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Model with damping
- Finite-time-singularity with noise and damping

In our studies so far we have ignored damping. It is, however, clear that in realistic physical situations friction or damping must enter on the same footing as the noise. This follows from the Einstein relation or more generally from the fluctuation-dissipation theorem relating the damping to the noise. In the present paper we attempt to amend this situation and thus summarize the results of an extension of the analysis in ref. [13] to the case of linear damping.

For this purpose we consider the following model for one degree of freedom:

In addition to the coupling parameter , and the noise parameter D, this model is also characterized by the damping constant g. Assuming for convenience a dimensionless variable x the coupling, noise strength, and damping, , D, and g have the dimension 1/time. The ratios /D and g/D are thus dimensionless parameters characterizing the behavior of the system. In terms of a free energy or potential F we can express Eq. (3) in the form

where F is given by

The free energy has a logarithmic sink and drives x to the absorbing state x = 0. For large x the free energy has the form of a harmonic well potential confining the motion. In Fig. 1 we have depicted the noiseless solution for h = 0 and the free energy in the various cases.  

In order to model an experimental situation the first-passage-time distribution W(t) is of direct interest. First-passage properties in fact underlie a large class of stochastic processes such as diffusion limited growth, neuron dynamics, self-organized criticality, and stochastic resonance [9].

In term of the distribution function P(x, t) the absorbing state distribution W(t) is defined as W(t) = . In the absence of noise P(x, t) = d(x - x(t)) and W(t) = d(t - t0), in accordance with the finite time singularity at t = t0. For weak noise we anticipate that W(t) will peak about t0 with vanishing tails for small t and large t.

In an analysis to be detailed elsewhere we have solved the Fokker-Planck equation associated with the Langevin equation (3),

analytically and have found for the probability distribution P(x, t)

and correspondingly for the first-passage-time distribution W(t)

In Eq. (7) In is the Bessel function of imaginary argument, In(z) = (-i)nJn(iz) [15] and we have introduced the time scaled variables

From an analysis of Eq. (8) it follows that the damping constant sets an inverse time scale 1/g. At intermediate time scales for gt 1 the distribution exhibits the same power law behavior as in the undamped case given by Eq. (2). At long times for gt >> the distribution falls off exponentially with time constant 1/g(1 + /D), i.e.,

In the short time limit W(t) vanishes exponentially and shows a maximum about the finite-time-singularity. In Fig. 2 we have depicted the first-passage-time-distribution as a function of t. In Fig. 3 we illustrate the behavior of W(t) in a log-log representation.

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