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Recently a technique has been introduced to -modify a Hamiltonian so that the -modified Hamiltonian thereby produced is isochronous: all its solutions are periodic in all degrees of freedom with the same period ˜ T = 2/ . In this paper—after briefly reviewing this approach— we focus in particular on the -modified version of the most general realistic many-body problem whose behavior, over time intervals much shorter than the isochrony period ˜ T , differs only marginally from the thermodynamically irreversible evolution of the corresponding, unmodified and realistic many-body system. We discuss the (apparently paradoxical) periodic recurrence of the irreversible processes occurring in this -modified model, implying a periodic reversal of its irreversible behavior. We then discuss the equilibrium statistical mechanics of this -modified model, including the compatibility of standard thermodynamic notions such as entropy with the peculiar phenomenology featured by its time evolution. The theoretical discussion is complemented by numerically simulated examples of the molecular dynamics yielded by the (standard and classical) Hamiltonian describing (many) particles interacting pairwise via potentials of Lennard-Jones type and via harmonic potentials in two-dimensional space, and by its -modified version. In the latter case, the simulation displays (approximate) returns to configurations away from thermodynamic equilibrium after relaxation to equilibrium had occurred.

Spontaneous reversal of irreversible processes in a many-body Hamiltonian evolution