Analogue simulation of quantum mechanical systems

An extension of the technique of analogue simulation to the treatment of quantum mechanical systems, based on an analogue variant of the method of stochastic quantization, is reported. The analogue stochastic quantization (ASQ) technique is introduced by application to the quantum harmonic oscillator, a particularly simple system for which all the answers are already known. ASQ measurements of the lowest eigenvalues and eigenfunctions of the latter system are presented and compared with theoretical predictions. The future potential of the ASQ technique in relation to some more complicated quantum systems of topical interest is discussed.     

The crossover from classical to quantum regime in the problem of the decay of the metastable state

The probability of the decay of the metastable state has been found as a function of viscosity and temperature. At low temperatures, a classical overbarrier transition changes to the quantum tunneling. At low viscosity in classical and transitive semiquantum region of temperatures a depopulation of the distribution function is significant. The distribution function is shown to satisfy the integral kinetic equation, the kernel of which equals the transition probability. The probability of transitions, induced by the red noise, with the frequency comparable to the transition frequency, is found.     

Inertial effects on the escape rate of a particle driven by colored noise: An instanton approach

A recent calculation, in the weak-noise limit, of the rate of escape of a particle over a one-dimensional potential barrier is extended by including an inertial term in the Langevin equation. Specifically, we consider a system described by the Langevin equation , where is a Gaussian colored noise with mean zero and correlator (t)(t')=(D/)exp(–|t–t'|/). A pathintegral formulation is augmented by a steepest descent calculation valid in the weak-noise (D0) limit. This yields an escape rateexp(–S/D), where the actionS is the minimum, over paths characterizing escape over the barrier, of a generalized Onsager-Machlup functional, the extremal path being an instanton of the theory. The extremal actionS is calculated analytically for smallm and for general potentials, and numerical results forS are displayed for various ranges ofm and for the typical case of the quartic potentialV(x)=–x ₂/2+x ₄/4.     

Decay of correlations. IV. Necessary and sufficient conditions for a rapid decay of correlations

We considerN-particle systems whose probability distributions obey the master equation. For these systems, we derive the necessary and sufficient conditions under which the reducedn-particle (n) probabilities also obey master equations and under which the Ursell functions decay to their equilibrium values faster than the probability distributions. These conditions impose restrictions on the form of the transition rate matrix and thus on the form of its eigenfunctions. We first consider systems in which the eigenfunctions of theN-particle transition rate matrix are completely factorized and demonstrate that for such systems, the reduced probabilities obey master equations and the Ursell functions decay rapidly if certain additional conditions are imposed. As an example of such a system, we discuss a random walk ofN pairwise interacting walkers. We then demonstrate that for systems whoseN-particle transition matrix can be written as a sum of one-particle, two-particle, etc. contributions, and for which the reduced probabilities obey master equations, the reduced master equations become, in the thermodynamic limit, those for independent particles, which have been discussed by us previously. As an example of suchN-particle systems, we discuss the relaxation of a gas of interacting harmonic oscillators.Supported in part (grants to D.B. and K.E.S.) by the Advanced Research Projects Agency of the Department of Defense as monitored by the U.S. Office of Naval Research under Contract N00014-69-A-0200-6018, and in part (grant to I.O.) by the National Science Foundation.     

Return to thermal equilibrium by the solution of a quantum Langevin equation

A quantum-mechanical treatment of the evolution of an anharmonic oscillator coupled to a heat bath is given. It is shown that for a certain class of anharmonic potentials the heat bath drives the oscillator to an equilibrium state, close to the quantum Gibbs state associated to the potential. Thus a partial proof is provided for a conjecture of R. Benguria and M. Kac.This paper contains part of the author's Ph.D. work, done at the Institute for Theoretical Physics of Groningen State University, Groningen, the Netherlands.     

On the decay of infinite energy solutions to the Navier-Stokes equations in the plane

Infinite energy solutions to the Navier-Stokes equations in R² may be constructed by decomposing the initial data into a finite energy piece and an infinite energy piece, which are then treated separately. We prove that the finite energy part of such solutions is bounded for all time and decays algebraically in time when the same can be said of heat energy starting from the same data. As a consequence, we describe the asymptotic behavior of the infinite energy solutions. Specifically, we consider the solutions of Gallagher and Planchon (2002) [2] as well as solutions constructed from a “radial energy decomposition”. Our proof uses the Fourier Splitting technique of M.E. Schonbek.     

An equation for the characteristic function of a Markov process and its application to a Langevin process

For Markov processes a “curtailed characteristic function” is defined. It obeys an equation similar to the master equation. Its solution provides the characteristic function of the process. By applying it to the radioactive decay process the stochastic properties of the corresponding Langevin force are determined.     

The cluster expansion for classical and quantum lattice systems

We develop a high-temperature expansion for general lattice systems which can be applied to classical as well as quantum systems. Applying the expansion we prove analyticity of correlation functions, uniqueness of equilibrium states, and cluster properties for classical and quantum lattice systems in the high-temperature region.     

Additive noise-induced Turing transitions in spatial systems with application to neural fields and the Swift-Hohenberg equation

This work studies the spatio-temporal dynamics of a generic integral-differential equation subject to additive random fluctuations. It introduces a combination of the stochastic center manifold approach for stochastic differential equations and the adiabatic elimination for Fokker-Planck equations, and studies analytically the systems’ stability near Turing bifurcations. In addition two types of fluctuation are studied, namely fluctuations uncorrelated in space and time, and global fluctuations, which are constant in space but uncorrelated in time. We show that the global fluctuations shift the Turing bifurcation threshold. This shift is proportional to the fluctuation variance. Applications to a neural field equation and the Swift-Hohenberg equation reveal the shift of the bifurcation to larger control parameters, which represents a stabilization of the system. All analytical results are confirmed by numerical simulations of the occurring mode equations and the full stochastic integral-differential equation. To gain some insight into experimental manifestations, the sum of uncorrelated and global additive fluctuations is studied numerically and the analytical results on global fluctuations are confirmed qualitatively.     

On the solutions and the steady states of a master equation

A complete characterization of the time behavior of the means and variance of a stochastic process which is generated by a finite number of independent systems is presented based on the master equation for the conditional probability. It is found that the means and variance relax to a steady state and that the steady state will be independent of the initial state if and only if a matrix related to the transition matrix is nonsingular. Finally, the result that the variance approaches its steady-state form at twice the rate of the means is shown to depend on the nonsingularity of the same matrix.     

The van Kampen expansion for the Fokker-Planck equation of a duffing oscillator

In Rodríguez and van Kampen's 1976 paper a method of extracting information from the Fokker-Planck equation without having to solve the equation is outlined. The Fokker-Planck equation for a Duffing oscillator excited by white noise is expanded about the intensity of the forcing function. This expansion is carried to order . However, no studies are made of the effects of the order of the expansion or variation of the parameters, nor are comparisons made to experimental results. In the present paper, the expansion is carried to a higher order, , results are presented and compared to Monte Carlo experiments using both white and colored noise, and parametric studies are performed on the intensity of the forcing function and the damping coefficient. It is found that the expansion method works well for the case of white noise and for colored noise where the correlation time is less than 0.1 sec, but fails to give certain details. It is also found that the system behaves as expected when the parameters are varied.     

The first-principle study on the equation of state of HMX under high pressure

The equation of states (EOS) of high energy explosive HMX (octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine) has been studied by using the first principle method. Our results include the lattice constants, elastic constants, and the dependence of total energy and pressure on volume for βand δ-HMX. The calculated elastic constants and the pressure-volume relationship of β-HMX are also compared with the experimental values. The theoretical tensile experiments are implemented on the β-HMX. The atomic-scale ana...     

The van Kampen expansion for the Fokker-Planck equation of a duffing oscillator excited by colored noise

In Rodríguez and van Kampen's 1976 paper a method of extracting information from the Fokker-Planck equation without having to solve the equation is outlined. The Fokker-Planck equation for a Duffing oscillator excited by white noise is expanded about the intensity of the forcing function. In Weinstein and Benaroya, the effect of the order of expansion is investigated by carrying the expansion to a higher order. The effect of varying the system parameters is also investigated. All results are verified by comparison to Monte Carlo experiments. In this paper, the van Kampen expansion is modified and applied to the case of a Duffing oscillator excited by colored noise. The effect of the correlation time is investigated. Again the results are compared to those of Monte Carlo experiments. It is found that the expansion compares closely with those of the Monte Carlo experiments as the correlation time _c is varied from 0.001 to 10 sec. Examination of the results reveals that the colored noise can be categorized in one of four ways: (1) for the noise can be considered as white for all intents and purposes, (2) for the noise can be considered white for some purposes, (3) for the correlated nature of the noise must be considered in an analysis, and (4) for the noise can be considered as deterministic.     

Dynamic Behavior of a Spin- 1 Ising Model. I. Relaxation of Order Parameters and the “Flatness” Property of Metastable States

The dynamic behavior of a spin-1 Ising system with arbitrary bilinear and biquadratic pair interactions is studied by using the path probability method, and approaches of the system toward the stable or metastable equilibrium states according to the ratio of interaction parameters and rate constants are presented. In particular, we investigate the relaxation of the order parameters for temperatures less than, equal to, and greater than the second-order and first-order phase transitions. From this investigation, the “flatness” property of metastable states is seen explicitly. We also show how a system freezes in a metastable state as well as how it escapes from one metastable state to the other.     

Comments on the mean spherical approximation for hard-core and soft potentials and an application to the one-component plasma

The thermodynamic properties of the mean spherical (MSA), Percus-Yevick (PY), and hypernetted-chain (HNC) approximations are derived by a simple and unified approach by considering the RPA free-energy functionalF and employing an Ewald-type identity. It is demonstrated that with decreasing relative contribution of the hard-core insertion to the thermodynamic functions, the MSA changes its nature from PY-like to HNC-like, withF changing its role from excess pressure to excess free energy, respectively. It is found that the condition of continuity of the MSA pair functions is equivalent to a stationarity condition forF and leads to thermodynamic consistency between the virial and energy equations of state for the (thus defined) soft-MSA (SMSA), withF playing the role of the excess free energy. It is shown that the PY-compressibility and virial equations of state forD-dimensional hard spheres may be simply obtained one from the other without knowing any details of the solution of the model. Using this relation we find an indication that the PY approximation for hard spheres becomes less accurate with increasing dimensionality. A general variational formulation is presented for the application of the MSA for soft potentials, and results for the one-component plasma are discussed and extended.On sabbatical leave from the Nuclear Research Center-Negev, P.O. Box 9001, Beer Sheva, Israel.     

The free energy and the equation of state of a hard-sphere system for homogeneous stable and metastable phases

A new free energy equation is deduced; it serves as a basis for the derivation of an equation for the homogeneous phase of a hard-sphere system. It exactly reproduces all the known virial coefficients and agrees with the molecular dynamics data within their accuracy. The absence of singular points during transition from the stable region to the metastable region is shown. According to the properties of the statistical integral, the free energy and the equation of state have a singularity at close packing.     

n维相对论粒子的量子态密度及其应用

利用半经典近似的方法,推导出了n维相对论自由粒子量子态密度以及黑体辐射中两类粒子的辐射公式的统一表达式,并且给出了低维情况下的表达式,所得结果与三维空间的预期结果一致.这对于深化低维体系的教学与研究具有重要的意义.     

The kinetic equation and the equation of momentum transfer for a plasma

An attempt is made to derive a simple form of the collision integral of the kinetic equation for a plasma, by using Rostoker’s equation which expresses the pair correlation function in terms of the distribution functions of the particles, and the conditional probability of one particle shielding the other. The conditional probability function is assumed to be given by its equilibrium value. By taking first order velocity-moment of the resulting kinetic equation, the equation of momentum transfer has been obtained.     

Effective quantum efficiency in the pulsed homodyne detection of a n-photon state

Homodyne detection can be used to perform measurements on various quantum states of the light, such as conditional single photon states produced by parametric fluorescence processes. In the pulsed regime, the time and frequency overlap between the single photon wave packet and the local oscillator field plays a crucial role. We show in this paper that this overlap can be characterized by an effective quantum efficiency, which is explicitly calculated in various situations of experimental interest. Received 27 July 2000 and Received in final form 29 November 2000