Time Evolution of a Modified Feynman Ratchet with Velocity-Dependent Fluctuations and the Second Law of Thermodynamics

It is shown that the randomness of Brownian motion at thermodynamic equilibrium can be spontaneously broken by velocity-dependence of fluctuations, i.e., by dependence of val- ues or probability distributions of fluctuating properties on Brownian-motional velocity. Such randomness-breaking can spontaneously obtain via interaction between Brownian-motional Doppler effects — which manifest the required velocity-dependence — and system geometrical asymmetry. A nonrandom walk is thereby spontaneously superposed on Brownian motion, resulting in a systematic net drift velocity despite thermodynamic equi- librium. (By contrast, the mere existence of fluctuations, without velocity-dependence thereof, cannot, in general, effect such randomness-breaking.) The time evolution of this systematic net drift velocity — implying acceleration — is derived for the velocity-dependent modification of Feynman's ratchet developed in this paper. On this basis, we derive the time evolution of the velocity probability density, of the accelerating force, and of power output = (accelerating force) × (systematic net drift velocity). Quantitative results are obtained for all times, includ- ing final steady-state values. We show that said (a) spontaneous randomness-breaking, and (b) consequent systematic net drift velocity, imply: (c) bias from the Maxwellian of the system's velocity probability density, (d) the force that tends to accelerate it, and (e) its power output. Maximization of (a) through (e) above, especially of (e) power output, is discussed. Uncom- pensated decreases in total entropy, challenging the second law of thermodynamics, are implied by (a) through (e) above.