Dissipative Gradient Dynamics: A Macroscopic Postulate and Its Mathematical Consequences

Abstract

Dissipative processes are usually described through individual entities exchanging energy or matter. A relational framework is developed whose fundamental quantities are differences between modes rather than the modes themselves. In an isolated system, under appropriate conditions, the total irreversible difference is non-increasing and decays to zero. Differences, dissipative forces, and the passage of irreversible time are shown to be equivalent manifestations of one underlying process, vanishing together at the terminal anchor point. The evolution is governed by a gradient flow of a dissipative potential acting as a Lyapunov functional. To isolate many-body coupling, the dynamics is decomposed exactly into two-body terms and a collective residual. In the pure reaction limit, this residual is controlled by a graph Hodge operator with explicitly determined spectrum. The resulting spectral decomposition gives an algebraic criterion distinguishing whether many-body effects assist or resist dissipation, and explains local reversals within global irreversibility. The reversal is proved analytically in the two-mode case and observed numerically for three modes. For the full model with spatial diffusion, weak solutions exist, are unique, and converge asymptotically. The disturbance function also exhibits macroscopic uncertainty: knowing all local derivatives of a single mode pair does not determine its value. The attainable set is the whole real line, but becomes bounded under a finite high-order energy constraint. This structural uncertainty arises from nonlocal many-body coupling rather than from noise.