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Register a new userSimilarity solutions of Fokker-Planck equation with time-dependent coefficients and fixed/moving boundaries
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Authors
C.-L. Ho
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We consider the solvability of the Fokker-Planck equation with both time-dependent drift and diffusion coefficients by means of the similarity method. By the introduction of the similarity variable, the Fokker-Planck equation is reduced to an ordinary differential equation. Adopting the natural requirement that the probability current density vanishes at the boundary, the resulting ordinary differential equation turns out to be integrable, and the probability density function can be given in closed form. New examples of exactly solvable Fokker-Planck equations are presented.
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We consider solvability of the generalized reaction-diffusion equation with both space- and time-dependent diffusion and reaction terms by means of the similarity method. By introducing the similarity variable, the reaction-diffusion equation is reduced to an ordinary differential equation. Matching the resulting ordinary differential equation with known exactly solvable equations, one can obtain corresponding exactly solvable reaction-diffusion systems. Several representative examples of exactly solvable reaction-diffusion equations are presented.
A procedure is presented for solving the Fokker-Planck equation with constant diffusion but non-stationary drift. It is based on the correspondence between the Fokker-Planck equation and the non-stationary Schrodinger equation. The formalism of supersymmetric quantum mechanics is extended by applying the Darboux transformation directly to the non-stationary Schrodinger equation. From a solution of a Fokker-Planck equation a solution of the Darboux partner is obtained. For drift coefficients satisfying the condition of shape invariance, a supersymmetric hierarchy of Fokker-Planck equations with solutions related by the Darboux transformation is obtained.
We consider similarity solutions of the generalized convection-diffusion-reaction equation with both space- and time-dependent convection, diffusion and reaction terms. By introducing the similarity variable, the reaction-diffusion equation is reduced to an ordinary differential equation. Matching the resulting ordinary differential equation with known exactly solvable equations, one can obtain corresponding exactly solvable convection-diffusion-reaction systems. Some representative examples of exactly solvable systems are presented. We also describe how an equivalent convection-diffusion-reaction system can be constructed which admits the same similarity solution of another convection-diffusion-reaction system.
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.
Molecular dynamics are extremely complex, yet understanding the slow components of their dynamics is essential to understanding their macroscopic properties. To achieve this, one models the molecular dynamics as a stochastic process and analyses the dominant eigenfunctions of the associated Fokker-Planck operator, or of closely related transfer operators. So far, the calculation of the discretized operators requires extensive molecular dynamics simulations. The Square-root approximation of the Fokker-Planck equation is a method to calculate transition rates as a ratio of the Boltzmann densities of neighboring grid cells times a flux, and can in principle be calculated without a simulation. In a previous work we still used molecular dynamics simulations to determine the flux. Here, we propose several methods to calculate the exact or approximate flux for various grid types, and thus estimate the rate matrix without a simulation. Using model potentials we test computational efficiency of the methods, and the accuracy with which they reproduce the dominant eigenfunctions and eigenvalues. For these model potentials, rate matrices with up to $mathcal{O}(10^6)$ states can be obtained within seconds on a single high-performance compute server if regular grids are used.
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