Influence of Gaussian fluctuations on a model kinetic system exhibiting explosive behavior

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Abstract
We examine the influence of fluctuations on a model kinetic equation which exhibits explosive behavior. The particular model kinetic equation ẋ=kx2−αx has fixed points at x=0 and x=x1=(α/k). For constant coefficients k and α, the trajectory x(t) diverges in a finite time if the initial value x0≳x1, and x(t) approaches zero if x0<x1; fluctuations in the coefficients may change this deterministic behavior. We consider three different models incorporating Gaussian fluctuations. In case A, the rate coefficient k(t) has a fluctuating part; in this case it is possible to determine exactly the entire probability distribution p(x,t‖x0) as well as N(t) the probability that the system has not exploded (for a given x0) at time t. In case B, the rate coefficients k(t) and α(t) contain identical fluctuating parts; we show that this type of fluctuating behavior has no influence on the transition between stable and unstable behavior. In case C, the rate coefficient α(t) contains a fluctuating part. An approximation solution leads to an effective kinetic equation which is identical to the deterministic equation except that α is replaced by αeff=α(1−αq), where q is a measure of the strength of the fluctuation in α(t). This model kinetic equation is helpful for illustrating how fluctuations may influence kinetic systems that contain explosive character, including the problem of passage over a barrier in the overdamped limit.

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