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      Anomalous diffusion associated with nonlinear fractional derivative Fokker-Planck-like equation: Exact time-dependent solutions

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          Abstract

          We consider the \(d=1\) nonlinear Fokker-Planck-like equation with fractional derivatives \(\frac{\partial}{\partial t}P(x,t)=D \frac{\partial^{\gamma}}{\partial x^{\gamma}}[P(x,t) ]^{\nu}\). Exact time-dependent solutions are found for \( \nu = \frac{2-\gamma}{1+ \gamma}\) (\(-\infty<\gamma \leq 2\)). By considering the long-distance {\it asymptotic} behavior of these solutions, a connection is established, namely \(q=\frac{\gamma+3}{\gamma+1}\) (\(0<\gamma \le 2\)), with the solutions optimizing the nonextensive entropy characterized by index \(q\) . Interestingly enough, this relation coincides with the one already known for L\'evy-like superdiffusion (i.e., \(\nu=1\) and \(0<\gamma \le 2\)). Finally, for \((\gamma,\nu)=(2, 0)\) we obtain \(q=5/3\) which differs from the value \(q=2\) corresponding to the \(\gamma=2\) solutions available in the literature (\(\nu<1\) porous medium equation), thus exhibiting nonuniform convergence.

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          Anomalous diffusion in the presence of external forces: exact time-dependent solutions and entropy

          The optimization of the usual entropy \(S_1[p]=-\int du p(u) ln p(u)\) under appropriate constraints is closely related to the Gaussian form of the exact time-dependent solution of the Fokker-Planck equation describing an important class of normal diffusions. We show here that the optimization of the generalized entropic form \(S_q[p]=\{1- \int du [p(u)]^q\}/(q-1)\) (with \(q=1+\mu-\nu \in {\bf \cal{R}}\)) is closely related to the calculation of the exact time-dependent solutions of a generalized, nonlinear, Fokker Planck equation, namely \(\frac{\partial}{\partial t}p^\mu= -\frac{\partial}{\partial x}[F(x)p^\mu]+D \frac{\partial^2} {\partial x^2}p^\nu\), associated with anomalous diffusion in the presence of the external force \(F(x)=k_1-k_2x\). Consequently, paradigmatic types of normal (\(q=1\)) and anomalous (\(q \neq 1\)) diffusions occurring in a great variety of physical situations become unified in a single picture.
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            Fractional Calculus as a Macroscopic Manifestation of Randomness

            We generalize the method of Van Hove so as to deal with the case of non-ordinary statistical mechanics, that being phenomena with no time-scale separation. We show that in the case of ordinary statistical mechanics, even if the adoption of the Van Hove method imposes randomness upon Hamiltonian dynamics, the resulting statistical process is described using normal calculus techniques. On the other hand, in the case where there is no time-scale separation, this generalized version of Van Hove's method not only imposes randomness upon the microscopic dynamics, but it also transmits randomness to the macroscopic level. As a result, the correct description of macroscopic dynamics has to be expressed in terms of the fractional calculus.
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              Fractional diffusion, waiting-time distributions, and Cattaneo-type equations

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                Author and article information

                Journal
                29 March 2000
                Article
                10.1103/PhysRevE.62.2213
                cond-mat/0003482
                9e8ae749-a846-49ea-9748-b99b181c2ae2
                History
                Custom metadata
                Phys.Rev.E62:2213-2218,2000
                3 figures
                cond-mat.stat-mech

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