Large Deviations and Ensembles of Trajectories in Stochastic Models

Jack, Robert L.; Sollich, Peter

doi:10.1143/ptps.184.304

Cited by 246 publications

(536 citation statements)

References 13 publications

(27 reference statements)

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“…A natural question is whether there is an alternative non-conditioned process which generates the same conditioned ensemble of trajectories but which is properly stochastic: this is the driven process (alternatively called "auxiliary process" [3]). As discussed in [6], the generator of the driven process can be obtained from the tilted operator L(s) through a (generalised) Doob transform.…”

Section: Gauge Invariance and Driven Process As Gauge Fixingmentioning

confidence: 99%

“…[3,4,6], that of a diagonal matrix whose entries are the coefficients of the leading left eigenvector of L(s) [the prefactor that depends on the initial conditions is irrelevant as it drops out of Eq. (46)].…”

Section: Gauge Invariance and Driven Process As Gauge Fixingmentioning

confidence: 99%

“…As we explain below, this approach enables to derive in a compact and concise way many (known) properties of trajectory ensembles associated to dynamical large-deviations [3][4][5][6] and non-equilibrium fluctuations [7,8].…”

Section: Introductionmentioning

confidence: 99%

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Classical stochastic dynamics and continuous matrix product states: gauge transformations, conditioned and driven processes, and equivalence of trajectory ensembles

Garrahan¹

2016

J. Stat. Mech.

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Borrowing ideas from open quantum systems, we describe a formalism to encode ensembles of trajectories of classical stochastic dynamics in terms of continuous matrix product states (cMPSs). We show how to define in this approach "biased" or "conditioned" ensembles where the probability of trajectories is biased from that of the natural dynamics by some condition on trajectory observables. In particular, we show that the generalised Doob transform which maps a conditioned process to an equivalent "auxiliary" or "driven" process (one where the same conditioned set of trajectories is generated by a proper stochastic dynamics) is just a gauge transformation of the corresponding cMPS. We also discuss how within this framework one can easily prove properties of the dynamics such as trajectory ensemble equivalence and fluctuation theorems.

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Section: Gauge Invariance and Driven Process As Gauge Fixingmentioning

confidence: 99%

Section: Gauge Invariance and Driven Process As Gauge Fixingmentioning

confidence: 99%

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Classical stochastic dynamics and continuous matrix product states: gauge transformations, conditioned and driven processes, and equivalence of trajectory ensembles

Garrahan¹

2016

J. Stat. Mech.

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“…A class of far-fromequilibrium system, that shares the Hamiltonian of an equilibrium system, can be defined by a subset of the equilibrium ensemble of phase-space trajectories, conditioned by a finite flux [2][3][4][5]. That is to say, those members of the equilibrium ensemble of systems, which exhibit a given flux during some time interval, are defined as belonging to the non-equilibrium constrained-flux ensemble.…”

Section: Introductionmentioning

confidence: 99%

“…Such an ensemble is appealing in that it shares many features of an equilibrium ensemble, and admits elegant techniques for investigation of it properties, both in the case where the constrained dynamical quantity is antisymmetric under time reversal (a flux) [4,5,9], and where it is symmetric (a "dynamical activity") [4,5,[10][11][12][13][14][15][16]. However, it remains unclear whether such ensembles are realized in practise, i.e.…”

Section: Introductionmentioning

confidence: 99%

Numerical comparison of a constrained path ensemble and a driven quasisteady state

Knežević

Evans

2014

Phys. Rev. E

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We investigate the correspondence between a non-equilibrium ensemble defined via the distribution of phase-space paths of a Hamiltonian system, and a system driven into a steady-state by non-equilibrium boundary conditions. To discover whether the non-equilibrium path ensemble adequately describes the physics of a driven system, we measure transition rates in a simple onedimensional model of rotors with Newtonian dynamics and purely conservative interactions. We compare those rates with known properties of the non-equilibrium path ensemble. In doing so, we establish effective protocols for the analysis of transition rates in non-equilibrium quasi-steady states. Transition rates between potential wells and also between phase-space elements are studied, and found to exhibit distinct properties, the more coarse-grained potential wells being effectively further from equilibrium. In all cases the results from the boundary-driven system are close to the path-ensemble predictions, but the question of equivalence of the two remains open.

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Large Deviation Approach to Nonequilibrium Systems

Touchette

Harris

2013

Nonequilibrium Statistical Physics of Small Systems

101

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The theory of large deviations has been applied successfully in the last 30 years or so to study the properties of equilibrium systems and to put the foundations of equilibrium statistical mechanics on a clearer and more rigorous footing. A similar approach has been followed more recently for nonequilibrium systems, especially in the context of interacting particle systems. We review here the basis of this approach, emphasizing the similarities and differences that exist between the application of large deviation theory for studying equilibrium systems on the one hand and nonequilibrium systems on the other. Of particular importance are the notions of macroscopic, hydrodynamic, and long-time limits, which are analogues of the equilibrium thermodynamic limit, and the notion of statistical ensembles which can be generalized to nonequilibrium systems. For the purpose of illustrating our discussion, we focus on applications to Markov processes, in particular to simple random walks.

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Large Deviations and Ensembles of Trajectories in Stochastic Models

Cited by 246 publications

References 13 publications

Classical stochastic dynamics and continuous matrix product states: gauge transformations, conditioned and driven processes, and equivalence of trajectory ensembles

Classical stochastic dynamics and continuous matrix product states: gauge transformations, conditioned and driven processes, and equivalence of trajectory ensembles

Numerical comparison of a constrained path ensemble and a driven quasisteady state

Large Deviation Approach to Nonequilibrium Systems

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