Current large deviations for partially asymmetric particle systems on a ring

Abstract

We study large deviations for the current of one-dimensional stochastic particle systems with periodic boundary conditions. Following a recent approach based on an earlier result by Jensen and Varadhan, we compare several candidates for atypical currents to travelling wave density profiles, which correspond to non-entropic weak solutions of the hyperbolic scaling limit of the process. We generalize previous results to partially asymmetric systems and systems with convex as well as concave current-density relations, including zero-range and inclusion processes. We provide predictions for the large deviation rate function covering the full range of current fluctuations using heuristic arguments, and support them by simulation results using cloning algorithms wherever they are computationally accessible. For partially asymmetric zero-range processes we identify an interesting dynamic phase transition between different strategies for atypical currents, which is of a generic nature and expected to apply to a large class of particle systems on a ring. arXiv:1804.07844v1 [cond-mat.stat-mech]

Rare Event Simulation for Stochastic Dynamics in Continuous Time

“…This leads to an obvious adaption of the integration interval in the estimator Λ N k (t) (3.2), but we do not alter the notation here to keep it simple. Both algorithms perform very well and agree with a simple theoretical estimate based on bias reversal, which is not the main concern in this paper and we refer the reader to [20]. Enlarged error bars indicating 5 standard deviations reveal that (3.7) is significantly more accurate than (3.22).…”

“…The above functional, which recently appeared in this form in [24], assigns a weight via the function g to jumps of the process, as well as to the local time via the function h. Dynamics conditioned on such a functional have been studied in many contexts [5], including driven diffusions on periodic continuous spaces E [25]. As mentioned before, the simplest examples covered by our setting are Markov chains with finite state space E. This includes stochastic particle systems on a finite lattice with periodic or closed boundary conditions such as zero-range or inclusion processes [23,20,26], and also processes with open boundaries and bounded local state space such as the exclusion process [3]. Choosing g appropriately and h ≡ 0 the functional A T can, for example, measure the empirical particle current across a bond of the lattice or within the whole system up to time T .…”

Dynamical criticality in open systems: Nonperturbative physics, microscopic origin, and direct observation

“…In contrast with standard critical phenomena [33,34], which occur at the configurational level, DPTs appear in trajectory space when conditioning the system to sustain an unlikely value of dynamical observables such as the time-integrated current [1,4,27,[35][36][37]. DPTs thus manifest as a peculiar change in the properties of trajectories responsible for such rare events, making these trajectories far more probable than anticipated due to the emergence of ordered structures such as traveling waves [2,9,11,21], condensates [3,12,29] or hyperuniform states [15,22,38]. In all these cases, the hallmark of the DPT is the appearance of a singularity in the so-called large deviation function (LDF), which controls the probability of fluctuations and plays the role of a thermodynamic potential for nonequilibrium systems [35,39,40].…”

Dynamical Phase Transitions for Flows on Finite Graphs

“…We will fix the function κ on the right hand side of (4.1) in such a way that (3.3) is strictly concave on a region and such that there is just one single globally attractive stationary solution to equation (2.7), which is natural in light of Remark 2.5. We refer to [10,11] for discussions of dynamical phase transitions for asymmetric dynamics of particle systems in the hydrodynamic rescaling. We stress again that the limit considered here is very different from the hydrodynamic one.…”