Chaotic behavior in nonlinear Hamiltonian systems and equilibrium statistical mechanics

The relation between chaotic dynamics of nonlinear Hamiltonian systems and equilibrium statistical mechanics in its canonical ensemble formulation has been investigated for two different nonlinear Hamiltonian systems. We have compared time averages obtained by means of numerical simulations of molecular dynamics type with analytically computed ensemble averages. The numerical simulation of the dynamic counterpart of the canonical ensemble is obtained by considering the behavior of a small part of a given system, described by a microcanonical ensemble, in order to have fluctuations of the energy of the subsystem. The results for the Fermi-Pasta-Ulam model (i.e., a one-dimensional anharmonic solid) show a substantial agreement between time and ensemble averages independent of the degree of stochasticity of the dynamics. On the other hand, a very different behavior is observed for a chain of weakly coupled rotators, where linear exchange effects are absent. In the high-temperature limit (weak coupling) we have a strong disagreement between time and ensemble averages for the specific heat even if the dynamics is chaotic. This behavior is related to the presence of spatially localized chaos, which prevents the complete filling of the accessible phase space of the system. Localized chaos is detected by the distribution of all the characteristic Liapunov exponents.     

Foundations of the quantum statistics of systems in equilibrium: A measuretheoretic approach

The fundamental equations of equilibrium quantum statistical mechanics are derived in the context of a measure-theoretic approach to the quantum mechanical ergodic problem. The method employed is an extension, to quantum mechanical systems, of the techniques developed by R. M. Lewis for establishing the foundations of classical statistical mechanics. The existence of a complete set of commuting observables is assumed, but no reference is made a priori to probability or statistical ensembles. Expressions for infinite-time averages in the microcanonical, canonical, and grand canonical ensembles are developed which reduce to conventional quantum statistical mechanics for systems in equilibrium when the total energy is the only conserved quantity. No attempt is made to extend the formalism at this time to deal with the difficult problem of the approach to equilibrium.     

Broad histogram relation is exact

The Broad Histogram is a method designed to calculate the energy degeneracy g(E) from microcanonical averages of certain macroscopic quantities N _up and N _dn. These particular quantities are defined within the method, and their averages must be measured at constant energy values, i.e. within the microcanonical ensemble. Monte-Carlo simulational methods are used in order to perform these measurements. Here, the mathematical relation allowing one to determine g(E) from these averages is shown to be exact for any statistical model, i.e. any energy spectrum, under completely general conditions. We also comment about some troubles concerning the measurement of the quoted microcanonical averages, when one uses a particular approach, namely the energy random walk dynamics. These troubles appear when movements corresponding to different energy jumps are performed using the same probability, and also when the correlations between successive averaging states are not adequately treated: they have nothing to do with the method itself. Received: 29 May 1998 / Received in final form: 12 June 1998 / Accepted: 13 June 1998     

Quantum microcanonical entropy of a pair of observables

The quantum analogue of the classical theory of the joint microcanonical entropy of a pair of observables is investigated for a system of a large number of identical non-interacting subsystems. It is shown that the quantum joint entropy coincides with the classical joint entropy of an appropriately chosen auxiliary classical system, and known results for classical systems are applied to prove the equivalence of the quantum microcanonical and quantum canonical ensembles.     

Rotators with long-range interactions: connection with the mean-field approximation

We analyze the equilibrium properties of a chain of ferromagnetically coupled rotators which interact through a force that decays as r(-alpha) where r is the interparticle distance and alpha>/=0. By integrating the equations of motion we obtain the microcanonical time averages of both the magnetization and the kinetic energy. We detect three different regimes depending on whether alpha belongs to the intervals [0,1), (1,2), or (2,infinity). For 0相似文献   

A Comparison Between Broad Histogram and Multicanonical Methods

We discuss the conceptual differences between the broad histogram (BHM) and reweighting methods in general, and particularly the so-called multicanonical (MUCA) approaches. The main difference is that BHM is based on microcanonical, fixed-energy averages which depend only on the good statistics taken inside each energy level. The detailed distribution of visits among different energy levels, determined by the particular dynamic rule one adopts, is irrelevant. Contrary to MUCA, where the results are extracted from the dynamic rule itself, within BHM any microcanonical dynamics could be adopted. As a numerical test, we have used both BHM and MUCA in order to obtain the spectral energy degeneracy of the Ising model in 4×4×4 and 32×32 lattices, for which exact results are known. We discuss why BHM gives more accurate results than MUCA, even using the same Markovian sequence of states. In addition, such an advantage increases for larger systems.     

Prequantum Classical Statistical Field Theory: Schrödinger Dynamics of Entangled Systems as a Classical Stochastic Process

The idea that quantum randomness can be reduced to randomness of classical fields (fluctuating at time and space scales which are essentially finer than scales approachable in modern quantum experiments) is rather old. Various models have been proposed, e.g., stochastic electrodynamics or the semiclassical model. Recently a new model, so called prequantum classical statistical field theory (PCSFT), was developed. By this model a “quantum system” is just a label for (so to say “prequantum”) classical random field. Quantum averages can be represented as classical field averages. Correlations between observables on subsystems of a composite system can be as well represented as classical correlations. In particular, it can be done for entangled systems. Creation of such classical field representation demystifies quantum entanglement. In this paper we show that quantum dynamics (given by Schrödinger’s equation) of entangled systems can be represented as the stochastic dynamics of classical random fields. The “effect of entanglement” is produced by classical correlations which were present at the initial moment of time, cf. views of Albert Einstein.     

Inhomogeneity of local temperature in small clusters in microcanonical equilibrium

Overall homogeneity of temperature is a condition for thermal equilibrium, but, as is demonstrated by classical molecular dynamics simulations, the local temperatures of atoms in small, isolated crystalline clusters in microcanonical equilibrium are not uniform. The effective temperature determined from individual atomic velocity decreases with distance from the cluster center. It is argued that these effects are due to the conservation of angular and translational momentum. A general microcanonical expression is derived for the spatial dependence of the statistics of the kinetic energies of individual atoms; this fits the numerical observations well.     

Microcanonical solution of the mean-field φ4 model: Comparison with time averages at finite size

We study the mean-field ${\phi }^4$ model in an external magnetic field in the microcanonical ensemble using two different methods. The first one is based on Rugh's microcanonical formalism and leads to express macroscopic observables, such as temperature, specific heat, magnetization and susceptibility, as time averages of convenient functions of the phase-space. The approach is applicable for any finite number of particles N. The second method uses large deviation techniques and allows us to derive explicit expressions for microcanonical entropy and for macroscopic observables in the $N\to \infty$ limit. Assuming ergodicity, we evaluate time averages in molecular dynamics simulations and, using Rugh's approach, we determine the value of macroscopic observables at finite N. These averages are affected by a slow time evolution, often observed in systems with long-range interactions. We then show how the finite N time averages of macroscopic observables converge to their corresponding $N\to \infty$ values as N is increased. As expected, finite size effects scale as $N^{-1}$.     

Are there any ergodic local relativistic field theories?

It is shown that in certain local relativistic field theories that are known to possess strong global solutions, time averages are not equal to the microcanonical averages computed with the Gibbs measure.     

Phase transitions in small systems: Microcanonical vs. canonical ensembles

We compare phase transition(-like) phenomena in small model systems for both microcanonical and canonical ensembles. The model systems correspond to a few classical (non-quantum) point particles confined in a one-dimensional box and interacting via Lennard-Jones-type pair potentials. By means of these simple examples it can be shown already that the microcanonical thermodynamic functions of a small system may exhibit rich oscillatory behavior and, in particular, singularities (non-analyticities) separating different microscopic phases. These microscopic phases may be identified as different microphysical dissociation states of the small system. The microscopic oscillations of microcanonical thermodynamic quantities (e.g., temperature, heat capacity, or pressure) should in principle be observable in suitably designed evaporation/dissociation experiments (which must realize the physical preconditions of the microcanonical ensemble). By contrast, singular phase transitions cannot occur, if a small system is embedded into an infinite heat bath (thermostat), corresponding to the canonical ensemble. For the simple model systems under consideration, it is nevertheless possible to identify a smooth canonical phase transition by studying the distribution of complex zeros of the canonical partition function.     

Conditional equilibrium and the equivalence of microcanonical and grandcanonical ensembles in the thermodynamic limit

Equivalence (allowing for convex combinations) of microcanonical, canonical and grandcanonical ensembles for states of classical systems is established under very mild assumptions on the limiting state. We introduce the notion of conditional equilibrium (C.E.), a property of states of infinite systems which characterizes convex combinations of limits of microcanonical ensembles. It is shown that C.E. states are, under quite general conditions, mixtures of Gibbs states.Supported in part by NSF Grant No. MCS 75-21684 A02Supported in part by NSF Grant No. MPS 72-04534Supported in part by NSF Grant No. Phy 77-22302     

A method for visualization of invariant sets of dynamical systems based on the ergodic partition

We provide an algorithm for visualization of invariant sets of dynamical systems with a smooth invariant measure. The algorithm is based on a constructive proof of the ergodic partition theorem for automorphisms of compact metric spaces. The ergodic partition of a compact metric space A, under the dynamics of a continuous automorphism T, is shown to be the product of measurable partitions of the space induced by the time averages of a set of functions on A. The numerical algorithm consists of computing the time averages of a chosen set of functions and partitioning the phase space into their level sets. The method is applied to the three-dimensional ABC map for which the dynamics was visualized by other methods in Feingold et al. [J. Stat. Phys. 50, 529 (1988)]. (c) 1999 American Institute of Physics.     

RS/6000机群系统中分子动力学并行算法的研究

在IBM　RS/6000工作站组成的机群系统和PVM（并行虚拟机）并行环境中,用消息传递机制方式和C++语言实现了微正则系综（NVE）的分子动力学并行计算程序,并对不同分子数（256～108000）组成的Lennard-jones理想流体进行了模拟研究。结果表明,在小型机群系统中,原子分解法具有理想的加速比,考虑到区域分解法的适用范围有限且实现困难,困此认为原子分解法是小型机群系统进行分子动力学并行模拟计算的理想选择。     

Sampling along reaction coordinates with the Wang-Landau method

The multiple range random walk algorithm recently proposed by Wang and Landau [2001, Phys. Rev. Lett., 86, 2050] is adapted to the computation of free energy profiles for molecular systems along reaction coordinates. More generally, we show how to extract partial averages in various statistical ensembles without invoking simulations with constraints, biasing potentials or unknown parameters. The method is illustrated on a model 10-dimensional potential energy surface, for which analytical results are obtained. It is then applied to the potential of mean force associated with the dihedral angle of the butane molecule in the gas phase and in carbon tetrachloride solvent. Finally, isomerization in a small rocksalt cluster, (NaF)₄, is investigated in the microcanonical ensemble, and the results are compared to those of parallel tempering Monte Carlo.     

Slow-down collisions and nonsequential double ionization in classical simulations

We use classical simulations to analyze the dynamics of nonsequential double-electron short-pulse photoionization. We utilize a microcanonical ensemble of 10(5) two-electron "trajectories," a number large enough to provide large subensembles and even sub-subensembles associated with double ionization. We focus on key events in the final doubly ionized subensemble and back-analyze the subensemble's history, revealing a classical slow-down scenario for nonsequential double ionization. We analyze the dynamics of these slow-down collisions and find that a good phase match between the motions of the electrons can lead to very effective energy transfer, followed by escape over a suppressed barrier.     

Classification of Phase Transitions and Ensemble Inequivalence, in Systems with Long Range Interactions

Systems with long range interactions in general are not additive, which can lead to an inequivalence of the microcanonical and canonical ensembles. The microcanonical ensemble may show richer behavior than the canonical one, including negative specific heats and other non-common behaviors. We propose a classification of microcanonical phase transitions, of their link to canonical ones, and of the possible situations of ensemble inequivalence. We discuss previously observed phase transitions and inequivalence in self-gravitating, two-dimensional fluid dynamics and non-neutral plasmas. We note a number of generic situations that have not yet been observed in such systems.     

Stable quantum computation of unstable classical chaos

We show on the example of the Arnold cat map that classical chaotic systems can be simulated with exponential efficiency on a quantum computer. Although classical computer errors grow exponentially with time, the quantum algorithm with moderate imperfections is able to simulate accurately the unstable chaotic classical nonlinear dynamics for long times. The algorithm can be easily implemented on systems of a few qubits.     

Canonical typicality

It is well known that a system weakly coupled to a heat bath is described by the canonical ensemble when the composite S + B is described by the microcanonical ensemble corresponding to a suitable energy shell. This is true for both classical distributions on the phase space and quantum density matrices. Here we show that a much stronger statement holds for quantum systems. Even if the state of the composite corresponds to a single wave function rather than a mixture, the reduced density matrix of the system is canonical, for the overwhelming majority of wave functions in the subspace corresponding to the energy interval encompassed by the microcanonical ensemble. This clarifies, expands, and justifies remarks made by Schr?dinger in 1952.