Time-irreversibility in the classical many-body dynamics

Presented by Gyula Toth - Interdisciplinary Centre for Mathematical Modelling, Department of Mathematical Sciences, Loughborough University

Abstract: One of the unsolved fundamental problems in physics is the origin of the thermodynamic arrow of time provided by the second law of thermodynamics. In essence, while the solutions of the governing equations of matter operating on microscopic scales are time reversible, macroscopic-scale order is known to solely decay in spontaneous spatiotemporal processes. In 1876, Loschmidt argued that time reversibility of the solution is a property of a mathematical model which is independent of the number of degrees of freedom. Consequently, time irreversibility should not emerge in models providing time reversible solutions. This is called the Loschmidt’s paradox.

Loschmidt’s paradox has been puzzling physicists for 145 years. Despite the variety of approaches (ranging from the “illusion” of macroscopic irreversibility to the special initial conditions of the early Universe and the incompleteness of fundamental microscopic theories), the ultimate resolution of the problem is yet unknown. The main problem is that statistical physics, the only known mathematical bridge between microscopic and macroscopic models of matter, is flexible enough to bear such a contradiction on the practical level. In particular, irreversibility in statistical physics is manually added during the derivation of equations addressing macroscopic scales, and therefore the exact microscopic origin of elementary irreversible physical processes, the diffusion of mass and momentum, remains hidden.

In this talk I present exact continuum equations to the Hamiltonian many-body dynamics of pair interacting particles in the limit of infinitely many particles. The derivation relies on a mathematical transformation lacking the utilisation of statistical mechanics or other approximations. The only assumption made here is that the sum of infinitely many, infinitely small amplitude Dirac-delta distributions located infinitely closely to each other is a bounded function. It is shown that the emerging scale-free equations are time reversible and universal for a certain class of interaction potentials. The existence of thermodynamic equilibrium and non-equilibrium relaxation processes are studied in numerical simulations for smooth random field initial conditions. It is shown that these seemingly time irreversible processes are not diffusional, providing evidence for the lack of the second law of thermodynamics in the classical many-body problem. Further directions of the research will also be discussed.