Approximate solution for Fokker-Planck equation

Abstract

In this paper, an approximate solution to a specific class of the Fokker-Planck equation is proposed. The solution is based on the relationship between the Schrödinger type equation with a partially confining and symmetrical potential. To estimate the accuracy of the solution, a function error obtained from the original Fokker-Planck equation is suggested. Two examples, a truncated harmonic potential and non-harmonic polynomial, are analyzed using the proposed method. For the truncated harmonic potential, the system behavior as a function of temperature is also discussed.

The solution of the truncated harmonic oscillator using Lie groups

“…We solve the Fokker-Planck equation using Lie groups. The founded solution was contrasted with the results in reference [4]. Both solutions match quite well, fig.…”

“…Where D is known as the diffusion constant [4], [6], k is the Hooke constant, and d is the position where the oscillator jumps.…”

“…The two integrals in N 2 have solutions that involve error functions, so their accuracy is as great as the number of summands of the error function can be. On the other hand, the solution given in [4], for the truncated harmonic oscillator, is: Figure 2: The continuous line corresponds to the error generated using the Lie group method. The dashed line is the error given in [4].…”

“…On the other hand, the solution given in [4], for the truncated harmonic oscillator, is: Figure 2: The continuous line corresponds to the error generated using the Lie group method. The dashed line is the error given in [4]. The discontinuity of the curves in d = 1, 55 is due to the potential jump of the oscillator at this point, [7].…”

“…This kind of series for the Fokker-Planck is already known in a very interesting work, given in [4]- [5]. In order to compare the Lies group solution with the one given in reference [4], we use the next values d = 1, 55, D = 1 and k = 1, 4. Also, we took till the tenth summatory term of the error function, plotting for t=1.…”