Entropy production in the nonequilibrium steady states of interacting many-body systems.
ABSTRACT Entropy production is one of the most important characteristics of nonequilibrium steady states. We study here the steady-state entropy production, both at short times as well as in the long-time limit, of two important classes of nonequilibrium systems: transport systems and reaction-diffusion systems. The usefulness of the mean entropy production rate and of the large deviation function of the entropy production for characterizing nonequilibrium steady states of interacting many-body systems is discussed. We show that the large deviation function displays a kink-like feature at zero entropy production that is similar to that observed for a single particle driven along a periodic potential. This kink is a direct consequence of the detailed fluctuation theorem fulfilled by the probability distribution of the entropy production and is therefore a generic feature of the corresponding large deviation function.
arXiv:1101.4566v1 [cond-mat.stat-mech] 24 Jan 2011
Entropy production in the non-equilibrium steady states of
interacting many-body systems
Sven Dorosz1,2and Michel Pleimling2
1Theory of Soft Condensed Matter,
Universit´ e du Luxembourg, Luxembourg, L-1511 Luxembourg
2Department of Physics, Virginia Polytechnic Institute and State University,
Blacksburg, Virginia 24061-0435, USA
(Dated: January 25, 2011)
Entropy production is one of the most important characteristics of non-equilibrium steady states.
We study here the steady-state entropy production, both at short times as well as in the long-time
limit, of two important classes of non-equilibrium systems: transport systems and reaction-diffusion
systems. The usefulness of the mean entropy production rate and of the large deviation function
of the entropy production for characterizing non-equilibrium steady states of interacting many-
body systems is discussed. We show that the large deviation function displays a kink-like feature
at zero entropy production that is similar to that observed for a single particle driven along a
periodic potential. This kink is a direct consequence of the detailed fluctuation theorem fulfilled
by the probability distribution of the entropy production and is therefore a generic feature of the
corresponding large deviation function.
PACS numbers: 05.40.-a,05.70.Ln,05.20.-y
The study of large deviation functions has been of increasing importance for the un-
derstanding of many-body systems. On the one hand large deviation functions form the
basis of a modern approach to equilibrium statistical mechanics [1–3], on the other hand
they are increasingly recognized of being of fundamental interest for the characterization
of non-equilibrium systems where they are sometimes considered to play a role similar to
that played by the free energy at equilibrium. Important examples are given by the large
deviations of the steady-state currents [4, 5] in systems that are far from equilibrium.
Recently, Mehl et al.  extended the study of large deviation functions far from equilib-
rium to the steady-state entropy production. Studying a one-dimensional system composed
of a single particle driven along a periodic potential, they reformulated the problem as a
time-independent eigenvalue problem  and showed that for this system the large devia-
tion function of the entropy production exhibits a kink at zero entropy production. Similar
kinks in the large deviation function of the entropy production or in related large deviation
functions of other quantities can also be found in a range of other systems [8–13]. These are
intriguing results that raise the important question whether the presence of this kink is a
universal feature of systems with non-equilibrium steady states.
In this paper we study the entropy production in the steady states of different non-
equilibrium interacting many-body systems. On the one hand we study the Partially Asym-
metric Simple Exclusion Process (PASEP) , a transport process in an open system where
particles can enter or leave only at the system boundaries. On the other hand we investigate
reaction-diffusion systems on a ring where the number of particles can change everywhere
in the system due to reactions between the particles. Our main emphasis thereby is on the
entropy production in the long-time limit where we study the large deviation function in
the same way as done by Mehl et al. in their study of the single particle system. For all
studied many-body systems we find a kink-like feature at zero entropy production, similar
to what has been observed for the single particle system. We show that this kink is a generic
feature in non-equilibrium systems obeying a detailed fluctuation theorem for the entropy
It is worth noting that large deviation functions of the current have been studied previ-
ously in some related systems. Studies of the Totally Asymmetric Simple Exclusion Process
(TASEP) [15, 16] and of the Symmetric Simple Exclusion Process (SSEP) [7, 17] revealed a
non-analytical behavior of the current large deviation function at the value 0 of the parame-
ter that is canonically conjugated to the particle current. Using dynamical renormalization
group techniques, it was later shown that the value of the corresponding exponent follows
from the noise renormalization . Very recently, Bodineau and Lagouge  studied the
current large deviations in a driven dissipative lattice gas model with creation and annihi-
lation processes, whereas Simon  investigated the large deviation function of the current
for the Weakly Asymmetric Simple Exclusion Process on a ring.
Our paper is organized in the following way. In the next Section we introduce our models,
before discussing in Section III our approaches for computing the entropy production at short
times as well as the mean entropy production rate and the large deviation function for the
entropy production in the long-time limit. Our results are presented and discussed in Section
IV. Finally, Section V gives our conclusions.
In order to elucidate the entropy production in interacting many-body systems, we discuss
in the following reaction-diffusion systems on a ring as well as open transport systems where
particles move through a system that they can enter or leave only at its boundaries. In the
past, due to the combination of their conceptual simplicity and highly non-trivial results,
diffusion-limited reaction systems [21, 22] and simple exclusion processes [23–25] have greatly
contributed to our understanding of processes far from equilibrium. For the same reasons,
these models are also the natural choices for our study of the entropy production in many-
As reaction-diffusion systems we consider simple cases where particles A diffuse on a one-
dimensional ring, under the condition that every lattice site can only be simply occupied.
This diffusion process is mimicked by the hopping of a particle to an empty nearest neighbor
site with rate D. In addition, we also allow for particle creation and annihilation. In the
creation process a new particle can be created at an empty site with rate h, whereas in the
annihilation process n particles on n connected sites are destroyed with rate λ, yielding n
empty sites. In the following we characterize our models by the number of particles involved
in the annihilation process and call Mn (with n = 2,3) the model where n particles are
destroyed. As a variant, we also study the situation (we call the resulting model M2′) where
in a two particles reaction only one particle is destroyed.
These models are the same as those studied in [26–28] in order to better understand
steady-state and transient properties of reaction-diffusion systems. All these systems have
non-equilibrium steady states. In addition, some or all reactions do not allow for a direct
back-reaction, and microscopic reversibility is broken. However, as discussed in the next
Section, the computation of the entropy production requires microscopic reversibility, i.e.
for every reaction the direct back-reaction must be possible. For that reason we are adding
to the reaction schemes the direct back-reactions of the reactions just described. Thus, any
particle can be destroyed with rate εhh, with 0 < εh< 1, irrespective of whether neighboring
sites are occupied or not, and a reversing of the annihilation process yields the creation of
particles with rate ελλ, where 0 < ελ< 1. It is important to note that even with these
expanded reaction schemes our systems are still characterized by non-equilibrium steady
states, as long as εh,ελ< 1. In the following we set εh= ελ = ε and focus on the case
ε ≪ 1. For the convenience of the reader we summarize the different reaction schemes in
A + A
0 + A nA
TABLE I: The different reaction schemes discussed in this work. The back-reactions are taking
place with rates εhh and ελλ, with 0 < εh< 1 and 0 < ελ< 1.
As transport process we consider the Partially Asymmetric Simple Exclusion Process
(PASEP) where every configuration is reversible. In the Total Asymmetric Simple Exclusion
Process (TASEP) in an open one-dimensional system, particles that are fed into the system
at, say, the left end with rate α can leave the system at the right end with rate β. Inside the
system particles can only jump to a right neighboring site, provided that site is not occupied.
Obviously, microscopic reversibility is always broken for this model. In the PASEP, however,
particles can always jump in both directions, provided that the chosen site is empty, with
the probability for a jump to the right being p, whereas the probability for a jump to the
left is 1−p. In addition, particles can exit the system at the left with probability εαα (with
0 < εα< 1) and enter at the right with probability εββ (with 0 < εβ< 1). It follows that
microscopic reversibility is always fulfilled. In order to keep the number of parameters as
small as possible, we set εα= εβ= ε, with ε ≪ 1.
Whereas our main focus in the following will be on the large deviation function of the
steady-state entropy production in the long time limit, we will also briefly discuss the entropy
production at short times.
Focusing on our lattice models, let us consider a path in configuration space C0 −→
C1 −→ ··· −→ CM−1 −→ CM that starts at some configuration C0 and ends at some
configuration CM after M diffusion or reaction steps. Every configuration Ciis uniquely
characterized by the occupation numbers (being 0 or 1) of all lattice sites. We will denote
by PS(Ci) the probability to find the configuration Ciin the steady state. With every step
i leading from configuration Ci−1to configuration Ciwe associate a time increment τigiven
where ω(Ci−1−→?Cj) is the rate with which we go from configuration Ci−1to any other
rations?Cj(including the configuration Ciin which the system will be at step i) that can be
a total time
to our trajectory in configuration space.
accessible configuration?Cj. The sum in the denominator is thereby a sum over all configu-
reached from the configuration Ci−1through diffusion or reaction. With that we can assign
Along the same trajectory the total entropy production is given by 
stot = lnPS(C0)
where smis the entropy produced in the particle bath connected to our system. The bound-
ary term, ln
PS(CM), which can be neglected in the long time limit, needs to be included
when investigating fluctuation relations of the entropy production at short times.