Approach to equilibrium of statistical systems: classical particles and quantum fields off-equilibrium

Abstract

Non-equilibrium evolution at absolute temperature T and approach to equilibrium of statistical systems in long-time (t) approximations, using both hierarchies and functional integrals, are reviewed. A classical non-relativistic particle in one spatial dimension, subject to a potential and a heat bath (ℎ𝑏), is described by the non-equilibrium reversible Liouville distribution (W) and equation, with a suitable initial condition. The Boltzmann equilibrium distribution 𝑊_(𝑒𝑞) generates orthogonal (Hermite) polynomials 𝐻_(𝑛) in momenta. Suitable moments 𝑊_(𝑛) of W (using the 𝐻_(𝑛)’s) yield a non-equilibrium three-term hierarchy (different from the standard Bogoliubov–Born–Green–Kirkwood–Yvon one), solved through operator continued fractions. After a long-t approximation, the 𝑊_(𝑛)’s yield irreversibly approach to equilibrium. The approach is extended (without ℎ𝑏) to: (i) a non-equilibrium system of N classical non-relativistic particles interacting through repulsive short range potentials and (ii) a classical 𝜙^(4) field theory (without ℎ𝑏). The extension to one non-relativistic quantum particle (with ℎ𝑏) employs the non-equilibrium Wigner function (𝑊_(𝑄)): difficulties related to non-positivity of 𝑊_(𝑄) are bypassed so as to formulate approximately approach to equilibrium. A non-equilibrium quantum anharmonic oscillator is analyzed differently, through functional integral methods. The latter allows an extension to relativistic quantum 𝜙^(4) field theory (a meson gas off-equilibrium, without ℎ𝑏), facing ultraviolet divergences and renormalization. Genuine simplifications of quantum 𝜙^(4) theory at high T and large distances and long t occur; then, through a new argument for the field-theoretic case, the theory can be approximated by a classical 𝜙^(4) one, yielding an approach to equilibrium.