Fluctuation relations for anomalous dynamics generated by time-fractional Fokker–Planck equations

Dieterich, Peter; Klages, Rainer; Chechkin, Aleksei V.

doi:10.1088/1367-2630/17/7/075004

Cited by 17 publications

(15 citation statements)

References 72 publications

(167 reference statements)

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“…which tends to zero at the power-law rate. The consistency between simulation and the theoretical results about the MSD of model (29) can be found in Fig. 2.…”

Section: B Force Acting On Subordinated Process Y(t)supporting

confidence: 74%

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Subdiffusion in an external force field

2019

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The phenomena of subdiffusion are widely observed in physical and biological systems. To investigate the effects of external potentials, say, harmonic potential, linear potential, and time dependent force, we study the subdiffusion described by subordinated Langevin equation with white Gaussian noise, or equivalently, by the single Langevin equation with compound noise. If the force acts on the subordinated process, it keeps working all the time; otherwise, the force just exerts an influence on the system at the moments of jump. Some common statistical quantities, such as, the ensemble and time averaged mean squared displacement, position autocorrelation function, correlation coefficient, generalized Einstein relation, are discussed to distinguish the effects of various forces and different patterns of acting. The corresponding Fokker-Planck equations are also presented. All the stochastic processes discussed here are non-stationary, non-ergodicity, and aging.

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“…which tends to zero at the power-law rate. The consistency between simulation and the theoretical results about the MSD of model (29) can be found in Fig. 2.…”

Section: B Force Acting On Subordinated Process Y(t)supporting

confidence: 74%

“…Let us pay attention to the critical time distinguishing two different scales in two models (15) and (29). It is t = (2γ) − 1 α in the first model, depending on the parameter α and influenced by the inverse subodinator s(t).…”

Section: B Force Acting On Subordinated Process Y(t)mentioning

confidence: 99%

Subdiffusion in an external force field

2019

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“…Our results are in particular applicable to the stochastic thermodynamics of anomalous processes. Previous studies of fluctuation theorems in anomalous subdiffusive systems focused on work induced by a constant force such that the work statistics are equivalent to that of the spatial coordinate itself [129][130][131]. On the other hand, the mechanical work imposed by a non-equilibrium driving in a generic situation is naturally captured by our framework.…”

Section: Conclusion and Open Questionsmentioning

confidence: 99%

Feynman–Kac equation for anomalous processes with space- and time-dependent forces

Cairoli¹,

Baule²

2017

J. Phys. A: Math. Theor.

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Functionals of a stochastic process Y (t) model many physical time-extensive observables, for instance particle positions, local and occupation times or accumulated mechanical work. When Y (t) is a normal diffusive process, their statistics are obtained as the solution of the celebrated Feynman-Kac equation. This equation provides the crucial link between the expected values of diffusion processes and the solutions of deterministic second-order partial differential equations. When Y (t) is non-Brownian, e.g., an anomalous diffusive process, generalizations of the Feynman-Kac equation that incorporate power-law or more general waiting time distributions of the underlying random walk have recently been derived. A general representation of such waiting times is provided in terms of a Lévy process whose Laplace exponent is directly related to the memory kernel appearing in the generalized Feynman-Kac equation. The corresponding anomalous processes have been shown to capture nonlinear mean square displacements exhibiting crossovers between different scaling regimes, which have been observed in numerous experiments on biological systems like migrating cells or diffusing macromolecules in intracellular environments. However, the case where both spaceand time-dependent forces drive the dynamics of the generalized anomalous process has not been solved yet. Here, we present the missing derivation of the Feynman-Kac equation in such general case by using the subordination technique. Furthermore, we discuss its extension to functionals explicitly depending on time, which are of particular relevance for the stochastic thermodynamics of anomalous diffusive systems. Exact results on the work fluctuations of a simple non-equilibrium model are obtained. An additional aim of this paper is to provide a pedagogical introduction to Lévy processes, semimartingales and their associated stochastic calculus, which underlie the mathematical formulation of anomalous diffusion as a subordinated process.

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“…Consequently, the definitions of work, heat and entropy production used within the recent theory of stochastic thermodynamics [64] have to be modified to account for the contribution of the external flow [65] highlighting fundamental similarities between normal and anomalous diffusive systems, even though the stochastic thermodynamics of the latter is so far not well understood [66]. Along these lines, connections between GI and the validity of fluctuation-dissipation relations on the one hand, and the celebrated fluctuation relations generalizing the second law of thermodynamics [64] on the other, have been suggested [66,67] and need to be investigated further. But our most important statement is that ignoring our weak GI rules can easily lead to unphysical models, as exemplified by the CTRW with an ad hoc advective term (Fig.…”

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confidence: 99%

Weak Galilean invariance as a selection principle for coarse-grained diffusive models

Cairoli

Klages

Baule

2018

Proc. Natl. Acad. Sci. U.S.A.

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How does the mathematical description of a system change in different reference frames? Galilei first addressed this fundamental question by formulating the famous principle of Galilean invariance. It prescribes that the equations of motion of closed systems remain the same in different inertial frames related by Galilean transformations, thus imposing strong constraints on the dynamical rules. However, real world systems are often described by coarse-grained models integrating complex internal and external interactions indistinguishably as friction and stochastic forces. Since Galilean invariance is then violated, there is seemingly no alternative principle to assess a priori the physical consistency of a given stochastic model in different inertial frames. Here, starting from the Kac-Zwanzig Hamiltonian model generating Brownian motion, we show how Galilean invariance is broken during the coarse-graining procedure when deriving stochastic equations. Our analysis leads to a set of rules characterizing systems in different inertial frames that have to be satisfied by general stochastic models, which we call "weak Galilean invariance." Several well-known stochastic processes are invariant in these terms, except the continuous-time random walk for which we derive the correct invariant description. Our results are particularly relevant for the modeling of biological systems, as they provide a theoretical principle to select physically consistent stochastic models before a validation against experimental data.

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Fluctuation relations for anomalous dynamics generated by time-fractional Fokker–Planck equations

Cited by 17 publications

References 72 publications

Subdiffusion in an external force field

Subdiffusion in an external force field

Feynman–Kac equation for anomalous processes with space- and time-dependent forces

Weak Galilean invariance as a selection principle for coarse-grained diffusive models

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