The Kramers problem: Fifty years of development

In the last fifty years the seminal work by Kramers of 1940 has been greatly extended both by elaboration on new theoretical approaches and through applications to new experimental systems. The most interesting case turns out to be the regime of weak-to-medium damping, in which case the Fokker-Planck equation can be reduced to an equation or to a system of integral equations of the Wiener-Hopf type. Exact solutions can then be given for the escape rate from single- and double-well potentials. This general scheme can be naturally extended to include quantum penetration through a semiclassical barrier and the effect of quantum noise. Finally, we consider the Brownian motion in titled washboard potential using Josephson junctions as an illustrative example. In that context we calculate (i) fluctuation-induced voltage-current characteristics; (ii) the lifetime of a zero-voltage state; (iii) the lifetime of the running state; (iv) partial probabilities of the phase jumps by 2π n ( n is an integer); and (v) retrapping current distribution in both the classical and quantum regimes.