Derivation of Langevin equation from microscopic theory

Langevin equation for the macrovariable is derived at the level of the generating functional starting from the field theoretical microscopic Hamiltonian. The macrovariable is precisely defined first, and then the generating functional is written by the path integral over the macrovariable. For the thermodynamically normal system, the path integral is dominated by the stationary phase in the macroscopic limit and the expression which is obtained by keeping the quadratic term in the fluctuation is shown to coincide with the Langevin theory. Technique of evaluating higher-order corrections is also presented.