On the Theory of the Brownian Motion

Abstract

The Brownian motion of an oscillator in a thermostat is considered assuming simple forms of interaction between them. No further assumption is made except that the thermostat is always in thermal equilibrium in itself. Solving the Liouville equation or its counterpart in quantum mechanics the long time evolution of the system is clarified. Thus the theory connects automatically the irreversible and the equilibrium behaviors of the system without any ad hoc assumption as in conventional theories. The results include, as a special case, the equation derived by Kramers and Chandrasekhar using the theory of stochastic processes. It is shown that the oscillating part of the distribution function or the density matrix plays an important role to which the peculiar way of damping of the oscillator is to be attributed.