^{\text{r}}_{l})\right)\left[({\Gamma^{s}}^{-1})_{ij}\dfrac{\partial\Gamma^{s}_{jk}}{\partial A_{k}}\right]_{\bar{\bm{A}}^{\text{r}}_{l}}+\epsilon\left[\dfrac{\partial\mathcal{F}^{\text{s}}_{i}}{\partial A_{i}}-\dfrac{\partial\mathcal{F}^{\text{a}}_{i}}{\partial A_{i}}\right]_{\bar{\bm{A}}^{\text{r}}_{l}}\right]\right]\text{det}\left(\bm{\Gamma^{\text{s}}}(\bar{\bm{A}}^{\text{r}}_{l})\right)^{-1/2}$
		$\displaystyle\times\left.\exp\left[\alpha^{2}\epsilon k_{\text{B}}T\left[\dfrac{\partial^{2}\Gamma^{s}_{ij}}{\partial A_{i}\partial A_{j}}-\dfrac{\partial\Gamma^{s}_{ik}}{\partial A_{k}}({\Gamma^{s}}^{-1})_{ij}\dfrac{\partial\Gamma^{s}_{jp}}{\partial A_{p}}\right]_{\bar{\bm{A}}^{\text{r}}_{l}}+\mathcal{O}(\epsilon^{3/2})\right]\right\}p_{0}(\bm{s}\circ\bm{A}_{N}).$

Appendix C Calculation of the ratio between the probability densities of a trajectory and its time-reversed trajectory

Using relations (86) & (87), expanding $N_{ij}(\bar{\bm{A}}^{\text{f}}_{l})$ and $N_{ij}(\bar{\bm{A}}^{\text{r}}_{l})$ around $\bm{A}=\bar{\bm{A}}_{l}\equiv(\bm{A}_{l}+\bm{A}_{l-1})/2$ :

	$\displaystyle N_{ij}(\bar{\bm{A}}^{\text{f}}_{l})$	$\displaystyle=$	$\displaystyle N_{ij}(\bar{\bm{A}}_{l})+\dfrac{2\alpha-1}{2}\left[\dfrac{\partial N_{ij}}{\partial A_{k}}\right]_{\bar{\bm{A}}_{l}}dA_{k}(l)+\dfrac{1}{2}\left(\dfrac{2\alpha-1}{2}\right)^{2}\left[\dfrac{\partial^{2}N_{ij}}{\partial A_{k}\partial A_{m}}\right]_{\bar{\bm{A}}_{l}}dA_{k}(l)dA_{m}(l)+\mathcal{O}(\epsilon^{3/2}),$		(81)
	$\displaystyle N_{ij}(\bar{\bm{A}}^{\text{r}}_{l})$	$\displaystyle=$	$\displaystyle N_{ij}(\bar{\bm{A}}_{l})-\dfrac{2\alpha-1}{2}\left[\dfrac{\partial N_{ij}}{\partial A_{k}}\right]_{\bar{\bm{A}}_{l}}dA_{k}(l)+\dfrac{1}{2}\left(\dfrac{2\alpha-1}{2}\right)^{2}\left[\dfrac{\partial^{2}N_{ij}}{\partial A_{k}\partial A_{m}}\right]_{\bar{\bm{A}}_{l}}dA_{k}(l)dA_{m}(l)+\mathcal{O}(\epsilon^{3/2}).$		(82)

Then, using the relation $N_{ik}N_{jk}=2k_{\text{B}}T\Gamma^{\text{s}}_{ij}$ and Eq. (61), we obtain

$\displaystyle\dfrac{\text{det}\left(\bm{\Gamma^{\text{s}}}(\bar{\bm{A}}^{\text{f}}_{l})\right)^{-1/2}}{\text{det}\left(\bm{\Gamma^{\text{s}}}(\bar{\bm{A}}^{\text{r}}_{l})\right)^{-1/2}}$	$\displaystyle=$	$\displaystyle\dfrac{\left\|\text{det}\left(\bm{N}^{-1}(\bar{\bm{A}}^{\text{f}}_{l})\right)\right\|\ }{\left\|\text{det}\left(\bm{N}^{-1}(\bar{\bm{A}}^{\text{r}}_{l})\right)\right\|}$	(83)
	$\displaystyle=$	$\displaystyle\exp\left[-(2\alpha-1)\left[{{\color[rgb]{0,0,0}(N^{-1})_{jm}}}\dfrac{\partial N_{mj}}{\partial A_{k}}\right]_{\bar{\bm{A}}_{l}}dA_{k}(l)+\mathcal{O}(\epsilon^{3/2})\right]$
	$\displaystyle=$	$\displaystyle\exp\left[-\dfrac{2\alpha-1}{2}\left[({\Gamma^{s}}^{-1})_{jm}\dfrac{\partial(\Gamma^{s})_{mj}}{\partial A_{k}}\right]_{\bar{\bm{A}}_{l}}dA_{k}(l)+\mathcal{O}(\epsilon^{3/2})\right].$

In $\epsilon\to 0$ limit, dividing (76) by (80) first, and using the above equation and the relations given in Appendix D, we get

	$\displaystyle\dfrac{P}{P_{r}}$	$\displaystyle=$	$\displaystyle\dfrac{p_{0}(\bm{A}_{0})}{p_{0}(\bm{s}\circ\bm{A}_{N})}\exp\left[-\dfrac{1}{k_{\text{B}}T}\sum^{N}_{l=1}\left(\mathcal{H}(\bm{A}_{l})-\mathcal{H}(\bm{A}_{l-1})\right)+\mathcal{O}(\epsilon^{3/2})\right]$		(84)
		$\displaystyle=$	$\displaystyle\dfrac{p_{0}(\bm{A}(0))}{p_{0}(\bm{s}\circ\bm{A}(\tau))}\exp\left[-\dfrac{1}{k_{\text{B}}T}\left(\mathcal{H}(\bm{A}(\tau))-\mathcal{H}(\bm{A}(0))\right)\right],$		(84)

where $\bm{A}(0)\equiv\bm{A}_{0}$ and $\bm{A}(\tau)\equiv\bm{A}_{N}$ .

Appendix D Various relations needed for the calculation in subsection II.2

Recalling Eq. (10) of the main text

dA_{i}(l)=\epsilon\mathcal{F}_{i}(\bar{\bm{A}}^{\text{f}}_{l})+\sqrt{\epsilon}N_{ij}(\bar{\bm{A}}^{\text{f}}_{l})\xi_{j}^{l}.

(85)

Note that the lowest order term in $dA_{i}(l)$ is a $\epsilon^{1/2}$ -term. Let us consider a function $G(\bm{A})$ ; expanding $G(\bar{\bm{A}}^{\text{f}}_{l})$ and $G(\bar{\bm{A}}^{\text{r}}_{l})$ around $\bm{A}=\bar{\bm{A}}_{l}\equiv(\bm{A}_{l}+\bm{A}_{l-1})/2$ :

	$\displaystyle G(\bar{\bm{A}}^{\text{f}}_{l})$	$\displaystyle=$	$\displaystyle G(\bar{\bm{A}}_{l}+\dfrac{2\alpha-1}{2}d\bm{A}_{l})$		(86)
		$\displaystyle=$	$\displaystyle G(\bar{\bm{A}}_{l})+\dfrac{2\alpha-1}{2}\left[\dfrac{\partial G}{\partial A_{k}}\right]_{\bar{\bm{A}}_{l}}dA_{k}(l)+\dfrac{1}{2}\left(\dfrac{2\alpha-1}{2}\right)^{2}\left[\dfrac{\partial^{2}G}{\partial A_{k}\partial A_{m}}\right]_{\bar{\bm{A}}_{l}}dA_{k}(l)dA_{m}(l)+\mathcal{O}(\epsilon^{3/2})$		(86)

and

	$\displaystyle G(\bar{\bm{A}}^{\text{r}}_{l})$	$\displaystyle=$	$\displaystyle G(\bar{\bm{A}}_{l}-\dfrac{2\alpha-1}{2}d\bm{A}_{l})$		(87)
		$\displaystyle=$	$\displaystyle G(\bar{\bm{A}}_{l})-\dfrac{2\alpha-1}{2}\left[\dfrac{\partial G}{\partial A_{k}}\right]_{\bar{\bm{A}}_{l}}dA_{k}(l)+\dfrac{1}{2}\left(\dfrac{2\alpha-1}{2}\right)^{2}\left[\dfrac{\partial^{2}G}{\partial A_{k}\partial A_{m}}\right]_{\bar{\bm{A}}_{l}}dA_{k}(l)dA_{m}(l)+\mathcal{O}(\epsilon^{3/2}),$		(87)

where $d\bm{A}_{l}=\bm{A}_{l}-\bm{A}_{l-1}$ . Then

	$\displaystyle dA_{i}(l)dA_{j}(l)G(\bar{\bm{A}}^{\text{f}}_{l})$	$\displaystyle=$	$\displaystyle dA_{i}(l)dA_{j}(l)G(\bar{\bm{A}}_{l})+\dfrac{2\alpha-1}{2}\left[\dfrac{\partial G}{\partial A_{k}}\right]_{\bar{\bm{A}}_{l}}dA_{i}(l)dA_{j}(l)dA_{k}(l)$		(88)
			$\displaystyle+\dfrac{1}{2}\left(\dfrac{2\alpha-1}{2}\right)^{2}\left[\dfrac{\partial^{2}G}{\partial A_{k}\partial A_{m}}\right]_{\bar{\bm{A}}_{l}}dA_{i}(l)dA_{j}(l)dA_{k}(l)dA_{m}(l)+\mathcal{O}(\epsilon^{5/2}).$		(88)

From Eq. (85),

dA_{i}(l)dA_{j}(l)dA_{k}(l)=\xi_{p}^{l}\xi_{q}^{l}\xi_{r}^{l}N_{ip}N_{jq}N_{kr}\epsilon^{3/2}+(\xi_{p}^{l}\xi_{q}^{l}N_{ip}N_{jq}\mathcal{F}_{k}+\xi_{p}^{l}\xi_{r}^{l}N_{ip}N_{kr}\mathcal{F}_{j}+\xi_{q}^{l}\xi_{r}^{l}N_{jq}N_{kr}\mathcal{F}_{i})\epsilon^{2}+\mathcal{O}(\epsilon^{5/2}).

(89)

Our final expressions will be written in integral form, and since $\xi_{j}(t)$ is the time derivative of a Wiener process, we can write:

$\displaystyle\xi_{p}^{l}\xi_{q}^{l}$	$\displaystyle\equiv$	$\displaystyle\delta_{pq}$	(90)
$\displaystyle\xi_{p}^{l}\xi_{q}^{l}\xi_{r}^{l}$	$\displaystyle\equiv$	$\displaystyle\delta_{pq}\xi_{r}^{l}+\delta_{pr}\xi_{q}^{l}+\delta_{rq}\xi_{p}^{l}$	(91)
$\displaystyle\xi_{p}^{l}\xi_{q}^{l}\xi_{r}^{l}\xi_{o}^{l}$	$\displaystyle\equiv$	$\displaystyle\delta_{pq}\delta_{ro}+\delta_{pr}\delta_{qo}+\delta_{qr}\delta_{po}.$	(92)

Using the above relations, Eq. (85) & Eq. (6) ( $N_{ik}N_{jk}=2k_{\text{B}}T\Gamma^{\text{s}}_{ij}$ ), Eq. (89) can be written as

dA_{i}(l)dA_{j}(l)dA_{k}(l)=2k_{\text{B}}T(\Gamma^{\text{s}}_{ij}dA_{k}(l)+\Gamma^{\text{s}}_{jk}dA_{i}(l)+\Gamma^{\text{s}}_{ki}dA_{j}(l))\epsilon+\mathcal{O}(\epsilon^{5/2}).

(93)

Similarly,

	$\displaystyle dA_{i}(l)dA_{j}(l)dA_{k}(l)dA_{m}(l)$	$\displaystyle=$	$\displaystyle\xi_{p}^{l}\xi_{q}^{l}\xi_{r}^{l}\xi_{o}^{l}N_{ip}N_{jq}N_{kr}N_{mo}\epsilon^{2}+\mathcal{O}(\epsilon^{5/2})$		(94)
		$\displaystyle=$	$\displaystyle(2k_{\text{B}}T)^{2}(\Gamma^{\text{s}}_{ij}\Gamma^{\text{s}}_{km}+\Gamma^{\text{s}}_{ik}\Gamma^{\text{s}}_{jm}++\Gamma^{\text{s}}_{im}\Gamma^{\text{s}}_{jk})\epsilon^{2}+\mathcal{O}(\epsilon^{5/2})$		(94)

Substituting (93) and (94) into Eq. (88), we obtain

	$\displaystyle dA_{i}(l)dA_{j}(l)G(\bar{\bm{A}}^{\text{f}}_{l})$	$\displaystyle=$	$\displaystyle dA_{i}(l)dA_{j}(l)G(\bar{\bm{A}}_{l})+(2\alpha-1)k_{\text{B}}T\left[\dfrac{\partial G}{\partial A_{k}}\right]_{\bar{\bm{A}}_{l}}(\Gamma^{\text{s}}_{ij}(\bar{\bm{A}}_{l})dA_{k}(l)+\Gamma^{\text{s}}_{jk}(\bar{\bm{A}}_{l})dA_{i}(l)+\Gamma^{\text{s}}_{ki}(\bar{\bm{A}}_{l})dA_{j}(l))\epsilon$		(95)
			$\displaystyle+\dfrac{1}{2}\left[(2\alpha-1)k_{\text{B}}T\right]^{2}\left[\dfrac{\partial^{2}G}{\partial A_{k}\partial A_{m}}(\Gamma^{\text{s}}_{ij}\Gamma^{\text{s}}_{km}+\Gamma^{\text{s}}_{ik}\Gamma^{\text{s}}_{jm}++\Gamma^{\text{s}}_{im}\Gamma^{\text{s}}_{jk})\right]_{\bar{\bm{A}}_{l}}\epsilon^{2}+\mathcal{O}(\epsilon^{5/2})$		(95)

Similarly, from Eq. (87), we readily obtain

	$\displaystyle dA_{i}(l)dA_{j}(l)G(\bar{\bm{A}}^{\text{r}}_{l})$	$\displaystyle=$	$\displaystyle dA_{i}(l)dA_{j}(l)G(\bar{\bm{A}}_{l})-(2\alpha-1)k_{\text{B}}T\left[\dfrac{\partial G}{\partial A_{k}}\right]_{\bar{\bm{A}}_{l}}(\Gamma^{\text{s}}_{ij}dA_{k}(l)+\Gamma^{\text{s}}_{jk}dA_{i}(l)+\Gamma^{\text{s}}_{ki}dA_{j}(l))\epsilon$		(96)
			$\displaystyle+\dfrac{1}{2}\left[(2\alpha-1)k_{\text{B}}T\right]^{2}\left[\dfrac{\partial^{2}G}{\partial A_{k}\partial A_{m}}\right]_{\bar{\bm{A}}_{l}}(\Gamma^{\text{s}}_{ij}\Gamma^{\text{s}}_{km}+\Gamma^{\text{s}}_{ik}\Gamma^{\text{s}}_{jm}++\Gamma^{\text{s}}_{im}\Gamma^{\text{s}}_{jk})\epsilon^{2}+\mathcal{O}(\epsilon^{5/2}).$		(96)

Likewise, we can easily derive the following relations:

$\displaystyle dA_{i}(l)G(\bar{\bm{A}}^{\text{f}}_{l})$	$\displaystyle=$	$\displaystyle dA_{i}(l)G(\bar{\bm{A}}_{l})+(2\alpha-1)k_{\text{B}}T\epsilon\left[\Gamma^{s}_{ij}\dfrac{\partial G}{\partial A_{j}}\right]_{\bar{\bm{A}}_{l}}+\mathcal{O}(\epsilon^{3/2})$	(97)
$\displaystyle dA_{i}(l)G(\bar{\bm{A}}^{\text{r}}_{l})$	$\displaystyle=$	$\displaystyle dA_{i}(l)G(\bar{\bm{A}}_{l})-(2\alpha-1)k_{\text{B}}T\epsilon\left[\Gamma^{s}_{ij}\dfrac{\partial G}{\partial A_{j}}\right]_{\bar{\bm{A}}_{l}}+\mathcal{O}(\epsilon^{3/2})$	(98)
$\displaystyle G(\bar{\bm{A}}^{\text{f}}_{l})$	$\displaystyle=$	$\displaystyle G(\bar{\bm{A}}_{l})+\mathcal{O}(\epsilon^{1/2})$	(99)
$\displaystyle G(\bar{\bm{A}}^{\text{r}}_{l})$	$\displaystyle=$	$\displaystyle G(\bar{\bm{A}}_{l})+\mathcal{O}(\epsilon^{1/2})$	(100)
$\displaystyle\left[\dfrac{\partial G}{\partial A_{i}}\right]_{\bar{\bm{A}}_{l}}dA_{i}(l)$	$\displaystyle=$	$\displaystyle G(\bm{A}_{l})-G(\bm{A}_{l-1})+\mathcal{O}(\epsilon^{3/2})$	(101)

Appendix E The relation between entropy production rate and $\dot{s}$

The free energy of the system at time $t$ would be

	$\displaystyle F(t)$	$\displaystyle=$	$\displaystyle\int\mathcal{H}(\bm{A})p_{t}(\bm{A})d\bm{A}-T\left[-k_{\text{B}}\int p_{t}(\bm{A})\ln p_{t}(\bm{A})d\bm{A}\right]$		(102)
		$\displaystyle=$	$\displaystyle\left\langle\left[\mathcal{H}(\bm{A})+k_{\text{B}}T\ln p_{t}(\bm{A})\right]\right\rangle,$		(102)

where $\left\langle\right\rangle$ stands for the ensemble average and $p_{t}(\bm{A})$ is the probability distribution of $\bm{A}$ at time $t$ . Let us define the free energy of a single trajectory of the system at time $t$ as

f(t)=\mathcal{H}(\bm{A}(t))+k_{\text{B}}T\ln p_{t}(\bm{A}(t))

(103)

Then it is straight forward to show that

\displaystyle\dfrac{d{F}(t)}{dt}=\left\langle\dfrac{df(t)}{dt}\right\rangle.

(104)

E.1 For passive systems

From Eq (32), we readily get

\left\langle\dfrac{df(t)}{dt}\right\rangle=-k_{\text{B}}T\left\langle\dot{s}(t)\right\rangle,

(105)

so from Eq. (104)

\displaystyle\dfrac{d{F}(t)}{dt}=-k_{\text{B}}T\left\langle\dot{s}(t)\right\rangle.

(106)

Assuming that the system is always in metastable thermal equilibrium with the reservoir, the rate of change total entropy of system and reservoir would be

\dfrac{dS(t)}{dt}=-\dfrac{1}{T}\dfrac{d{F}(t)}{dt}=k_{\text{B}}\left\langle\dot{{s}}(t)\right\rangle.

(107)

E.2 For active systems

For active systems, from Eq. (46), one can trivially prove that

\left\langle\dfrac{df(t)}{dt}\right\rangle=-k_{\text{B}}T\left\langle\dot{s}(t)\right\rangle+\left\langle w(t)\right\rangle,

(108)

where $\left\langle w(t)\right\rangle$ is average rate of the work perfomed by active forces. Hence, from Eq. (104),

\dfrac{d{F}(t)}{dt}=-k_{\text{B}}T\left\langle\dot{{s}}(t)\right\rangle+\left\langle w(t)\right\rangle.

(109)

Therefore, the rate of total enetropy production of the system and the reservoir:

	$\displaystyle\dfrac{dS(t)}{dt}$	$\displaystyle=$	$\displaystyle-\dfrac{1}{T}\dfrac{d{F}(t)}{dt}+\dfrac{1}{T}\left\langle w(t)\right\rangle$		(110)
		$\displaystyle=$	$\displaystyle k_{\text{B}}\left\langle\dot{{s}}(t)\right\rangle.$		(110)

Appendix F The dissipation function defined by Seifert et al. [5]

Seifert et al. [5] used the following form of the dissipation function:

\mathcal{S}_{\tau}=\ln\left[\dfrac{P}{P^{\prime}_{r}}\right],

(111)

where $P$ is the probability density of a trajectory between $t=0$ and $t=\tau$ which is given by Eq. (A), and $P^{\prime}_{r}$ is the probability density of the time-reversed trajectory, considering that the time-reserved trajectory starts at $t=\tau$ , not at $t=0$ . Thus, the epxression of $\mathcal{S}_{\tau}$ is readily obtained by replacing $p_{0}(\bm{s}\circ\bm{A}_{N})$ with $p_{t}(\bm{s}\circ\bm{A}_{N})$ in the expression of $\mathcal{R}_{\tau}$ (see Eq. (25)), that is

\mathcal{S}_{\tau}=\ln\dfrac{p_{0}(\bm{A}(0))}{p_{\tau}(\bm{s}\circ\bm{A}(\tau))}-\dfrac{1}{k_{\text{B}}T}\left[\mathcal{H}(\bm{A}(\tau))-\mathcal{H}(\bm{A}(0))\right]

(112)

This dissipation function follows the fluctuation relation (24) in steady states only, not in general. However, as discussed by [5], it does always follow the integrated fluctuation relation (28) and thereofore $\left\langle\mathcal{S}_{\tau}\right\rangle\geq 0$ .

Appendix G Calculation of $\dot{s}$ for the active systems

Recalling equations of motion for the active systems

\dfrac{dA_{i}}{dt}=\mathcal{F}_{i}+\mathcal{X}_{i}+N_{ij}\xi_{j}(t),

(113)

where

\mathcal{F}_{i}\equiv-\Gamma_{ij}\dfrac{\partial\mathcal{H}}{\partial A_{j}}+k_{\text{B}}T\dfrac{\partial\Gamma_{ij}}{\partial A_{j}}-\alpha N_{lj}\dfrac{\partial N_{ij}}{\partial A_{l}}

(114)

and $\mathcal{X}_{i}$ is the active term. Writing $\bm{\mathcal{X}}$ as $\bm{\mathcal{X}}=\bm{\mathcal{X}}^{\text{s}}+\bm{\mathcal{X}}^{\text{a}}$ , where

	$\displaystyle\bm{\mathcal{X}}^{\text{s}}(\bm{A})$	$\displaystyle=$	$\displaystyle\dfrac{1}{2}\left(\bm{\mathcal{X}}(\bm{A})+\bm{s}\circ\bm{\mathcal{X}}(\bm{s}\circ\bm{A})\right),$		(115)
	$\displaystyle\bm{\mathcal{X}}^{\text{a}}(\bm{A})$	$\displaystyle=$	$\displaystyle\dfrac{1}{2}\left(\bm{\mathcal{X}}(\bm{A})-\bm{s}\circ\bm{\mathcal{X}}(\bm{s}\circ\bm{A})\right)$		(116)

follow the properties $\bm{\mathcal{X}}^{\text{s}}(\bm{s}\circ\bm{A})=\bm{s}\circ\bm{\mathcal{X}}^{\text{s}}(\bm{A})$ and $\bm{\mathcal{X}}^{\text{a}}(\bm{s}\circ\bm{A})=-\bm{s}\circ\bm{\mathcal{X}}^{\text{a}}(\bm{A})$ . The Fokker-Planck equation for the probability density $p_{t}(\bm{A})$ of $\bm{A}$ reads

\dfrac{\partial p_{t}(\bm{A})}{\partial t}=-\dfrac{\partial J_{i}(\bm{A},t)}{\partial A_{i}},

(117)

where

	$\displaystyle J_{i}(\bm{A},t)$	$\displaystyle=$	$\displaystyle\left(-{\Gamma^{\text{s}}}_{ij}(\bm{A})\dfrac{\partial\mathcal{H}(\bm{A})}{\partial A_{j}}+\mathcal{Y}^{\text{a}}_{i}(\bm{A})+\mathcal{X}^{\text{s}}_{j}(\bm{A})\right)p_{t}(\bm{A})$		(118)
			$\displaystyle\qquad\qquad\qquad\qquad-k_{\text{B}}T\Gamma^{\text{s}}_{ij}(\bm{A})\dfrac{\partial p_{t}(\bm{A})}{dA_{j}}$		(118)

is the probability current [23] and

\mathcal{Y}^{\text{a}}_{i}=\mathcal{X}^{\text{a}}_{i}+\dfrac{\partial\mathcal{H}}{\partial A_{k}}\Gamma^{\text{a}}_{ki}-k_{\text{B}}T\dfrac{\partial\Gamma^{\text{a}}_{ki}}{\partial A_{k}}.

(119)

Since $\bm{\mathcal{X}}^{\text{a}}(\bm{s}\circ\bm{A})=-\bm{s}\circ\bm{\mathcal{X}}^{\text{a}}(\bm{A})$ , using the relation $\Gamma^{\text{a}}_{ij}=-\Gamma^{\text{a}}_{ij}s_{i}s_{j}$ (Eq. (16)), it is easy to show that

\bm{\mathcal{Y}}^{\text{a}}(\bm{s}\circ\bm{A})=-\bm{s}\circ\bm{\mathcal{Y}}^{\text{a}}(\bm{A}).

(120)

As Eq. (117) has no term with $N_{ij}$ , $p_{t}(\bm{A})$ would be independent of the choice of $N_{ij}$ . Recalling Eq. (III)

\dot{s}(t)=w(t)-\left.\dfrac{d}{dt^{\prime}}\left(\ln p_{t}(\bm{A}(t^{\prime}))+\dfrac{1}{k_{\text{B}}T}\mathcal{H}(\bm{A}(t^{\prime}))\right)\right|_{t^{\prime}=t}.

(121)

where

		$\displaystyle w(t)$	$\displaystyle=\dfrac{\partial\mathcal{H}(\bm{A}(t))}{\partial A_{i}}\mathcal{X}^{\text{a}}_{i}(\bm{A}(t))-k_{\text{B}}T\dfrac{\partial\mathcal{X}^{\text{a}}_{i}(\bm{A}(t))}{\partial A_{i}}$		(122)
		$\displaystyle+$	$\displaystyle{{\color[rgb]{0,0,0}({\Gamma^{\text{s}}}^{-1})_{ij}}}(\bm{A}(t))\mathcal{X}^{\text{s}}_{j}(\bm{A}(t))\left[\dfrac{dA_{i}}{dt}-\mathcal{Y}^{\text{a}}_{i}(\bm{A}(t))\right].$		(122)

Using Eqs. (117) & (118), one can write Eq. (III) in the following form:

	$\displaystyle\dot{{s}}(t)$	$\displaystyle=$	$\displaystyle\dfrac{1}{k_{\text{B}}T}\left(\dfrac{J_{i}(\bm{A},t)}{p_{t}(\bm{A})}-\mathcal{Y}^{\text{a}}_{i}(\bm{A})\right){{\color[rgb]{0,0,0}({\Gamma^{\text{s}}}^{-1})_{ij}}}(\bm{A})$		(123)
			$\displaystyle\times\left(\dfrac{J_{j}(\bm{A},t)}{p_{t}(\bm{A})}-\mathcal{Y}^{\text{a}}_{j}(\bm{A})\right)+R_{s}+R_{0},$		(123)

where

R_{s}=-\dfrac{1}{p_{t}(\bm{A})}\dfrac{\partial}{\partial A_{i}}\left(p_{t}(\bm{A})\mathcal{Y}^{\text{a}}_{i}(\bm{A})\right)

(124)

and

		$\displaystyle R_{0}$	$\displaystyle=\left(\dfrac{dA_{i}}{dt}-\dfrac{J_{i}(\bm{A},t)}{p_{t}(\bm{A})}\right)\left[\dfrac{1}{p_{t}(\bm{A})}\dfrac{\partial p_{t}(\bm{A})}{\partial A_{i}}\right.$		(125)
			$\displaystyle\left.-\dfrac{1}{k_{\text{B}}T}\left({{\color[rgb]{0,0,0}({\Gamma^{\text{s}}}^{-1})_{ij}}}(\bm{A})\mathcal{X}^{\text{s}}_{j}(\bm{A})-\dfrac{\partial\mathcal{H}(\bm{A})}{\partial A_{j}}\right)\right].$		(125)

The average of $R_{s}$ reads

	$\displaystyle\left\langle R_{s}\right\rangle$	$\displaystyle=$	$\displaystyle-\int\dfrac{1}{p_{t}(\bm{A})}\dfrac{\partial}{\partial A_{i}}\left(p_{t}(\bm{A})\mathcal{Y}^{\text{a}}_{i}(\bm{A})\right)p_{t}(\bm{A})d^{n}A$		(126)
		$\displaystyle=$	$\displaystyle-\int\dfrac{\partial}{\partial A_{i}}\left(p_{t}(\bm{A})\mathcal{Y}^{\text{a}}_{i}(\bm{A})\right)d^{n}A$		(126)

The above expression can be written as a surface integral with the integrand $\bm{I}_{\text{s}}=-p_{t}(\bm{A})\bm{\mathcal{Y}}^{\text{a}}(\bm{A})$ . If the system is periodic in $A_{i}$ (e.g., $A_{i}$ is an angle), the surface integral is already zero. If $A_{i}$ lies in the infinite interval $(-\infty,\infty)$ , in $A_{i}\to\pm\infty$ limit, $p_{t}(\bm{A})\to 0$ , given that $\bm{A}$ are physical variables. Assuming that $||\bm{I}_{\text{s}}||$ converges faster than $||\bm{A}||^{1-n}$ , the surface integral is again zero. For $p_{t}(\bm{s}\circ\bm{A})=p_{t}(\bm{A})$ case, $\left\langle R_{s}\right\rangle$ is always zero as follows: setting $\bm{A}=\bm{s}\circ\bm{A}^{\prime}$ gives $d^{n}A=d^{n}A^{\prime}$ , then using Eq. (120), we get

	$\displaystyle\left\langle R_{s}\right\rangle$	$\displaystyle=$	$\displaystyle\int\dfrac{\partial}{\partial A^{\prime}_{i}}\left(p_{t}(\bm{A}^{\prime})\mathcal{Y}^{\text{a}}_{i}(\bm{A}^{\prime})\right)d^{n}A^{\prime}$		(127)
		$\displaystyle=$	$\displaystyle-\left\langle R_{s}\right\rangle,$		(127)

so $\left\langle R_{s}\right\rangle=0$ . Since the ensemble average of $dA_{i}/dt$ for given $\bm{A}$ and $t$ is just $J_{i}(\bm{A},t)/p_{t}(\bm{A})$ , $\left\langle R_{o}\right\rangle=0$ . Therefore, the ensemble average of Eq. (123) is given by

	$\displaystyle\left\langle\dot{{s}}(t)\right\rangle$	$\displaystyle=$	$\displaystyle\dfrac{1}{k_{\text{B}}T}\left\langle\left(\dfrac{J_{i}(\bm{A},t)}{p_{t}(\bm{A})}-\mathcal{Y}^{\text{a}}_{i}(\bm{A})\right){{\color[rgb]{0,0,0}({\Gamma^{\text{s}}}^{-1})_{ij}}}(\bm{A})\right.$		(128)
			$\displaystyle\times\left.\left(\dfrac{J_{j}(\bm{A},t)}{p_{t}(\bm{A})}-\mathcal{Y}^{\text{a}}_{j}(\bm{A})\right)\right\rangle.$		(128)

Similarly, $\left\langle w(t)\right\rangle$ given by Eq. (122) can be written in the following form

	$\displaystyle\left\langle w(t)\right\rangle$	$\displaystyle=$	$\displaystyle\left\langle\left(\dfrac{dA_{i}}{dt}-\dfrac{\partial\mathcal{H}}{\partial A_{k}}\Gamma^{\text{a}}_{ki}+k_{\text{B}}T\dfrac{\partial\Gamma^{\text{a}}_{ki}}{\partial A_{k}}\right){{\color[rgb]{0,0,0}({\Gamma^{\text{s}}}^{-1})_{ij}}}\mathcal{X}^{\text{s}}_{j}\right\rangle$		(129)
			$\displaystyle-\left\langle\left(\dfrac{dA_{i}}{dt}-\mathcal{Y}^{\text{a}}_{i}\right){{\color[rgb]{0,0,0}({\Gamma^{\text{s}}}^{-1})_{ij}}}\mathcal{X}^{\text{a}}_{j}\right\rangle.$		(129)

Here, if $p_{t}(\bm{s}\circ\bm{A})\neq p_{t}(\bm{A})$ , we must assume that $||p_{t}(\bm{A})\bm{\mathcal{X}}^{\text{a}}(\bm{A})||$ converges faster than $||\bm{A}||^{1-n}$ .

Appendix H Stationary solution of the Fokker-Planck equation associated with Eq. (8)

The Fokker-Planck equation for the probability distribution $p_{t}(\bm{A})$ of the solution of Eq (8) is given by

\dfrac{\partial p_{t}(\bm{A})}{\partial t}=-\dfrac{\partial J_{i}(\bm{A},t)}{\partial A_{i}};

(130)

the expression of the probability current $J_{i}(\bm{A},t)$ reads [23]

	$\displaystyle J_{i}(\bm{A},t)$	$\displaystyle=$	$\displaystyle\left(-{\Gamma^{\text{s}}}_{ij}(\bm{A})\dfrac{\partial\mathcal{H}(\bm{A})}{\partial A_{j}}+\dfrac{\partial\mathcal{H}}{\partial A_{k}}\Gamma^{\text{a}}_{ki}-k_{\text{B}}T\dfrac{\partial\Gamma^{\text{a}}_{ki}}{\partial A_{k}}\right)$		(131)
			$\displaystyle\times p_{t}(\bm{A})-k_{\text{B}}T\Gamma^{\text{s}}_{ij}(\bm{A})\dfrac{\partial p_{t}(\bm{A})}{dA_{j}}.$		(131)

Undoubtedly, the dynamics of $p_{t}(\bm{A})$ is independent of the choice of $N_{ij}$ and $\alpha$ . Since

\dfrac{\partial\mathcal{H}}{\partial A_{k}}\Gamma^{\text{a}}_{ki}-k_{\text{B}}T\dfrac{\partial\Gamma^{\text{a}}_{ki}}{\partial A_{k}}

(132)

is the Poisson bracket term, the stationary solution of the above equation is given by [25]

p_{\text{s}}(\bm{A})=\dfrac{1}{\mathcal{Z}}\exp\left[-\dfrac{\mathcal{H}(\bm{A})}{k_{\text{B}}T}\right],

(133)

where $Z$ is the normalizing constant.

Appendix I Dependence of $\left\langle\mathcal{R}_{\tau}\right\rangle$ on $\alpha$ for the passive systems

For the passive systems, $\mathcal{R}_{\tau}$ depends only on the initial and final states of the system $\bm{A}_{0}$ and $\bm{A}_{\tau}$ , so its average can be calculated using the formula [35]

\left\langle\mathcal{R}_{\tau}\right\rangle=\int\mathcal{R}_{\tau}p_{0}(\bm{A}_{0})G(\bm{A}_{0},\bm{A}_{\tau};\tau)d^{n}A_{0}d^{n}A_{\tau},

(134)

where $p_{0}(\bm{A})$ is the probability distribution of $\bm{A}$ at $t=0$ , and $G(\bm{A},\bm{A}^{\prime};\tau)$ is the probability distribution of state $\bm{A}^{\prime}$ at $t=\tau$ given that the system was in the state $\bm{A}$ at $t=0$ ; it is the solution of Eq. (130) with the initial condition $G(\bm{A},\bm{A}^{\prime};\tau=0)=\bm{\delta}(\bm{A}-\bm{A}^{\prime})$ . As the solution of Eq. (130) is independent of $\alpha$ , $\left\langle\mathcal{R}_{\tau}\right\rangle$ would be constant in $\alpha$ .

References

Evans et al. [1993] D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Probability of second law violations in shearing steady states, Phys. Rev. Lett. 71, 2401 (1993).
Evans and Searles [2002] D. J. Evans and D. J. Searles, The fluctuation theorem, Advances in Physics 51, 1529 (2002).
Searles and Evans [2004] D. J. Searles and D. J. Evans, Fluctuations relations for nonequilibrium systems, Australian Journal of Chemistry 57, 1119 (2004).
Liphardt et al. [2002] J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, and C. Bustamante, Equilibrium information from nonequilibrium measurements in an experimental test of jarzynski’s equality, 296, 1832 (2002).
Seifert [2005] U. Seifert, Entropy production along a stochastic trajectory and an integral fluctuation theorem, Phys. Rev. Lett. 95, 040602 (2005).
Blickle et al. [2006] V. Blickle, T. Speck, L. Helden, U. Seifert, and C. Bechinger, Thermodynamics of a colloidal particle in a time-dependent nonharmonic potential, Phys. Rev. Lett. 96, 070603 (2006).
Seifert [2012] U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Reports on progress in physics 75, 126001 (2012).
Gallavotti [2008] G. Gallavotti, Heat and fluctuations from order to chaos, The European Physical Journal B 61, 1 (2008).
Jarzynski [2007] C. Jarzynski, Comparison of far-from-equilibrium work relations, Comptes Rendus Physique 8, 495 (2007).
Hurtado et al. [2011] P. I. Hurtado, C. Pèrez-Espigares, J. J. del Pozo, and P. L. Garrido, Symmetries in fluctuations far from equilibrium, Proceedings of the National Academy of Sciences (2011).
Villavicencio-Sanchez et al. [2014] R. Villavicencio-Sanchez, R. J. Harris, and H. Touchette, Fluctuation relations for anisotropic systems, EPL 105, 30009 (2014).
Soni [2019] H. Soni, Flocks, Flow and Fluctuations in Inanimate Matter: Simulations and Theory, Ph.D. thesis (2019).
Wang et al. [2002] G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales, Phys. Rev. Lett. 89, 050601 (2002).
Collin et al. [2005] D. Collin, F. Ritort, C. Jarzynski, S. B. Smith, I. Tinoco, and C. Bustamante, Verification of the crooks fluctuation theorem and recovery of rna folding free energies, Nature 437, 231 (2005).
Carberry et al. [2004] D. M. Carberry, J. C. Reid, G. M. Wang, E. M. Sevick, D. J. Searles, and D. J. Evans, Fluctuations and irreversibility: An experimental demonstration of a second-law-like theorem using a colloidal particle held in an optical trap, Phys. Rev. Lett. 92, 140601 (2004).
Evans and Searles [1994] D. J. Evans and D. J. Searles, Equilibrium microstates which generate second law violating steady states, Phys. Rev. E 50, 1645 (1994).
Khan and Sood [2010] M. Khan and A. Sood, Out-of-equilibrium microrheology using optical tweezers to probe directional viscoelastic properties under shear, EPL (Europhysics Letters) 92, 48001 (2010).
Kumar et al. [2011] N. Kumar, S. Ramaswamy, and A. K. Sood, Symmetry properties of the large-deviation function of the velocity of a self-propelled polar particle, Phys. Rev. Lett. 106, 118001 (2011).
Kumar et al. [2015] N. Kumar, H. Soni, S. Ramaswamy, and A. K. Sood, Anisotropic isometric fluctuation relations in experiment and theory on a self-propelled rod, Physical Review E 91, 030102 (2015).
Aron et al. [2016] C. Aron, D. G. Barci, L. F. Cugliandolo, Z. G. Arenas, and G. S. Lozano, Dynamical symmetries of markov processes with multiplicative white noise, Journal of Statistical Mechanics: Theory and Experiment 2016, 053207 (2016).
Dengler [2015] R. Dengler, Another derivation of generalized langevin equations, arXiv preprint arXiv:1506.02650 (2015).
Mazenko [2008] G. F. Mazenko, Nonequilibrium statistical mechanics (John Wiley & Sons, 2008).
Lau and Lubensky [2007] A. W. Lau and T. C. Lubensky, State-dependent diffusion: Thermodynamic consistency and its path integral formulation, Physical Review E 76, 011123 (2007).
Cugliandolo and Lecomte [2017] L. F. Cugliandolo and V. Lecomte, Rules of calculus in the path integral representation of white noise langevin equations: the onsager–machlup approach, Journal of Physics A: Mathematical and Theoretical 50, 345001 (2017).
Chaikin and Lubensky [1995] P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, 1995) cambridge Books Online.
Onsager [1931] L. Onsager, Reciprocal relations in irreversible processes. i., Phys. Rev. 37, 405 (1931).
Gard [1988] T. C. Gard, Introduction to stochastic differential equations (New York : M. Dekker, 1988).
Sevick et al. [2008] E. Sevick, R. Prabhakar, S. R. Williams, and D. J. Searles, Fluctuation theorems, Annual Review of Physical Chemistry 59, 603 (2008), pMID: 18393680.
Reid et al. [2004] J. Reid, D. Carberry, G. Wang, E. M. Sevick, D. J. Evans, and D. J. Searles, Reversibility in nonequilibrium trajectories of an optically trapped particle, Physical Review E 70, 016111 (2004).
Marchetti et al. [2013] M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrodynamics of soft active matter, Rev. Mod. Phys. 85, 1143 (2013).
Bricard et al. [2013] A. Bricard, J.-B. Caussin, N. Desreumaux, O. Dauchot, and D. Bartolo, Emergence of macroscopic directed motion in populations of motile colloids, Nature 503, 95 (2013).
Kumar et al. [2014] N. Kumar, H. Soni, S. Ramaswamy, and A. K. Sood, Flocking at a distance in active granular matter, Nat Commun 5 (2014).
Weber et al. [2013] C. A. Weber, T. Hanke, J. Deseigne, S. Léonard, O. Dauchot, E. Frey, and H. Chaté, Long-range ordering of vibrated polar disks, Phys. Rev. Lett. 110, 208001 (2013).
Chatterjee et al. [2021] R. Chatterjee, N. Rana, R. A. Simha, P. Perlekar, and S. Ramaswamy, Inertia drives a flocking phase transition in viscous active fluids, Physical Review X 11, 031063 (2021).
Doi et al. [1988] M. Doi, S. F. Edwards, and S. F. Edwards, The theory of polymer dynamics, Vol. 73 (oxford university press, 1988).

Properties of the dissipation functions for passive and active systems

Abstract

I Introduction

II Passive systems

II.1 Generic Langevin equations

II.2 The ratio between the probability densities of a trajectory and its time-reversed trajectory

II.3 Fluctuation relations and the dissipation function for the passive systems

II.4 The dissipation function for quenched systems

II.4.1 Colloidal particle in a harmonic potential well

III The dissipation function for active systems

IV Conclusion

Appendix A The probability density of a trajectory for passive systems

Appendix B The probability density for the time-reversed trajectory

Appendix C Calculation of the ratio between the probability densities of a trajectory and its time-reversed trajectory

Appendix D Various relations needed for the calculation in subsection II.2

Appendix E The relation between entropy production rate and s˙\dot{s}

E.1 For passive systems

E.2 For active systems

Appendix F The dissipation function defined by Seifert et al. [5]

Appendix G Calculation of s˙\dot{s} for the active systems

Appendix H Stationary solution of the Fokker-Planck equation associated with Eq. (8)

Appendix I Dependence of ⟨ℛτ⟩\left\langle\mathcal{R}_{\tau}\right\rangle on α\alpha for the passive systems

References

Appendix E The relation between entropy production rate and $\dot{s}$

Appendix G Calculation of $\dot{s}$ for the active systems

Appendix I Dependence of $\left\langle\mathcal{R}_{\tau}\right\rangle$ on $\alpha$ for the passive systems