Towards a General Theory of Molecular-Scale Energy Transfer

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Specialized proteins inside cells work on a molecular scale to convert chemical energy into mechanical motion. Brownian motion on a multi-dimensional tilted periodic free-energy potential is a possible unifying theory for the description of far-from-equilibrium molecular-scale energy transfer. In this framework each degree of freedom is a generalized coordinate representing displacements in real space or along reaction coordinates. The minima of the periodic potential define microscopic (meta-)stable states and the potential barriers between minima determine the probability of hopping transitions. The linear potential represents a constant macroscopic thermodynamic force driving the system out of thermal equilibrium. This framework provides a compelling physical picture of a molecular-scale system undergoing Brownian motion on a potential landscape that directs the average behavior of the system thereby enabling energy coupling between degrees of freedom.In this work we derive analytic solutions to the continuous stochastic diffusion equation describing overdamped Brownian motion on a multi-dimensional tilted periodic potential in the limit of deep potential wells. using an approach similar to the tight-binding model of quantum mechanics, we derive a master equation describing discrete hopping between localized (meta-)stable states of the untilted periodic potential. This discrete master equation represents a significant simplification of the system valid in a well-defined parameter regime and can be solved analytically. For non-separable potentials the master equation describes energy transfer between degrees of freedom. We derive expressions for the dependence of the rate and efficiency of energy transfer on the tilt valid beyond the near-equilibrium limit. These results provide an opportunity to connect with experiments, phenomenological models, and other general results from non-equilibrium thermodynamics.