Abstract
We report timedependent probability density functions (PDFs) for a nonlinear stochastic process with a cubic force using analytical and computational studies. Analytically, a transition probability is formulated by using a path integral and is computed by the saddlepoint solution (instanton method) and a new nonlinear transformation of time. The predicted PDF p(x,t) in general involves a time integral, and useful PDFs with explicit dependence on x and t are presented in certain limits (e.g., in the short and long time limits). Numerical simulations of the FokkerPlanck equation provide exact time evolution of the PDFs and confirm analytical predictions in the limit of weak noise. In particular, we show that transient PDFs behave drastically differently from the stationary PDFs in regard to the asymmetry (skewness) and kurtosis. Specifically, while stationary PDFs are symmetric with the kurtosis smaller than 3, transient PDFs are skewed with the kurtosis larger than 3; transient PDFs are much broader than stationary PDFs. We elucidate the effect of nonlinear interaction on the strong fluctuations and intermittency in the relaxation process.
Original language  English 

Article number  052118 
Journal  Physical Review E 
Volume  94 
Issue number  5 
DOIs  
Publication status  Published  10 Nov 2016 
Externally published  Yes 
ASJC Scopus subject areas
 Statistical and Nonlinear Physics
 Statistics and Probability
 Condensed Matter Physics
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Eunjin Kim
Person: Teaching and Research