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

Generalizations of the microcanonical and canonical ensembles for paths of Markov processes have been proposed recently to describe the statistical properties of nonequilibrium systems driven in steady states. Here, we propose a theory of these ensembles that unifies and generalizes earlier results and show how it is fundamentally related to the large deviation properties of nonequilibrium systems. Using this theory, we provide conditions for the equivalence of nonequilibrium ensembles, generalizing those found for equilibrium systems, construct driven physical processes that generate these ensembles, and rederive in a simple way known and new product rules for their transition rates. A nonequilibrium diffusion model is used to illustrate these results.

The paper presents a unified approach to different fluctuation relations for classical nonequilibrium dynamics described by diffusion processes. Such relations compare the statistics of fluctuations of the entropy production or work in the original process to the similar statistics in the time-reversed process. The origin of a variety of fluctuation relations is traced to the use of different time reversals. It is also shown how the application of the presented approach to the tangent process describing the joint evolution of infinitesimally close trajectories of the original process leads to a multiplicative extension of the fluctuation relations.

We have shown recently that a Markov process conditioned on rare events involving timeintegrated random variables can be described in the long-time limit by an effective Markov process, called the driven process, which is given mathematically by a generalization of Doob's h-transform. We show here that this driven process can be represented in two other ways: first, as a process satisfying various variational principles involving large deviation functions and relative entropies and, second, as an optimal stochastic control process minimizing a cost function also related to large deviation functions. These interpretations of the driven process generalize and unify many previous results on maximum entropy approaches to nonequilibrium systems, spectral characterizations of positive operators, and control approaches to large deviation theory. They also lead, as briefly discussed, to new methods for analytically or numerically approximating large deviation functions.

We obtain the rate function for the level 2.5 of large deviations for pure jump and diffusion processes. This result is proved by two methods: tilting, for which a tilted process with an appropriate typical behavior is considered, and a spectral method, for which the scaled cumulant generating function is used. We also briefly discuss fluctuation relations, pointing out their connection with large deviations at the level 2.5.

We discuss fluctuation relations in simple cases of non-equilibrium Langevin dynamics. In particular, we show that close to non-equilibrium steady states with non-vanishing probability currents some of these relations reduce to a modified version of the fluctuation-dissipation theorem. The latter may be interpreted as the equilibrium-like relation in the reference frame moving with the mean local velocity determined by the probability current.

A modified fluctuation-dissipation-theorem (MFDT) for a non-equilibrium steady state (NESS) is experimentally checked by studying the position fluctuations of a colloidal particle whose motion is confined in a toroidal optical trap. The NESS is generated by means of a rotating laser beam which exerts on the particle a sinusoidal conservative force plus a constant non-conservative one. The MFDT is shown to be perfectly verified by the experimental data. It can be interpreted as an equilibrium-like fluctuation-dissipation relation in the Lagrangian frame of the mean local velocity of the particle.The validity of the fluctuation-dissipation theorem (FDT) in systems out of thermal equilibrium has been the subject of intensive study during the last years. We recall that for a system in equilibrium with a thermal bath at temperature T the FDT establishes a simple relation between the 2-time correlation function C(t − s) of a given observable and the linear response function R(t − s) of this observable to a weak external perturbationHowever, Eq. (1) is not necessarily fulfilled out of equilibrium and violations are observed in a variety of systems such as glassy materials [1,2,3,4,5], granular matter [6], and biophysical systems [7]. This motivated a theoretical work devoted to a search of a general framework describing FD relations, see the review [8] or [9,10,11,12,13,14] for recent attempts in simple stochastic systems. In the same spirit, a modified fluctuation-dissipation theorem (MFDT) has been recently formulated for a non-equilibrium steady dynamics governed by the Langevin equation with nonconservative forces [15]. In particular, this MFDT holds for the overdamped motion of a particle on a circle, with angular position θ, in the presence of a periodic potential H(θ) = H(θ + 2π) and a constant non-conservative force Fθwhere ζ is a white noise term of mean ζ t = 0 and covariance ζ t ζ s = 2Dδ(t − s), with D the (bare) diffusivity. This is a system that may exhibit an increase in the effective diffusivity [16,17]. Here, we shall study the dynamical non-equilibrium steady state (NESS) reached for observables that depend only on the particle position on the circle so are periodic functions of the angle θ. Such a state corresponds to a constant non-vanishing probability current j along the circle and a periodic invariant probability density function ρ 0 (θ) that allow us to define a mean local velocity v 0 (θ) = j/ρ 0 (θ). This is the average velocity of the particle at θ. For a stochastic system in NESS evolving according to Eq. (2), the MFDT reads for t ≥ swhere the 2-time correlation of a given observable O(θ) is defined byand the linear response function to a δ-perturbation of the conjugated variable h t is given by the functional derivativeIn Eq. (5), ... h denotes the average in the perturbed time-dependent state obtained from the NESS by replacing H(θ) in Eq. (2) by H(θ) − h t O(θ). It reduces for h = 0 to the NESS average ... 0 . In Eq. (3), the correlation b(t − s) is given byThis new term takes into accoun...

An open quantum system interacting with its environment can be modeled under suitable assumptions as a Markov process, described by a Lindblad master equation. In this work, we derive a general set of fluctuation relations for systems governed by a Lindblad equation. These identities provide quantum versions of Jarzynski-Hatano-Sasa and Crooks relations. In the linear response regime, these fluctuation relations yield a fluctuation-dissipation theorem (FDT) valid for a stationary state arbitrarily far from equilibrium. For a closed system, this FDT reduces to the celebrated Callen-Welton-Kubo formula