---
abstract: '[Using the Feynman-Kac and Cameron-Martin-Girsanov formulas, we obtain a generalized integral fluctuation theorem (GIFT) for discrete jump processes by constructing a time-invariable inner product. The existing discrete IFTs can be derived as its specific cases. A connection between our approach and the conventional time-reversal method is also established. Different from the latter approach that were extensively employed in existing literature, our approach can naturally bring out the definition of a time-reversal for a Markovian stochastic system. Intriguingly, we find the robust GIFT usually does not result into a detailed fluctuation theorem. ]{}'
author:
- Fei Liu
- 'Yu-Pin Luo'
- 'Ming-Chang Huang'
- 'Zhong-can Ou-Yang'
title: A generalized integral fluctuation theorem for general jump processes 
---

Introduction
============

One of important progresses in nonequilibrium statistic physics in the past two decades is the discovery of a various of fluctuation theorems. They are thought of to be a nonperturbative extension of the fluctuation-dissipation theorems in near-equilibrium region to far-from equilibrium region. According to their mathematical expressions, these theorems are loosely divided into two types. One is called the integral fluctuation theorems (IFT) [@JarzynskiPRE97; @JarzynskiPRL97; @Crooks99; @Crooks00; @Maes; @SeifertPRL05; @HatanoSasa; @Speck; @Schmiedl; @QianJPC; @Harris], and the other is called the detailed fluctuation theorems (DFT) [@Evans; @Gallavotti; @Kurchan; @Lebowitz; @Crooks00]. The former follows a unified expression $$\begin{aligned}
\langle \exp[-{\cal A}]\rangle=1, \label{IFT}\end{aligned}$$ where $\cal{A}$ is a functional of a stochastic trajectory of a concerned stochastic system, and angular brackets denote an average over the ensemble of the trajectories that start from an given initial distribution. For instance, $\cal{A}$ may be the dissipated work along a trajectory and eq. (\[IFT\]) is the celebrated Jarzynski equality (JE) [@JarzynskiPRE97; @JarzynskiPRL97].

Due to the insightful work of Hummer and Szabo [@Hummer01], we now know that these IFTs have an intimate connection with the famous Feynman-Kac formula (FK) [@Feynman; @Kac] in the stochastic theory of diffusion processes [@Stroock]. Recently, several works including us reinvestigated this issue from mathematic generalization and rigors [@QianJPC; @Ge; @Chetrite; @Ao; @LiuF]. One of findings is that the application of the FK formula in proving the IFTs is based on a construction of a time-reversed process of a diffusion process [@Chetrite; @LiuF]. Because the definition of a time-reversal has some certain arbitrariness [@Chetrite], we have obtained a generalized IFT (GIFT) by constructing a time-invariable integral and employing the FK and Cameron-Martin-Girsanov (CMG) formulas [@Cameron; @Girsanov] simultaneously, and the several IFTs [@JarzynskiPRE97; @SeifertPRL05; @HatanoSasa; @QianJPC] were specific cases of the GIFT [@LiuF]. We should emphasize that all of the works were concerning with continuous diffusion processes described by Fokker-Planck (FK) equation.

In addition to continuous case, there are still another kind of stochastic jump processes described by Markovian discrete master equations. In many practical physical systems, a description of discrete jump process is more satisfactory than a description using continuous diffusion process, e.g., the systems only involving few individual objects [@Gardiner]. One may naturally think of that there exists a GIFT in discrete version, and the discrete IFTs in literature [@SeifertPRL05; @Harris; @SeifertJPA04] are specific cases of it. At a first sight, this effort seems trivial since a continuous diffusion process can be always discretized to a discrete jump process. However, In addition that one hardly ensures that the “discrete" GIFT achieved in this way is really exact, we know that a jump process is not always equivalent to a discretization of a certain continuous process [@Gardiner]. Additionally, to our knowledge, fewer works have formally studied the IFTs for general jump processes employing the FK and CMG formulas, though several authors have mentioned this possibility [@QianJP; @GeJMP] earlier. Therefore, in our opinion a rigorous derivation of an exact GIFT for discrete jump processes is essential and meaningful. In this work we present this effort. Because we focus on the general Markovian jump processes, fewer physics are mentioned here. The detailed discussions about the specific IFTs in previous literature [@Harris] should make it up.

Generalized integral fluctuations for jump processes
====================================================

We start with a Markovian jumping process described by a discrete master equation $$\begin{aligned}
\frac{dp_n(t)}{dt}=\left[{\textbf H}(t){\textbf p}(t)\right]_n,
\label{forwardeq}\end{aligned}$$ where the $N$-dimension column vector ${\textbf
p}(t)=(p_1,\cdots,p_N)^{\rm T}$ is the probabilities of the system at individual states at time $t$ (the state index $n$ may be a vector), the matrix element of the time-dependent (or time-independent) rate ${\textbf H}_{mn}>0$ ($m\neq n$) and ${\textbf H}_{nn}=-\sum_{m\neq n}{\textbf H}_{mn}$. Given a normalized positive column vector ${\textbf
f}(t)=(f_1,\cdots,f_N)^{\rm T}$ and a $N\times N$ matrix ${\textbf
A}$ that satisfies conditions $f_n{\textbf H_{mn}+\textbf
A}_{mn}>0$ ($m\neq n$) and ${\textbf A}_{nn}=-\sum_{m\neq
n}{\textbf A}_{mn}$, we state that an inner product ${\textbf
f}^{\rm T}(t'){\textbf v}(t')$ is time-invariable if the column vector $\textbf v(t')=(v_1,\cdots,v_N)^{\rm T}$ satisfies $$\begin{aligned}
\frac{dv_n(t')}{dt'}=-\left[{\textbf H}^{\rm T}{\textbf
v}\right]_n-f_n^{-1}\left[\partial_{t'}{ \textbf f}-{\textbf H}{
\textbf f}\right]_nv_n+f_n^{-1} \left[\left({ \textbf A} \textbf
1\right)_nv_n-\left({\textbf A}^{\rm T} {\bf\text
v}\right)_n\right], \label{backwardeq}\end{aligned}$$ where the final condition of $v_n(t)$ is $q_n$ ($t'$$<$$t$), and the column vector ${\textbf 1}=(1,\cdots,1)^{\rm T}$. This is easily proved by noting a time differential $d_{t'}\left[{\textbf
f}^{\rm T}(t'){\textbf v}(t')\right]= d_{t'}({\textbf f}^{\rm
T}){\textbf v}+{\textbf f}^{\rm T}d_{t'}({\textbf v})$ and the transpose property of a matrix. Employing the Feynman-Kac and Cameron-Martin-Girsanov formulas for jump processes (a simple derivation about the latter see the Appendix I), eq. (\[backwardeq\]) has a stochastic representation given by $$\begin{aligned}
v_n(t')=E^{n,t'}\left[e^{-{\cal J}[{\textbf x},\textbf f,\textbf
A]} q_{s(t)}\right] \label{stochrep}\end{aligned}$$ and $$\begin{aligned}
{\cal J}[{\textbf x},\textbf f,\textbf A]=\int_{t'}^tf_{\textbf
x(\tau)}^{-1}\left[-\partial_{\tau}\textbf f+\textbf H\textbf
f+\textbf A\textbf 1\right]_{{\textbf
x}(\tau)}d\tau-\int_{t'}^tf_{{\textbf x}(\tau)}^{-1}\textbf
A_{{\textbf x}(\tau){\textbf x}(\tau)}d\tau
-\sum_{i=1}^k\ln\left[1+\frac{{\textbf A}_{{\textbf
x}(t_i^+){\textbf x}(t_{i}^-)}(t_i)}{f_{{\textbf
x}(t_i^-)}(t_i)\textbf H_{{\textbf x}(t_i^+){\textbf
x}(t_{i}^-)}(t_i)} \right],\label{functional}\end{aligned}$$ the expectation $E^{n,t'}$ is over all trajectories $\textbf x$ generated from eq.(\[forwardeq\]) with fixed initial state $n$ at time $t'$, $\textbf x(t')$ is the discrete state at time $t'$, $\textbf x(t_i^{-})$ and $\textbf x(t_i^{+})$ represent the states just before and after a jump occurs at time $t_i$, respectively, and we assumed the jumps occur $k$ times for a process. The readers are reminded that the first and last two terms of the functional are the consequences of the FK and GCM formulas, respectively. We see that the last term is significantly different from that in the continuous processes \[eq. (11) in ref. [@LiuF]\]. Combining the stochastic representation and the time-invariable quantity and choosing $t'=0$, we obtain the exact discrete GIFT for a jump process, $$\begin{aligned}
\sum_{m=1}^N f_m(0) E^{m,0}\left\{e^{-J[{\textbf x},\textbf
f,\textbf A]} q_{{\textbf x}(t)}]\right\}={\textbf f}^{\rm T}(t)
{\textbf q} \label{GIFT}\end{aligned}$$ Particulary, the right hand side of the equation become $1$ if $\textbf q=1$.

Relationship between the GIFT and existing IFTs for jump processes {#secIII}
==================================================================

The abstract eq. (\[GIFT\]) includes several discrete IFTs in literature. First we investigate the case in which the discrete system has a transient steady-state solution ${\textbf H}(t) {\textbf p}^{\rm ss}(t)=0$. Choosing the matrix $\textbf A$=0 and the vector $\textbf f(t)={\textbf p}^{\rm
ss}(t)$, eq. (\[functional\]) is immediately simplified into $$\begin{aligned}
{\cal J}=-\int_0^t\partial_\tau p^{\rm ss}_{\textbf
x(\tau)}(\tau)d\tau \label{JarzynskiHantanFunctional}\end{aligned}$$ If one further thinks of ${\textbf p}^{\rm ss}$ satisfying a time-dependent detailed balance condition ${\textbf
H}_{mn}(t)p^{ss}_n(t)={\textbf H}_{nm}(t)p^{ss}_m(t)$, the above functional may be analogous to the dissipated work and eq. (\[GIFT\]) is the discrete version of the JE [@JarzynskiPRE97; @JarzynskiPRL97]. On the other hand, if ${\textbf p}^{\rm ss}$ is a transient nonequilibrium steady-state without detailed balance, eq. (\[JarzynskiHantanFunctional\]) could be rewritten to $$\begin{aligned}
{\cal J}=\ln\frac{p^{\rm ss}_{\textbf x(0)}(0)}{p^{\rm
ss}_{\textbf x(t)}(t)}+\sum_{i=1}^k\ln\frac{p^{\rm ss}_{\textbf
x(t_i^+)}(t_i)}{p^{\rm ss}_{\textbf x(t_i^-)}(t_i)}.
\label{HantanFunctional}\end{aligned}$$ where we used the following relationship $$\begin{aligned}
{d_t}\ln p^{\rm ss}_{\textbf x(t)}(t)=\partial_t\ln p^{\rm
ss}_{\textbf x(t)}(t)+\sum_{i=1}^{k}\delta(t-t_i)
 \ln\left[p^{\rm ss}_{\textbf x(t_i^+)}(t_i)/p^{\rm
 ss}_{\textbf x(t_i^-)}(t_i)\right].
\label{differentialrelation}\end{aligned}$$ Then we may interpret the first term in eq. (\[HantanFunctional\]) as the entropy change of system and the second term as the “excess" heat of the driven jump process. Under this circumstance eq. (\[GIFT\]) is the discrete version of the Hatano-Sasa equality [@HatanoSasa].

The last case is about nonvanishing $\textbf A(t)$. Choosing the matrix element ${\textbf A}_{mn}(t)={\textbf
H}_{nm}(t)f_m(t)-{\textbf H}_{mn}(t)f_n(t)$ ($m\neq n$), or the flux $J_{mn}(t)$ between states $m$ and $n$ for a distribution $\textbf f(t)$. Obviously, the condition of $f_n\textbf
H_{mn}+\textbf A_{mn}$$>0$. Substituting this matrix into eq. (\[functional\]), we obtain $$\begin{aligned}
{\cal J}=-\int_0^t \partial_{\tau}\ln p_{{\textbf
x}(\tau)}(\tau)d\tau+\sum_{i=1}^{k}\ln\frac{{\textbf H}_{{\textbf
x}(t_i^-)\textbf x(t_i^+)}(t_i)p_{\textbf x(t_i^+)}(t_i)}{{\textbf
H}_{\textbf x(t_i^+)\textbf x(t_i^-)}(t_i)p_{\textbf
x(t_i^-)}(t_i)} \label{orgtotentropy}\end{aligned}$$ To achieve obvious physical meaning of the above expression, we employ eq. (\[differentialrelation\]) again and have $$\begin{aligned}
 {\cal
J}=\ln\frac{f_{\textbf x(0)}(0)}{f_{\textbf
x(t)}(t)}+\sum_{i=1}^k\ln\frac{{\textbf H}_{\textbf
x(t_i^+)\textbf x(t_i^-)}(t_i)}{{\textbf H}_{\textbf
x(t_i^-)\textbf x(t_i^+)}(t_i)}. \label{totalentropy}\end{aligned}$$ Hence, if $\textbf f(t)$ is the distribution of the system itself satisfying the evolution eq. (\[forwardeq\]), the first term in the equation is just the entropy change of the system and the second term is interpreted as entropy change of environment [@SeifertPRL05; @Harris]. In other words, the GIFT with eq. (\[totalentropy\]) is about the total entropy change of a stochastic jump process.

The GIFT and time reversal for jump processes
=============================================

Like the case of continuous diffusion processes, we can connect the time-invariable inner product to be a jump process that is regarded to be a time-reversal of the original jump process [@LiuF]. Multiplying $f_n(t')$ and rearranging on both sides of eq. (\[backwardeq\]), we have $$\begin{aligned}
\frac{d}{dt'}\left[f_n(t')v_n(t')\right]=-\sum_{m=1}^N
f_m^{-1}\left[\textbf H_{mn}f_n+{\textbf
A}_{mn}\right]f_mv_m+f_nv_n\sum_{m=1}^Nf_n^{-1}\left[\textbf
H_{nm}f_m+{\textbf A}_{nm}\right] \label{orgtimereversal}\end{aligned}$$ Then we define a new function $q_{\bar n}(s)=f_n(t')v_n(t')$, where $s=t-t'$ and $\bar n$ represents an index whose components are the same or the minus of the components of the index $n$ depending on whether they are even or odd under time reversal ($t\to -t$). We also define a new rate matrix $\overline{\textbf
H}(s)$ whose elements are $$\begin{aligned}
\overline{\textbf H}_{{\bar n}{\bar
m}}(s)=f_m^{-1}(t')\left[\textbf H_{mn}(t')f_n(t')+\textbf
A_{mn}(t')\right] \label{ratetimereversal}\end{aligned}$$ for $m\neq n$, and $\overline{\textbf H}_{mm}(s)=-\sum_{n\neq
m}\overline{\textbf H}_{nm}(s)$, respectively. Then eq. (\[orgtimereversal\]) is rewritten as $$\begin{aligned}
\frac{dq_{\bar n}(s)}{ds}=[\overline{\textbf H}(s)\textbf
q(s)]_{\bar n}. \label{timereversal}\end{aligned}$$ Because of the variable $s=t-t'$, we interpret $\overline{\textbf
H}(t)$ to be a time-reversal of the original $\textbf H(t)$. Equation (\[timereversal\]) directly presents the reason of the time-invariable inner product $\textbf f^{\rm T}(t')\textbf v(t')$ that equals $\textbf 1^{\rm T}\textbf q(s)$; the latter is a constant due to probability conservation.

The generalized time-reversal (\[ratetimereversal\]) includes several types of time-reversal in literature [@HatanoSasa; @Chernyak; @Harris]. For convenience, we only consider even components only in the state-index $n$. First, if the matrix $\textbf A=0$ and $\textbf f(t')=\textbf p^{\rm
ss}(t')$ satisfying the detailed balance condition, the time-reversed rate matrix $\overline{\textbf H}(t')={\textbf
H}(s)$ simply. The process determined by this rate matrix was termed backward process [@Harris] (or a reversed protocol in Ref. [@Chernyak]). In contrast, if $\textbf p^{\rm ss}(t')$ is transient nonequilibrium steady-state, a process determined by $\overline{\textbf H}_{mn}(t')=f_n(s){\textbf H}_{nm}(s)/f_m(s)$ was termed an adjoint process [@Harris] (or the current reversal in Ref. [@Chetrite]). Intriguingly, if we choose $\textbf A_{mn}(s)$ to be the flux $J_{mn}(t)$ between the states $m$ and $n$ for a distribution $\textbf f(s)$, we reobtain $\overline{\textbf H}(t')={\textbf H}(s)$ that is the same with case of the detailed balance condition. Considering that these choices of $\textbf f$ and $\textbf A$ here are corresponding to those in Sec. \[secIII\], respectively, we see that the JE and the IFT of the total entropy have the same physical origin. It is expected in physics since a realization of a reversed protocol is usually possible and does not depend on whether the system satisfies detailed balance condition. We should point out that one may construct infinite time-reversals, because $\textbf f$ and $\textbf A$ are almost completely arbitrary, e.g., $\textbf
A_{mn}(s)=\alpha J_{mn}(s)$ and $0\leq\alpha\leq 1$. Before ending this section, we give two comments about the relationship $q_{\bar
n}(s)=f_n(t')v_n(t')$. First, for a time-independent $\textbf H$, if $f_n$ is the equilibrium solution of the rate matrix, eq. (\[backwardeq\]) with zero $\textbf A$ is just the backward master equation [@Gardiner]. Second, employing the relationship repeatedly, we may obtain the detailed DFTs for the specific vectors $\textbf f(t')$ and matrixes $\textbf A(t')$ in Sec. \[secIII\] (the details see the Appendix II).

Conclusion
==========

In this work we derived a GIFT for general jump processes. The existing IFTs for discrete master equations are its special cases. We see that, in form the GIFT for the jump cases is apparently distinct from that for the continuous diffusions that we obtained earlier [@LiuF]. Additionally, we also find this robust GIFT usually does not result into a detailed fluctuation theorem. Compared to other approaches, the major advantage of the current and previous works is that the time-reversal can come out automatically during the constructions of the time-invariable integral or the inner product, which should be direct and obvious, at least from point of view of us. Of course, A limit of our two works is that we did not show some applications of the two GIFTs in concrete physical systems. We hope that this point would be remedied in near future.\
[This work was supported in part by Tsinghua Basic Research Foundation and by the National Science Foundation of China under Grant No. 10704045 and No. 10547002.]{}

Appendix I: The Cameron-Martin-Girsanov formula for jump processes {#appendix-i-the-cameron-martin-girsanov-formula-for-jump-processes .unnumbered}
==================================================================

Compared to the CMG formula for continuous diffusion processes, little literature discussed the CMG formula for discrete jump processes. For the convenience of the readers, we give a simple derivation of the formula here. Given a master equation with rate matrix ${\textbf H}$. The probability observing a trajectory ${\textbf x}(\cdot)$ which starts state $n_1$ at time $t_0=0$, jumps at time $t_1$ to state $n_2$,$\cdots$, finally jumps at time $t_k$ to $n_{k+1}$ and stay till time $t_{k+1}=t$ is $$\begin{aligned}
{\rm prob}[{\textbf
x}(\cdot)]=&&\prod_{i=1}^{k}\exp\left[\int_{t_{i-1}}^{t_i}{\textbf
H}_{\textbf x(t_i^-)\textbf x(t_i^-)}(\tau)d\tau\right]{\textbf
H}_{\textbf x(t_i^-)\textbf x(t_i^+)}\times
\exp\left[\int_{t_{k}}^{t}{\textbf
H}_{n_{k+1}n_{k+1}}(\tau)d\tau\right],\end{aligned}$$ where $\textbf x(t_i^{-})=n_i$ and $\textbf x(t_i^{+})=n_{i+1}$ ($i=1,\cdots,k$). Assuming that there is another master equation with a different rate matrix ${\textbf H'}={\textbf H}+{\textbf
A}$, where the matrix elements of ${\textbf A}$ may be negative. Then the ration of the probabilities observing the same trajectory in these two equations is simply $$\begin{aligned}
\frac{{\rm prob}'[{\textbf x}(\cdot)]}{{\rm prob}[{\textbf
x}(\cdot)]}=e^{-Q[{\textbf x}(\cdot)]}. \label{CGMformula}\end{aligned}$$ where $$\begin{aligned}
Q[{\textbf x}(\cdot)]=-\int_{0}^{t}{\textbf A}_{\textbf
x(\tau)\textbf
x({\tau})}(\tau)d\tau-\sum_{i=1}^{k}\ln\left(1+\frac{ {\textbf
A}_{\textbf x(t_i^-)\textbf x(t_i^+)}}{{\textbf H}_{\textbf
x(t_i^-)\textbf x(t_i^+)}}\right)\end{aligned}$$ Obviously, eq. (\[CGMformula\]) results into a IFT $$\begin{aligned}
\langle e^{-Q[{\textbf x}(\cdot)]}\rangle =1,\end{aligned}$$ where the average is over an ensemble of trajectories generated from the stochastic system with the rate matrix $\textbf H$ and with any initial distribution. Choosing a specific $$\begin{aligned}
{\textbf A}_{mn}(t)=p^{\rm ss}_n(t)^{-1}\left[{\textbf
H}_{nm}(t)p^{\rm ss}_m(t)-{\textbf H}_{mn}(t)p^{\rm
ss}_n(t)\right]{\text (m\neq n)}\end{aligned}$$ and ${\textbf A}_{nn}=-\sum_{m\neq n}{\textbf A}_{mn}(t)=0$, we obtain the IFT of the house-keeping heat [@Speck; @Harris] in discrete version, where $$\begin{aligned}
Q_{\rm hk}[{\textbf x}(\cdot)]=\sum_{i=1}^{k}\frac{{\textbf
H}_{\textbf x(t_i^-)\textbf x(t_i^+)}(t_i)p^{\rm ss}_{\textbf
x(t_i^+)}(t_i)}{{\textbf H}_{\textbf x(t_i^+)\textbf x(t_i^-)}
(t_i)p^{\rm ss}_{\textbf x(t_i^-)}(t_i)}\end{aligned}$$ Intriguingly, replacing $p_m^{\rm ss}$ above by the real probability distribution $p_m(t)$ of the system ${\textbf H}$, one obtains an IFT with $$\begin{aligned}
Q[{\textbf x}(\cdot)]=\int_0^t \partial_{\tau}\ln p_{{\textbf
x}(\tau)}(\tau)d\tau+\sum_{i=1}^{k}\frac{{\textbf H}_{\textbf
x(t_i^-)\textbf x(t_i^+)}(t_i)p_{\textbf x(t_i^+)}(t_i)}{{\textbf
H}_{\textbf x(t_i^+)\textbf x(t_i^-)}(t_i)p_{\textbf
x(t_i^-)}(t_i)} \label{psudoentropy}\end{aligned}$$ We notice that this functional is the almost same with that of the IFT of total entropy eq. (\[orgtotentropy\]) except that the first term becomes minus here. In addition, the average of $\langle Q[{\textbf s}(\cdot)]\rangle$ is the same with the average of the total entropy since the first terms of eqs. (\[totalentropy\]) and (\[orgtotentropy\]) vanish. We are not very clear whether eq. (\[psudoentropy\]) has new physical interpretation.

Appendix II: the detailed fluctuation theorem {#appendix-ii-the-detailed-fluctuation-theorem .unnumbered}
=============================================

Given the transition probability of eq. (\[timereversal\]) to be $q_{\bar n}(s'|m,s)$ (0$<$$s$$<$$s'$$<$$t$), the previous relationship implies $$\begin{aligned}
q_{\bar n}(s'|m,s)f_{\bar
m}(t-s)=f_{n}(t-s')E^{n,t-s'}\left[e^{-{{\cal
J}(t-s',t-s)}}\delta_{{\textbf x}(t-s),\bar m}\right]
\label{Gdetailedbalance}\end{aligned}$$ if one notices the initial condition $q_{\bar
n}(s|m,s)=\delta_{\bar n,m}$, where we use ${\cal J}(t-s',t-s)$ to denote the functional eq. (\[functional\]) with the lower and upper limits $t-s'$ and $t-s$, respectively, and $\delta$ is the Kronecker’s. Now we consider a mean of a $(k+1)$-point function over the time-reversed system (\[timereversal\]), $$\begin{aligned}
\langle{F}[{\bar {\textbf x}}(s_k),\cdots,{\bar {\textbf
x}}(s_0)]\rangle_{\rm TR}=\sum_{n_0,\cdots,n_k}
q_{n_k}(s_k|n_{k-1},s_{k-1})\cdots
q_{n_1}(s_1|n_0,s_0)q_{n_0}(s_0){F}(\bar n_k,\cdots,\bar n_0)\end{aligned}$$ where $0$=$s_0$$<$$s_1$$<$$\cdots$$<$$s_k$$=$$t$ and $\textbf
q(s_0)$ is the initial distribution. If we choose a specific $\textbf q_{n_0}(s_0)= f_{\bar n_0}(t-s_0)$ and employ eq. (\[Gdetailedbalance\]) repeatedly, the right hand side of the above equation becomes $$\begin{aligned}
\langle{e^{-{\cal J}[{\textbf x},\textbf f,\textbf A]}F}[{{\textbf
x}}(t_0),\cdots,{{\textbf x}}(t_k)]\rangle=\sum_{\bar n_k}f_{\bar
n_k}(t-s_k)E^{\bar n_k,t-s_k}\left\{ e^{-{\cal
J}(0,t)}{F}[{{\textbf x}}(t-s_k),\cdots,{{\textbf
x}}(t-s_0)]\right\}.\end{aligned}$$ Here we define $t_i=t-s_{k-i}$ and $0$$=$$t_0$$<$$t_1$$<$$\cdots$$<$$t_k$$=$$t$. On the basis of the above discussion, if $k\to\infty$ the function $F$ becomes a functional ${\cal F}$ over the space of all trajectories $\textbf
x$, and we get an identity $$\begin{aligned}
\langle{\bar{\cal F}}\rangle_{\rm TR}= \langle e^{-{{\cal
J}[{\textbf x},\textbf f,\textbf A]}}{\cal F}\rangle,
\label{GCrook}\end{aligned}$$ where ${\cal {\bar F}}(\textbf x)={\cal F}({\bar {\textbf x}})$ and $\bar {\textbf x}$ is simply the time-reversed trajectory of $\textbf x$. This is a generalization of Crooks’ relation [@Crooks00]. Obviously, choosing ${\cal F}$ constant, one obtains the GIFT (\[GIFT\]). An important following question is whether the GIFT results into a DFT. For the specific matrixes $\textbf A(t)$ and vectors $\textbf f(t)$ in sec. \[secIII\], we indeed obtain several DFTs $$\begin{aligned}
P_{\rm TR}(-J)=P(J)e^{-J}, \label{DFT}\end{aligned}$$ by choosing ${\cal F}(\textbf x)=\delta({\cal J}[{\textbf
x},\textbf f,\textbf A]-J)$, where $P(J)$ is the probability distribution for the quantity $\cal J$ achieved from the jump process (\[forwardeq\]) and $P_{\rm TR}(J)$ is the corresponding distribution from the time-revered system (\[timereversal\]). For any a pair of $\textbf A$ and $\textbf f$, eq. (\[DFT\]) usually does not hold.

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