The quartic oscillator in an external field and the statistical physics of highly anisotropic solids

The statistical mechanics of one- and two-dimensional Ginzburg–Landau systems is evaluated analytically, via the transfer matrix method, using an expression of the ground state energy of the quartic anharmonic oscillator in an external field. In the two-dimensional case, the critical temperature of the order–disorder phase transition is expressed as a Lambert function of the inverse inter-chain coupling constant.     

Elimination of Fast Chaotic Degrees of Freedom: On the Accuracy of the Born Approximation

We apply standard projection operator techniques known from nonequilibrium statistical mechanics to eliminate fast chaotic degrees of freedom in a low-dimensional dynamical system. Through the usual perturbative approach we end up in second order with a stochastic system where the fast chaotic degrees of freedom are modelled by Gaussian white noise. The accuracy of the perturbation expansion is analysed in detail by the discussion of an exactly solvable model.     

Ergodic Properties and Thermodynamics in Anharmonic Oscillator Systems

Ergodic properties of anharmonic oscillator systems with only a few degrees of freedom are both theoretically and numerically investigated. Statistical mechanics and thermodynamics of ergodic Hamiltonian systems are built. The Henon-Heiles systems with reflecting walls and the diamagnetic Kepler problem are studied     

Weyl-Wigner-Moyal formalism for Fermi classical systems

The Weyl-Wigner-Moyal formalism of fermionic classical systems with a finite number of degrees of freedom is considered. The Weyl correspondence is studied by computing the relevant Stratonovich-Weyl quantizer. The Moyal -product, Wigner functions and normal ordering are obtained for generic fermionic systems. Finally, this formalism is used to perform the deformation quantization of the Fermi oscillator and the supersymmetric quantum mechanics.     

Relational Evolution of the Degrees of Freedom of Generally Covariant Quantum Theories

We study the classical and quantum dynamics of generally covariant theories with vanishing Hamiltonian and with a finite number of degrees of freedom. In particular, the geometric meaning of the full solution of the relational evolution of the degrees of freedom is displayed, which means the determination of the total number of evolving constants of motion required. Also a method to find evolving constants is proposed. The generalized Heisenberg picture needs M time variables, as opposed to the Heisenberg picture of standard quantum mechanics where one time variable t is enough. As an application, we study the parametrized harmonic oscillator and the SL(2, R) model with one physical degree of freedom that mimics the constraint structure of general relativity where a Schrödinger equation emerges in its quantum dynamics.     

Classical dissipation and asymptotic equilibrium via interaction with chaotic systems

We study the energy flow between a one-dimensional oscillator and a chaotic system with two degrees of freedom in the weak coupling limit. The oscillator's observables are averaged over an initially microcanonical ensemble of trajectories of the chaotic system, which plays the role of an environment for the oscillator. We show numerically that the oscillator's average energy exhibits irreversible dynamics and ‘thermal’ equilibrium at long times. We use linear response theory to describe the dynamics at short times and we derive a condition for the absorption or dissipation of energy by the oscillator from the chaotic system. The equilibrium properties at long times, including the average equilibrium energies and the energy distributions, are explained with the help of statistical arguments. We also check that the concept of temperature defined in terms of the ‘volume entropy’ agrees very well with these energy distributions.     

A new kind of representations on noncommutative phase space

We introduce new representations to formulate quantum mechanics on noncommutative phase space, in which both coordinate-coordinate and momentum-momentum are noncommutative. These representations explicitly display entanglement properties between degrees of freedom of different coordinate and momentum components. To show their potential applications, we derive explicit expressions of Wigner function and Wigner operator in the new representations, as well as solve exactly a two-dimensional harmonic oscillator on the noncommutative phase plane with both kinetic coupling and elastic coupling.     

On the Foundations of Statistical Mechanics：Ergodicity,Many Degrees of Freedom and Inference

We review the main aspects of the foundations of statistical mechanics. In particular we explain why many degrees of freedom are necessary, while chaos (in the sense of positive Lyapunov exponents) is only marginally relevant, for the emergence of statistical laws in macroscopic systems.     

Quantum Nonlinear Oscillator with Two Degrees of Freedom in a Laser Field

The semiclassical dynamics of a quantum nonlinear oscillator with two degrees of freedom and anharmonicity of the fourth order in a periodic laser field is studied both analytically and numerically. In the absence of external excitation and dissipation, the equations of motion for the mean values of the coordinate and momentum operators of both degrees of freedom reduce to the equation of a onedimensional nonlinear pendulum. The general solution of this equation is written in terms of the Jacobian elliptic functions. As can be expected, the energy of the free oscillator is redistributed periodically between degrees of freedom. The periodic excitation of the nonlinear oscillator may substantially change its motion pattern. Using as an example an oscillator with two coupled vibrational degrees of freedom, it is numerically shown that the amount of laser photons absorbed depending on the parameter values and initial conditions may vary with time in a rather complex manner, including chaotic oscillations. A nonlinear oscillator is capable of manifesting bistable behavior with allowance for dissipation. The analytical condition for the origination of bistability is found. Examples of the bistable dependence of the number of quanta in the oscillator vibrational mode on the level of laser excitation are presented.     

Rigged Hilbert spaces and time asymmetry: The case of the upside-down simple harmonic oscillator

The upside-down simple harmonic oscillator system is studied in the contexts of quantum mechanics and classical statistical mechanics. It is shown that in order to study in a simple manner the creation and decay of a physical system by way of Gamow vectors we must formulate the theory in a time-asymmetric fashion, namely using two different rigged Hilbert spaces to describe states evolving toward the past and the future. The spaces defined in the contexts of quantum and classical statistical mechanics are shown to be directly related by the Wigner function.     

The KAM theory of systems with short range interactions,I

The existence of quasiperiodic trajectories for Hamiltonian systems consisting of long chains of nearly identical subsystems, with interactions which decay rapidly with increasing distance between the interacting components, is studied. Such models are of interest in statistical mechanics. It is shown that nonergodic motions persist for much larger perturbations than prior work indicated. If the number of degrees of freedom of the system isN, the allowed perturbation decreases only as an inverse power ofN, as the number of degrees of freedom increases, rather than the inverse power ofN! which previous estimates yielded.This work was completed while the author was at the Institute for Mathematics and Its Applictations, University of Minnesota, Minneapolis, MN, 55455, USASupported in part by NSF Grant DMS-8403664     

Testing Monte Carlo methods for path integrals in some quantum mechanical problems

The metropolis algorithm for numerical simulation of configurations, widely used nowadays for lattice gauge theories, is applied to the evaluation of path integrals of quantum mechanics. Our aim is both to test the method in the case of well-known systems (nonlinear oscillator, H and He atoms) and also to consider its ability to treat more complicated systems with more degrees of freedom (light nuclei up to ⁴He). We also show how effective the method is for the evaluation of Green functions in arbitrary (multidimensional) potentials.     

Statistical mechanics of warm and cold unfolding in proteins

We present a statistical mechanics treatment of the stability of globular proteins which takes explicitly into account the coupling between the protein and water degrees of freedom. This allows us to describe both the cold and the warm unfolding, thus qualitatively reproducing the known thermodynamics of proteins. Received: 19 March 1998 / Revised and Accepted: 25 May 1998     

Complex probabilities on R(N) as real probabilities on C(N) and an application to path integrals

We establish a necessary and sufficient condition for averages over complex-valued weight functions on R(N) to be represented as statistical averages over real, non-negative probability weights on C(N). Using this result, we show that many path integrals for time-ordered expectation values of bosonic degrees of freedom in real-valued time can be expressed as statistical averages over ensembles of paths with complex-valued coordinates, and then speculate on possible consequences of this result for the relation between quantum and classical mechanics.     

Non-existence of long-range order in a model of polyacetylene

Polyacetylene (CH)_χ, a one-dimensional polymer, is studied as a model of quantum statistical mechanics. The effective Gibbs measure of phonons in (CH)_χ is obtained by integrating out the fermion degrees of freedom, and it is shown that the effective interactions of phonons are rather complicated many-body ones but of short range (at least for small coupling constant g or for small inverse temperature β). Thus the Araki-Ruelle theory implies non-existence of long-range orders in this system.     

Realization of holonomic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory. Part I

The so-called problem of the realization of the holonomic constraints of classical mechanics is here revisited, in the light of Nekhoroshev-like classical perturbation theory. Precisely, if constraints are physically represented by very steep potential wells, with associated high frequency transversal vibrations, then one shows that (within suitable assumptions) the vibrational energy and the energy associated to the constrained motion are separately almost constant, for a very long time scale growing exponentially with the frequency (i.e., with the rigidity of the constraint one aims to realize). This result can also be applied to microscopic physics, providing a possible entirely classical mechanism for the freezing of the high-frequency degrees of freedom, in terms of non-equilibrium statistical mechanics, according to some ideas expressed by Boltzmann and Jeans at the turn of the century. In this Part I we introduce the problem and prove a first theorem concerning the realization of a single constraint (within a system of any number of degrees of freedom). The problem of the realization of many constraints will be considered in a forthcoming Part II.     

统计物理的微观动力学基础

统计的基本出发点是研究系统具有的随机性,不同系统在不同情形下的宏观热力学性质起源于系统内部随机性的差异,通过对宏观热力学系统的微观非线性动力学进行研究探索，我们可以进一步更为深入地理解物态方程、相变等诸多的宏观热力学现象。本文通过哈密顿系统的非线性动力学研究，以及遍历性理论的动力学随机性研究对此问题进行了分析，研究表明，动力学系统的全局性混沌是系统统计成立的根本要素，系统的无限大自由度（热力学极限）已不是决定性的因素，人们可以在此基础上建立少自由度系统的统计力学及热力学。     

Interacting Bosons at Finite Temperature: How Bogolubov Visited a Black Hole and Came Home Again

The structure of the thermal equilibrium state of a weakly interacting Bose gas is of current interest. We calculate the density matrix of that state in two ways. The most effective method, in terms of yielding a simple, explicit answer, is to construct a generating function within the traditional framework of quantum statistical mechanics. The alternative method, arguably more interesting, is to construct the thermal state as a vector state in an artificial system with twice as many degrees of freedom. It is well known that this construction has an actual physical realization in the quantum thermodynamics of black holes, where the added degrees of freedom correspond to the second sheet of the Kruskal manifold and the thermal vector state is a state of the Unruh or the Hartle–Hawking type. What is unusual about the present work is that the Bogolubov transformation used to construct the thermal state combines in a rather symmetrical way with Bogolubov's original transformation of the same form, used to implement the interaction of the nonideal gas in linear approximation. In addition to providing a density matrix, the method makes it possible to calculate efficiently certain expectation values directly in terms of the thermal vector state of the doubled system.