Order-Parameter Field-Theoretic Foundation of the Maxwell Distribution — Information Dissipation, Fluctuation-Dissipation Theorem, and the Necessity of Statistical Equilibrium

Abstract

The central result of statistical mechanics — the Maxwell velocity distribution — is derived in standard textbooks from the maximum entropy principle or the molecular chaos hypothesis. These derivations take a statistical postulate as the logical starting point and fail to answer why microscopically reversible mechanical equations lead to macroscopically irreversible statistical behavior. Starting from the Order Parameter Spacetime Theory, this paper argues that the Maxwell distribution is not a statistical postulate but an inevitable consequence of the combined action of information dissipation and the fluctuation-dissipation theorem. The argument proceeds in three steps. First, in the classical sector (Υ > 0), the motion of an order-parameter soliton is described by a Langevin equation; the dissipative force is determined by the information dissipation tensor Υ_μν, and the fluctuating force is rigorously given by the fluctuation-dissipation theorem. Second, the corresponding Fokker-Planck equation possesses a unique stationary solution — the Maxwell-Boltzmann distribution. Its universality is guaranteed by the algebraic structure of the fluctuation-dissipation theorem, independent of the specific values of the relaxation time or spectral density. Third, the conditions for deviations are precisely given, defining new statistical paradigms beyond standard statistical mechanics — including power-law tails in non-Ohmic baths, quantum statistical corrections at low temperatures, and a tiny skewness induced by Machian forces. In particular, the stationary velocity distribution maintains zero skewness under all conditions, constituting a testable exclusive prediction. Together with the information bottleneck theorem, this paper establishes the complete picture of the quantum-classical transition: quantum coherence is preserved when Υ = 0, and the system irreversibly tends toward the Maxwell distribution when Υ > 0.