Two Refreshing Views of Fluctuation Theorems Through Kinematics Elements and Exponential Martingale

Chétrite, Raphaël; Gupta, Shamik

doi:10.1007/s10955-011-0184-0

Cited by 79 publications

(138 citation statements)

References 88 publications

(219 reference statements)

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“…The key property of cMPSs is that of gauge invariance from which the equivalences follow. Other dynamical properties can be proved via gauge transformations, as for example certain fluctuation relations [7].…”

Section: Discussionmentioning

confidence: 99%

“…A general discussion on the connection between gauge invariance in stochastic dynamics and fluctuation theorems was given in Ref. [7].…”

Section: Integral Fluctuation Theorems From Gauge Transformationsmentioning

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: Discussionmentioning

confidence: 99%

“…A general discussion on the connection between gauge invariance in stochastic dynamics and fluctuation theorems was given in Ref. [7].…”

Section: Integral Fluctuation Theorems From Gauge Transformationsmentioning

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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“…In the following, we will also use time-dependent functions h t and f t to transform L [109]. In this case, the generalized Doob transform is a nonhomogeneous process with path measure given by…”

Section: Definitionmentioning

confidence: 99%

“…From the point of view of probability theory, the Radon-Nikodym derivative associated with this transform is an example of exponential martingale. The generalized Doob transform also has interesting applications in physics: it appears in the stochastic mechanics of Nelson [114] and underlies, as shown in [109], the classical fluctuation-dissipation relations of near-equilibrium systems [54,[115][116][117] and recent generalizations of these relations obtained for nonequilibrium systems [91,[118][119][120][121][122][123][124]. The work of [109] shows moreover that the exponential martingale (76) verifies a non-perturbative general version of these relations, which also include the fluctuation relations of Jarzynski [125] and Gallavotti-Cohen [126][127][128].…”

Section: Definitionmentioning

confidence: 99%

Nonequilibrium Markov Processes Conditioned on Large Deviations

Chétrite

Touchette

2014

Ann. Henri Poincaré

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333

762

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International audienceWe consider the problem of conditioning a Markov process on a rare event and of representing this conditioned process by a conditioning-free process, called the effective or driven process. The basic assumption is that the rare event used in the conditioning is a large-deviation-type event, characterized by a convex rate function. Under this assumption, we construct the driven process via a generalization of Doob's h-transform, used in the context of bridge processes, and show that this process is equivalent to the conditioned process in the long-time limit. The notion of equivalence that we consider is based on the logarithmic equivalence of path measures, and implies that the two processes have the same typical states. In constructing the driven process, we also prove equivalence with the so-called exponential tilting of the Markov process, often used with importance sampling to simulate rare events and giving rise, from the point of view of statistical mechanics, to a nonequilibrium version of the canonical ensemble. Other links between our results and the topics of bridge processes, quasi-stationary distributions, stochastic control, and conditional limit theorems are mentioned

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Heat Release by Controlled Continuous-Time Markov Jump Processes

2013

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We derive the equations governing the protocols minimizing the heat released by a continuous-time Markov jump process on a one-dimensional countable state space during a transition between assigned initial and final probability distributions in a finite time horizon. In particular, we identify the hypotheses on the transition rates under which the optimal control strategy and the probability distribution of the Markov jump problem obey a system of differential equations of Hamilton-Jacobi-Bellman-type. As the state-space mesh tends to zero, these equations converge to those satisfied by the diffusion process minimizing the heat released in the Langevin formulation of the same problem. We also show that in full analogy with the continuum case, heat minimization is equivalent to entropy production minimization. Thus, our results may be interpreted as a refined version of the second law of thermodynamics.

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Two Refreshing Views of Fluctuation Theorems Through Kinematics Elements and Exponential Martingale

Cited by 79 publications

References 88 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

Nonequilibrium Markov Processes Conditioned on Large Deviations

Heat Release by Controlled Continuous-Time Markov Jump Processes

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