Stochastic functionals and fluctuation theorem for the multi-kangaroo process

We introduce multi-kangaroo Markov processes and provide a general procedure for evaluating a certain type of stochastic functionals. We calculate analytically the large deviation properties. Applications include zero-crossing statistics and stochastic thermodynamics.


I. INTRODUCTION
A wide range of physical phenomena can be described in terms of a stochastic process, and in particular by a Markov process. One special class of such a process is the so-called kangaroo process, characterized by a transition rate that factorizes [1] in terms of the variables of the initial and final states. Such a model has been extensively applied in the context of kinetic theory [2]. In this letter, we introduce a generalized kangaroo process, with the transition rate being the sum of factorizable contributions. This model has a much wider range of applicability, while keeping, as we will see, the mathematical simplicity of the simple kangaroo process. In particular, we show how it is possible to obtain explicit analytic results for some functionals of such a stochastic process. In particular the evaluation of large deviation properties is reduced to finding the largest eigenvalue of a finite matrix. As illustrations, we discuss zero-crossing statistics, we evaluate the entropy production for a model from stochastic thermodynamics, and show that the fluctuation theorem is verified [3].

II. KANGAROO PROCESS AND ITS GENERALIZATION
A kangaroo process is a Markov process in which the transition rate per unit time to go from state x to x, [1]. For simplicity, we will consider here the case of φ(x) independent of x, so that the transition rate has the following form: The physical interpretation is as follows: k is the constant rate at which transitions take place; whenever a transition occurs, the new state x is chosen from the probability density P ss (x), which is in fact the steady state distribution of the Markov process. The master equation for the probability P (x; t) to be in state x at time t: simplifies as follows for the kangaroo process defined by (1): We conclude that the "kangaroo master equation" (2) describes a pure exponential relaxation towards the steady state P ss (x). For the problems we want to discuss, we need to introduce a more general type of kangaroo process: a transition from x to x can be realized by several distinct physical processes ν, each with corresponding rates W (ν) (x → x). The resulting total rate is given by To keep the simplicity of the kangaroo process, we assume that each of the rates have the same form as in (1): Here, k (ν) is the (constant) rate of transitions associated to process the ν, and P (ν) ss the corresponding steady state distribution. Despite the fact that, in general, the process defined by the resulting rate W (x → x) is no longer a standard kangaroo process, one can verify that the master equation is still given by (3), but with the following total rate k and overall steady state distribution P ss : arXiv:1401.2154v1 [cond-mat.stat-mech] 9 Jan 2014

III. LARGE DEVIATIONS
We want to evaluate the probability distribution for an "incremental cumulative" quantity X = X(t) associated to the stochastic trajectory generated by the Markovian stochastic process over a time interval of length t. A simple example is the cumulative number of transitions: the value of X changes by +1 whenever a transition takes place, i.e., the increment of X is ∆(x → x) = 1, independent of x and x . Another example is the net number of "net zero-crossings" (or flux through 0), corresponding to ∆(x → x) = [sgn(x) − sgn(x )]/2, with sgn(x) representing the sign of x. As a third example we mention the cumulative energy exchanged between a system and a bath, with ∆(x → x) = (x) − (x ), and (x) the energy of the system in state x.
For a generalized kangaroo process, the increments of X will depend on the states between which transition takes place, but also on the process responsible for them, i.e., the increments are given by ∆ (ν) (x → x) for a transition x → x due to process ν. One example is the entropy production for a system in contact with different heat baths at . Another example is the cumulated effect due to one of the processes, say ν 0 , implying ∆ (ν) (x → x) is zero for all processes except for ν = ν 0 .
In all the above cited examples, the increments have a specific feature in common: they can be written as the sum (or difference) of a function of x and x , i.e.: It turns out that this feature, combined with the kangaroo property of the transition rate, greatly reduces the mathematical complexity of the problem, as we now proceed to show.
Since the increases of X are supposed to be a deterministic function of the transitions, the combined pair x , X obeys a master equation, which obviously reads as follows: It is convenient to perform the following generating function transformation on the X variable: This function obeys the following evolution equation: It is now clear why a significant simplification takes place for the "generalized kangaroo scenario" considered here. By taking the transition rates and increments given by (4) and (6), respectively, the above equation (9) reduces to: Here, we introduced the integrals I The quantity of prime interest is the generating function: where X k is the cumulant of order k. Combination of the former three equations leads to the following closed set of linear equations for F λ (t) and the I (ν) Here A (ν,ν ) λ and B (ν) λ are the following time-independent quantities: Note that the above set of equations (13) and (14) can be written under a matrix form:V λ (t) = M λ V λ (t) where the vector V λ (t) has components F λ (t), I λ (t), ..., I (N ) λ (t), N being the number of processes ν. M λ is a time-independent (N + 1) × (N + 1) matrix, whose elements can be identified by inspection of the equations (13) and (14).
In order to obtain explicit analytic results, we next focus on the large deviation properties of X in the asymptotic limit t → ∞. As the Markov process x does not possess long-time correlations, the increments of X cumulated over time periods longer than the correlation time, are essentially independent. Hence the behaviour of X for t → ∞ is described by the asymptotic cumulant generating function φ λ [4]: Note that φ λ is the Legendre transform of the so-called large deviation function describing the asymptotic behavior for the probability for a current j = X/t. This transform can be performed in several of the cases discussed below, but this discussion is omitted here for lack of space. From the formal solution V λ (t) = e tM λ V λ (0), it is clear that φ λ has to be identified with the largest eigenvalue of the matrix M λ . The analysis is further simplified by the observation that the equations for the I (ν) λ components do not couple to F λ . We hence identify an eigenvalue equal to −k associated to the F λ component while φ λ is the largest eigenvalue of the block-matrix related to the I (ν) λ components, i.e., the largest eigenvalue of the N × N matrix For the case of a single process, N = 1, we can drop the superscripts ν and ν in the above formulas. We conclude that φ λ = k(A λ − 1) (note that A λ ≥ 0), or explicitly: This result can also be obtained directly by noting that the number n of transitions during a time t obeys a Poisson distribution and that the contributions ∆ (i) = b(x (i) ) − a(x (i) ) (x (i) , i = 0, ..., n being the successive states of the system) are independent and identically distributed random variables with probability distribution given by P (∆) = dxδ(∆ − b(x) + a(x))P ss (x). For the case of two processes, the asymptotic cumulant generating function is: The asymptotic cumulant generating function can also be obtained for three processes, but the expression is too lengthy to be reproduced here.

IV. NET ZERO-CROSSINGS
For the special choice a (ν) (x) = b (ν) (x) ≡ q(x), ∀ν, one finds A (ν,ν ) λ ≡ 1, implying φ λ = 0, and all normalized cumulants t −1 X k vanish in the long time limit. This is however no longer the case when a (ν) (x) = b (ν) (x) = q (ν) (x), ∀ν, but with functions q (ν) (x) that are not identical. As an illustration, we evaluate the asymptotic cumulant generating function for the cumulated net zero-crossings of a process ν = (1), in the presence of another "resetting" process ν = (2), whose zero-crossings are not counted. More precisely we set: The asymptotic cumulant generating function is given by (15), with the following values: The result becomes particularly transparent for equal rates k (1) = k (2) = k/2 and P (1) ± = P (2) ± = 1/2, namely φ λ = k(e λ + e −λ − 2)/4, which is the asymptotic cumulant generating function of an unbiased random walk with jump rate k/4. Indeed, the probability to be in + or − state is equal to 1/2 at all times, and the probability per unit time to select process (1) for a jump is k (1) = k/2, hence k/4 is the rate of zero-crossings by process (1) for both + → − and − → +.

V. FLUCTUATION THEOREM
As second example, we consider a system in contact with different heat baths ν = 1, . . . , N with corresponding temperatures T (ν) . The transitions between different states are due to the contact with the heat baths. In particular, a transition x → x requires the following amount of heat Q(x → x) = (x) − (x ), being (x) the energy of the system when in state x. If this transition is produced by contact with heat bath ν, the corresponding entropy change in the bath is given by (minus sign as we are monitoring the entropy change of the bath): The sum X of all these contributions over a time t is equal to the total entropy change of the reservoirs. With respect to the application of stochastic thermodynamics [3], we note that the kangaroo transition rates, associated to bath ν, automatically satisfy the required condition of (local) detailed balance, ss (x ), because we assumed that the rates k (ν) do not depend on the state. The corresponding steady distribution P (ν) ss should however also reproduce the equilibrium distribution when in contact with this bath, hence it is given by: where we introduced the partition function Z(β): Here g( ) is the density of states and β has the usual definition 1/β = k B T . With the identification a (ν) (x) = b (ν) (x) = − (x)/T (ν) , one finds that (15) simplifies as follows: We note that these quantities satisfy the following symmetry property: The cumulant generating function φ λ , being the largest eigenvalue of the matrix k (ν ) A (ν,ν ) λ −k δ (ν,ν ) Kr , therefore obeys φ λ = φ (−λ−1/k B ) , implying the following asymptotic behavior for X: Since X is the cumulated entropy production in the reservoirs, the above result is nothing but the celebrated asymptotic or steady state fluctuation theorem [3,5]. We also mention the result for the asymptotic average rate of entropy production for N = 2, obtained from (18,24): where (ν) is the average energy calculated with respect to the canonical distribution (22). To obtain more explicit results, one needs to specify the density of states g( ) of the system under consideration. The simplest situation corresponds to a discrete spectrum with two energy states = 0 and = 0 . The λ dependence of φ λ is then similar to the one in a general two-state problem [6]. For a spectrum of the form g( ) ∼ ( − 0 ) α , > 0 , such as encountered for a gas of ideal Fermions or ideal Bosons, the calculations can still be done analytically. Here, we just reproduce the cumulant generating function φ λ for a few representative situations in Fig. 1. Figure 1: Asymptotic cumulant generating function φ λ for a spectrum of the form g( ) ∼ ( − 0) α , > 0 in the cases α = −1/2 (outer curve) → 5/2 (inner curve) in steps of 1/2, with T (1) = 1, T (2) = 2, 0 = 0. The symmetry of φ λ about the point λ = −1/2, imposed by the fluctuation theorem, is clearly visible . Note also the divergences at kBλ = 1/(T1/T2 − 1) and kBλ = 1/(T2/T1 − 1). kB has been set equal to 1.

VI. PERSPECTIVES
We have presented two illustrations of the large deviation theory for a generalized "kangaroo scenario". It should however be clear that the above results can be applied to a wide range of problems. The variables x could be vectors (for example the speed of a particle), functions or fields (for example probability distributions or density profiles), matrices or operators (with the quantity of interest for example its largest eigenvalue), or more abstract quantities (for example symbols or processes), with the corresponding probability distributions P ss , processes ν and increments ∆ having widely different interpretations. While the true stochastic dynamics will typically be more involved, the analysis of a generalized kangaroo model leads to analytic results that can serve as a guideline for the properties of the original system. It also provides an alternative to a description in terms of a two-state Ising-type model, with which it shares the mathematical simplicity, while keeping the richness associated to an arbitrary spectrum of states and steady state distribution.

VII. ACKNOWLEDGMENTS
This work was supported by the research network program "Exploring the Physics of Small Devices" from the European Science Foundation. RT acknowledges financial support from MINECO (Spain), Comunitat Autònoma de les Illes Balears, FEDER, and the European Commission under project FIS2012-30634.