Macrostates of classical stochastic systems

Abstract

The thermodynamic and dynamic properties of a stochastic system can be determined from the underlying microscopic description once appropriate macroscopic states (‘‘macrostates’’) have been identified. Macrostates correspond to temperature-dependent regions of conformation space that are effectively isolated by potential energy barriers. However, there is no rigorous procedure for defining them and they are generally specified by ad hoc temperature-independent prescriptions. This is inadequate for complicated multidimensional systems like proteins. Here we provide a rigorous definition of macrostates of diffusive stochastic systems by relating the eigenfunction expansion of the Smoluchowski equation to a macrostate expansion via a ‘‘Minimum Uncertainty Condition.’’ We develop a general computational bootstrap procedure for identifying macrostates in multiple dimensions and computing their thermal and dynamic properties. This employs nonlinear ‘‘characteristic packet equations’’ to identify anisotropic Gaussian packets that provide a coarse-grained representation of the equilibrium probability distribution. These provide starting points for a variational method for calculating transition rates between macrostates and for a perturbative method for describing relaxations within macrostates.