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Model without damping - Finite-time-singularity with noise and damping

In a recent paper [13] we investigated a simple generic model system with one degree of freedom governed by a nonlinear Langevin equation driven by Gaussian white noise,

The model is characterized by the coupling parameter , determining the amplitude of the singular term and the noise parameter D, determining the strength of the noise correlations. Specifically, in the case of a thermal environment at temperature T the noise strength D ยต T.

In the absence of noise this model exhibits a finite-time-singularity at a time t0, where the variable x vanishes with a square law dependence. When noise is added the finite-time-singularity event at t0 becomes a statistical event and is conveniently characterized by a first-passage-time distribution W(t) [9]. For vanishing noise we have W(t) = d(t - t0), restating the presence of the finite-time-singularity. In the presence of noise W(t) develops a peak about t = t0, vanishes at short times, and acquires a long time tail.

The model in Eq. (1) has also been studied in the context of persistence distributions related to the nonequilibrium critical dynamics of the two-dimensional XY model [10] and in the context of non-Gaussian Markov processes [11]. Finally, regularized for small x, the model enters in connection with an analysis of long-range correlated stationary processes [12].

From our analysis in ref. [13] it followed that the distribution at long times is given by the power law behavior

For vanishing nonlinearity, i.e., = 0, the finite-time-singularity is absent and the Langevin equation (1) describes a simple random walk of the reaction coordinate, yielding the well-known exponent a = 3/2 [14]. In the nonlinear case with a finite-time-singularity the exponent attains a non-universal correction, depending on the ratio of the nonlinear strength to the strength of the noise; for a thermal environment the correction is proportional to 1/T.

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