Kinetic theory of hard spheres

Kinetic equations for the hard-sphere system are derived by diagrammatic techniques. A linear equation is obtained for the one-particle-one particle equilibrium time correlation function and a nonlinear equation for the one-particle distribution function in nonequilibrium. Both equations are nonlocal, noninstantaneous, and extremely complicated. They are valid for general density, since statistical correlations are taken into account systematically. This method derives several known and new results from a unified point of view. Simple approximations lead to the Boltzmann equation for low densities and to a modified form of the Enskog equation for higher densities.     

Effects of the Form of Correlation Between Noises on Dynamic Properties of Linear System

We have derived exact expressions for the correlation function, power spectrum and correlation time of a one-dimensional non-Markovian linear system driven by correlated noises with correlation forms of delta function and different-time-region exponential function, respectively. We find that these dynamic properties of system for delta form are much different from those for different-time-region exponential form. Especially, the power spectrum for the former only has one peak but for the latter can have one, two or three peaks.     

Yukawa fluid at a hard wall: field theory description

We adopt a field-theoretical approach to study the structure and thermodynamics of a spatially confined fluid interacting with the Yukawa potential. We derive analytic expressions for the pressure, the Helmholtz free energy, the correlation function, the density profile, and the adsorption. Different simple analytic expressions of the density profile are compared with the numerical estimation of the mean field results. Beyond the mean field approximation, we show that fluctuations can contribute significantly to the properties of the system. Notably they lead to a desorption phenomenon regardless of the sign of the interaction. As a consequence, a non-monotonous density profile at the wall and adsorption curves as a function of the density are found for some systems. This behaviour rationalizes the ionic depletion phenomenon responsible for the anomalous behaviour of the electric capacitance as a function of temperature. Particular attention is given to the contact theorem condition.     

Canonical symmetry of a constrained Hamiltonian system and canonical ward identity

An algorithm for the construction of the generators of the gauge transformation of a constrained Hamiltonian system is given. The relationships among the coefficients connecting the first constraints in the generator are made clear. Starting from the phase space generating function of the Green function, the Ward identity in canonical formalism is deduced. We point out that the quantum equations of motion in canonical form for a system with singular Lagrangian differ from the classical ones whether Dirac's conjecture holds true or not. Applications of the present formulation to the Abelian and non-Abelian gauge theories are given. The expressions for PCAC and generalized PCAC of the AVV vertex are derived exactly from another point of view. A new form of the Ward identity for gauge-ghost proper vertices is obtained which differs from the usual Ward-Takahashi identity arising from the BRS invariance.     

Spectral Densities and Partition Functions of Modular Quantum ystems as Derived from a Central Limit Theorem

Using a central limit theorem for arrays of interacting quantum systems, we give analytical expressions for the density of states and the partition function at finite temperature of such a system, which are valid in the limit of infinite number of subsystems. Even for only small numbers of subsystems we find good accordance with some known, exact results.     

Instantaneously Dynamical Response and Correlation Functions in Independent-Oscillator Model

For the independent-oscillator model (IOM), the explicit expressions of the instantaneously dynamical response functions (Χ₁ + Υ,t₁), X(t₁ + Υ,t₁)) and correlation functions (φ(t₁ + Υ,t₁)+Φ(t₁ + Υ,t₁)) for both the system and environment are derived. It is shown that for external perturbation the environment of IOM is dynamically stable and the system is oscillatory. Although the external perturbation exists, there is still strongly temporal correlation for the system, for the environment, however, the temporal correlation quickly decays with the second time variable Υ and is almost independent of the first time variable t₁, nevertheless is significantly affected by dissipation. It is shown that the environment behaves as a heat bath from the dynamical point of view.     

Multiple-order derivatives of a waveguide acoustic field with respect to sound speed, density, and frequency

An adjoint perturbative method is used to derive expressions for the first- through third-order derivatives of a pressure field with respect to sound speed, density, and frequency, for the restricted case of a laterally homogenous waveguide in which environmental parameters are only a function of depth. By using a normal-mode Green's function, the three-dimensional spatial correlation required by the standard acoustic adjoint equation can be reduced to a set of one-dimensional depth integrals. The resulting expressions for the first-order derivative are similar to those obtained by previous perturbative approaches based on the depth-separated wave equation, but the approach followed here permits straightforward extension to higher-order derivatives. Explicit evaluations of the expressions for a representative shallow-water waveguide model are in excellent agreement with numerical finite-difference computations. An analysis of the expressions as a function of source-receiver range finds the contributions to the mode amplitude derivatives to be non-negligible at ranges less than a few modal interference lengths, for parameters associated with the ocean bottom. Therefore, linear perturbative inversion methods that perturb only horizontal wavenumbers and not mode amplitudes should either be used with caution or modified to incorporate the expressions presented here.     

A statistical-mechanical theory of slow dynamics near the glass transition

A statistical-mechanical theory of slow dynamics near the glass transition in two kinds of glass-forming systems, (M) molecular systems and (S) suspensions of colloids, is presented from a unified point of view based on the Tokuyama-Mori projection operator method. The exact diffusion equations for the coherent- and the incoherent-intermediate scattering functions are first derived, whose memory functions are convolutionless in time and contain the correlation effects due to the hydrodynamic interactions in (S). The analytic expressions of the memory functions are then calculated within the mode-coupling theory (MCT) approximation and are shown to coincide with the conventional ones obtained by MCT. Alternative mode-coupling equations are thus obtained in (M) and (S) separately. Self-diffusion is also discussed. Alternative equations for the mean-square displacement and the non-Gaussian parameter are also derived within MCT approximation. All results in both the systems are compared with those obtained by MCT.     

关联噪声对单模激光动力学性质的影响

应用线性近似方法,计算了具有指数形式关联的两白噪声驱动下的单模激光光强的关联函数、功率谱及关联时间。根据计算结果讨论了噪声间的关联程度和噪声间互联时间对以上各量的影响,并和噪声间具有δ函数关联形式的情况进行了比较     

Suppression of Smale horseshoe structure via secondary perturbations in pendulum systems

We analyse the use of parametric and quasiperiodic modulations in suppressing horseshoe structure in the phase plane of perturbed pendulum systems. Taking the Froude pendulum as a typical system, four different modulation mechanisms are studied by deriving analytic expressions for the window of the strength of modulation giving suppression in each case. A comparison of the four cases from the point of view of flexibility and efficiency is also given.     

Recurrence Times in Quasi-Periodic Motion: Statistical Properties,Role of Cell Size,Parameter Dependence

We investigate the probabilistic properties of recurrence times for the simplest form of aperiodic deterministic dynamics, quasi-periodic motion. Previous results using number theory techniques predict two fundamental recurrence times for uniform quasi-periodic motion on a two-dimensional torus, while no analogous analytic result seems to exist for higher dimensional tori. The two-dimensional uniform case is reanalyzed from a more geometric point of view and new, workable expressions are derived that enable us fully to understand and predict the recurrence phenomenon and to analyze its parameter dependence. Emphasis is placed on the statistical properties and, in particular, on the variability of recurrence times around their mean, in relation to local Farey tree structure. Higher-dimensional tori are considered, and seen to also display a high variability in their finite-time recurrence behavior. The results are finally extended to the non-uniform quasi-periodic case.     

Chaotifying a stable linear controllable system by single input state feedback

In this paper, an approach for chaotifying a stable controllable linear system via single input state-feedback is presented. The overflow function of the system states is designed as the feedback controller, which can make the fixed point of the closed-loop system to be a snap-back repeller, thereby yields chaotic dynamics. Based on the Marotto theorem, it proves theoretically that the closed-loop system is chaotic in the sense of Li and Yorke. Finally, the simulation results are used to illustrate the effectiveness of the proposed method.     

One self-consistent closure for chains of equations for Green's functions. An anisotropic heisenberg model

This article present a self-consistent approach to computation of the correlation function in the method of Green's functions. The basis for the approach is representation of the desired Green's function in the form of a chain of fractions that is subsequently closed. The closure is based on the use of concrete relations imposed on the higher-order correlation functions. General expressions for the correlation functions and conditions for self-consistency of the computations are presented. The method has been tested by computing the magnetization and critical temperature in a Heisenberg model with arbitrary anisotropy parameter. We obtain general expressions for these quantities. The critical temperature obtained is less than the corresponding value given by the molecular-field approximation. The latter approximation also leads to an overestimate of the magnetization values. It is shown that no critical transition is possible for any value of the anisotropy parameter. The corresponding inequalities are obtained. The method is compared with a method for self-consistent computation of correlation functions that was proposed earlier by the author.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 1, pp. 46–52, January, 1996.     

Renormalisation group approach to the ideal Bose gas ind dimensions

Critical behaviour of ad-dimensional ideal Bose gas is investigated from the point of view of the renormalisation-group approach. Rescaling of quantum-field amplitudes is avoided by introducing a scaling variable inversely proportional to the thermal momentum of the particles. The scaling properties of various thermodynamic quantities are seen to emerge as a consequence of the irrelevant nature of this variable. Critical behaviour is discussed at fixed particle density as well as at fixed pressure. Connection between susceptibility and correlation function of the order-parameter for a quantum system is elucidated.     

Non-equilibrium thermo-field dynamics for a non-bilinear Hamiltonian

The non-equilibrium thermo-field dynamics proposed by Arimitsu and Umezawa are generalized to the case of a non-bilinear unperturbed Hamiltonian which includes not only a bilinear part but also a non-bilinear part with momentum mixing. The forms of the quasi-particle operators for a semi-free boson field are derived. The form of the two-point Green’s function for the semi-free boson field is evaluated. A form of the admittance for a boson system interacting with its heat reservoir, which includes effects of the initial correlation and memory, is derived using the TCLE method formulated in terms of generalized non-equilibrium thermo-field dynamics. The expressions of the zeroth-order, first-order and second-order parts of the admittance in powers of the boson-boson interaction, are derived.     

Adiabatic Theorems and Reversible Isothermal Processes

Isothermal processes of a finitely extended, driven quantum system in contact with an infinite heat bath are studied from the point of view of quantum statistical mechanics. Notions like heat flux, work and entropy are defined for trajectories of states close to, but distinct from states of joint thermal equilibrium. A theorem characterizing reversible isothermal processes as quasi-static processes (“isothermal theorem”) is described. Corollaries concerning the changes of entropy and free energy in reversible isothermal processes and on the 0th law of thermodynamics are outlined.*Supported by the Swiss National Foundation.