Quasiequilibrium directed hopping in a time-dependent two-well periodic potential

We consider the directed motion of a Brownian particle in a two-well periodic potential with time-varying barriers and wells described by arbitrary periodic functions of time, v(t) and u(t), alternating with the period τ. In the framework of the low-temperature kinetic approach, we obtain explicit formulas for the probabilities of finding the particle in potential wells, average velocity of directed motion, input energy P(in) and useful work P(out) against additionally introduced stationary load force f. These formulas are considerably simplified by the assumption of the quasiequilibrium regime of motion corresponding to small values of u(t) and f. It is shown that depending on the same or opposite parity of the functions v(t) and u(t) with respect to time reversal, the motion direction of a Brownian particle is retained or reversed under the reversal of the direction of movement along the (v-u) loop in the phase space of the functions v(t) and u(t), and the nondiagonal kinetic coefficients are mutually symmetric or antisymmetric. In the adiabatic limit τ→∞, the average velocity is proportional to τ(-1) in two cases: (i) the above loop has a nonzero area, (ii) the functions v(t) and u(t) are proportional to each other (zero loop area) and include intervals of fast changes with small durations τ(0) on the period τ of their variations. In both of these cases, the efficiency of energy conversion, η=P(out)/P(in), tends to unity at large variations of the barriers v(t). In the second case, the deviation of η from unity can be split into two contributions: The former decreases exponentially with increasing amplitude v(0) of v(t), while the latter is a small nonadiabatic correction proportional to v(0)(-3/2). It is the nonadiabatic correction that limits high efficiencies at large variations of barriers.