The relaxation time exact value of a dynamical
system
with noise, described by an arbitrary symmetric potential
profile
A.L. Pankratov
Radiophysical Dept.,
Nizhny Novgorod State University,
Nizhny Novgorod, Russia
The investigation of temporal scales of any relaxation process in a various polystable systems is a subject of great theoretical and practical importance in many physical aspects, e.g. phase transitions, chaotic systems, stochastic resonance, solid body physics, Josephson electronics, etc.
The pioneering work of this problem was carried out by Kramers [1], who used the Fokker-Planck equation to obtain several approximate expressions for the desired time characteristics. The main approach of the Kramers' method is the assumption that the probability current over a potential barrier is constant. This condition is valid only if a potential barrier is sufficiently high in comparison with noise intensity. For obtaining exact time characteristics it is nesessary to solve the Fokker-Planck equation, that is the main difficulty of the common problem to find the time scales of diffusion relaxation processes.
In recent years this problem has been reexamined by some authors (see e.g. [2-4]) but almost all analytical results obtained are approximate and again, only hold true in the limit of a large barrier height.
Our approach based on Laplace transform method gave us an opportunity to prove the new principle [5], which allows us to obtain exact relaxation time of the system described by an arbitrary symmetric potential profile. According to this principle (let us call it as the principle of correspondence), the relaxation time of the "symmetric system" may be written in quadrature simply from the dimensionless potential profile.
Using this principle, we have obtained the relaxation time of the system, described by the symmetric potential profile which is very important in the problem of stochastic resonance [6], i.e. by so-called quartic potential. This relaxation time holds true for any height of the potential barrier, i.e. for any noise intensity. The asymptotic representation of the relaxation time for a large potential barrier coincides with the approximate formula, obtained in paper [3] by Larson and Kostin.