Irreversibility and spontaneous appearance of coherent behavior in reversible systems

There is empirical evidence that long time numerical simulations of conservative and reversible partial differential equations evolve, as a general rule (exceptions are the integrable models), towards an equilibrium state that is mainly a coherent structure plus small fluctuations inherent in the conservative and reversible character of the original system. The fluctuations account for the energy difference between the initial configuration and the one of the coherent structure. If the energy is not small enough, then the intrinsic fluctuations may destroy the coherent structure. Thus we arrive to the conclusion that a transition arises from a non-coherent state to a coherent structure as we decrease the initial energy below a critical value. This phenomenon has been successfully observed in various numerical simulations. In this article, we stress that this general behavior is also observed in reversible and conservative cellular automata as in the Q2R model. We point out that this conservative and reversible cellular automata is ab initio deterministic and therefore all our numerical computations are not affected by an approximation of any kind.     

Ideal magnetofluid turbulence in two dimensions

A continuum model of coherent structures in two-dimensional magnetohydro-dynamic turbulence is developed. These structures are macroscopic states which persist among the turbulent microscopic fluctuations, typically as magnetic islands with flow. They are modeled as statistical equilibrium states for the non-dissipative dynamics, which conserves energy and families of cross-helicity and flux integrals. The model predicts that from a given initial state an ideal magnetofluid will evolve into a final state having steady mean magnetic and velocity fields, and Gaussian local fluctuations in these fields. Excellent qualitative and quantitative agreement is found with the known results of direct numerical simulations. A rigorous justification of the theory is also provided, in the sense that the continuum model is derived from a lattice model in a fixed-volume, small-spacing limit. This construction uses the discrete Fourier transform to link the discretization ofx-space with the truncation ofk-space. Under the ergodic hypothesis and a separation-of-scales hypothesis, the lattice model is defined by a mean-field approximation to the Gibbs measure on the discretized phase space. A concentration property shows that this measure is equivalent to the microcanonical measure in the continuum limit.     

Phase transitions of quasistationary states in the Hamiltonian Mean Field model

The out-of equilibrium dynamics of the Hamiltonian Mean Field (HMF) model is studied in presence of an externally imposed magnetic field h. Lynden-Bell’s theory of violent relaxation is revisited and shown to adequately capture the system dynamics, as revealed by direct Vlasov based numerical simulations in the limit of vanishing field. This includes the existence of an out-of-equilibrium phase transition separating magnetized and non magnetized phases. We also monitor the fluctuations in time of the magnetization, which allows us to elaborate on the choice of the correct order parameter when challenging the performance of Lynden-Bell’s theory. The presence of the field h removes the phase transition, as it happens at equilibrium. Moreover, regions with negative susceptibility are numerically found to occur, in agreement with the predictions of the theory.     

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.     

Free energy and the relative entropy

Under the assumption of an identity determining the free energy of a state of a statistical mechanical system relative to a given equilibrium state by means of the relative entropy, it is shown: first, that there is in any physically definable convex set of states a unique state of minimum free energy measured relative to a given equilibrium state; second, that if a state has finite free energy relative to an equilibrium state, then the set of its time translates is a weakly relatively compact set; and third, that a unique perturbed equilibrium state exists following a change in Hamiltonian that is bounded below.     

Breakdown of weak-turbulence and nonlinear wave condensation

The formation of a large-scale coherent structure (a condensate) as a result of the long time evolution of the initial value problem of a classical partial differential nonlinear wave equation is considered. We consider the nonintegrable and unforced defocusing NonLinear Schrödinger (NLS) equation as a representative model. In spite of the formal reversibility of the NLS equation, the nonlinear wave exhibits an irreversible evolution towards a thermodynamic equilibrium state. The equilibrium state is characterized by a homogeneous solution (condensate), with small-scale fluctuations superposed (uncondensed particles), which store the information necessary for “time reversal”. We analyze the evolution of the cumulants of the random wave as originally formulated by D.J. Benney and P.G. Saffman [D.J. Benney, P.G. Saffman, Proc. Roy. Soc. London A 289 (1966) 301] and A.C. Newell [A.C. Newell, Rev. Geophys. 6 (1968) 1]. This allows us to provide a self-consistent weak-turbulence theory of the condensation process, in which the nonequilibrium formation of the condensate is a natural consequence of the spontaneous regeneration of a non-vanishing first-order cumulant in the hierarchy of the cumulants’ equations. More precisely, we show that in the presence of a small condensate amplitude, all relevant statistical information is contained in the off-diagonal second order cumulant, as described by the usual weak-turbulence theory. Conversely, in the presence of a high-amplitude condensate, the diagonal second-order cumulants no longer vanish in the long time limit, which signals a breakdown of the weak-turbulence theory. However, we show that an asymptotic closure of the hierarchy of the cumulants’ equations is still possible provided one considers the Bogoliubov’s basis rather than the standard Fourier’s (free particle) basis. The nonequilibrium dynamics turns out to be governed by the Bogoliubov’s off-diagonal second order cumulant, while the corresponding diagonal cumulants, as well as the higher order cumulants, are shown to vanish asymptotically. The numerical discretization of the NLS equation implicitly introduces an ultraviolet frequency cut-off. The simulations are in quantitative agreement with the weak turbulence theory without adjustable parameters, despite the fact that the theory is expected to breakdown nearby the transition to condensation. The fraction of condensed particles vs energy is characterized by two distinct regimes: For small energies (H?H_c) the Bogoliubov’s regime is established, whereas for H?H_c the small-amplitude condensate regime is described by the weak-turbulence theory. In both regimes we derive coupled kinetic equations that describe the coupled evolution of the condensate amplitude and the incoherent field component. The influence of finite size effects and of the dimensionality of the system are also considered. It is shown that, beyond the thermodynamic limit, wave condensation is reestablished in two spatial dimensions, in complete analogy with uniform and ideal 2D Bose gases.     

Quantum statistical properties of the radiation field in the degenerate two-photon emission process

The degenerate two-photon emission process is treated on the basis of quantum theory. Neglecting relaxation mechanisms, solutions in short time approximation are given. Relevant quantities as the mean photon number, second order correlation, field fluctuations and the uncertainty product are analyzed. It is shown, that an initial coherent state does not tend to a two-photon coherent state by two-photon emission as it can be expected from the results given by Yuen.     

Big entropy fluctuations in statistical equilibrium: The macroscopic kinetics

Large entropy fluctuations in the equilibrium steady state of classical mechanics are studied in extensive numerical experiments in a simple strongly chaotic Hamiltonian model with two degrees of freedom described by the modified Arnold cat map. The rise and fall of a large separated fluctuation is shown to be described by the (regular and stable) “macroscopic” kinetics, both fast (ballistic) and slow (diffusive). We abandon a vague problem of the “appropriate” initial conditions by observing (in a long run) a spontaneous birth and death of arbitrarily big fluctuations for any initial state of our dynamical model. Statistics of the infinite chain of fluctuations similar to the Poincaré recurrences is shown to be Poissonian. A simple empirical relationship for the mean period between the fluctuations (the Poincaré “cycle”) is found and confirmed in numerical experiments. We propose a new representation of the entropy via the variance of only a few trajectories (“particles”) that greatly facilitates the computation and at the same time is sufficiently accurate for big fluctuations. The relation of our results to long-standing debates over the statistical “irreversibility” and the “time arrow” is briefly discussed.     

Imprint of non-standard interactions on the CP violation measurements at long baseline experiments

By introducing an additional state feedback into classic Rikitake system, a new hyperchaotic system without equilibrium is derived. The proposed system is investigated through numerical simulations and analyses including time phase portraits, Lyapunov exponents, and Poincaré maps. Based on adaptive control and Lyapunov stability theory, we design a reduced-order projective synchronization scheme for synchronizing the hyperchaotic Rikitake system coexisting without equilibria and the original classic Rikitake system coexisting with two non-hyperbolic equilibria. Finally, numerical simulations are given to illustrate the effectiveness of the proposed synchronization scheme.     

Statistical Mechanical Theory of a Closed Oscillating Universe

Based on Newton’s laws reformulated in the Hamiltonian dynamics combined with statistical mechanics, we formulate a statistical mechanical theory supporting the hypothesis of a closed universe oscillating in phase-space. We find that the behavior of this universe as a whole can be represented by a free entropic oscillator whose lifespan is nonhomogeneous, thus implying that time is shorter or longer according to the state of this universe given through its entropy. We conclude that time reduces to the entropy production of this universe and that a nonzero entropy production means that local fluctuations could exist giving rise to the appearance of masses and to the curvature of the space.     

Extensive Adiabatic Invariants for Nonlinear Chains

We look for extensive adiabatic invariants in nonlinear chains in the thermodynamic limit. Considering the quadratic part of the Klein-Gordon Hamiltonian, by a linear change of variables we transform it into a sum of two parts in involution. At variance with the usual method of introducing normal modes, our constructive procedure allows us to exploit the complete resonance, while keeping the extensive nature of the system. Next we construct a nonlinear approximation of an extensive adiabatic invariant for a perturbation of the discrete nonlinear Schrödinger model. The fluctuations of this quantity are controlled via Gibbs measure estimates independent of the system size, for a large set of initial data at low specific energy. Finally, by numerical calculations we show that our adiabatic invariant is well conserved for times much longer than predicted by our first order theory, with fluctuation much smaller than expected according to standard statistical estimates.     

Evolution and Generation of Dynamics of Two-Mode Squeezed States of Time-Dependent Coupled Boson System

The system of the Hamiltonian involving a driving part, a single mode part, and a two-mode squeezed one and a two-mode coupled one is discussed using the Lewis-Riesenfeld invariant theory. The time evolution operator is obtained. When the initial state is a coherent state, the quantum fluctuation of the system is calculated, and it is dependent on the squeezed part and the two-mode coupled part, but not dependent on the driving one.     

Thermodynamics under Dynamic Forcing

Beginning with a system that is governed by an arbitrary time-dependent Hamiltonian, we exhibit an existence proof for a unitary generator that has an arbitrary initial value and yet contact transforms the representation to one governed by any given kinematically equivalent Hamiltonian. By choosing the initial value of the unitary operator to be unity, we are able to compare the behaviour of the same system under two different Hamiltonians and the same initial state vector. We thus are able to establish that the eventual physical states evolving from two distinct initial quantum state vectors will become practically indistinguishable under one of the two Hamiltonians if and only if they do so under the other. For the restricted class of systems for which one of the two Hamiltonians is a time-independent energy operator, and also generates equilibrium thermodynamics, then the condition for merging under the time-dependent Hamiltonian is the same as under the time-independent one. The two states must have the same initial energy. As a special case of the above, we choose the time-independent Hamiltonian to be the relativistic energy measuring operator for the time-dependent Hamiltonian, as associated with the chosen initial time. If the system under the time-dependent Hamiltonian is such that its relativistic energy measuring operator for any fixed time generates equilibrium thermodynamics, then we are led rigorously to the conclusion that the instantaneous relativistic energy for the system under the time-dependent Hamiltonian is simply a well-defined function of time and depends only on the initial energy and not on any other initial conditions. For a composite system that is of the above type, and in addition consists of one very small system in contact with a very large one, which is called a generalized reservoir, we consider a specific initial physical state for the large system, and various states for the small one. The eventual dynamic state of the composite system is essentially independent of the initial state of the small system which has almost no influence on the total composite energy. Hence the eventual dynamic state of the small system is shown rigorously to be independent of its initial state. For a forced system with a time-dependent Hamiltonian, we discuss the assignment of equilibrium thermodynamic potentials to a representation with a time-independent Hamiltonian. We discuss the concept of a process under a time-dependent Hamiltonian. Such a process is a natural generalization of the static and quasi-static processes. Also, we verify all of the theory with both general and specific examples of electromagnetic interactions.     

Colored noise in a two-dimensional nonlinear system

A two-dimensional decoupling theory is developed when colored noise is included in a nonlinear dynamical system. By a functional analysis, the colored noise is transformed to an effective noise that includes the noise correlation time, the mean dynamical variable, and the original noise strength. When the two-dimensional decoupling theory is applied to single-mode and two-mode dye laser systems, the mean, variance, and effective eigenvalue of laser intensity are calculated. Excellent agreement between theoretical analysis, numerical simulations, and experimental measurements are obtained. It is seen that the increase of noise correlation time can reduce the fluctuations in the laser system. It is also shown that there is relatively large fluctuation in the phase when the laser undergoes from thermal light to coherent light when the theory is applied to a single mode dye laser. Received 20 August 2001 and Received in final form 4 December 2001     

Statistical relevance of vorticity conservation in the Hamiltonian particle-mesh method

We conduct long-time simulations with a Hamiltonian particle-mesh method for ideal fluid flow, to determine the statistical mean vorticity field of the discretization. Lagrangian and Eulerian statistical models are proposed for the discrete dynamics, and these are compared against numerical experiments. The observed results are in excellent agreement with the theoretical models, as well as with the continuum statistical mechanical theory for ideal fluid flow developed by Ellis et al. (2002) [10]. In particular the results verify that the apparently trivial conservation of potential vorticity along particle paths within the HPM method significantly influences the mean state. As a side note, the numerical experiments show that a nonzero fourth moment of potential vorticity can influence the statistical mean.     

原子玻色-爱因斯坦凝聚体对V型三能级原子激光压缩性质的影响

利用格子液体方法对V型三能级原子玻色-爱因斯坦凝聚体与双模压缩相干态光场相互作用系统的哈密顿量进行分析,发现文献中对原子间相互作用部分的处理有不合理之处,从而对该哈密顿量作出了改进并研究了V型三能级原子玻色-爱因斯坦凝聚体与双模压缩相干态光场相互作用系统中原子激光的两个正交分量的压缩性质.研究表明:V型三能级原子玻色-爱因斯坦凝聚体中光场-原子相互作用强度对原子激光的两正交分量的涨落有明显的影响. 关键词： 玻色-爱因斯坦凝聚 V型三能级原子 压缩相干态 压缩原子激光     

Conservative, unconditionally stable discretization methods for Hamiltonian equations, applied to wave motion in lattice equations modeling protein molecules

A new approach is described for generating exactly energy-momentum conserving time discretizations for a wide class of Hamiltonian systems of DEs with quadratic momenta, including mechanical systems with central forces; it is well-suited in particular to the large systems that arise in both spatial discretizations of nonlinear wave equations and lattice equations such as the Davydov System modeling energetic pulse propagation in protein molecules. The method is unconditionally stable, making it well-suited to equations of broadly “Discrete NLS form”, including many arising in nonlinear optics.Key features of the resulting discretizations are exact conservation of both the Hamiltonian and quadratic conserved quantities related to continuous linear symmetries, preservation of time reversal symmetry, unconditional stability, and respecting the linearity of certain terms. The last feature allows a simple, efficient iterative solution of the resulting nonlinear algebraic systems that retain unconditional stability, avoiding the need for full Newton-type solvers. One distinction from earlier work on conservative discretizations is a new and more straightforward nearly canonical procedure for constructing the discretizations, based on a “discrete gradient calculus with product rule” that mimics the essential properties of partial derivatives.This numerical method is then used to study the Davydov system, revealing that previously conjectured continuum limit approximations by NLS do not hold, but that sech-like pulses related to NLS solitons can nevertheless sometimes arise.