A stochastic model of long-range interacting particles
We introduce a model of long-range interacting particles evolving under a stochastic Monte Carlo dynamics, in which possible increase or decrease in the values of the dynamical variables is accepted with preassigned probabilities. For symmetric increments, the system at long times settles to the Gibbs equilibrium state, while for asymmetric updates, the steady state is out of equilibrium. For the associated Fokker-Planck dynamics in the thermodynamic limit, we compute exactly the phase space distribution in the nonequilibrium steady state, and find that it has a nontrivial form that reduces to the familiar Gibbsian measure in the equilibrium limit.
Keywords: stochastic particle dynamics (theory), stationary states
Nonequilibrium systems abound in nature, with examples encompassing different branches of science. Although there has been much recent progress in characterizing and understanding some features of nonequilibrium steady states , developing a general principle akin to the one due to Gibbs-Boltzmann for equilibrium has been one of the greatest challenges of modern statistical physics. In this respect, it is instructive to develop and analyze simple models in order to gain insights into features of nonequilibrium steady states that make them distinct from those in equilibrium. Often, even for simple models, the steady state distribution has been nontrivial to obtain , and in many cases, has even remained elusive, thereby requiring one to resort to numerical simulations and approximation methods as only possible tools to analyze the steady states .
Here, we develop a model of particles interacting via long-range interactions and evolving under a stochastic Monte Carlo dynamics, for which we could characterize exactly the steady state distribution. This is one of the first examples of engineering such a dynamics in the arena of long-range models interacting with an external heat bath, where all previous studies, to the best of our knowledge, have been based on Langevin equations with noise terms that mimic the effect of the heat bath (see, e.g., ). The model effectively simulates driven motion of particles in one dimension under the action of a mean-field.
Long-range interactions have generated considerable interest in recent years, with examples ranging from plasma physics to gravitational systems . These systems are characterized by an interparticle potential which in dimensions decays at large separation, , as , with . Long-range interacting systems are different from the short-range ones in that they are generically non-additive, whereby thermodynamics quantities scale superlinearly with the system size. This latter feature manifests in properties, both static and dynamic, which are unusual for short-range systems .
In this work, we introduce a model of long-range interacting systems involving globally coupled particles moving on a circle, in presence of an external field acting individually on the particles, and in contact with an external heat bath at inverse temperature . This inverse temperature will coincide with the steady state temperature of the system only in equilibrium. The system evolves under a stochastic Monte Carlo dynamics. Thus, a randomly chosen particle decides to move either to the left or to the right of its present position to take up a new location on the circle. The displacement from the initial position of the particle is a quenched random variable sampled independently from a common distribution for all the particles. The new location of the particle is accepted with a preassigned transition rate which is chosen in the following way. In the case the particle jumps symmetrically to the left and to the right, the transition rate is such that the stationary state of the system is the Gibbs equilibrium state at the inverse temperature . In the case of asymmetric particle jump to the left and to the right, the system at long times reaches a nonequilibrium stationary state. Considering the dynamics in the Fokker-Planck limit (i.e., only small jumps are allowed), we find that even with asymmetric jumps, if the external field is turned off and the jump distribution is a delta function so that all particles jump by the same amount, the steady state is in equilibrium in a suitable comoving frame obtained by performing a Galilean transformation. In all other cases, the steady state is out of equilibrium. In the thermodynamic limit , the system is characterized by a single-particle distribution, that we compute exactly in the nonequilibrium steady state. We find that the distribution has a non-trivial form that reduces to the usual Gibbsian distribution in the equilibrium limit. Our results show excellent agreement with -particle Monte Carlo simulations of the dynamics.
The paper is organized as follows. In Section 2, we give a precise definition of the model and discuss the Master equation for the evolution of the -particle phase space distribution. In Section 3, we consider the Fokker-Planck limit of the dynamics and obtain the exact steady state single-particle distribution in the limit . In Section 4, we compare -particle Monte Carlo simulation results for the steady state single-particle distribution with our theoretical predictions in the Fokker-Planck limit, and demonstrate an excellent agreement between the two. The paper ends with conclusions.
Consider a system of interacting particles moving on a unit circle, with the particles labelled as . Let the angle denote the location of the th particle on the circle. A microscopic configuration of the system is denoted by . The particles interact via a long-range potential , where is the coupling constant; we consider in the following. Application of an external field of strength produces a potential , so that the net potential energy is . The interaction has the same form as in the Hamiltonian mean-field (HMF) model, a paradigmatic example of systems with long-range interactions . The fields ’s may be considered as quenched random variables with a common distribution .
We now specify the dynamics of the system. We take hints from one of the first models devoted to studying characteristics of nonequilibrium steady states, the celebrated Katz-Lebowitz-Spohn model . The configuration evolves according to a stochastic Monte Carlo dynamics. In discrete time, the dynamics in a small time involves every particle attempting to hop to a new position on the circle. The th particle attempts with probability to move forward (in the counter clockwise sense) by an amount on the circle, , while with probability , it attempts to move backward by the amount , so that . In either case, the particle takes up the new position with probability . Here, is a quenched random variable which for each particle is distributed according to a common distribution , while the quantity is the change in the potential energy due to the attempted hop from to : . The function is of the form , where is the inverse temperature. The dynamics models the overdamped motion of the particles in contact with an external heat bath at inverse temperature and in presence of an external field. The case for which the particles move asymmetrically forward and backward mimics the action of an external drive that makes the particles to move in one preferential direction along the circle. Note that in the dynamics, the initial ordering of particles on the circle is not conserved in time. Taking ’s as quenched random variables introduces in the dynamics a different source of noise than the one due to the Monte Carlo update scheme which is annealed in nature.
There are two sources of quenched randomness in the model through the presence of (i) jump lengths ’s, and (ii) field strengths ’s. Later, we will consider specifically the first of the two sources of randomness, and take the ’s to be the same for all particles. In the conclusions, we will comment on how our analytical approach may be easily adapted to consider the randomness due to the ’s.
Let be the probability to observe the configuration at time . In the limit of continuous time, the evolution of is given by the Master equation, which may be derived by considering the change in in a small time according to the dynamical evolution rules given above, and then taking the limit while keeping ’s fixed and finite. Defining , we get the Master equation as
At long times, the system settles into a stationary state corresponding to the time-independent probability . For , the particles attempt to move forward and backward in a symmetric manner, and the system has an equilibrium stationary state in which the condition of detailed balance is satisfied with the measure . On the other hand, for , the particles have a preferred direction to hop on the circle, and the system at long times settles into a nonequilibrium stationary state, characterized by a violation of detailed balance leading to nonzero probability currents in phase space. In the absence of the external field, the dynamics with samples the equilibrium measure of the Brownian mean-field model of long-range interacting systems, developed as an extension of the microcanonical dynamics of the HMF model to a canonical dynamics that mimics the interaction of the system with an external heat bath .
3 Fokker-Planck limit
Here, we analyze the Fokker-Planck limit of the dynamics of our model. To this end, we first obtain the Fokker-Planck equation corresponding to the Master equation (1) by making the assumption that , so that we may Taylor expand functions in powers of ’s . As shown in Appendix A, retaining terms to second-order in ’s, we get the Fokker-Planck equation for as
where the probability current for the th particle is given by
The corresponding Langevin equation is easily written down as
where the dot denotes derivative with respect to time, and is a random noise with
From the Fokker-Planck equation (2), it is evident that, as in the finite- dynamics, the system for settles into the equilibrium stationary state with which makes individually for each . On the other hand, for , the system at long times reaches a non-equilibrium stationary state. However, in the special case when the jump length is the same for all the particles and there is no external field ( and ), one may make a Galilean transformation, , so that in the frame moving with the velocity , the Langevin equation (4) takes a form identical to the one for , and the stationary state has the equilibrium measure .
3.1 Limit and constant field: Single-particle distribution
In the thermodynamic limit , when the external field is the same for all the particles, , let us define the single-particle distribution such that gives the density of particles with jump length which are at location on the circle at time . We have , and also the normalization
In terms of , the Langevin equation (4) in the limit for a particle with jump length and at position reads
Let us note that the dynamics (7) is similar with to that of the Kuramoto model of synchronization  and with to that of its extension considered in Ref.  that includes noise. However, in the latter case, a crucial difference is that in Eq. (7), the noise term and the drift term (the first term on the right hand side) contain the same factor , and are therefore, related, unlike the model in Ref. .
The single-particle Fokker-Planck equation satisfied by may be obtained from the Langevin equation (7) as
where the probability current is given by
The stationary solution of the Fokker-Planck equation (10) is given by (see Appendix B)
where , and the constant is fixed by the normalization condition (6).
When the jump length is the same for all particles, , we have
where the constant is fixed by normalization: . For , we obtain the equilibrium single-particle distribution as
4 Numerical simulations
Choosing the jump length to be the same for all particles, we show in Fig. 1 a comparison of the -particle Monte Carlo simulation results for , obtained for , and and its theoretical form, Eq. (14), applicable in the Fokker-Planck approximation and in the limit . We observe an excellent agreement between simulations and theory. For the case and , Fig. 2(a),(b) compare simulation results and theory (Eq. (13)) for for two values of , again illustrating an excellent agreement. Figures 2(c),(d) compare simulation results for with the Fokker-Planck-limit theory valid for ; in (c), we see a reasonable agreement, while in (d), the disagreement is quite large. For the latter case, we have checked that for the same parameter values, the mismatch between theory and simulations does not reduce with larger , which implies that it is due to the large value of used as compared to the Fokker-Planck limit, and not due to finiteness of .
In this work, we introduced a model of long-range interacting systems involving globally coupled particles moving on a circle. The system evolves under a stochastic Monte Carlo dynamics consisting of particle jumps, either symmetrically to the left and to the right, or, asymmetrically, by quenched random amounts sampled from a common distribution. The attempted new locations of the particles are accepted with transition rates chosen in such a way that for symmetric jumps, the stationary state of the system is the Gibbs equilibrium state. For asymmetric jumps, the system at long times reaches a nonequilibrium steady state characterized by nonzero probability currents in phase space. For the associated Fokker-Planck dynamics in the thermodynamic limit , we computed exactly the steady state distribution and found that it has a nontrivial form that reduces to the Gibbs distribution in the equilibrium limit, see Eqs. (13) and (14). We compared our theoretical predictions with -particle Monte Carlo simulations, and found an excellent agreement between the two in the Fokker-Planck limit. The observed disagreement when the limit is not satisfied opens up the very interesting scope of analyzing and obtaining corrections to the Fokker-Planck answer. It is also of interest to treat the external fields ’s in Eq. (1) as quenched random variables sampled from a common distribution. It is easy to generalize our analytical framework to treat this case in the Fokker-Planck limit by considering instead of the distribution giving the density of particles with jump length and under the action of field with strength , which are at location at time . One would then have to obtain the Fokker-Planck equation that satisfies, by considering and in the Master equation (1) and performing suitable Taylor series expansion of functions of ’s and ’s. With quenched ’s, one may consider the canonical dynamics introduced in this paper and its grand canonical counterpart (where, say, the number of particles is not conserved), and investigate the issue of equivalence of the nonequilibrium steady state under the two dynamics. This issue is particularly relevant, since long-range interacting systems are known to show inequivalence in equilibrium in presence of random fields .
SG acknowledges the support of the CEFIPRA Grant 4604-3 and the contract ANR-10-CEXC-010-01, and the hospitality of ENS-Lyon. We acknowledge fruitful discussions with Sergio Ciliberto.
7 Appendix A: Derivation of the Fokker-Planck equation (2)
Considering the Master equation (1) for , we expand all functions of ’s in powers of ’s. Retaining terms to second-order in ’s, we get
where the probability current for the th particle is given by
8 Appendix B: Stationary solution of Eq. (10)
where is a constant independent of , and we have defined
Multiplying both sides of Eq. (17) by , and then integrating over , we get
where is the initial condition at time . Requiring that fixes to be
Redefining , and reverting to the variables and , we get
where is fixed by the normalization: .
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