Probability amplitude structure of Fokker–Plank equation

Torres, Maíra Soares; Figueiredo, J. M. A.

doi:10.1016/s0378-4371(03)00615-0

Cited by 6 publications

(1 citation statement)

References 11 publications

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“…Many practical problems in statistical mechanics and other areas of science and engineering are described as multi-degree-of-freedom (MDOF) or high-dimensional nonlinear stochastic dynamic (NSD) systems [1][2][3][4][5][6][7][8][9][10][11][12][13]. Hence, the investigation on the probabilistic solutions of MDOF systems attracted many researchers in the last decades.…”

Section: Introductionmentioning

confidence: 99%

Methodology for the solutions of some reduced Fokker‐Planck equations in high dimensions

Er¹

2011

Annalen der Physik

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In this paper, a new methodology is formulated for solving the reduced Fokker-Planck (FP) equations in high dimensions based on the idea that the state space of large-scale nonlinear stochastic dynamic system is split into two subspaces. The FP equation relevant to the nonlinear stochastic dynamic system is then integrated over one of the subspaces. The FP equation for the joint probability density function of the state variables in another subspace is formulated with some techniques. Therefore, the FP equation in high-dimensional state space is reduced to some FP equations in low-dimensional state spaces, which are solvable with exponential polynomial closure method. Numerical results are presented and compared with the results from Monte Carlo simulation and those from equivalent linearization to show the effectiveness of the presented solution procedure. It attempts to provide an analytical tool for the probabilistic solutions of the nonlinear stochastic dynamics systems arising from statistical mechanics and other areas of science and engineering.

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Section: Introductionmentioning

confidence: 99%

Methodology for the solutions of some reduced Fokker‐Planck equations in high dimensions

Er¹

2011

Annalen der Physik

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Stochastic Quantization of Time-Dependent Systems by the Haba and Kleinert Method

Haas

2005

Int J Theor Phys

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The stochastic quantization method recently developed by Haba and Kleinert is extended to non-autonomous mechanical systems, in the case of the time-dependent harmonic oscillator. In comparison with the autonomous case, the quantization procedure involves the solution of a nonlinear, auxiliary equation. Using a rescaling transformation, the Schrödinger equation for the time-dependent harmonic oscillator is obtained after averaging of a classical stochastic differential equation.KEY WORDS: stochastic quantization; time-dependent harmonic oscillator; Schrödinger equation; classical simulation of quantum systems.

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Stochastic Quantization of Time-Dependent Systems by the Haba and Kleinert Method

Haas

2005

Int J Theor Phys

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Probability amplitude structure of Fokker–Plank equation

Cited by 6 publications

References 11 publications

Methodology for the solutions of some reduced Fokker‐Planck equations in high dimensions

Methodology for the solutions of some reduced Fokker‐Planck equations in high dimensions

Stochastic Quantization of Time-Dependent Systems by the Haba and Kleinert Method

Stochastic Quantization of Time-Dependent Systems by the Haba and Kleinert Method

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