Fluctuations in interacting particle systems with memory

Abstract

We consider the effects of long-range temporal correlations in manyparticle systems, focusing particularly on fluctuations about the typical behaviour. For a specific class of memory dependence we discuss the modification of the large deviation principle describing the probability of rare currents and show how superdiffusive behaviour can emerge. We illustrate the general framework with detailed calculations for a memory-dependent version of the totally asymmetric simple exclusion process as well as indicating connections to other recent work.

Ergodicity and large deviations in physical systems with stochastic dynamics

“…In (3,4) the speed is τ , which is an assumption of the theory presented so far. There are physical systems where time-averaged quantities obey LDPs with other speeds (for example [47,48]) but we focus here on LDPs with speed τ , which is the most common situation. In simple cases (see Sec II B for examples), the rate function I is analytic and strictly convex, with a unique minimum at b = b , and I( b ) = 0.…”

Large deviations in models of growing clusters with symmetry-breaking transitions

“…This formulation is related to the elephant random walk [27], where M k = km k is viewed as the position of a particle, which obeys a dynamical rule similar to (4) except that the probabilities of s k+1 = ±1 are linear in m k . Within this non-Markovian formulation, the irreversible growth model falls in the broad class considered by Harris and Touchette [20,21]. They derived several general results for large deviations in models within this class.…”

“…We restrict our analysis to paths for which m(k) − m * and b(k) − m * are both small compared to unity. This allows us to apply the results of [21]. Under these assumptions the action (42) may be expanded as…”

“…We refer to Ref. [21] for all details of the minimisation procedure. The important physical conclusion is that the solutions do not have constant forces, so one finds in general that A con < A ind .…”

“…(c) The negative of the asymptotic boundH ∞ K that appears in (33), for various K. As an illustrative comparison, this is compared with the (numerically) exact distribution at K = 10 5 from panel (b), shown as a thin black line. We emphasise that the asymptotic bound is not a bound on this finite-K distribution Since m is small, one may linearise(21), leading to (dm/du) = (J − 1)m and hence m k = m k * (k/k * ) J−1 . We choose k * in such a way that this deterministic equation gives m K = O(1).…”