The Characteristics Of The Escape Rate Of Lévy Particles In Truncated Quartic Potential Well And The Underlying Physical Mechanism

In recent decades,Lévy flight as an important class of anomalous diffusion has received much attention, which has been widely used in the fields of physics,chemistry, paleoclimatology, economics, etc. Because of the big jump behavior with a long-tailed distribution, the second moment of Lévy particles’ distribution is divergent,which makes the Lévy flight different from general Brownian motion. At present,people can only get the accurate solution of probability density of the Lévy particles in free field, linear potential field and harmonic potential field, which lay the foundation for solving the escape rate of Lévy particles in truncated quartic potential well.This paper mainly studies the escape rate of Lévy particles and the corresponding physical mechanism in truncated quartic potential well in the case of inertial and overdamped. Taking into account the complexity of the fractional Fokker-Planck equation in quartic potential well, we use Monte Carlo simulation method and get extensive simulation results: In the overdamped case, the escape rate for Lévy particles from a truncated confined quartic potential well presents qualitatively different characteristics compared with that from a truncated unconfined harmonic potential well; We have analyzed the contribution of the inertial term for the escape rate of Lévy particles and discussed the corresponding physical mechanism.In addition, we get the analytical expression of the escape rate in Cauchy circumstances by solving fractional Fokker-Planck equation. In the overdamped case,two kinds of different escape mechanism for low noise and high noise intensities are found: For low noise intensities, equivalent to a high barrier, the escape of Lévy particles in potential well can be described with "zero-flux model" approximately; by reason of the active particles, the escape can be described approximately with "constant-flux model" for high noise intensities; As the noise intensity increases, the distribution of escaping particles in quasi-stationary state shows a noise-induced phase transition phenomenon, with transiting from a bimodal narrow distribution to a unimodal wide distribution.In this paper, we develop a kind of approximate analytical method for the inertial Cauchy case and get better results consistent with numerical simulation under the condition of certain parameters.Finally we summarize the work of the paper, and put forward some issues worthyof further discussion.