Fokker-Planck equation with arbitrary dc and ac fields: continued fraction method.

Area of Science:

• Computational Physics
• Nonlinear Dynamics
• Statistical Mechanics

Background:

• The Fokker-Planck equation describes systems with many degrees of freedom.
• Solving this equation under arbitrary DC and AC fields is computationally challenging.
• Understanding nonlinear responses in driven systems is crucial for various physical phenomena.

Purpose of the Study:

• To introduce and validate the continued fraction method (CFM) for solving the Fokker-Planck equation.
• To investigate the nonlinear response of driven systems using both perturbative and exact CFM.
• To compare the efficiency and validity of the perturbative CFM approach.

Main Methods:

• Transforming the Fokker-Planck equation into linear algebraic equations using basis functions.
• Implementing the continued fraction method (CFM) for numerical solutions.
• Utilizing both a perturbative CFM and a numerically exact matrix CFM.
• Analyzing nonlinear effects through hysteresis loops for molecular dipoles and particles in periodic potentials.

Main Results:

• The CFM provides efficient and accurate numerical solutions for the Fokker-Planck equation.
• The perturbative CFM is efficient within its validity regime, requiring only scalar quantities.
• Nonlinear effects, such as hysteresis loop deformation, become pronounced with increasing AC field strength.
• Both molecular dipole and particle in a periodic potential systems exhibit significant nonlinear behavior.

Conclusions:

• The continued fraction method is a powerful tool for solving complex Fokker-Planck equations.
• The perturbative CFM offers an efficient alternative for studying nonlinear dynamics when applicable.
• AC field strength is a key parameter influencing the manifestation of nonlinear effects in driven systems.