From Eq. This implies the convergence of the following integrals. Consequently, the STM-entropy results to be positive definite, continuous, symmetric, expandible, decisive, maximal, concave and Lesche stable . We conclude by recalling that several generalized entropies, which have been currently used in the study of anomalous statistical systems, belong to the STM family. In figure 1 , we depicted the loci of points representing some of these one-parameter entropies: the q -entropy. The thermostatistics theory based on the STM-entropy fulfills the Legendre structure .
The main proprieties of a statistical system described by this entropy, in the microcanonical formalism, has been investigated in . Within these settings, the nonlinear current becomes. It is worthy to note that this equation embodies two well-known special cases: the linear Fokker-Planck equation. This kind of nonlinearity is substantially different from the one appearing in Eq.
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Let us now consider a linear combination of two q -NFPE 25 with two power indexes q 1 and q 2 , respectively, according to. Clearly, this equation coincides with Eq. In other words, the STM-NFPE describes kinetic processes occurring in anomalous media where both super-diffusive and sub-diffusive mechanisms arise contemporarily and competitively.
In this way, from Eq. Differently, out of equilibrium, it can be shown that Eq. In fact, from Eqs. A way to obtain special solutions of a PDE is based on the determination of its Lie symmetries. The determination of the Lie symmetries can be accomplished by following well-known techniques described in standard textbooks [21, 22, 34] which we remind for the details. For the sake of clarity, it is useful to consider the three cases represented by the Eqs. Firstly, we consider the linear Fokker-Planck equation 24 which has been widely studied in the past .
Its maximal symmetry group is composed by the following seven operators. Finally, the last two generators reflect the linearity of Eq.
The Fokker-Planck Equation: Methods of Solution and Applications (Springer Series in Synergetics)
The maximal symmetry group of equation 25 is formed by the following four operators. The first two generators are identical to those of the linear case and produce time and velocity translations whilst the last two generators have a q -dependence and produce dilations. The maximal symmetry group of Eq. Any other symmetry is destroyed by the particular expression of the nonlinearity of the diffusive term.
Group-invariant solutions. Having classifiied the classical Lie symmetries, we derive now several physically meaningful solutions characterized by their invariance under some of the above symmetries transformations. It is merely a constant. The invariant corresponding to this symmetry is. Rewriting Eq. Its solution, given by. Notwithstanding, the Lyapunov function decreases in time since the system dissipates energy during the evolution toward the equilibrium. Let us now consider the two symmetries generated by the operators 3 and 4 which are typical of the q -NFPE.
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Quite interesting, the first of these transformation introduces the following scaling. Unfortunately, the related GIS. More in general, starting from suitable linear combinations of the generators 31 we can obtain some physically significative solutions. Among the many, let us consider the following case. Introducing the global invariants. Z q the normalization constant and the q -exponential.
From the previous analysis it follows that, in general, the self-similar function. In order to clarify this role let us recall that, according to the following time-dependent transformation. Correspondingly, the self-similar q -Gaussian solution 46 is changed into 2. By applying such transformation on Eq. We recall now that any localized solution of STM-NFPE 23 asymptotically approaches the stationary state given by the generalized Gaussian 29 which is, therefore, transformed by means of equation 54 in another one that well approximates the solution of Eq. In order to confirm this asymptotic behavior, we numerically study the approach to the equilibrium of a given Cauchy problem for the diffusive equation In fact, the asymptotic behavior of the solutions of Eq.
We have run several simulations with different initial shapes to confirm this asymptotic behavior. In this work we have studied the classical Lie symmetries and the related group invariant solutions of a nonlinear Fokker-Planck equation based on the Sharma-Taneja-Mittal entropy.
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The analysis showed that the generalized Gaussian function, obtained by replacing the standard exponential with its generalized version, is recurrent in the expression of several GIS. In fact, it models the stationary state 29 as well as the traveling wave 36 and, limiting to the q -case, also the self-similar solution Pedron, R. Mendes, L. Malacarne and E. Lenzi, Phys. E 65, Malacarne, R.
The Fokker-Planck Equation - Methods of Solution and Applications | Hannes Risken | Springer
Mendes, I. Pedron and E. E 63, R Spohn, J. France I 3, 69 Berryman, J. Bardou, J. Bouchand, O. Emile, A. Aspect and C. Cohen-Tannoudji, Phys. Ott et al. Solomon, E. Weeks and H. Swinney, Phys. Bychuk and B. O' Shaughnessy, Phys. Curado and F. Nobre, Eur. B 58, Tsallis, J. Kaniadakis, Phys.
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