Integral relaxation time

The aim of this chapter is to describe approaches of obtaining exact time characteristics of diffusion stochastic processes (Markov processes) that are in fact a generalization of FPT approach and are based on the definition of characteristic timescale of evolution of an observable as integral relaxation time [5,6,30—41]. These approaches allow us to express the required timescales and to obtain almost exactly the evolution of probability and averages of stochastic processes in really wide range of parameters. We will not present the comparison of these methods because all of them lead to the same result due to the utilization of the same basic definition of the characteristic timescales, but we will describe these approaches in detail and outline their advantages in comparison with the FPT approach. 

It should be noted that besides being widely used in the literature definition of characteristic timescale as integral relaxation time, recently intrawell relaxation time has been proposed [42] that represents some effective averaging of the MFPT over steady-state probability distribution and therefore gives the slowest timescale of a transition to a steady state, but a description of this approach is not within the scope of the present review. 

In the forthcoming sections we will consider several methods that have been used to derive different integral relaxation times for cases where both drift and diffusion coefficients do not depend on time, ranging from the considered mean transition time and to correlation times and time scales of evolution of different averages. 

In this section we consider the notion of an effective eigenvalue and the approach for calculation of correlation time by Risken and Jung [2,31], A similar approach has been used for the calculation of integral relaxation time of magnetization by Garanin et al. [5,6]. 

A quite different approach from all other presented in this review has been recently proposed by Coffey [41]. This approach allows both the MFPT and the integral relaxation time to be exactly calculated irrespective of the number of degrees of freedom from the differential recurrence relations generated by the Floquet representation of the FPE. 

Let us define the characteristic scale of time evolution of the average m.f(t) as an integral relaxation time ... 

In the frame of the present review, we discussed different approaches for description of an overdamped Brownian motion based on the notion of integral relaxation time. As we have demonstrated, these approaches allow one to analytically derive exact time characteristics of one-dimensional Brownian diffusion for the case of time constant drift and diffusion coefficients in arbitrary potentials and for arbitrary noise intensity. The advantage of the use of integral relaxation times is that on one hand they may be calculated for a wide variety of desirable characteristics, such as transition probabilities, correlation functions, and different averages, and, on the other hand, they are naturally accessible from experiments. 

The decrements A/ or, equivalently, relaxation times r/, which are characteristic of the eigenfunctions of the distribution function, are not observable if taken as separate quantities. However, in combination they are involved in a useful, directly measurable quantity, the so-called integral relaxation time. In terms of correlation functions this characteristic is defined as... 

In summary, in this section (Section III.B) we give a consistent procedure yielding the integral relaxation time and initial nonlinear susceptibilities for an... 

Figure 4.20. Integral relaxation time as a function of the dimensionless temperature 1 /a and bias field strength 8.

Garanin DA (1996) Integral relaxation time of single-domain ferromagnetic particles. Phys Rev E 54 3250-3256... 

Garanin DA (1997) Quantum thermoactivation of nanoscale magnets. Phys Rev E 55 2569-2572 Garanin DA (1999) New integral relaxation time for thermal activation of magnetic particles. EuroPhys Lett 48 486-490... 

For normal diffusion, a = 1, these parameters correspond to the integral relaxation time xint [the area under the corresponding relaxation function... 

Equation (159), which involves the integral relaxation time x(nf, the effective relaxation time xef, and the smallest nonvanishing eigenvalue /., correctly predicts /Jot) both at low (co —> 0) and high (co ocj frequencies. Moreover, for a particular form of the potential V, x( ) ma> be determined in the entire frequency range 0 < co < oo as we shall presently see for a double-well periodic potential representing the internal field due to neighboring molecules. 

In order that a meaningful accuracy can be obtained at all, it is mandatory that is so large that AA t) has already decayed to zero for t [Pg.466]

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