Directed diffusion and logarithmic susceptibility of the activated escape problem

The logarithmic susceptibility (LS) theory of the activated escape predicts resonant behavior and changes in the direction of the current with the change in the frequency of the driving field. According to the LS theory, such logarithmic susceptibilities (LS) χ± are given by the Fourier transform of the velocity along the MPEP (most probable escape path) in the absence of the driving field. According to the LS theory, the activation energy Ea is linear in the field amplitude. The driving field resonantly decreases the activation barrier for a periodic potential, which corresponds to resonant peaks in the logarithmic susceptibilities. We report numerical results for the underdamped case for a broad range of frequencies and amplitudes of the driving force. The field-induced changes in the activation energies are found to depend linearly on the field amplitude for moderate fields. We obtained the activation energy Ea from lnW, where W is the average escape rate, obtained from the transition times. The proportionality coefficient of the variation of Eawith the field amplitude defines the logarithmic susceptibility (LS). We studied the dependence of the pre-factor in the activation law on the noise intensity and the field amplitude, and we numerically demonstrated the occurrence of resonant peaks in the logarithmic susceptibility, as predicted by the LS theory, for some ranges of the field spectrum. We also present numerical demonstration of the directed diffusion (DD) to the right or to the left from the minima of a symmetric periodic potential when the driving has both even and odd harmonics. Directed diffusion is a unidirectional motion of the systems far from equilibrium due to their fluctuations in a symmetric or asymmetric periodic potentials. The rate of directed diffusion is determined by the difference between the transition probabilities to the right and to the left. The numerical results show that the current reverses direction to that for which the activation energy is smaller, as expected by the LS theory.