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Register a new userGeneralized Fokker-Planck equation: Derivation and exact solutions
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S. I. Denisov
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We derive the generalized Fokker-Planck equation associated with the Langevin equation (in the Ito sense) for an overdamped particle in an external potential driven by multiplicative noise with an arbitrary distribution of the increments of the noise generating process. We explicitly consider this equation for various specific types of noises, including Poisson white noise and L{e}vy stable noise, and show that it reproduces all Fokker-Planck equations that are known for these noises. Exact analytical, time-dependent and stationary solutions of the generalized Fokker-Planck equation are derived and analyzed in detail for the cases of a linear, a quadratic, and a tailored potential.
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Microscopic theory of Brownian motion of a particle of mass $M$ in a bath of molecules of mass $mll M$ is considered beyond lowest order in the mass ratio $m/M$. The corresponding Langevin equation contains nonlinear corrections to the dissipative force, and the generalized Fokker-Planck equation involves derivatives of order higher than two. These equations are derived from first principles with coefficients expressed in terms of correlation functions of microscopic force on the particle. The coefficients are evaluated explicitly for a generalized Rayleigh model with a finite time of molecule-particle collisions. In the limit of a low-density bath, we recover the results obtained previously for a model with instantaneous binary collisions. In general case, the equations contain additional corrections, quadratic in bath density, originating from a finite collision time. These corrections survive to order $(m/M)^2$ and are found to make the stationary distribution non-Maxwellian. Some relevant numerical simulations are also presented.
We consider the $d=1$ nonlinear Fokker-Planck-like equation with fractional derivatives $frac{partial}{partial t}P(x,t)=D frac{partial^{gamma}}{partial x^{gamma}}[P(x,t) ]^{ u}$. Exact time-dependent solutions are found for $ u = 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 Levy-like superdiffusion (i.e., $ u=1$ and $0<gamma le 2$). Finally, for $(gamma, u)=(2, 0)$ we obtain $q=5/3$ which differs from the value $q=2$ corresponding to the $gamma=2$ solutions available in the literature ($ u<1$ porous medium equation), thus exhibiting nonuniform convergence.
We obtain exact results for fractional equations of Fokker-Planck type using evolution operator method. We employ exact forms of one-sided Levy stable distributions to generate a set of self-reproducing solutions. Explicit cases are reported and studied for various fractional order of derivatives, different initial conditions, and for differe
We study the connection between the parameters of the fractional Fokker-Planck equation, which is associated with the overdamped Langevin equation driven by noise with heavy-tailed increments, and the transition probability density of the noise generating process. Explicit expressions for these parameters are derived both for finite and infinite variance of the rescaled transition probability density.
In this paper we statistically analyze the Fokker-Planck (FP) equation of Schramm-Loewner evolution (SLE) and its variant SLE($kappa,rho_c$). After exploring the derivation and the properties of the Langevin equation of the tip of the SLE trace, we obtain the long and short time behaviors of the chordal SLE traces. We analyze the solutions of the FP and the corresponding Langevin equations and connect it to the conformal field theory (CFT) and present some exact results. We find the perturbative FP equation of the SLE($kappa,rho_c$) traces and show that it is related to the higher order correlation functions. Using the Langevin equation we find the long-time behaviors in this case. The CFT correspondence of this case is established and some exact results are presented.
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