Abstract
An analogy of the Fokker-Planck equation (FPE) with the Schr\"odinger equation allows us to use quantum mechanics technique to find the analytical solution of the FPE in a number of cases. However, previous studies have been limited to the Schr\"odinger potential with a discrete eigenvalue spectrum. Here, we will show how this approach can be also applied to a mixed eigenvalue spectrum with bounded and free states. We solve the FPE with boundaries located at x=\pm L/2 and take the limit L\rightarrow\infty, considering the examples with constant Schr\"{o}dinger potential and with P\"{o}schl-Teller potential. An oversimplified approach was proposed earlier by M.T. Araujo and E. Drigo Filho. A detailed investigation of the two examples shows that the correct solution, obtained in this paper, is consistent with the expected Fokker-Planck dynamics.
Highlights
IntroductionThe one–dimensional Fokker-Planck equation (FPE) for the probability density p(x, t ), depending on variable x and time t , assumes the generic form [1,2,3,4,5,6,7]
The one–dimensional Fokker-Planck equation (FPE) for the probability density p(x, t ), depending on variable x and time t, assumes the generic form [1,2,3,4,5,6,7] ∂p(x, t) ∂ ∂2 D(x, t ) ∂t=− ∂x f (x, t)p(x, t) + ∂x2 p(x,t) . 2 (1.1)Here, the drift coefficient or force f (x, t ) and the diffusion coefficient D(x, t ) depend on x and t in general
The aim of our work is to show how the problem with mixed eigenvalue spectrum can be treated correctly
Summary
The one–dimensional Fokker-Planck equation (FPE) for the probability density p(x, t ), depending on variable x and time t , assumes the generic form [1,2,3,4,5,6,7]. The FPE (1.1) with a general time-dependence and a special x-dependence of the drift and diffusion coefficients has been studied analytically in [7] using Lie algebra This method is applicable when the Fokker-Planck equation has a definite algebraic structure, which makes it possible to employ the Lie algebra and the Wei-Norman theorem. For a general Schrödinger potential, numerical treatments used in quantum mechanics, such as the Crank-Nicolson time propagation with implicit Numerov’s method for second order derivatives [9], are very useful. To apply it to Schrödinger-type equation, we just need to replace the real time step ∆t by an imaginary time step ∆t → −i∆t. To avoid any confusion one has to note that the Pöschl-Teller potential is referred to as Rosen-Morse potential in [10]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
