Dynamical Phase Transitions for Flows on Finite Graphs
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Dynamical Phase Transitions for Flows on Finite Graphs Davide Gabrielli1 · D. R. Michiel Renger2 Received: 12 August 2020 / Accepted: 27 October 2020 / Published online: 17 November 2020 © The Author(s) 2020
Abstract We study the time-averaged flow in a model of particles that randomly hop on a finite directed graph. In the limit as the number of particles and the time window go to infinity but the graph remains finite, the large-deviation rate functional of the average flow is given by a variational formulation involving paths of the density and flow. We give sufficient conditions under which the large deviations of a given time averaged flow is determined by paths that are constant in time. We then consider a class of models on a discrete ring for which it is possible to show that a better strategy is obtained producing a time-dependent path. This phenomenon, called a dynamical phase transition, is known to occur for some particle systems in the hydrodynamic scaling limit, which is thus extended to the setting of a finite graph. Keywords Large deviations · Particle systems · Phase transitions Mathematical Subject Classification 60F10 · 05C21 · 82C22 · 82C26
1 Introduction One of the main challenges of statistical mechanics is to understand the thermodynamics of particle systems that are not in detailed balance. A violation of detailed balance means that even in a steady state, there can be non-trivial net flows. It is therefore natural to study flows, currents and their corresponding large deviations, which is the basis of macroscopic fluctuation theory [4]. Various limits and large deviations of flows and currents have been studied in the literature, e.g. steady-state or pathwise large deviations as the number N of particles and/or lattice sites goes to infinity, e.g. [4,13], or large deviations of time-averaged flows as time T goes to infinity, e.g. [5,6]. In this work we are concerned with large deviations
Communicated by Eric A. Carlen.
B
D. R. Michiel Renger [email protected] Davide Gabrielli [email protected]
1
Università dell’Aquila, Via Vetoio, Loc. Coppito, 67010 L’Aquila, Italia
2
WIAS Berlin, Mohrenstrasse 39, 10117 Berlin, Germany
123
2354
D. Gabrielli, D.R. M. Renger
T of time-averaged flows T1 0 Q N (t) dt, where first the number N of particles and then the time horizon T is sent to infinity, as for example in [1–3,8,9]. This yields a large deviation principle of the type: T Prob T1 0 Q N (t) dt ≈ q
with (q) := lim
T →∞
inf
(ρ,q): T T 0 q(t) dt=q, ρ(t)+div ˙ q(t)=0 1
1 T
N →∞ T →∞ −N T (q)
∼
T
e
,
L(ρ(t), q(t)) dt,
0
where the infimum ranges over paths of particle densities ρ(t) and particle flows q(t), and L is some non-negative cost function. In many cases, the infimum over paths is attained or approximated by constant paths, i.e. ρ(t) constant and q(t) constant and divergence free. In this case the above expression simplifies to (q) = inf ρ L(ρ, q) [8]. Such simplification will fail if a time-dependent flow q(t) is significantly less costly
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