Exact relaxation times and metastable state decay times for
Brownian diffusion in arbitrary potential profiles
A.N. Malakhov
Radiophysical Dept., Nizhny Novgorod State University,
Nizhny Novgorod, Russia
One-dimensional chaotic motion of the Brownian particles in the overdumped
regime (Brownian diffusion) in potential profiles is considered. Exact
relaxation times and metastable state decay times are obtained. The
potential profiles 
may be arbitrary except its behaviour at 
. The initial probability distribution of the particles
density is 
.
Three types of potential profiles are investigated. The first: 
at ; the second: 
at ; the third: at 
and at 
(or at and at ).
For the first type of a potential profile there is the equilibrium
probability distribution 
where 
is a dimensionless potential profile, A;SPMgt;0 is the
normalization factor. The relaxation time is a characteristic time of a
variation probability distribution from the initial probability
distribution to a final one 
.
For the second and third types of potential profiles the final probability
distribution is zero: 
.
Let in potential profiles exist local minima (metastable states) where the
initial probability distribution is placed. The total probability near a
local minimum is varied from unity to final. A characteristic time of this
variation is the metastable state decay time. Besides that it may be very
interesting to know the life time of an unstable state, when the potential
profile at the initial point 
does not have a local minimum.
By the new method of attack it has been demonstrated that all exact characteristic times in question are represented in terms of quadrature of the dimensionless potential profile, as well as it was found by L.A.Pontryagin et. al. for the mean first passage time.
The results following from common obtained formulae are shown to coinside with the time characteristics derived for different particular cases of the potential profiles, which is known from the literature (see e.g. M.Mörsch, H.Risken, V.Vollmer, Z. Phys. B32, 245 (1979), H.Frisch, V.Privman, C.Nicolis, G.Nicolis, J.Phys. A23, L1147 (1990), V.Privman, H.Frisch, J.Chem.Phys. 94, 8216 (1991), N.V. Agudov, A.N.Malakhov, Radiophys. and Quant. Electr. 36, 97 (1993), etc.).