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Register a new userAnomalous fluctuation relations
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R. Klages Queen Mary
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We study Fluctuation Relations (FRs) for dynamics that are anomalous, in the sense that the diffusive properties strongly deviate from the ones of standard Brownian motion. We first briefly review the concept of transient work FRs for stochastic dynamics modeled by the ordinary Langevin equation. We then introduce three generic types of dynamics generating anomalous diffusion: Levy flights, long-time correlated Gaussian stochastic processes and time-fractional kinetics. By combining Langevin and kinetic approaches we calculate the work probability distributions in the simple nonequilibrium situation of a particle subject to a constant force. This allows us to check the transient FR for anomalous dynamics. We find a new form of FRs, which is intimately related to the validity of fluctuation-dissipation relations. Analogous results are obtained for a particle in a harmonic potential dragged by a constant force. We argue that these findings are important for understanding fluctuations in experimentally accessible systems. As an example, we discuss the anomalous dynamics of biological cell migration both in equilibrium and in nonequilibrium under chemical gradients.
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Anomalous dynamics characterized by non-Gaussian probability distributions (PDFs) and/or temporal long-range correlations can cause subtle modifications of conventional fluctuation relations. As prototypes we study three variants of a generic time-fractional Fokker-Planck equation with constant force. Type A generates superdiffusion, type B subdiffusion and type C both super- and subdiffusion depending on parameter variation. Furthermore type C obeys a fluctuation-dissipation relation whereas A and B do not. We calculate analytically the position PDFs for all three cases and explore numerically their strongly non-Gaussian shapes. While for type C we obtain the conventional transient work fluctuation relation, type A and type B both yield deviations by featuring a coefficient that depends on time and by a nonlinear dependence on the work. We discuss possible applications of these types of dynamics and fluctuation relations to experiments.
We consider the application of fluctuation relations to the dynamics of coarse-grained systems, as might arise in a hypothetical experiment in which a system is monitored with a low-resolution measuring apparatus. We analyze a stochastic, Markovian jump process with a specific structure that lends itself naturally to coarse-graining. A perturbative analysis yields a reduced stochastic jump process that approximates the coarse-grained dynamics of the original system. This leads to a non-trivial fluctuation relation that is approximately satisfied by the coarse-grained dynamics. We illustrate our results by computing the large deviations of a particular stochastic jump process. Our results highlight the possibility that observed deviations from fluctuation relations might be due to the presence of unobserved degrees of freedom.
The fluctuation relations have received considerable attention since their emergence and development in the 1990s. We present a summary of the main results and suggest ways to interpret this material. Starting with a consideration of the under-determined time evolution of a simple open system, formulated using continuous Markovian stochastic dy- namics, an expression for the entropy generated over a time interval is developed in terms of the probability of observing a trajectory associated with a prescribed driving protocol, and the probability of its time-reverse. This forms the basis for a general theoretical description of non-equilibrium thermodynamic pro- cesses. Having established a connection between entropy production and an inequivalence in probability for forward and time-reversed events, we proceed in the manner of Sekimoto and Seifert, in particular, to derive results in stochastic thermodynamics: a description of the evolution of a system between equilibrium states that ties in with well-established thermodynamic expectations. We derive fluctuation relations, state conditions for their validity, and illustrate their op- eration in some simple cases, thereby providing some introductory insight into the various celebrated symmetry relations that have emerged in this field.
A stochastic dynamics has a natural decomposition into a drift capturing mean rate of change and a martingale increment capturing randomness. They are two statistically uncorrelated, but not necessarily independent mechanisms contributing to the overall fluctuations of the dynamics, representing the uncertainties in the past and in the future. A generalized Einstein relation is a consequence solely because the dynamics being stationary; and the Green-Kubo formula reflects a balance between the two mechanisms. Equilibrium with reversibility is characterized by a novel covariance symmetry.
Previously, we have derived a generalization of the canonical fluctuation relation between heat capacity and energy fluctuations $C=beta^{2}<delta U^{2}>$, which is able to describe the existence of macrostates with negative heat capacities $C<0$. In this work, we extend our previous results for an equilibrium situation with several control parameters to account for the existence of states with anomalous values in other response functions. Our analysis leads to the derivation of three different equilibrium fluctuation theorems: the textit{fundamental and the complementary fluctuation theorems}, which represent the generalization of two fluctuation identities already obtained in previous works, and the textit{associated fluctuation theorem}, a result that has no counterpart in the framework of Boltzmann-Gibbs distributions. These results are applied to study the anomalous susceptibility of a ferromagnetic system, in particular, the case of 2D Ising model.
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