Generalized Theory of Thermal Fluctuations

A general theory of thermal fluctuations, or Brownian motions, in a system in thermal equilibrium is developed on the basis of Gibbs's classical statistical mechanics. Taking advantage of the deterministic character of classical systems and applying fluctuation formula for a canonical ensemble, a formula for the cross-correlation function for two quantities u and X is given as [Formula: see text] where \((τ)\) is the after-effect function for u responding to a unitfunction change of x , while X is the force associated with the generalized coordinate x . Thus the formula indicates the identity of the statistical after-effect to the dynamical after-effect. It may be shown to comprise virtually all the known results pertaining to thermal fluctuation derivable from classical statistical mechanics. A generalization of Onsager's reciprocity relation to a system with after-effect is also obtained.