Large Deviation Approach to Nonequilibrium Systems

Touchette, Hugo; Harris, Rosemary J.

doi:10.1002/9783527658701.ch11

Cited by 68 publications

(103 citation statements)

References 62 publications

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“…Indeed, the cusp in ψ ω (λ) when N → ∞ implies that the region of the large deviation function between the entropy rates σ * h and σ * l is connected by a Maxwell construction (or a tie-line) with a slope λ * . The LF transform of ψ ω (λ) only provides the convex envelope of I(σ), but in the limit of large τ , I(σ) must converge to that envelope [22]. Further, as illustrated in Fig.…”

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confidence: 99%

Dynamic phase transitions in simple driven kinetic networks

2014

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We analyze the probability distribution for entropy production rates of trajectories evolving on a class of out-of-equilibrium kinetic networks. These networks can serve as simple models for driven dynamical systems, where energy fluxes typically result in non-equilibrium dynamics. By analyzing the fluctuations in the entropy production, we demonstrate the emergence, in a large system size limit, of a dynamic phase transition between two distinct dynamical regimes.The study of fluctuation phenomena is one of the central endeavors of non-equilibrium statistical mechanics. Analysis of fluctuations in non-equilibrium processes have, for example, led to the discovery of the fluctuation theorems, which have helped elucidate how macroscopic notions of irreversibility emerge from microscopic laws [1][2][3]. More recently, theoretical and numerical analysis of the statistics of rare fluctuations in driven lattice gas models [4,5], exclusion processes [6], zero-range processes [7], 1D models of transport [8], and models of glass formers [9,10] have revealed the presence of coexisting ensembles of trajectories and so-called dynamic phase transitions between them [4,5,8,11]. In this paper, we analyze the statistics of rare fluctuations in entropy production rates for certain model non-equilibrium, or driven, kinetic networks (see Fig. 1). While this Markovian system, with effectively one-particle dynamics, lacks much of the complexity of previously studied driven systems [4-8], we show -numerically and analyticallythe presence of two dynamical phases, each with a characteristic entropy production rate. This demonstration shows that singularities in trajectory space can in fact arise even in very simple driven kinetic networks with a single degree of freedom. Driven kinetic networks of this general flavor are used to model a variety of physical, chemical, and biological systems including molecular motor dynamics [12,13]; cellular feedback, control, and regulation [14]; and kinetic proofreading mechanisms [15,16]. Physically, the dynamic phase transition serves to enhance the probability of observing large fluctuations in the dynamical behavior of these hopping processes.We study dynamical fluctuations of a system evolving on cyclical or periodic driven kinetic networks with some heterogeneity in the transition rates. We consider two types of cyclic networks, which we hereafter refer to as the ring network and the triangle network. The ring network connects N states in a circle with transition rates x in the clockwise direction and 1 in the reverse direction. The network has translational symmetry, but we also construct a variation of the ring network with that symmetry broken by a link we call the heterogeneous link, or h-link. As shown in Fig.

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confidence: 99%

Dynamic phase transitions in simple driven kinetic networks

2014

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“…Moreover, this 'Level 2.5' formulation allows to reconstruct any time-additive observable of the dynamical trajectory via its decomposition in terms of the empirical densities and of the empirical flows, and is thus closely related to the studies focusing on the generating functions of time-additive observables via deformed Markov operators, that have attracted a lot of interest recently in various models [16][17][18][19][20][21][22][23][24][25][26][27][28][29]. More generally, the idea that the theory of large deviations is the appropriate language to analyze non-equilibrium dynamics has been emphasized from various perspectives (see the reviews [16,25,26,[30][31][32][33] and the PhD Theses [4,13,34,35] or HDR Thesis [36]).…”

Section: Introductionmentioning

confidence: 99%

Statistical physics of long dynamical trajectories for a system in contact with several thermal reservoirs

Monthus¹

2019

J. Phys. A: Math. Theor.

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For a system in contact with several reservoirs r at different inverse-temperatures βr, we describe how the Markov jump dynamics with the generalized detailed balance condition can be analyzed via a statistical physics approach of dynamical trajectories [C(t)] 0≤t≤T over a long time interval T → +∞. The relevant intensive variables are the time-empirical density ρ(C), that measures the fractions of time spent in the various configurations C, and the time-empirical jump densities kr(C , C), that measure the frequencies of jumps from configuration C to configuration C when it is the reservoir r that furnishes or absorbs the corresponding energy difference (E(C ) − E(C)). I.arXiv:1906.03606v1 [cond-mat.stat-mech]

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“…While the region (ii) of universal typical fluctuations has been traditionally the main focus of studies for various physical observables, the theory of large deviations (iii) is nowadays considered as the unifying language for the statistical physics of equilibrium, non-equilibrium and dynamical systems (see the reviews [8][9][10] and references therein). In particular, the large deviations with respect to the large time limit of dynamical trajectories has produced an appropriate statistical physics approach for various Markovian processes (see the reviews [11][12][13][14][15][16][17] and the PhD Theses [18][19][20][21] and the HDR Thesis [22]). However the recent huge activity on large deviations in the field of random matrices has shown that the maximal eigenvalue [23][24][25][26][27][28] and many other observables involving the eigenvalues [29][30][31][32][33][34][35][36][37][38][39] display asymmetric scaling in large deviations : the probability P N (u) to observe bigger values than typical u > u typ and smaller values than typical u < u typ are governed by two different scalings D ± N (for instance two different power-laws D ± N = N θ± ) and two rate functions I ± (u)…”

Section: Introductionmentioning

confidence: 99%

Asymmetric scaling in large deviations for rare values bigger or smaller than the typical value

Monthus¹

2019

J. Stat. Mech.

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In various disordered systems or non-equilibrium dynamical models, the large deviations of some observables have been found to display different scalings for rare values bigger or smaller than the typical value. In the present paper, we revisit the simpler observables based on independent random variables, namely the empirical maximum, the empirical average, the empirical non-integer moments or other additive empirical observables, in order to describe the cases where asymmetric large deviations already occur. The unifying starting point to analyze the large deviations of these various empirical observables is given by the Sanov theorem for the large deviations of the empirical histogram : the rate function corresponds to the relative entropy with respect to the true probability distribution and it can be optimized in the presence of the appropriate constraints. Finally, the physical meaning of large deviations rate functions is discussed from the renormalization perspective.I.

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Large Deviation Approach to Nonequilibrium Systems

Cited by 68 publications

References 62 publications

Dynamic phase transitions in simple driven kinetic networks

Dynamic phase transitions in simple driven kinetic networks

Statistical physics of long dynamical trajectories for a system in contact with several thermal reservoirs

Asymmetric scaling in large deviations for rare values bigger or smaller than the typical value

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