Continuous-time random walk for open systems: Fluctuation theorems and counting statistics

Continuous-time random walk for open systems: Fluctuation theorems and counting statistics

Massimiliano Esposito

*

and Katja Lindenberg

Department of Chemistry and Biochemistry and Institute for Nonlinear Science, University of California, San Diego, La Jolla, California 92093-0340, USA

共Received 5 February 2008; published 19 May 2008兲

We consider continuous-time random walks共CTRW兲 for open systems that exchange energy and matter with multiple reservoirs. Each waiting time distribution 共WTD兲 for times between steps is characterized by a positive parameter␣, which is set to ␣=1 if it decays at least as fast as t−2_{at long times and therefore has a}

finite first moment. A WTD with␣⬍1 decays as t−␣−1_{. A fluctuation theorem for the trajectory quantity R,}

defined as the logarithm of the ratio of the probability of a trajectory and the probability of the time reversed trajectory, holds for any CTRW. However, R can be identified as a trajectory entropy change only if the WTDs have␣=1 and satisfy separability 共also called “direction time independence”兲. For nonseparable WTDs with ␣=1, R can only be identified as a trajectory entropy change at long times, and a fluctuation theorem for the entropy change then only holds at long times. For WTDs with 0⬍␣⬍1 no meaningful fluctuation theorem can be derived. We also show that the共experimentally accessible兲 nth moments of the energy and matter transfers between the system and a given reservoir grow as tn␣_{at long times.}

DOI:10.1103/PhysRevE.77.051119 PACS number共s兲: 05.70.Ln, 05.40.Fb, 05.40.⫺a

I. INTRODUCTION

It has long been clearly understood that the statement of the second law of thermodynamics concerning the increase in entropy in an isolated system as it goes to equilibrium refers only to the average behavior, but this was sufficient as long as one dealt only with macroscopic systems character-ized by extremely narrow ensemble distributions with fluc-tuations that were essentially never observed. More recently, with the ability to quantitatively monitor systems on the ex-tremely small scales of single molecules and quantum dots, it is possible to study fluctuations around the average behavior.

Fluctuation theorems that hold arbitrarily far from

equilib-rium have thus become subject to experimental verification 关1–7兴. These theorems in general deal with the ratio of the

probabilities of a given system trajectory and that of its time reversed trajectory, either as the system goes to equilibrium or as it evolves to a steady state under the action of nonequi-librium constraints imposed on the system. From this one can calculate, for example, the relative probabilities that the entropy of an isolated system away from thermodynamic equilibrium will spontaneously increase or decrease over a given period of time. The ratio is essentially infinite in a macroscopic system away from equilibrium and is unity due to fluctuations in equilibrium, but in sufficiently small sys-tems away from equilibrium it is merely large 共and experi-mentally accessible兲 rather than infinite.

Fluctuation theorems can take different forms depending on the specific problem under consideration, but they are all ultimately connected to the probabilistic asymmetry of sys-tem trajectories and time reversed trajectories. Equilibrium corresponds to the situation where the symmetry is restored as stated by the principle of microreversibility. Fluctuation

theorems have been formulated for a wide range of dynamics such as driven Hamiltonian dynamics 关8–10兴, stochastic

dy-namics 关11–16兴, deterministic thermostated dynamics

关17,18兴, and even quantum dynamics 关19–24兴. Here we focus

on stochastic dynamics in an effort to explore the validity of fluctuation theorems beyond the stochastic dynamics that have been considered to date.

In this narrower context of stochastic dynamics, most pre-vious studies of fluctuation theorems have focused on sys-tems described by Markovian master equations or Fokker-Planck equations. Recently there have been some efforts to investigate fluctuation theorems for systems described by nonlinear generalized Langevin equations 关25,26兴 with an

external driving force as a nonequilibrium constraint. Our focus is on nonequilibrium systems described by continuous-time random walks 共CTRW兲 关27–32兴 in which transitions

between microscopic states may be caused by more than one mechanism. The nonequilibirum constraint is imposed when these mechanisms have different statistical properties such as, for example, through contact with two heat baths main-tained at different temperatures. In general, identifying such nonequilibrium constraints may itself be complicated 关15,33兴, and we will here explicitly point to these

differ-ences.

We pose the following question: What properties of a CTRW are necessary for an entropy fluctuation theorem to be valid under such nonequilibrium constraints? We note, for example, that CTRWs are known to display aging关34–36兴 as

well as nonergodic phenomena 关37,38兴 which may

signifi-cantly alter the behavior of the system under time reversal and prevent a fluctuation theorem from being satisfied. At the same time, CTRWs under certain conditions reduce to Mar-kovian master equations which are known to satisfy fluctua-tion theorems. CTRWs therefore provide a good framework to study the validity of fluctuation theorems. In particular, our results will hopefully contribute clarification to recent observations of anomalous statistics in the nonequilibrium fluctuations of single molecules and quantum dots关39–42兴.

*Also at Center for Nonlinear Phenomena and Complex Systems, Universite Libre de Bruxelles, Code Postal 231, Campus Plaine, B-1050 Brussels, Belgium.

A second purpose of this paper is the formulation of a general framework for the calculation of共experimentally ac-cessible兲 counting statistics of events associated with a given mechanism. Examples of such events might involve particle or energy transfer. To accomplish this we use a method based on the propagation of the generating function associated with the probability distribution of the events, in the spirit of the method used for Markovian master equations 关12,16兴. This

will allow us to investigate the long-time behavior of the moments of the distribution associated with the counting sta-tistics.

Our basic CTRW model is constructed as follows. We consider a stochastic dynamics between microscopic states m of a system with a finite number of states. The transitions between states may be due to different mechanisms ␯. For example, we will subsequently consider a system in which each microscopic state m is characterized by a number Nmof

particles and an energy⑀m, and where the transitions between

the m’s are triggered by different reservoirs 共heat baths兲 ␯. Suppose that the system arrives at state m

⬘

at a given time and that its next jump is to state m at a time t later via mechanism ␯. The distribution of waiting times共WTD兲 for this to occur is denoted by␺_mm共␯兲_⬘共t兲, with other related quan-tities specified in more detail subsequently. We focus on waiting time distributions whose long-time behavior is re-flected in the small-s Laplace transform

␺ ˜ mm⬘ 共␯兲 _{共s兲 =} s→0 P_mm共␯兲_⬘− B_mm共␯兲_⬘s␣, 共1兲 where f˜共s兲⬅兰0⬁dte−stf共t兲 and 0⬍␣ⱕ1. The Bmm⬘

共␯兲 _are ele-ments of an arbitrary matrix. A detailed discussion surround-ing this choice can be found in 关43兴. When 0⬍␣⬍1 the

long-time decay of the WTDs is then of the power-law form

␺mm⬘

共␯兲 _共t兲⬃t−␣−1_{. When ␣= 1 the decay is at least as fast as} 1/t2but may be faster.

In Sec. II we present the CTRW model for an open system driven by different mechanisms described by different statis-tical properties, and formally express the probability that the system is in state m at time t. In Sec. III we derive the generalized master equation satisfied by this probability and study its long-time behavior. In Sec. IV we present a gener-ating function formalism to calculate the probability distri-bution of heat and matter transfers which is used to study the long-time behavior of the moments of the distribution. The principal results of this work, namely, the conditions for the validity of fluctuation theorems, are presented in Sec. V. In particular, we show that fluctuation theorems for the entropy change can only be obtained if ␣= 1, that is, if the WTDs decay at least as fast as t−2_{at long times. Furthermore, even} in this case the entropy change can be expressed as a familiar ratio of the probability of a trajectory and its time reversed trajectory only if the WTDs satisfy constraints of separabil-ity. A summary of results and some concluding remarks are presented in Sec. VI.

II. CONTINUOUS-TIME RANDOM WALKS FOR OPEN SYSTEMS

Our goal in this section is to construct the probability that the system will be found in a particular state at time t.

Sup-pose that the system arrives at state m

⬘

at time zero, and that its next jump is to state m at time t via mechanism ␯. The waiting time distribution ␺_mm共␯兲_⬘共t兲 for this event introduced earlier satisfies the normalization condition

冕

0 ⬁ d␶

兺

m,␯ ␺mm⬘ 共␯兲 _{共␶兲 = 1. 共2兲}

For convenience, we define␺mm共␯兲共␶兲⬅0. The probability that

no transition occurs up to time t after arrival at m

⬘

at time zero is ␾m⬘共t兲 =

兺

m,␯

冕

t ⬁ d␶␺_mm共␯兲_⬘共␶兲 = 1 −

兺

m,␯

冕

0 t d␶␺_mm共␯兲_⬘共␶兲. 共3兲

We see that by construction␾m⬘共⬁兲=0, that is, a jump

even-tually occurs with certainty. We define the auxiliary distribu-tions ␺m⬘ 共␯兲_{共t兲 ⬅}

_兺

m ␺mm⬘ 共␯兲 _共t兲, ␺m⬘共t兲 ⬅

兺

␯ ␺m⬘ 共␯兲_{共t兲. 共4兲} ␺m⬘

共␯兲_{共t兲 is the waiting time distribution of the first jump from} state m

⬘

to any other state via a mechanism␯, and ␺m⬘共t兲 is

the waiting time distribution of the first jump from state m

⬘

to any other state regardless of the mechanism, given that arrival at m

⬘

occurred at time zero. We also define

P_mm共␯兲_⬘⬅

冕

0 ⬁ d␶␺_mm共␯兲_⬘共␶兲, Pmm⬘⬅

兺

␯ P_mm共␯兲_⬘. 共5兲 Here P_mm共␯兲_⬘is the probability that, being at m

⬘

, the next jump will be from state m

⬘

to m via mechanism␯, and Pmm⬘is the

probability that the jump will be from m

⬘

to m irrespective of the mechanism. Note that by definition Pmm⬅0. We finally

define the probability f_m共␯兲_⬘that the next jump from m

⬘

will be due to mechanism ␯,

f_m共␯兲_⬘⬅

兺

m

P_mm共␯兲_⬘. 共6兲 The normalization condition共2兲 implies

infi-nite. This first moment is just the average time that the sys-tem remains in state m

⬘

before jumping to m via mechanism

␯:

t_mm共␯兲_⬘⬅

冕

0 ⬁

d␶␶␺_mm共␯兲_⬘共␶兲. 共8兲

Associated mean waiting times are given by

t_m共␯兲_⬘⬅

兺

m t_mm共␯兲_⬘, tm⬘⬅

兺

␯ tm⬘ 共␯兲 . 共9兲

The first is the average time that the system remains in state

m

⬘

without jumping anywhere else via mechanism ␯. The second is the average time that the system remains at m

⬘

without making any jumps at all by any mechanism. If ␣ ⬍1, all moments of the waiting time distribution 共including the first moments or mean waiting times兲 are divergent, whereas for ␣= 1 the first moments t_mm共␯兲_⬘ are finite. In this case, from Eqs.共1兲 and 共8兲 it follows that B_mm共␯兲_⬘= t_mm共␯兲_⬘.

Suppose that we begin our observations at time t = 0, at which time we find the system in state m

⬘

. Since in general we may not know when the jump occurred that brought the system to that state, we need to distinguish the waiting time distribution of the first jump after time zero from that of subsequent jumps. We mark this first waiting time distribu-tion with a prime,␺_mm

⬘

共␯兲_⬘共t兲. The probability that no transition away from m

⬘

occurs up to time t must be similarly distin-guished, ␾_m

⬘

共t兲. The primed functions are equal to the unprimed ones only if a jump occurred exactly at time zero or if the WTDs decay exponentially. If there is no informa-tion about when the last jump before time t = 0 occurred, then the primed functions can be related to the unprimed ones

only if the mean waiting times t_mm共␯兲_⬘are finite. In this case, an average over the uncertain past yields

␺mm

⬘

⬘ 共␯兲_{共t兲 =}

冕

−⬁ 0 d␶␺_mm共␯兲_⬘共t −␶兲

兺

m,␯

冕

0 ⬁ dt

冕

−⬁ 0 d␶␺mm⬘ 共␯兲 _{共t −␶兲} =

冕

−⬁ 0 d␶␺_mm共␯兲_⬘共t −␶兲 tm⬘ =

冕

t ⬁ d␶␺_mm共␯兲_⬘共␶兲 tm⬘ . 共10兲 Note that summing Eq.共10兲 over m and␯and using Eq.共3兲

leads to ␺m

⬘

⬘共t兲 = ␾m⬘共t兲 tm⬘ . 共11兲 Similarly, ␾m

⬘

⬘共t兲 =

冕

t ⬁ d␶␾m⬘共␶兲 tm⬘ . 共12兲

It is straightforward to construct an integral equation for

␳m共t兲, the probability that the system is in state m at time t.

For this purpose we also define␩m共t兲, the probability that the

system jumps onto state m at time t. The following integral CTRW relations are evident关27–30,32兴:

␩m共t兲 =

兺

m⬘,␯ ␺mm

⬘

⬘ 共␯兲_共t兲␳ m⬘共0兲 +

兺

m⬘,␯

冕

0 t d␶␺_mm共␯兲_⬘共t −␶兲␩m⬘共␶兲, 共13兲 and ␳m共t兲 =␾m

⬘

共t兲␳m共0兲 +

冕

0 t d␶␾m共t −␶兲␩m共␶兲. 共14兲

Since these are convolutions, it is easiest to solve for the Laplace transform␳˜m共s兲 of the desired probability␳m共t兲. The

Laplace transforms of these two relations are

␩ ˜m共s兲 =

兺

m⬘,␯ 关␺˜ mm⬘

⬘

共␯兲_共s兲␳ m⬘共0兲 +␺˜_m,m共␯兲_⬘共s兲␩˜m⬘共s兲兴, 共15兲 and ␳ ˜m共s兲 =␾˜m

⬘

共s兲␳m共0兲 +␾˜m共s兲␩˜m共s兲. 共16兲

The solution is most neatly expressed in terms of the matri-ces ⌽ and ⌿ with matrix elements

关⌿˜ 共s兲兴mm⬘⬅␺˜m,m

⬘

⬘共s兲,

关⌽˜ 共s兲兴mm⬘⬅␦mm⬘␾˜m共s兲, 共17兲

and the vectors 兩␩˜共s兲典 and 兩˜␳共s兲典 with elements ␩˜m共s兲 and

␳

˜_m共s兲. The solution of Eq. 共15兲 is

兩␩˜共s兲典 = 关I − ⌿˜ 共s兲兴−1_⌿˜

_⬘

_{共s兲兩␳共0兲典, 共18兲}

where I is the identity matrix. Using Eq. 共16兲, the formal

solution of the CTRW then is

兩˜␳共s兲典 = 兵⌽˜

⬘

共s兲 + ⌽˜ 共s兲关I − ⌿˜ 共s兲兴−1_⌿˜

_⬘

_{共s兲其兩␳共0兲典. 共19兲}

III. GENERALIZED MASTER EQUATION

In this section we construct the generalized master equa-tion for the probability that the system is in state m at time t. This equation facilitates the exploration of the behavior of this probability at long times. Using the Laplace transform of Eq. 共3兲, ␾˜_{m共s兲 =}1 s

兺

_m⬘关␦mm⬘−␺ ˜ m⬘m共s兲兴 = 1 s关1 −␺ ˜ m共s兲兴, 共20兲

␳ ˜m共s兲 =␾˜

⬘

共s兲␳m共0兲 + 1 s␩˜m共s兲 − 1 s_m

兺

_⬘,␯␺ ˜ m⬘m 共␯兲 _共s兲␩˜ m共s兲. 共21兲 Inserting Eq. 共15兲 in the second term of Eq. 共21兲 and using

Eq. 共16兲, we obtain s˜␳m共s兲 −␳m共0兲 = I˜共s兲 +

兺

m⬘,␯ 关W˜ mm⬘ 共␯兲 _共s兲˜␳ m⬘共s兲 − W˜m⬘m 共␯兲 _共s兲˜␳ m共s兲兴, 共22兲 where the transition matrix elements are given by

W˜_mm共␯兲_⬘共s兲 =␺ ˜ mm⬘ 共␯兲 _共s兲 ␾ ˜_m ⬘共s兲 共23兲 and I˜共s兲 ⬅

兺

m⬘,␯ 兵W˜ mm⬘ 共␯兲 _{共s兲关␾˜} m⬘共s兲 −␾˜m

⬘

⬘共s兲兴␳m⬘共0兲 − W˜_m共␯兲_⬘m共s兲关␾˜m共s兲 −␾˜m

⬘

共s兲兴␳m共0兲其. 共24兲

The inhomogeneous term I˜共s兲 in Eq. 共22兲 thus depends on

the initial condition. It vanishes if a jump occurs at time zero or if the waiting time distributions are exponential because then␾m

⬘

共t兲=␾m共t兲. Upon inverse Laplace transformation we

arrive at the generalized master equation

␳˙m共t兲 = I共t兲 +

兺

m⬘,␯

冕

0

t

d␶关W_mm共␯兲_⬘共␶兲␳m⬘共t −␶兲

− W_m共␯兲_⬘m共␶兲␳m共t −␶兲兴. 共25兲

The generalized master equation is clearly non-Markovian unless W˜_mm共␯兲_⬘共s兲 is independent of s. This occurs, for example, for separable distributions ␺_mm共␯兲_⬘共t兲= P_mm共␯兲_⬘␺m⬘共t兲 with ␺m共t兲

= e−t/tm/t

m. Indeed, in this case W˜mm⬘

共␯兲 _{共s兲= P}

mm⬘

共␯兲 _/t

m⬘.

Of interest for our purposes is the long-time behavior of the probability␳m共t兲. Here we distinguish the case␣= 1,

as-sociated with finite mean waiting times, from the case 0 ⬍␣⬍1, associated with divergent mean waiting times.

A. Long-time behavior for␣=1

Consider first the long-time behavior of the generalized master equation in the case ␣= 1. From Eq. 共20兲 it follows

that ␾ ˜_{m共s兲 =} s_→0 Bms␣−1, 共26兲 where Bm=兺m⬘,␯Bm⬘m

共␯兲 _{. Since it follows from this result and} from Eq. 共1兲 that Bmm⬘= tmm⬘and Bm= tm, Eq.共23兲 then

im-mediately leads to W˜_mm共␯兲_⬘共0兲 = lim s_→0 W˜_mm共␯兲_⬘共s兲 =␺ ˜ mm⬘ 共␯兲 _共0兲 ␾˜_m ⬘共0兲 = Pmm⬘ 共␯兲 tm⬘ . 共27兲

Since lims_→0(␾˜m⬘共s兲−␾˜m

⬘

⬘共s兲) is constant when␣= 1, we find

that lims→0I共s兲 is also constant 关see Eq. 共24兲兴. Therefore,

using the final value theorem f共⬁兲=lims→0sf˜共s兲 关32,45兴, we

find that lim t_→⬁I共t兲 = 0, lim t_→⬁ W_mm共␯兲_⬘共t兲 = 0. 共28兲

This means that at long times, the generalized master equa-tion behaves like the Markovian master equaequa-tion

␳˙m共t兲 =

兺

m⬘,␯ 关W˜ mm⬘ 共␯兲 _共0兲␳ m⬘共t兲 − W˜m⬘m 共␯兲 _共0兲␳ m共t兲兴. 共29兲

Defining the rate matrix V⬅兺_␯V共␯兲where

V_mm共␯兲_⬘⬅Pmm⬘ 共␯兲 Bm⬘ for m⫽ m

⬘

, Vmm共␯兲⬅ −

兺

m⬘共⫽m兲 V_m共␯兲_⬘m, 共30兲 we can rewrite Eq.共29兲 as 兩␳˙共t兲典=V兩␳共t兲典. Using the

Perron-Frobenious theorem, all eigenvalues of V are negative aside from one which is zero. The probability ␳m共t兲 therefore

de-cays exponentially to a steady state solution ␳_mss that obeys the condition

兺

m⬘,␯ 关W˜ mm⬘ 共␯兲 _共0兲␳ m⬘ ss − W˜_m共␯兲_⬘m共0兲␳m ss_{兴 = 0. 共31兲} The steady state solution corresponds to equilibrium if the detailed balance condition

W˜_mm共␯兲_⬘共0兲␳_meq_⬘= W˜_m共␯兲_⬘m共0兲␳meq 共32兲

is satisfied, that is, if all fluxes between pairs of states asso-ciated with the different mechanisms␯become zero at equi-librium. This would not be possible if the long-time statistics of the different mechanisms were different. If the detailed balance condition is not satisfied, then the solution␳mm

ss is a nonequilibrium steady state.

A useful connection to thermodynamics is provided if we consider that each state m has a given energy⑀mand number

of particles Nmand that each different mechanism␯inducing

W˜_mm共␯兲_⬘共0兲 W˜_m共␯兲_⬘m共0兲

= exp兵−␤_␯共⑀m−⑀m⬘兲 +␤␯␮␯共Nm− Nm⬘兲其.

共33兲 Equation共32兲 holds only if all the␤_␯’s and␮_␯’s are equal. In this case the equilibrium distribution corresponds to the grand canonical ensemble. If the␤_␯’s and␮_␯’s are different, the steady state is a nonequilibrium steady state that obeys Eq. 共31兲.

B. Long-time behavior for 0⬍␣⬍1

When 0⬍␣⬍1, we must separately specify the statistical properties of the waiting time distribution for the first jump after t = 0. We choose ␾˜m

⬘

共s兲=␾˜m共s兲 because other choices

add only further complications but little of general interest to our specific problem. The inhomogeneous initial condition term then drops out, and for small s, using Eq.共30兲, Eq. 共22兲

becomes

s˜␳m共s兲 −␳m共0兲 =

兺

m⬘,␯

V_mm共␯兲_⬘s1−␣˜␳m⬘共s兲. 共34兲

Using the rules of fractional calculus关44兴, this equation can

be written in the time domain as

d

dt␳m共t兲 =_m

兺

⬘,␯

V_mm共␯兲_⬘0D_t1−␣␳m⬘共t兲, 共35兲

where the Riemann-Liouville fractional integral is given by

0Dt1−␣␳m⬘共t兲 = 1 ⌫共␣兲 d dt

冕

₀ t d␶共t −␶兲␣−1␳m⬘共␶兲. 共36兲

If the matrices with elements V_mm共␯兲_⬘can be diagonalized and the eigenvalues vi共which are all negative or zero兲 are

non-degenerate, the solution of Eq.共35兲 can be written as a linear

combination of solutions of

d

dt␳i共t兲 = vi 0Dt

1−␣_␳

i共t兲. 共37兲

The solution of this equation is the Mittag-Leffler function

E_␣关−共t/␶兲␣兴, where ␶=共−vi兲1/␣ 共see Appendix B of Ref.

关44兴兲. At long times the Mittag-Leffler function decays as a

power law,

E_␣关− 共t/␶兲␣兴 ⬃ 关共t/␶兲␣⌫共1 −␣兲兴−1 _共38兲 共at short times it behaves as a stretched exponential,

E_␣关−共t/␶兲␣兴⬃exp共−_{⌫共1+␣兲}共t/␶兲␣兲兲. Thus the general solution of the

generalized master equation is a linear combination of Mittag-Leffler functions, and the probability␳m共t兲 decays

to-ward the zero eigenvalue mode as a power law.

IV. COUNTING STATISTICS

Although our ultimate goal is to establish conditions un-der which fluctuation theorems are valid for systems whose dynamics are described by CTRWs, we first consider the

counting statistics for such a system. These statistics are in-teresting because they are experimentally accessible, and this analysis leads to some definitions that are useful in the dis-cussion of fluctuation theorems.

Consider a system described by the CTRW of Sec. II where each microscopic state m has a given number of par-ticles Nmand an energy⑀m, and where the allowed transitions

between pairs of states are due to different mechanisms ␯, each corresponding to a reservoir␯. We want to calculate the probability P共兵⌬E共␯兲_其,兵⌬N共␯兲_{其,t兲 that an energy transfer} ⌬E共␯兲_{and a matter transfer⌬N}共␯兲_{occurs between the system} and the reservoir ␯ during time t. We define the set of pa-rameters␥⬅共兵␥_e共␯兲其,兵␥_m共␯兲其兲 and the generating function

G共i␥,t兲 =

冕

−⬁ ⬁ 兵d⌬E共␯兲_其

兺

兵⌬N共␯兲其=−⬁ ⬁ P共兵⌬E共␯兲其,兵⌬N共␯兲其,t兲 ⫻exp关i共␥e共␯兲⌬E共␯兲+␥m共␯兲⌬N共␯兲兲兴. 共39兲 The probability can be recovered from the generating func-tion using P共兵⌬E共␯兲其,兵⌬N共␯兲其,t兲 =

冕

−⬁ ⬁

再

_d␥ e 共␯兲 2␲

冎

冕

₀ 2␲

再

_d␥ m 共␯兲 2␲

冎

⫻exp兵− i共⌬E共␯兲_␥ e 共␯兲_+⌬N共␯兲_␥ m 共␯兲_兲其 ⫻G共i␥,t兲. 共40兲

Derivatives of the generating function with respect to the elements of ␥ evaluated at ␥= 0 gives the moments of the distribution of this process.

We define ␺mm⬘共␥,t兲 ⬅

兺

␯ e ␥m共␯兲共Nm−Nm⬘兲_e␥e共␯兲共⑀m−⑀m⬘兲␺ mm⬘ 共␯兲 _{共t兲, 共41兲} which is the WTD matrix whose elements associated with a transition caused by mechanism␯ are weighted by the expo-nential of␥_e共␯兲共␥_m共␯兲兲 times the change of energy 共matter兲 that this transition induces. We note that ⌿共␥= 0 , t兲=⌿共t兲. To evaluate the generating function and the associated moments, we replace the WTD ␺_mm共␯兲_⬘共t兲 by Eq. 共41兲 in the CTRW 共13兲

and共14兲, and thus obtain

␩m共␥,t兲 =

兺

m⬘␯ ␺mm

⬘

⬘ 共␯兲_{共␥,t兲␳} m⬘共0兲 +

兺

m⬘␯

冕

0 t d␶␺_mm共␯兲_⬘共␥,t −␶兲␩m⬘共␥,␶兲 共42兲 and ␳m共␥,t兲 =␾m

⬘

共t兲␳m共0兲 +

冕

0 t d␶␾m共t −␶兲␩m共␥,␶兲. 共43兲

By doing so, we are weighting the probability of all the trajectories of length t ending up in the state m and along which a transfer of energy共matter兲 between the system and the reservoir␯occurs, by the exponential of␥_e共␯兲共␥_m共␯兲兲 times this energy共matter兲 transfer. By summing over all final states

G共␥,t兲 = 具兩I兩␳共␥,t兲典 =

兺

m

␳m共␥,t兲, 共44兲

where 兩I典 is the unit vector. Proceeding as in Sec. II, the formal solution of Eqs.共42兲 and 共43兲 in Laplace space leads

to the general solution for the generating function

G˜ 共␥,s兲 = 具I兩关⌽˜ˆ

⬘

共s兲 + ⌽˜ˆ 共s兲共Iˆ − ⌿˜ˆ 共␥,s兲兲−1⌿˜ˆ

⬘

共␥,s兲兴兩␳共0兲典.

共45兲 If Eq. 共45兲 can be evaluated and inverse Laplace

trans-formed, it provides the full statistics of the energy and matter transfer for a finite time interval. This is often a difficult task, and one therefore often focuses on the long-time behavior. This behavior is accessed through the solution共45兲 and also

through the equation of motion for the generating function, which can be deduced by proceeding in the same way as in Sec. III when deriving the generalized master equation from Eqs. 共42兲 and 共43兲. Defining

W˜_mm共␯兲_⬘共␥,s兲 ⬅␺ ˜ mm⬘ 共␯兲 _共␥,s兲 ␾ ˜_m ⬘共s兲 , 共46兲 we find ␳˙m共␥,t兲 = I共␥,t兲 +

兺

m⬘,␯

冕

0 t d␶关W_mm共␯兲_⬘共␥,␶兲␳m⬘共␥,t −␶兲 − W_m共␯兲_⬘m共␶兲␳m共␥,t −␶兲兴, 共47兲 where I˜共␥,s兲 ⬅

兺

m⬘,␯ 兵W˜ mm⬘ 共␯兲 _{共␥,s兲关␾˜} m⬘共s兲 −␾˜m⬘共s兲兴␳m⬘共0兲 − W˜_m共␯兲_⬘m共s兲关␾˜m共s兲 −␾˜m

⬘

共s兲兴␳m共0兲其. 共48兲

To calculate the long-time behavior of the moments of the probability distribution of heat and matter transfer between the system and its reservoir, we first consider the situation when a jump occurred at time zero. We will comment later on situations in which this is not the case. In the long-time limit, Eq. 共45兲 diverges at ␥= 0 because 关Iˆ−⌿˜ˆ 共s兲兴−1 =

s→0

共Iˆ − Pˆ 兲−1_{, a limit which is singular because the determinant of}

Iˆ − Pˆ is zero. This can be seen by considering the transpose of

the matrix共transposition does not affect the determinant兲 and by replacing its first column by the sum of all the columns of the matrix 共the determinant is not affected by replacing a column by a linear combination of it with other columns兲, which only contains zeros since兺m⬘⫽mPm⬘m= 1共the

determi-nant of a matrix with a zero column is zero兲. To lowest order in s, the determinant behaves like ⬃s␣. This means that at small s,关Iˆ−⌿˜ˆ 共s兲兴−1⬃

s→0

s−␣. Using the long-time behavior of the WTD as expressed in Eqs. 共1兲 and 共26兲, and using Eq.

共45兲, we thus see that G˜ 共␥= 0 , s兲 ⬃

s→0

s−1. We could have

ar-rived at this directly because G共␥= 0 , t兲=1 implies G˜ 共␥

= 0 , s兲=s−1_{. This reasoning makes it clear that in order to} calculate moments, the derivatives with respect to one of the

␥’s evaluated at ␥= 0 must be calculated from Eq.共45兲

be-fore the long-time limit is taken.

Upon taking the nth derivative of Eq.共45兲 with respect to

one of the␥’s at␥= 0, the dominant contribution at small s is

⳵␥nG˜ 共␥= 0,s兲 = s→0具I兩Bˆs ␣−1_兵⳵ ␥ n_{关Iˆ − ⌿˜ˆ 共␥= 0,s兲兴}−1_{其Pˆ共␥= 0兲} ⫻兩␳共0兲典 ⬃ s−n␣−1_{. 共49兲}

We used the fact that at small s, ⳵_␥n关Iˆ−⌿˜ˆ 共␥= 0 , s兲兴−1 ⬃s−␣共n+1兲_{. Bˆ is a diagonal matrix with diagonal elements B}

m.

Using Tauberian theorems 关32,45,46兴, we conclude that the

moments of the energy and matter transfer between the sys-tem and the reservoir ␯ will behave at long times as

具共⌬E共␯兲_兲n_{典,具共⌬N共␯兲兲}n_{典 ⬃} t_→⬁

tn␣. 共50兲 As noted earlier, these moments are experimentally acces-sible.

Let us now turn back to the case where no jump occurred at time zero. We only comment on WTDs with ␣= 1 since only then is it possible to carry out the averaging procedure described in Sec. II. Since ⌿˜_mm

⬘

_⬘共s兲 =

s_→0

tmm⬘/tm⬘ and

⌿˜

mm⬘共s兲 = s→0

Pmm⬘, Eq. 共50兲 with ␣= 1 is still valid, perhaps

with a different proportionality factor. In the Appendix we explicitly implement these ideas by calculating the moments for a two level quantum dot.

V. FLUCTUATION THEOREMS

We now explore the conditions under which a fluctuation theorem holds for a CTRW. Two types of derivations have been used to obtain steady-state fluctuation theorems for sto-chastic dynamics. The first relies on a symmetry of the gen-erating function, which translates into a fluctuation theorem using large deviation theory as in Refs. 关12,15兴. The other

exploits the specific form of the logarithm of the ratio of the probability of a trajectory and the probability of its time reversed trajectory. We will use both for CTRWs and find that they can only be considered equivalent under specific conditions.

A. Using large deviation

This approach to arrive at a fluctuation theorem can only be implemented for ␣= 1 because it requires a finite mean waiting time between transitions. We thus restrict our discus-sion to this case. We have seen in Sec. III A that as long as one considers WTDs with␣= 1, the generalized master equa-tion at long times behaves like a Markovian master equaequa-tion and thus reaches a steady state. Using the same arguments, we can show that the equation of motion 共47兲 for the

⳵ ⳵t兩G共␥,t兲典 =t→⬁ W共␥兲兩G共␥,t兲典, 共51兲 where 关W共␥兲兴mm⬘⬅ W˜mm⬘共␥,0兲 for m ⫽ m

⬘

, 关W共␥兲兴mm⬅ −

兺

n W˜nm共0兲. 共52兲

This implies that for long times

G共␥,t兲 = C共␥,t兲eS共␥兲t, 共53兲 where limt→⬁

1

tln C共␥, t兲=0 and where S共␥兲 is the dominant

eigenvalue of W˜ 共␥, 0兲. This dominant eigenvalue gives the cumulant generating function because

S共␥兲 = lim

t→⬁

1

tln G共␥,t兲. 共54兲

Note that the derivation to follow does not explicitly require the generalized master equation to be equivalent to a Mar-kovian master equation at long times; what is required is the limiting behavior 共54兲. Furthermore, using Eq. 共33兲, we can

verify that Eq.共52兲 satisfies

W共␥兲 = Wt共A −␥兲, 共55兲 where A =共兵␤_␯其,兵−␤_␯␮_␯其兲. Since these two matrices have the same eigenvalues, this implies the symmetry

S共␥兲 = S共A −␥兲. 共56兲

Large deviation theory关12,15兴 can now be applied. The

lim-iting behavior 共54兲 and the symmetry 共56兲 then imply the

fluctuation theorem for the probability P共兵−1_t⌬E共␯兲其,

兵−1

t⌬N共␯兲其兲 for the energy and matter currents 共cf. below兲

between the system and the reservoir␯at long times关12,15兴,

P

共兵

1_t⌬E共␯兲

其

,

兵

1_t⌬N共␯兲

其兲

P

共兵

−1_t⌬E共␯兲

其

,

兵

−1_t⌬N共␯兲

其兲

= e ⌬Sr _{for t}→ ⬁. 共57兲 Here ⌬Sr=

兺

␯ 共␤␯⌬E 共␯兲_−␤ ␯␮␯⌬N共␯兲兲 共58兲 represents the change of entropy due to the exchange pro-cesses with the reservoirs. We note that the change in energy and matter can be written in terms of energy and matter currents as ⌬E共␯兲_⬅

冕

0 t d␶I_e共␯兲共␶兲, 共59兲 ⌬N共␯兲_⬅

冕

0 t d␶I_m共␯兲共␶兲, 共60兲

so that in the long-time limit ⌬E共␯兲/t and 1_t⌬N共␯兲/t corre-spond to steady-state currents. This is why Eq.共57兲 is called

a current fluctuation theorem关15兴.

We now define the matrix W共¯␥兲 as W共␥兲 where ␥ is re-placed by ␥=共兵␥e␤␯其,兵−␥m␤␯␮␯其兲. Obviously, when

replac-ing W共␥兲 by W共␥¯兲 in our previous results, we calculate the

statistics of ⌬Sr. The symmetry 共55兲 now implies that

W共␥¯兲=Wt_{共1−¯␥兲 so that}

S共¯␥兲 = S共1 −¯␥兲. 共61兲

Using again large deviation theory, we get the fluctuation theorem

P

共

1_t⌬Sr

兲

P

共

−1_t⌬Sr

兲

= e⌬Sr _{for t}→ ⬁. 共62兲

In the steady state, the average energy and matter transferred with a reservoir ␯ can be obtained by taking the derivative with respect to␥_␯at␥= 0 of the formal solution of Eq.共51兲.

We get 具⌬E共␯兲_{典/t ⬅ 具I} e 共␯兲_{典 =}

兺

m,m⬘ 共⑀m−⑀m⬘兲Wmm⬘ 共␯兲 _␳ m⬘ ss , 具⌬N共␯兲_{典/t ⬅ 具Im}共␯兲_{典 =}

兺

m,m⬘ 共Nm− Nm⬘兲Wmm⬘ 共␯兲 _␳ m⬘ ss . 共63兲 Current conservation at steady state follows from

兺

_␯ 具Ie共␯兲典 =

兺

␯ 具Im

共␯兲_{典 = 0. 共64兲}

If we assume that all M reservoirs共␯= 1 , . . . , M兲 have differ-ent temperatures and chemical potdiffer-entials, we have M − 1 in-dependent nonequilibrium forces associated with energy and matter transfer that can be defined as

Xe共i兲=␤1−␤i+1,

X_m共i兲= −␤1␮1+␤i+1␮i+1, 共65兲

where i = 1 , . . . , M − 1. Using Eq.共64兲, the average change in

the entropy due to exchange processes with the reservoirs 具⌬Sr典 can be written in the familiar thermodynamical form of

the entropy production in a steady state

具⌬Sr典/t =

兺

i=1 M−1

共Xe共i兲具Im共i兲典 + Xm共i兲具Im共i兲典兲. 共66兲 When 0⬍␣⬍1, we have seen in Sec. III B that the solution of the generalized master equation behaves at long times as a power law t−␣. This will also be the case for the solution of the equation of motion for the generating function 共47兲.

Therefore, contrary to Eq. 共54兲, the cumulant generating

function limt→⬁

1

tln G共␥, t兲 is zero. This indicates that the

cumulants decay more slowly than t. The fluctuation theorem symmetry is only present in the eigenvalues of the generator of the generalized master equation共51兲 but not in the

B. Using time reversal symmetry

We denote a forward trajectory of the system between times t = 0 and t = T by m_␶. As illustrated in Fig.1, at t = 0 the system is in state m0and stays there until it jumps to state m1 at time␶1 via mechanism␯1. It remains there until time␶2, when it jumps to state m2via mechanism␯2. The trajectory continues in this fashion; at time␶N there is a jump to state

mN, where the system remains at least until time T. The total

number of jumps in this trajectory is N. The probability of this trajectory is

P关m_␶兴 =␳m₀共0兲␺mm

⬘

⬘

共␯1兲共␶1兲

冉

兿

i=1 N−1

␺m共␯_i+1i+1m兲_i共␶i+1−␶i兲

冊

␾m_N共T −␶N兲.

共67兲 The time-reversed trajectory m¯_␶starts in state mN at time T,

jumps to state mN−1at time T −␶Nvia mechanism␯N, and so

on. At time T −␶1a jump to state m1 occurs via mechanism

␯1, and the system remains there until at least time 0. The probability of this time reversed trajectory is

P关m¯_␶兴=␳m_N共T兲␺m

⬘

共␯_N−1N−1m_N兲共T−␶N兲

冉

兿

i=0 N−2

␺m共␯_iim兲_i+1共␶i+2−␶i+1兲

冊

␾m₀共␶1兲.

共68兲 Next we consider the quantity

R关m_␶兴 ⬅ lnP关m␶兴

P关m¯_␶兴, 共69兲

whose explicit form reads

R关m_␶兴 = ln␳m0共0兲 ␳m_N共T兲 + ln

兿

i=1 N−1_␺ m_i+1m_i 共␯i+1兲 共␶ i+1−␶i兲 ␺m共␯_i−1i兲m_i共␶i+1−␶i兲 + ln ␺m

⬘

1m0 共␯1兲共␶1兲␾ m_N共T −␶N兲 ␺m

⬘

共␯_N−1N兲m_N共T −␶N兲␾m₀共␶1兲 . 共70兲

Because each m¯_␶ is the unique mirror image of m_␶, a sum over all possible forward trajectories is equivalent to a sum over all possible time-reversed trajectories. Therefore since normalization implies that 兺m¯_␶P关m¯_␶兴=1, an integral

fluctua-tion theorem follows immediately from the definifluctua-tion共69兲,

具e−R_{典 =}

兺

m_␶

e−R关m␶兴_{P关m␶兴 = 1. 共71兲} The positivity of the ensemble average of R关m_␶兴, 具R典ⱖ0, then follows from Jensen’s inequality. Using the important property R关m␶兴=−R关m¯␶兴, which follows from Eq. 共69兲 to-gether with the fact that by taking twice the time reversal of a trajectory we get back to the original trajectory m¯¯_␶= m_␶, we can also derive a detailed fluctuation theorem for R,

P共R兲 =

兺

m_␶ ␦共R − R关m␶兴兲P关m␶兴 =

兺

m_␶ ␦共R − R关m␶兴兲eR关m␶兴P关m¯␶兴 = eR

兺

m ¯_␶ ␦共R − R关m␶兴兲P关m¯␶兴 = eR

兺

m ¯_␶ ␦共R + R关m¯_␶兴兲P关m¯_␶兴 = eR P共− R兲. 共72兲

The integral fluctuation theorem 共71兲 and the detailed

fluc-tuation theorem 共72兲 for R are thus completely general and

valid for any continuous-time random walk. It is worth men-tioning that a similar derivation can be done for any dynam-ics as long as each trajectory has a corresponding time-reversed trajectory with a nonzero probability 关10兴.

To make these fluctuation theorems useful, we need to give a physical interpretation to R关m_␶兴. In particular, we will argue that R can only be interpreted as a change of entropy if two conditions are satisfied. One is that the WTDs have a finite first moment共␣= 1兲. The other is that they be separable 关47,48兴, that is, that the waiting time distributions can be

written as the product of a waiting time portion that depends only on the originating state, and a transition matrix that connects given initial and final states,

␺mm⬘

共␯兲 _{共t兲 = P}

mm⬘

共␯兲 _␺

m⬘共t兲. 共73兲

The separability condition is called “direction time indepen-dence” by Qian and Wang关47,48兴. The first term of Eq. 共70兲,

⌬S关m␶兴 ⬅ ln

␳m₀共0兲

␳m_N共T兲

= Sm_N共T兲 − Sm₀共0兲, 共74兲

where Sm共t兲=−ln␳m共t兲, can be interpreted as a change of

system 共Gibbs兲 entropy along the trajectory because it de-pends only on the initial and final microscopic states of the system for that trajectory, and the average over trajectories is then simply 具⌬S典=S共t兲−S共0兲, where S共t兲=兺m␳m共t兲Sm共t兲 is

just a straightforward average over states.

In order to interpret the two remaining terms in Eq.共70兲,

we implement separability of the WTDs so that

R关m_␶兴 = ln␳m0共0兲 ␳m_N共T兲 + ln

冉

兿

i=0 N−1_P m_i+1m_i 共␯i+1兲 Pm共␯_ii+1m_i+1兲

冊

+ ln␺m

⬘

0共␶1兲␾mN共T −␶N兲 ␺m

⬘

_N共T −␶N兲␾m₀共␶1兲 . 共75兲

Now only the WTDs of the first jumps remain. In Eq. 共72兲,

τ τ τ τ τ T 0 m m m m m m m m m(τ) ν ν ν ν _ν 1 2 j-1 j N 0 1 j-2 2 j-1 j N-1 N 1 2 j-1 j N

the path summation runs over all possible trajectories includ-ing those with or without a jump at time zero. One way to handle the problem of having to treat the first jump differ-ently from the others is via the time averaging procedure 共10兲 in Eq. 共67兲 as well as in Eq. 共68兲. This can only be done

if the first moments are finite. The third term in Eq.共75兲 now

becomes equal to ln共tm_N/tm₀兲, so that using Eq. 共27兲 the

sec-ond and third terms in Eq.共75兲 can be combined and R关m_␶兴

can be written in terms of the transition matrix elements as

R关m_␶兴 = ln␳m0共0兲 ␳m_N共T兲 + ln

冉

兿

i=0 N−1_W˜ m_i+1m_i 共␯i+1兲 共0兲 W˜m共␯_ii+1m_i+1兲 共0兲

冊

. 共76兲

This form of R关m␶兴 is now exactly the same as the one de-rived for the Markovian master equation共29兲 共see 关14,23兴兲.

This is a manifestation of the so-called corresponding Mar-kov process of a CTRW关48兴.

We denote the second term on the right-hand side of Eq. 共76兲 as ⌬Sr关m␶兴 and call it the reservoir part of the trajectory

entropy because it can be interpreted as the change in en-tropy along the trajectory due to exchange processes between the system and the reservoirs. Indeed, using Eq. 共33兲, this

term can be expressed as

⌬Sr关m␶兴 ⬅ ln

冉

兿

i=0 N−1_W˜ m_i+1m_i 共␯i+1兲 共0兲 W˜m共␯_ii+1m_i+1兲 共0兲

冊

=

兺

␯ 共−␤␯⌬E 共␯兲_关m ␶兴 +␤␯␮␯⌬N共␯兲关m␶兴兲, 共77兲 where the change of energy and number of particles along the trajectory due to the mechanism ␯ can be expressed in terms of heat and matter currents along the trajectory as

⌬E共␯兲_关m ␶兴 ⬅

冕

0 t d␶Ie共␯兲关m␶兴, 共78兲 ⌬N共␯兲_关m ␶兴 ⬅

冕

0 t d␶Im共␯兲关m␶兴 共79兲

关cf. Eqs. 共59兲 and 共60兲兴. These currents are sequences of ␦

functions centered at the times of the jumps and multiplied by the corresponding energy or matter change. They are positive共negative兲 if energy or matter increases 共decreses兲 in the system.

We have thus arrived at the important result that for sepa-rable WTDs with ␣= 1, R关m_␶兴=⌬S关m_␶兴+⌬Sr关m␶兴. Since

⌬S关m␶兴 and ⌬Sr关m␶兴 are interpreted, respectively, as the

change in the system entropy along the trajectory and the change in entropy due to the exchange processes between the system and its reservoirs, it is natural to interpret R关m␶兴 as the total change in entropy along the trajectory 共also called the total change in the trajectory entropy production兲 关14,23兴.

Note that in this case the principle of microreversibility, im-plying that at equilibrium the probability of a forward tory is identical to the probability of its time reversed trajec-tory共R关m_␶兴=0兲, is satisfied if the detailed balance condition 共32兲 is satisfied and if the system is initially at equilibrium

关so that ␳m₀共0兲=␳m₀

eq

and␳m_N共T兲=␳m_T

eq_{兴. This result is}

consis-tent with the findings of Ref.关48兴 stating that separability of

the WTDs and detailed balance are sufficient conditions for microreversibility to be satisfied in a CTRW.

In Sec. V A we showed that a fluctuation theorem for⌬Sr

can be derived for long times 关see Eq. 共62兲兴. In fact,

consid-ering separable WTDs with ␣= 1, the fluctuation theorem 共62兲 can be seen as resulting from the fluctuation theorem

共72兲, as follows. The quantity ⌬S关m_␶兴 is a bounded quantity

which only depends on the probability distribution of the initial and final states of the trajectory. On the contrary, ⌬Sr关m␶兴 changes each time a jump occurs along the system

trajectory. It is therefore reasonable to assume that for very long trajectories and for the huge majority of realizations, the contribution from ⌬Sr关m␶兴 in R关m␶兴=⌬S关m␶兴+⌬Sr关m␶兴 will

be significantly dominant so that in the long-time limit Eq. 共72兲 reduces to Eq. 共62兲. However, the derivation of Eq. 共62兲

only required WTDs with ␣= 1, but did not require separa-bility. The reason for this requirement in order to identify Eq. 共72兲 as a useful fluctuation theorem for all times is that

oth-erwise R关m_␶兴 has no clear physical interpretation. The prob-lem is that R关m␶兴 关and, more specifically, the second term in Eq. 共70兲, which according to our previous argument

domi-nates at long times兴 depends on the time intervals between the jumps along the trajectory. This implies that at the level of a single trajectory, R关m␶兴 cannot be expressed in terms of exchange processes with the reservoirs 共more precisely, in terms of time integrated currents兲. However, the fluctuation theorem共62兲 indicates that at long times the probability P共R兲

to observe a trajectory such that R关m_␶兴=R becomes equiva-lent to the probability to observe a trajectory with a set of time integrated currents so that, via Eq. 共58兲, ⌬Sr⬇R. One

way to understand this result is to coarse-grain the trajecto-ries in time. Instead of specifying exactly the time at which each jump occurs, we define small time intervals of equal size dt共sufficiently small so that the probability of observing two transitions in one is interval negligible, i.e., small com-pared to the mean time for a transition to occur兲. We then specify whether a transition between two states occurred or not in this interval. In this way, we define coarse-grained trajectories 共denoted by m˜_␶兲, and we note that different

mi-croscopic trajectories can lead to the same coarse-grained trajectory. To calculate the probabilities of these trajectories, we use the fact that at long times the dynamics is described by Eq. 共29兲, which can be discretized in time intervals dt.

This discretized form allows us to identify the probability to stay in a given state m during a given time interval with 1 −兺m⬘,␯W˜m⬘m

␯ _{共0兲dt, and the probability to jump from m to m}

_⬘

by mechanism ␯ with W˜_mm␯ _⬘共0兲dt. Using these probabilities, we can construct the probability of a coarse-grained trajec-tory and that of its time reversed coarse-grained trajectrajec-tory. The logarithm of their ratio gives⌬S关m˜_␶兴+⌬Sr关m˜␶兴, with the

same definitions as in Eqs.共74兲 and 共77兲, but for m˜_␶instead of m_␶. With the argument that at long times⌬Sr关m˜␶兴 will be

dominant, we understand why the fluctuation theorem 共62兲

holds at long times. One should note that long times are also needed in this case to get rid of the contribution from the initial part of the trajectory where Eq.共29兲 is not valid. Also,

that such a dynamics can satisfy “microscopic” reversibility. The fluctuation theorem共62兲 does not result from a

probabi-listic asymmetry under time reversal at the microscopic tra-jectory level as in the case of separable WTDs, but at a coarse-grained level and only for long times.

We can finally comment on the case of WTDs with di-verging mean waiting times. Even if separability is satisfied, because mean waiting times diverge, Eq.共33兲 cannot be used

to express R关m␶兴 共or parts of it兲 in terms of exchange pro-cesses with the reservoirs. The fluctuation theorem 共72兲 is

thus valid but seemingly useless.

VI. CONCLUSIONS

In this paper we have considered continuous time random walks 共CTRWs兲 in which multiple mechanisms that may have different statistical properties induce transitions be-tween pairs of states. A given energy and number of particles can be attributed to each state, so that each mechanism can be thought of as corresponding to a given reservoir. If these reservoirs have different statistical properties, such a CTRW therefore describes the stochastic dynamics of an open sys-tem. The statistics of the transitions between states associ-ated with each mechanism are described by waiting time distributions共WTDs兲. If a WTD decays at long times at least as fast as t−2, the distribution has a finite first moment and we say that ␣= 1. If the decay is slower, as t−␣−1 where 0⬍␣ ⬍1, all moments diverge.

We have analyzed the long-time behavior of the probabil-ity␳m共t兲 that the system is in state m at time t via the

gen-eralized master equation for this probability. When ␣= 1 the generalized master equation leads at long times to an expo-nential decay of the probability to a steady state, exactly as it would for an ordinary Markovian master equation. If the WTDs corresponding to the different mechanisms are differ-ent, the steady state is a nonequilibrium steady state. If they are identical, the steady state distribution corresponds to equilibrium and satisfies detailed balance. On the other hand, if 0⬍␣⬍1, the probability␳m共t兲 evolves as a linear

combi-nation of power-law decays t−␣ toward the zero eigenvalue mode, and the system never reaches a steady state.

We have presented a formalism to calculate the counting statistics of the energy and matter transfer in the CTRW based on a generating function propagation method. By con-sidering systems exchanging energy and matter with differ-ent reservoirs, we have shown that the nth momdiffer-ent of the probability distribution to exchange a certain amount of en-ergy or matter between the system and a given reservoir during a time interval t behaves as tn␣ at long times. This result holds for 0⬍␣ⱕ1 and reflects the subordination prin-ciple关49,50兴.

Using our generating function formalism together with large deviation theory, for WTDs with ␣= 1 we derived a fluctuation theorem for the trajectory quantity ⌬Sr关m␶兴

rep-resenting the change of entropy along the trajectory due to exchange processes with the reservoirs关which can be explic-itly related to the energy and matter currents between the system and the reservoirs 共58兲兴. If P共⌬Sr兲 is the probability

to observe a trajectory along which occurs a change

⌬Sr关m␶兴=⌬Sr, the fluctuation theorem reads P共⌬Sr兲/

P共−⌬Sr兲=exp兵⌬Sr其 and is only valid at long times. For

WTDs with diverging first moments, 0⬍␣⬍1, this fluctua-tion theorem does not hold.

The trajectory quantity R关m␶兴 is defined as the logarithm of the ratio of the probability of a given trajectory to the time reversed trajectory. We have shown that for any CTRW, the ensemble average of R关m_␶兴 is always positive and a fluctua-tion theorem stating that the probability P共R兲 to observe a trajectory such that R关m␶兴=R is exponentially more likely than to observe a trajectory such that R关m␶兴=−R, i.e.,

P共R兲/ P共−R兲=exp兵R其 always holds. Separable WTDs are

ones for which the state-directional part and the temporal part of the WTD factorize 关see Eq. 共73兲兴. We have shown

that it is only for separable WTDs with␣= 1 that R关m_␶兴 can be interpreted as the change of entropy along the trajectory. In this case, this fluctuation theorem is related to the previous one because R关m␶兴=⌬S关m␶兴+⌬Sr关m␶兴, where ⌬S关m␶兴 is the

change of system entropy along the trajectory and is bounded, contrary to⌬Sr关m␶兴 which typically grows for long

trajectories. Therefore for most realizations R关m_␶兴

⬇⌬Sr关m␶兴 at long times, thus providing an interpretation of

the fluctuation theorem for⌬Sr关m␶兴 in terms of probabilistic

asymmetry under time reversal of the probability of a trajec-tory. At equilibrium, the symmetry is restored and R关m␶兴=0 as expected from the principle of microreversibility. For non-separable WTDs with ␣= 1, R关m_␶兴 can only be related to a change of entropy after a coarse graining of the trajectories in time. The fluctuation theorem for ⌬Sr关m␶兴 still holds, but

can only be interpreted as a measure of a probabilistic asym-metry in time at the level of coarse-grained trajectory 共and also only for long times兲, and not at the level of the micro-scopic trajectories as in the case of separable WTDs.

ACKNOWLEDGMENTS

M. E. would like to thank František Šanda and Shaul Mukamel for early discussions which motivated this work. M. E. was supported by the FNRS Belgium 共chargé de re-cherches兲 and by the Luxemburgish government 共Bourse de formation recherches兲. This research was supported in part by the National Science Foundation under Grant No. PHY-0354937.

APPENDIX: SINGLE LEVEL QUANTUM DOT

As an application of the counting statistics results of Sec. IV, we consider a single level quantum dot between left and right leads ␯= L , R. There are two states in the system, one corresponding to the empty level m = 0 and the other to the filled level m = 1. Since the energy transfer between the sys-tem and a reservoir is directly proportional to the particle transfer in this model, we only consider particle transfer. We take separable WTDs, specifically

␺00共␥,t兲 =␺11共␥,t兲 = 0,

␺10共␥,t兲 = P10共␥兲␺0共t兲,

where P10共␥兲 = e␥共L兲P₁₀共L兲+ e␥共R兲P₁₀共R兲, P01共␥兲 = e−␥共L兲P共L兲₀₁ + e−␥共R兲P₀₁共R兲. 共A2兲 Note that P10共␥= 0兲 = P₁₀共L兲+ P₁₀共R兲= 1, P₀₁共␥= 0兲 = P₀₁共L兲+ P₀₁共R兲= 1. 共A3兲 We define the affinities Ax⬅ln P01共x兲/ P10共x兲, where x = L , R, and ⌬A⬅AL− AR.

From now on, we focus on the net particle transfer be-tween the left lead and the dot so that␥共L兲=␥and␥共R兲= 0. We also note that

P01共␥兲 ⬅ eAR_P10共⌬A −␥兲,

P10共␥兲 ⬅ e−AR_P01共⌬A −␥兲. 共A4兲

The generating function 共45兲 for this model reads

G ˜ 共␥,s兲 =

冉

␾˜₀

⬘

共s兲 +␺˜₀

⬘

共s兲␾˜1共s兲P10共␥兲 +␺ ˜_1共s兲␾˜_{0共s兲P01共␥兲P10共␥兲} 1 − P01共␥兲P10共␥兲␺˜0共s兲␺˜1共s兲

冊

⫻␳0共0兲 + 共1 ↔ 0兲. 共A5兲

We can verify that the generating function at␥= 0 diverge as

s−1in the long-time limit, and, because of Eq.共A4兲, the same

is true at␥=⌬A. The first moment reads

兩⳵␥G˜ 共␥,s兲兩␥=0=␾ ˜_0共s兲␺˜_1共s兲共P 10 共L兲_{− P} 01 共L兲_{兲 +␾}˜_1共s兲关P 10 共L兲_−␺˜_0共s兲␺˜_1共s兲P 01 共L兲_兴 关␺˜_0共s兲␺˜_{1共s兲 − 1兴}2 ␺ ˜ 0

⬘

共s兲␳0共0兲 − 共1 ↔ 0兲. 共A6兲

The WTDs are taken to be of the form

␺

˜_{0共s兲 = 1 − B0s}␣_,

␺

˜_{1共s兲 = 1 − B1s}␣_{, 共A7兲} where 0ⱕ␣ⱕ1. If␣= 1, B0= t0and B1= t1.

We can now confirm that if 0⬍␣⬍1 and a jump occurred at time zero关␺m

⬘

共t兲=␺m共t兲兴, or if␣= 1 and a jump occurred at

time zero关␺m

⬘

共t兲=␺m共t兲兴, or if ␣= 1 and we do not know at

time zero when the last jump occurred关␺˜_m

⬘

共s兲=␾˜m共s兲/tm兴, we

always get 兩⳵␥nG˜ 共␥,s兲兩␥=0 = s→0 n !共P10共L兲− P₀₁共L兲兲n 共B0+ B1兲n s −n␣−1_{, 共A8兲} where n = 1 , 2 , 3 , . . .. Using the Tauberian theorem, this leads to 兩⳵␥nG共␥,t兲兩␥=0 = t→⬁ n !共P10共L兲− P01共L兲兲n ⌫共n␣+ 1兲共B0+ B1兲nt n␣ . 共A9兲

When␣= 1, Eq.共A9兲 can be written as

兩⳵␥nG共␥,t兲兩␥=0 = t→⬁

冉

W₁₀LW01− W01 L W10 W10+ W01

冊

n tn␣, 共A10兲 where W_mm共␯兲_⬘= P_mm共␯兲_⬘/tm⬘and Wmm⬘= Wmm⬘ 共L兲 _{+ W} mm⬘ 共R兲 _{. This is the} same result as obtained in Ref.关23兴 using a Markovian

mas-ter equation. In can easily be seen that Eq. 共A10兲 vanishes

when detailed balance is satisfied.

Finally, we note that the cumulants calculated from these moments all vanish. This indicates that we should calculate not only the leading time dependence of the moments as we do when applying Tauberian theorems, but must retain higher orders to extract information about the time dependence of the cumulants.

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