Thermodynamics and fractional Fokker-Planck equations

Im, Sokolov

doi:10.1103/physreve.63.056111

Cited by 107 publications

(90 citation statements)

References 22 publications

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“…The situations are considered in detail in Refs. [3,8]. The growing kernels are definitely unphysical.…”

Section: Examplementioning

confidence: 99%

“…The subdiffusive processes described by fractional Fokker-Planck equations with 1 < α < 2 are known to be subordinated to a Wiener process [8,9], so that the solution of this equation can be obtained through an integral transformation of the solution of a usual (Markovian) FPEs with the same potential, initial and boundary conditions. As we proceed to show, some analogue statements can be done also for the general version of the non-Markovian FPE.…”

mentioning

confidence: 99%

“…Parallel to Ref. [8] we shall call τ the internal variable of decomposition, and x and t its external variables. Moreover, we show that the Laplace-transformT (τ, u) of T (τ, t) in its external variable,T (τ, u) = ∞ 0 T (τ, t)e −ut dt reads:…”

mentioning

confidence: 99%

“…and for the same initial and boundary conditions, and the function T (τ, t) is connected with the memory kernel K(t) [8,9]. Parallel to Ref.…”

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confidence: 99%

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Solutions of a class of non-Markovian Fokker-Planck equations

Sokolov

2002

Phys. Rev. E

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We show that a formal solution of a rather general non-Markovian Fokker-Planck equation can be represented in a form of an integral decomposition and thus can be expressed through the solution of the Markovian equation with the same Fokker-Planck operator. This allows us to classify memory kernels into safe ones, for which the solution is always a probability density, and dangerous ones, when this is not guaranteed. The first situation describes random processes subordinated to a Wiener process, while the second one typically corresponds to random processes showing a strong ballistic component. In this case the non-Markovian Fokker-Planck equation is only valid in a restricted range of parameters, initial and boundary conditions. Many physical phenomena related to relaxation in complex systems are described by non-Markovian Fokker-Planck equations in a formwhere K(t) is a memory kernel and where L is a linear operator acting on variable(s) x. Such equations are often postulated on the basis of linear-response considerations for different physical situations and in several cases can be more or less rigorously derived based on a microscopic description. In the symmetric case, the usual form of the operator L reads:

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“…The situations are considered in detail in Refs. [3,8]. The growing kernels are definitely unphysical.…”

Section: Examplementioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…and for the same initial and boundary conditions, and the function T (τ, t) is connected with the memory kernel K(t) [8,9]. Parallel to Ref.…”

mentioning

confidence: 99%

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Solutions of a class of non-Markovian Fokker-Planck equations

Sokolov

2002

Phys. Rev. E

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“…(2) is a PDF. The derivation here parallels to the method used in [Sok01]. Its aim is to show that the random process whose PDF obeys Eq.…”

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confidence: 99%

Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations

2002

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We propose diffusion-like equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena, whose diffusion exponent varies with time and which, correspondingly, can not be viewed as self-affine random processes possessing a unique Hurst exponent. We prove the positivity of the solutions of the proposed equations and establish the relation to the Continuous Time Random Walk theory. We show that the distributed order time fractional diffusion equation describes the sub-diffusion random process which is subordinated to the Wiener process and whose diffusion exponent diminishes in time (retarding sub-diffusion) leading to superslow diffusion, for which the square displacement grows logarithmically in time. We also demonstrate that the distributed order space fractional diffusion equation describes super-diffusion phenomena when the diffusion exponent grows in time (accelerating super-diffusion).

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