Statistical Field Theory of Interacting Nambu Dynamics

Abstract We develop a statistical field theory for classical Nambu dynamics by partially employing the quantum field-theory method. One of the unsolved problems in Nambu dynamics has been to extend it to interacting systems without violating a generalized canonical structure characterized by the presence of multiple Hamiltonians, which together govern the dynamics of time evolution on an equal footing through a Nambu bracket as an analog to the Poisson bracket of ordinary Hamiltonian dynamics with a single Hamiltonian. In the present paper, we propose to include interactions from the standpoint of classical statistical dynamics by formulating it as a field theory on Nambu’s generalized phase space in an operator formalism. We first construct a general framework for such a field theory and its probabilistic interpretation. Then, on the basis of this framework, we give an exactly solvable model of self-interactions in a many-body Nambu system treated as a closed dynamical system satisfying the H-theorem. It is shown that a generalized microcanonical ensemble and a generalized canonical ensemble characterized by many temperatures are reached dynamically as equilibrium states starting with certain classes of initial nonequilibrium states via a continuous Markov process, which lies beyond the usual framework based on the Liouville theorem. Compared with the usual classical statistical mechanics on the basis of standard Hamiltonian dynamics, some important new features associated with Nambu dynamics will emerge, with respect to the symmetries underlying the dynamics of the nonequilibrium as well as the equilibrium states and also to some conceptual properties, such as a formulation of a generalized Kubo–Martin–Schwinger-like condition characterizing the generalized canonical equilibrium states and the “relative” nature of the temperatures.