Incoherent Fermi-Pasta-Ulam Recurrences and Unconstrained Thermalization Mediated by Strong Phase Correlations

Abstract

The long-standing and controversial Fermi-Pasta-Ulam problem addresses fundamental issues of statistical physics, and the attempt to resolve the mystery of the recurrences has led to many great discoveries, such as chaos, integrable systems, and soliton theory. From a general perspective, the recurrence is commonly considered as a coherent phase-sensitive effect that originates in the property of integrability of the system. In contrast to this interpretation, we show that convection among a pair of waves is responsible for a new recurrence phenomenon that takes place for strongly incoherent waves far from integrability. We explain the incoherent recurrence by developing a nonequilibrium spatiotemporal kinetic formulation that accounts for the existence of phase correlations among incoherent waves. The theory reveals that the recurrence originates in a novel form of modulational instability, which shows that strongly correlated fluctuations are spontaneously created among the random waves. Contrary to conventional incoherent modulational instabilities, we find that Landau damping can be completely suppressed, which unexpectedly removes the threshold of the instability. Consequently, the recurrence can take place for strongly incoherent waves and is thus characterized by a reduction of nonequilibrium entropy that violates the H theorem of entropy growth. In its long-term evolution, the system enters a secondary turbulent regime characterized by an irreversible process of relaxation to equilibrium. At variance with the expected thermalization described by standard Gibbsian statistical mechanics, our thermalization process is not dictated by the usual constraints of energy and momentum conservation: The inverse temperatures associated with energy and momentum are zero. This unveils a previously unrecognized scenario of unconstrained thermalization, which is relevant to a variety of weakly dispersive wave systems. Our work should stimulate the development of new experiments aimed at observing recurrence behaviors with random waves. From a broader perspective, the spatiotemporal kinetic formulation we develop here paves the way to the study of novel forms of global incoherent collective behaviors in wave turbulence, such as the formation of incoherent breather structures.