Optimal modulation of a Brownian ratchet and enhanced sensitivity to a weak external force

Optimal modulation of a Brownian ratchet and enhanced sensitivity to a weak external force

Martin B. Tarlie* and
R. Dean Astumian†,‡

^*James Franck Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637; and ^†Departments of Surgery and of Biochemistry, University of Chicago, 5841 South Maryland Avenue, Chicago, IL 60637

Communicated by Robert K. Adair, Yale University, New Haven, CT (received for review June 15, 1997)

Abstract

We studied the dynamics of a Brownian particle moving in a spatially anisotropic potential acted on by multiplicative temporal modulations so that V(x,t) = g(t)U(x). Using the concept of the “thermodynamic action,” we show that the class of modulation that maximizes the flow is a square-wave in time. We also show that adding a weak, homogenous force F in synergy with the square-wave modulation can cause particles of slightly different size to move in opposite directions. The synergetic change in velocity caused by F can be much greater than the drift velocity that would be caused by F alone.

Footnotes

↵ ‡ To whom reprint requests should be addressed.
↵ § These approximate inequalities become exact in the limit that T → ∞.
↵ ¶ For g(t) = 1, S _c ⁺ and S _c ⁻ are equal and are monotonically decreasing functions of t_f − t_i that approach ΔU as t_f − t_i → ∞.
↵ ‖ In general, we could have taken g to switch between one positive and one negative constant. This alters t _a and t _b but does not affect the basic results.
↵ ** Although we were able to obtain an approximate analytic formula for v(τ₋,τ₊) in the entire τ₋ − τ₊ plane, for clarity of presentation we have concentrated on the regime where n ⁺ and n ⁻ are approximately unity.