Thermodynamic Formalism for Systems with Markov Dynamics

The thermodynamic formalism allows one to access the chaotic properties of equilibrium and out-of-equilibrium systems, by deriving those from a dynamical partition function. The definition that has been given for this partition function within the framework of discrete time Markov chains was not suitable for continuous time Markov dynamics. Here we propose another interpretation of the definition that allows us to apply the thermodynamic formalism to continuous time. We also generalize the formalism—a dynamical Gibbs ensemble construction—to a whole family of observables and their associated large deviation functions. This allows us to make the connection between the thermodynamic formalism and the observable involved in the much-studied fluctuation theorem. We illustrate our approach on various physical systems: random walks, exclusion processes, an Ising model and the contact process. In the latter cases, we identify a signature of the occurrence of dynamical phase transitions. We show that this signature can already be unraveled using the simplest dynamical ensemble one could define, based on the number of configuration changes a system has undergone over an asymptotically large time window.     

Multifractal Analysis of Weak Gibbs Measures for Intermittent Systems

 In this paper, we establish a multifractal formalism of weak Gibbs measures associated to potentials of weak bounded variation for certain nonhyperbolic systems. We apply our results to Manneville-Pomeau type maps and a piecewise conformal two-dimensional countable Markov map with indifferent periodic points which is related to a complex continued fraction. Received: 6 September 2001 / Accepted: 21 May 2002 Published online: 12 August 2002     

Mean square relaxation times for evolution of random fields

We consider the problem of relaxation times for Markov evolution of systems composed of a countable number of locally interacting particles, each one of which has a finite phase space. We give a theorem for comparison of mean square relaxation times of evolutions possessing the same ergodic stationary state. We give a reduction theorem for “attractive” evolutions. The results are applied to a generalization of the Glauber evolution of the one dimensional Ising chain.     

A Simple Discrete Model of Brownian Motors: Time-periodic Markov Chains

In this paper, we consider periodically inhomogeneous Markov chains, which can be regarded as a simple version of physical model—Brownian motors. We introduce for them the concepts of periodical reversibility, detailed balance, entropy production rate and circulation distribution. We prove the equivalence of the following statements: The time-periodic Markov chain is periodically reversible; It is in detailed balance; Kolmogorov's cycle condition is satisfied; Its entropy production rate vanishes; Every circuit and its reversed circuit have the same circulation weight. Hence, in our model of Markov chains, the directed transport phenomenon of Brownian motors, i.e. the existence of net circulation, can occur only in nonequilibrium and irreversible systems. Moreover, we verify the large deviation property and the Gallavotti-Cohen fluctuation theorem of sample entropy production rates of the Markov chain.     

A Gallavotti–Cohen-Type Symmetry in the Large Deviation Functional for Stochastic Dynamics

We extend the work of Kurchan on the Gallavotti–Cohen fluctuation theorem, which yields a symmetry property of the large deviation function, to general Markov processes. These include jump processes describing the evolution of stochastic lattice gases driven in the bulk or through particle reservoirs, general diffusive processes in physical and/or velocity space, as well as Hamiltonian systems with stochastic boundary conditions. For dynamics satisfying local detailed balance we establish a link between the average of the action functional in the fluctuation theorem and the macroscopic entropy production. This gives, in the linear regime, an alternative derivation of the Green–Kubo formula and the Onsager reciprocity relations. In the nonlinear regime consequences of the new symmetry are harder to come by and the large deviation functional difficult to compute. For the asymmetric simple exclusion process the latter is determined explicitly using the Bethe ansatz in the limit of large N.     

Hamiltonian and Lagrangian for the Trajectory of the Empirical Distribution and the Empirical Measure of Markov Processes

We compute the Hamiltonian and Lagrangian associated to the large deviations of the trajectory of the empirical distribution for independent Markov processes, and of the empirical measure for translation invariant interacting Markov processes. We treat both the case of jump processes (continuous-time Markov chains and interacting particle systems) as well as diffusion processes. For diffusion processes, the Lagrangian is a quadratic form of the deviation of the trajectory from the solution of the Kolmogorov forward equation. In all cases, the Lagrangian can be interpreted as a relative entropy or relative entropy density per unit time.     

Suspension Flows Over Countable Markov Shifts

We introduce the notion of topological pressure for suspension flows over countable Markov shifts, and we develop the associated thermodynamic formalism. In particular, we establish a variational principle for the topological pressure, and an approximation property in terms of the pressure on compact invariant sets. As an application we present a multifractal analysis for the entropy spectrum of Birkhoff averages for suspension flows over countable Markov shifts. The domain of the spectrum may be unbounded and the spectrum may not be analytic. We provide explicit examples where this happens. We also discuss the existence of full measures on the level sets of the multifractal decomposition.     

Finite Size Corrections to the Large Deviation Function of the Density in the One Dimensional Symmetric Simple Exclusion Process

The symmetric simple exclusion process is one of the simplest out-of-equilibrium systems for which the steady state is known. Its large deviation functional of the density has been computed in the past both by microscopic and macroscopic approaches. Here we obtain the leading finite size correction to this large deviation functional. The result is compared to the similar corrections for equilibrium systems.     

Landscapes of non-gradient dynamics without detailed balance: stable limit cycles and multiple attractors

Landscape is one of the key notions in literature on biological processes and physics of complex systems with both deterministic and stochastic dynamics. The large deviation theory (LDT) provides a possible mathematical basis for the scientists' intuition. In terms of Freidlin-Wentzell's LDT, we discuss explicitly two issues in singularly perturbed stationary diffusion processes arisen from nonlinear differential equations: (1) For a process whose corresponding ordinary differential equation has a stable limit cycle, the stationary solution exhibits a clear separation of time scales: an exponential terms and an algebraic prefactor. The large deviation rate function attains its minimum zero on the entire stable limit cycle, while the leading term of the prefactor is inversely proportional to the velocity of the non-uniform periodic oscillation on the cycle. (2) For dynamics with multiple stable fixed points and saddles, there is in general a breakdown of detailed balance among the corresponding attractors. Two landscapes, a local and a global, arise in LDT, and a Markov jumping process with cycle flux emerges in the low-noise limit. A local landscape is pertinent to the transition rates between neighboring stable fixed points; and the global landscape defines a nonequilibrium steady state. There would be nondifferentiable points in the latter for a stationary dynamics with cycle flux. LDT serving as the mathematical foundation for emergent landscapes deserves further investigations.     

Tunneling and Metastability of Continuous Time Markov Chains

We propose a definition of tunneling and of metastability for a continuous time Markov process on countable state spaces. We obtain sufficient conditions for a irreducible positive recurrent Markov process to exhibit a tunneling behaviour. In the reversible case these conditions can be expressed in terms of the capacities and of the stationary measure of the Markov process.     

A Gallavotti-Cohen-Evans-Morriss Like Symmetry for a Class of Markov Jump Processes

We investigate a new symmetry of the large deviation function of certain time-integrated currents in non-equilibrium systems. The symmetry is similar to the well-known Gallavotti-Cohen-Evans-Morriss-symmetry for the entropy production, but it concerns a different functional of the stochastic trajectory. The symmetry can be found in a restricted class of Markov jump processes, where the network of microscopic transitions has a particular structure and the transition rates satisfy certain constraints. We provide three physical examples, where time-integrated observables display such a symmetry. Moreover, we argue that the origin of the symmetry can be traced back to time-reversal if stochastic trajectories are grouped appropriately.     

On multiple phase transitions for branching Markov chains

We consider branching Markov chains on a countable set. We give a necessary and sufficient condition in terms of the transition kernel of the underlying Markov chain to have two phase transitions. We compute the critical values. We apply this result to prove that asymmetric branching random walks onZ have two phase transitions.     

Power-law distributions in random multiplicative processes with non-Gaussian colored multipliers

One class of universal mechanisms that generate power-law probability distributions is that of random multiplicative processes. In this paper, we consider a multiplicative Langevin equation driven by non-Gaussian colored multipliers. We analytically derive a formula that relates the power-law exponent to the statistics of the multipliers and numerically confirm its validity using multiplicative noise generated by chaotic dynamical systems and by a two-valued Markov process. We also investigate the relationship between our treatment and the large deviation analysis of time series, and demonstrate the appearance of log-periodic fluctuations superimposed on the power-law distribution due to the non-Gaussian nature of the multipliers.     

Fluctuations and large deviations in non-equilibrium systems

For systems in contact with two reservoirs at different densities or with two thermostats at different temperatures, the large deviation function of the density gives a possible way of extending the notion of free energy to non-equilibrium systems. This large deviation function of the density can be calculated explicitly for exclusion models in one dimension with open boundary conditions. For these models, one can also obtain the distribution of the current of particles flowing through the system and the results lead to a simple conjecture for the large deviation function of the current of more general diffusive systems.     

Zero Temperature Limits of Gibbs States for Almost-Additive Potentials

This paper is devoted to study ergodic optimisation problems for almost-additive sequences of functions (rather than a fixed potential) defined over countable Markov shifts (that is a non-compact space). Under certain assumptions we prove that any accumulation point of a family of Gibbs equilibrium states is a maximising measure. Applications are given in the study of the joint spectral radius and to multifractal analysis of Lyapunov exponent of non-conformal maps.     

Spectral decomposition of expanding probabilistic dynamical systems

We study probabilistic combinations of expanding dynamical systems, which we call expanding probabilistic dynamical systems, in one dimension. If the system is composed by exact endomorphisms we prove that the probabilistic dynamical system is an exact Markov semigroup, and we determine a generalized spectral decomposition of the associated Markov operator on densities for an example of the tent map coupled with the 2-Renyi map.     

Rare Events in Stochastic Partial Differential Equations on Large Spatial Domains

A methodology is proposed for studying rare events in stochastic partial differential equations in systems that are so large that standard large deviation theory does not apply. The idea is to deduce the behavior of the original model by breaking the system into appropriately scaled subsystems that are sufficiently small for large deviation theory to apply but sufficiently large to be asymptotically independent from one another. The methodology is illustrated in the context of a simple one-dimensional stochastic partial differential equation. The application reveals a connection between the dynamics of the partial differential equation and the classical Johnson–Mehl–Avrami–Kolmogorov nucleation and growth model. It also illustrates that rare events are much more likely and predictable in large systems than in small ones due to the extra entropy provided by space.     

Phase Transitions for Countable Markov Shifts

We study the analyticity of the topological pressure for some one-parameter families of potentials on countable Markov shifts. We relate the non-analyticity of the pressure to changes in the recurrence properties of the system. We give sufficient conditions for such changes to exist and not to exist. We apply these results to the Manneville–Pomeau map, and use them to construct examples with different critical behavior. Received: 6 March 2000 / Accepted: 17 October 2000     

Small random perturbations of dynamical systems: Exponential loss of memory of the initial condition

We consider Markov processes arising from small random perturbations of non-chaotic dynamical systems. Under rather general conditions we prove that, with large probability, the distance between two arbitrary paths starting close to a same attractor of the unperturbed system decreases exponentially fast in time. The case of paths starting in different basins of attraction is also considered as well as some applications to the analysis of the invariant measure and to elliptic problems with small parameter in front to the second derivatives. The proof is based on a multiscale analysis of the typical trajectories of the Markov process; this analysis is done using techniques involved in the proof of Anderson localization for disordered quantum systems.     

Large Deviation Principle for Benedicks-Carleson Quadratic Maps

Since the pioneering works of Jakobson and Benedicks &; Carleson and others, it has been known that a positive measure set of quadratic maps admit invariant probability measures absolutely continuous with respect to Lebesgue. These measures allow one to statistically predict the asymptotic fate of Lebesgue almost every initial condition. Estimating fluctuations of empirical distributions before they settle to equilibrium requires a fairly good control over large parts of the phase space. We use the sub-exponential slow recurrence condition of Benedicks &; Carleson to build induced Markov maps of arbitrarily small scale and associated towers, to which the absolutely continuous measures can be lifted. These various lifts together enable us to obtain a control of recurrence that is sufficient to establish a level 2 large deviation principle, for the absolutely continuous measures. This result encompasses dynamics far from equilibrium, and thus significantly extends presently known local large deviations results for quadratic maps.