Structure of stochastic dynamics near fixed points

Abstract

We analyze the structure of stochastic dynamics near either a stable or unstable fixed point, where the force can be approximated by linearization. We find that a cost function that determines a Boltzmann-like stationary distribution can always be defined near it. Such a stationary distribution does not need to satisfy the usual detailed balance condition but might have instead a divergence-free probability current. In the linear case, the force can be split into two parts, one of which gives detailed balance with the diffusive motion, whereas the other induces cyclic motion on surfaces of constant cost function. By using the Jordan transformation for the force matrix, we find an explicit construction of the cost function. We discuss singularities of the transformation and their consequences for the stationary distribution. This Boltzmann-like distribution may be not unique, and nonlinear effects and boundary conditions may change the distribution and induce additional currents even in the neighborhood of a fixed point.Boltzmann distribution ͉ cost function ͉ detailed balance ͉ cyclic motion I n equilibrium statistical mechanics, the principle of detailed balance and the related fluctuation-dissipation theorem play important roles. Einstein used the principle that the excess energy that is put into each mode of an equilibrium system in the course of thermal fluctuations is also removed from the same mode by dissipative forces. This principle is implicit in his work on Brownian movement (1), and explicit in later works on the photoelectric effect (2), and on the relation between spontaneous and induced emission of electromagnetic radiation (3). It was formulated as the principle of detailed balance by Bridgman (4) and used to explain Johnson noise in electrical circuits by Nyquist (5). It is related to the fact that the same processes that drive fluctuations in the neighborhood of a typical equilibrium configuration also drive the configuration back towards a typical equilibrium or steady-state configuration when it is displaced from equilibrium by an amount that is small, but large compared with the fluctuations in thermal equilibrium. In this situation, the equilibrium distribution in phase space is just the Boltzmann distribution, proportional to e Ϫ␤E , where ␤ is inversely proportional to temperature, and E is the energy of the point in phase space. Configurations that differ significantly from those that contribute to the minimum of the free energy are driven back to the neighborhood of this minimum by dissipative effects such as thermal or electrical conduction or viscosity, and the magnitude of these effects is related to the equilibrium fluctuations of related variables.In many situations, there is no thermodynamic equilibrium, but external steady and fluctuating forces drive the system into a steady or very slowly varying state for which the principle of detailed balance does not hold. A light bulb powered by an external battery or a chemical reaction in which the reactants are introduced at a steady ra...

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