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arXiv:2512.09965 · Progress of Theoretical and Experimental Physics

Statistical Field Theory of Interacting Nambu Dynamics

Tamiaki Yoneya

Published Apr 2, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1093/ptep/ptag061

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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.
It has recently been shown, by application of statistical mechanical methods to determine the canonical ensemble governing the equilibrium distribution of operator initial values, that complex quantum field theory can emerge as a statistical approximation to an underlying generalized quantum dynamics. This result was obtained by an argument based on a Ward identity analogous to the equipartition theorem of classical statistical mechanics. We construct here a microcanonical ensemble which forms the basis of this canonical ensemble. This construction enables us to define the microcanonical entropy and free energy of the field configuration of the equilibrium distribution and to study the stability of the canonical ensemble. We also study the algebraic structure of the conserved generators from which the microcanonical and canonical ensembles are constructed, and the flows they induce on the phase space.
Abstract Nambu mechanics is a generalized Hamiltonian dynamics characterized by an extended phase space and multiple Hamiltonians. In a previous paper [Prog. Theor. Exp. Phys. 2013, 073A01 (2013)] we revealed that the Nambu mechanical structure is hidden in Hamiltonian dynamics, that is, the classical time evolution of variables including redundant degrees of freedom can be formulated as Nambu mechanics. In the present paper we show that the Nambu mechanical structure is also hidden in some quantum or semiclassical dynamics, that is, in some cases the quantum or semiclassical time evolution of expectation values of quantum mechanical operators, including composite operators, can be formulated as Nambu mechanics. We present a procedure to find hidden Nambu structures in quantum/semiclassical systems of one degree of freedom, and give two examples: the exact quantum dynamics of a harmonic oscillator, and semiclassical wave packet dynamics. Our formalism can be extended to many-degrees-of-freedom systems; however, there is a serious difficulty in this case due to interactions between degrees of freedom. To illustrate our formalism we present two sets of numerical results on semiclassical dynamics: from a one-dimensional metastable potential model and a simplified Henon–Heiles model of two interacting oscillators.
We explore some aspects of nonequilibrium statistical mechanics of classical and quantum systems. Two chapters are devoted to fluctuation theorems which were originally derived for classical systems. The main challenge in formulating them in quantum mechanics is the fact that fundamental quantities of interest, like work, are defined via the classical concept of a phase space trajectory. We utilize the decoherent histories conceptual framework, in which classical trajectories emerge in quantum mechanics as a result of coarse graining, and provide a first-principles analysis of the nonequilibrium work relation of Jarzynski and Crooks's fluctuation theorem for a quantum system interacting with a general environment based on the quantum Brownian motion (QBM) model. We indicate a parameter range at low temperatures where the theorems might fail in their original form. Fluctuation theorems of Jarzynski and Crooks for systems obeying classical Hamiltonian dynamics are derived under the assumption that the initial conditions are sampled from a canonical ensemble, even though the equilibrium state of an isolated system is typically associated with the microcanonical ensemble. We address this issue through an exact analysis of the classical Brownian motion model. We argue that a stronger form of ensemble equivalence than usually discussed in equilibrium statistical mechanics is required for these theorems to hold in the infinite environment limit irrespective of the ensemble used, and proceed to prove it for this model. An exact expression for the probability distribution of work is obtained for finite environments. Intuitively one expects a system to relax to an equilibrium state when brought into contact with a thermal environment. Yet it is important to have rigorous results that provide conditions for equilibration and characterize the equilibrium state. We consider the dynamics of open quantum systems using the Langevin and master equations and rigorously show that under fairly general conditions quantum systems interacting with a heat bath relax to the equilibrium state defined as the reduced thermal state of the system plus environment, even in the strong coupling regime. Our proof is valid to second-order in interaction strength for general systems and exact for the linear QBM model, for which we also show the equivalence of multi-time correlations. In the final chapter we give a sampling of our investigations into macroscopic quantum phenomena. We work out in detail a specific example of how and under what conditions the center of mass (CoM) coordinate of a macroscopic object emerges as the relevant degree of freedom. Interaction patterns are studied in terms of the couplings they induce between the CoM and relative coordinates of two macroscopic objects. We discuss the implications of these interaction patterns on macroscopic entanglement.
Quantum dynamics (i.e. the Schrödinger equation) and classical dynamics (i.e. Hamilton equations) can both be formulated in equal geometric terms: a Poisson bracket defined on a manifold. In this paper, we first show that the hybrid quantum-classical dynamics prescribed by the Ehrenfest equations can also be formulated within this general framework, what has been used in the literature to construct propagation schemes for Ehrenfest dynamics. Then, the existence of a well-defined Poisson bracket allows us to arrive to a Liouville equation for a statistical ensemble of Ehrenfest systems. The study of a generic toy model shows that the evolution produced by Ehrenfest dynamics is ergodic and therefore the only constants of motion are functions of the Hamiltonian. The emergence of the canonical ensemble characterized by the Boltzmann distribution follows after an appropriate application of the principle of equal a priori probabilities to this case. Once we know the canonical distribution of an Ehrenfest system, it is straightforward to extend the formalism of Nosé (invented to do constant temperature molecular dynamics by a non-stochastic method) to our Ehrenfest formalism. This work also provides the basis for extending stochastic methods to Ehrenfest dynamics.
Quantum dynamics (e.g., the Schrödinger equation) and classical dynamics (e.g., Hamilton equations) can both be formulated in equal geometric terms: a Poisson bracket defined on a manifold. The difference between both worlds is due to the presence of extra structure in the quantum case, that leads to the appearance of the probabilistic nature of the measurements and the indetermination and superposition principles. In this paper we first show that the quantum-classical dynamics prescribed by the Ehrenfest equations can also be formulated within this general framework, what has been used in the literature to construct propagation schemes for Ehrenfest dynamics. Then, the existence of a well defined Poisson bracket allows to arrive to a Liouville equation for a statistical ensemble of Ehrenfest systems. The study of a generic toy model shows that the evolution produced by Ehrenfest dynamics is ergodic and therefore the only constants of motion are functions of the Hamiltonian. The emergence of the canonical ensemble characterized by the Boltzmann's distribution follows after an appropriate application of the principle of equal a priori probabilities to this case. This work provides the basis for extending stochastic methods to Ehrenfest dynamics.
A theoretical framework for the nonequilibrium statistical mechanics of open systems is constructed. This is concerned with a formulation of a generalized master equation governing the evolution of an arbitrary system S in interaction with a ``large'' reservoir R. The dynamics of S are analyzed on the basis of a precise quantum-mechanical treatment of the microscopic equations of motion for the combined system S + R. On proceeding to the thermodynamical limit for R we obtain a generalized master equation for S, subject to specified conditions on the many-particle structure of R, its initial state, and its coupling to S. This master equation corresponds to a self-contained law of motion for S, in which the R variables appear only in the forms of certain thermal averages, taken over the initial state. This dynamical law is a generalization of the quantum-mechanical Liouville equation to a form appropriate to open systems.