The methods of quantum field theory, originally designed to study quantum fluctuations, are also the tool for studying the thermal fluctuations of statistical physics, for example in connection with understanding critical phenomena. In fact, the methods and formalism of quantum fields are the universal language of fluctuations. In this chapter we shall capitalize on the universality of the methods of field theory as introduced in Chapters 9 and 10, and use them to study non-equilibrium phenomena in classical statistical physics where the fluctuations are those of a classical stochastic variable. We shall show that the developed non-equilibrium real-time formalism in the classical limit provides the theory of classical stochastic dynamics.

Newton's law, which governs the motion of the heavenly bodies, is not the law that seems to govern earthly ones. They sadly seem to lack inertia, get stuck and feebly ramble around according to Brownian dynamics as described by the Langevin equation. Their dynamics show transient effects, but if they are on short time scale too fast to observe, dissipative dynamics is typically specified by the equation v ∝ F where the proportionality constant could be called the friction coefficient. This is Aristotelian dynamics, average velocity proportional to force, believed to be correct before Galileo came along and did thorough experimentation. If a sponge is dropped from the tower of Pisa, it will almost instantly reach its saturation final velocity. If a heavier sponge is dropped simultaneously, it will fall faster reaching the ground first.