A Theoretical Study of Brownian Motors with Self-Induced Temperature Gradients

Abstract

Brownian motors surmount energy barriers by absorbing and emitting heat to and from their local environment. In most theoretical treatments, the temperature gradients created in these heat exchanges are assumed to dissipate instantaneously. Here we relax this assumption to consider the case where Brownian dynamics can lead to self-induced temperature gradients in the local environment. We explore self-induced temperature gradients by considering an explicit equation of motion for the local temperature field. The equation of motion for the temperature contains a heat source term arising from the Brownian dynamics of the motor. In the same way that externally-imposed temperature gradients can cause directed motion, these self-induced gradients effect the Brownian dynamics of the system. The result is a two-way coupling between the local environment and the Brownian motor. We explore the resulting dynamics and thermodynamics of these novel coupled systems both in the steady state and in the dynamical state. We develop two numerical methods for solving the resulting equations of motion, both methods exploit the physical structure of our model. We check that our numerical methods converge to an analytical solution when available. We also check that the numerical methods converge as the discretization size is reduced. We consider self-induced temperature gradients on potentials that are relevant to Brownian motors. In particular, we show that self-induced temperature gradients can reduce barrier crossing rates in both one and two dimensions. We also implement heat engines and heat pumps based on temperature gradients induced by a Brownian motor on a non-equilibrium potential.