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- Finite-time-singularity with noise and damping

In this paper we have reviewed the problem of the combined influence of white Gaussian noise of strength D and a linear damping of strength g on a finite-time-singularity of strength . We have for simplicity considered only a single degree of freedom. We find that the first-passage-time distribution W(t) displays a peak about the finite-time-singularity and at intermediate times shorter than 1/g a power law dependence µ t-a, characterized by the scaling exponent a = 3/2 + /2D. The exponent is nonuniversal and depends on the ratio between the singularity strength and the noise strength D. In the case where the noise originates from a thermal environment at temperature T we have D µ T and the scaling exponent depends on the temperature, a = 3/2 + const./T. At long times later than 1/g the behavior of W(t) crosses over to a an exponential fall-off. To the extent that the character of a finite-time-singularity in the vicinity of threshold can be modelled by a single degree of freedom the present study should hold as regard the influence of noise on the time distribution. We note in particular that in the case of a thermal environment at temperature T the change of the scaling exponent becomes large in the limit of low temperatures as the distribution narrows around the noiseless threshold time. The present study also suggests generalizations to the case of several coupled variable subject to a finite-time-singularity.

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