Nonideal Bose Gases: Correlation Inequalities and Bose Condensation

We consider two simple model systems describing effective repulsion in a nonideal Bose gas. The interaction Hamiltonians in these systems can be analytically represented as functions of the occupation number operators for modes with nonzero momenta (p0). One of these models contains an interaction term corresponding to repulsion of bosons with the mode p=0 and ensuring the thermodynamic superstability of the system; the other model does not contain such a term. We use the Bogoliubov–Dirac–Ginibre approximation and the method of correlation inequalities to prove that a Bose condensate can exist in these model systems. Because of the character of interaction, the condensate can be formed in the superstable case for any values of the spatial dimensions, temperature, and positive chemical potentials.     

Evolution in a Gaussian Random Field

We consider an evolution process in a Gaussian random field V(q) with the mean ‹V(q)› = 0 and the correlation function W(|q – q|) ‹V(q)V(q)›, where q ^d and d is the dimension of the Euclidean space ^d . For the value ‹G(q,t;q ₀)›, t > 0, of the Green's function of the evolution equation averaged over all realizations of the random field, we use the Feynman–Kac formula to establish an integral equation that is invariant with respect to a continuous renormalization group. This invariance property allows using the renormalization group method to find an asymptotic expression for ‹G(q,t;q ₀)› as |q – q ₀| and t .     

The infrared bounds method in the study of boson systems

This is a study of the equilibrium thermodynamics of a lattice boson gas with on-site repulsion and nearest-neighbor site attraction. For this system, the existence of a Bose condensate is proved and an equation for the lower estimate of the critical temperature is obtained. Moreover, the upper and lower bounds for the structure factor are derived. Finally, in the framework of the infrared bounds method, a Huang-Davies-type Bose system is studied.Republished from Teoreticheskaya i Matematicheskaya Fizika, Vol. 108, No. 3, pp. 421–430, September, 1996.     

Superconductivity in two-band systems with a small concentration of carriers

Superconductors with two characteristic features of high-temperature superconducting materials, i.e., a small concentration of charge carriers and overlapping energy bands on the Fermi surface, are considered. At T=0, the order parameters _n and the chemical potential for the Bose condensate of local pairs (<0) is determined in the mean field approximation. The equation for the bound state energy _b is obtained and the relationship _b=2 is established. An application of the path integral method to the two-band model is developed and on this basis, the transition from the Fermi to the Bose pattern of elementary excitations in the presence of a two-particle bound state in the system is demonstrated. The expression for the temperature of Bose condensation T_k is obtained and the contribution of the residual boson interactions is estimated for systems with different dimensions.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 1, pp. 137–151, October, 1996.     

Heat Flow for the Yang–Mills–Higgs Field and the Hermitian Yang–Mills–Higgs Metric

For a parameter > 0, we study a type of vortex equations, which generalize the well-known Hermitian–Einstein equation, for a connection A and a section of a holomorphic vector bundle E over a Kähler manifold X. We establish a global existence of smooth solutions to heat flow for a self-dual Yang–Mills–Higgs field on E. Assuming the -stability of (E, ), we prove the existence of the Hermitian Yang–Mills–Higgs metric on the holomorphic bundle E by studying the limiting behaviour of the gauge flow.     

Closer estimators of a common mean in the sense of Pitman

Consider the problem of estimating the common mean of two normal populations with different unknown variances. Suppose a random sample of sizem is drawn from the first population and a random sample of sizen is drawn from the second population. The paper gives a family of estimators closer than the sample mean of the first population in the sense of Pitman (1937,Proc. Cambridge Phil. Soc.,33, 212–222). In particular, the Graybill-Deal estimator (1959,Biometrics,15, 543–550) is shown to be closer than each of the sample means ifm5 andn5.     

Generating Random Elements in SL n (F q ) by Random Transvections

This paper studies a random walk based on random transvections in SL _n(F _q ) and shows that, given > 0, there is a constant c such that after n + c steps the walk is within a distance from uniform and that after n – c steps the walk is a distance at least 1 – from uniform. This paper uses results of Diaconis and Shahshahani to get the upper bound, uses results of Rudvalis to get the lower bound, and briefly considers some other random walks on SL _n(F _q ) to compare them with random transvections.     

Large deviations from a macroscopic scaling limit for particle systems with Kac interaction and random potential

We consider a lattice gas in a periodic d-dimensional lattice of width γ⁻¹, γ>0, interacting via a Kac's type interaction, with range and strength γ^d, and under the influence of a random one body potential given by independent, bounded, random variables with translational invariant distribution. The system evolves through a conservative dynamics, i.e. particles jump to nearest neighbor empty sites, with rates satisfying detailed balance with respect to the equilibrium measures. In [M. Mourragui, E. Orlandi, E. Saada, Macroscopic evolution of particles systems with random field Kac interactions, Nonlinearity 16 (2003) 2123–2147] it has been shown that rescaling space as γ⁻¹ and time as γ⁻², in the limit γ→0, for dimensions d3, the macroscopic density profile ρ satisfies, a.s. with respect to the random field, a non-linear integral partial differential equation, having the diffusion matrix determined by the statistical properties of the external random field. Here we show an almost sure (with respect to the random field) large deviations principle for the empirical measures of such a process. The rate function, which depends on the statistical properties of the external random field, is lower semicontinuous and has compact level sets.     

Some examples of random walks on free products of discrete groups

Summary We consider the random walk (X_n) associated with a probability p on a free product of discrete groups. Knowledge of the resolvent (or Green's function) of p yields theorems about the asymptotic behaviour of the n-step transition probabilities p^*n(x)=P(X_n= x¦ X₀=e) as n. Woess [15], Cartwright and Soardi [3] and others have shown that under quite general conditions there is behaviour of the type p^*n(x)C_x^– n n^– 3/2. Here we show on the other hand that if G is a free product of m copies ofZ ^r and if (X_n) is the « average » of the classical nearest neighbour random walk on each of the factorsZ ^r, then while it satisfies an « n^–3/2 — law » for r small relative to m, it switches to an n^– r/2 -law for large r. Using the same techniques, we give examples of irreducible probabilities (of infinite support) on the free groupZ ^*m which satisfyn ^– for .     

On the Lattice Approximation for Certain Generalized Vector Markov Fields in Four Spacetime Dimensions

We extend previous results by Albeverio, Iwata and Schmidt on the construction of a convergent lattice approximation for invariant scalar 3-vector generalized random fields F of an infinitely divisible type and apply them to the construction of convergent lattice approximation for the generalized random vector field A determined by the stochastic quaternionic Cauchy–Riemann equation A = F.     

A New Microscopic Theory of Superfluidity at All Temperatures

Following the program suggested in [1], we propose a new microscopic theory of superfluidity for all temperatures and densities. In particular, the corresponding phase diagram of this theory exhibits: (i) a thermodynamic behavior corresponding to the Perfect Bose Gas for small densities or high temperatures, (ii) the Landau-type excitation spectrum in the presence of non-conventional Bose condensation for high densities or small temperatures, (iii) a depletion of the Bose condensate with the formation of Cooper-type pairs, even at zero-temperature (experimentally, an estimate of the fraction of condensate in liquid ⁴He at T=0 K is 9%, see [2, 3]). In contrast to Bogoliubovs last approach and while warning that the full interacting Hamiltonian is truncated, the analysis performed here is rigorous by involving a complete thermodynamic analysis of a non-trivial continuous gas in the canonical ensemble. Communicated by Vincent Pasquier Submitted 31/03/03, accepted 30/01/04     

On the approximation of continuous time threshold ARMA processes

Threshold autoregressive (AR) and autoregressive moving average (ARMA) processes with continuous time parameter have been discussed in several recent papers by Brockwellet al. (1991,Statist. Sinica,1, 401–410), Tong and Yeung (1991,Statist. Sinica,1, 411–430), Brockwell and Hyndman (1992,International Journal Forecasting,8, 157–173) and Brockwell (1994,J. Statist. Plann. Inference,39, 291–304). A threshold ARMA process with boundary width 2>0 is easy to define in terms of the unique strong solution of a stochastic differential equation whose coefficients are piecewise linear and Lipschitz. The positive boundary-width is a convenient mathematical device to smooth out the coefficient changes at the boundary and hence to ensure the existence and uniqueness of the strong solution of the stochastic differential equation from which the process is derived. In this paper we give a direct definition of a threshold ARMA processes with =0 in the important case when only the autoregressive coefficients change with the level of the process. (This of course includes all threshold AR processes with constant scale parameter.) The idea is to express the distributions of the process in terms of the weak solution of a certain stochastic differential equation. It is shown that the joint distributions of this solution with =0 are the weak limits as 0 of the distributions of the solution with >0. The sense in which the approximating sequence of processes used by Brockwell and Hyndman (1992,International Journal Forecasting,8, 157–173) converges to this weak solution is also investigated. Some numerical examples illustrate the value of the latter approximation in comparison with the more direct representation of the process obtained from the Cameron-Martin-Girsanov formula. It is used in particular to fit continuous-time threshold models to the sunspot and Canadian lynx series.Research partially supported by National Science Foundation Research Grants DMS 9105745 and 9243648.     

A Singular Equation of Length Four Over Groups

Let G be a group and t an unknown. In this paper we prove that the equation atbtct^–1dt^–1 = 1 (a,b,c,d G, a² 1, c² 1, bd 1) has a solution over G. This forms part of a program to investigate precisely when an equation, whose associated star graph contains no admissible paths of length less than 3, fails to have a solution over G.2000 Mathematics Subject Classification: 20E06     

Measurable solutions of a (2, 2)-type sum form functional equation

Summary In this paper we find the general measurable solutions of the functional equationF(xy) + F(x(1 – y)) – F((1 – x)y) – F((1 – x)(1 – y)) = G(x)H(y) (x, y ]0, 1[) whereF, G, H:]0, 1[ C are unknown functions. The solution of this equation is part of our program to determine the measurable solutions of the functional equationF ₁₁ (xy) + F ₁₂ (x(1 – y)) + F ₂₁ ((1 – x)y) + F ₂₂ ((1 – x)(1 – y)) = G(x)H(y) (x, y ]0, 1[). Our method of solution is based on the structure theorem of sum form equations of (2, 2)-type and on a result of B. Ebanks and the author concerning the linear independence of certain functions.     

Diffusions in random environment and ballistic behavior

In this article we investigate the ballistic behavior of diffusions in random environment. We introduce conditions in the spirit of (T) and (T^′) of the discrete setting, cf. [A.-S. Sznitman, On a class of transient random walks in random environment, Ann. Probab. 29 (2) (2001) 723–764; A.-S. Sznitman, An effective criterion for ballistic behavior of random walks in random environment, Probab. Theory Related Fields 122 (4) (2002) 509–544], that imply, when d2, a law of large numbers with non-vanishing limiting velocity (which we refer to as ‘ballistic behavior’) and a central limit theorem with non-degenerate covariance matrix. As an application of our results, we consider the class of diffusions where the diffusion matrix is the identity, and give a concrete criterion on the drift term under which the diffusion in random environment exhibits ballistic behavior. This criterion provides examples of diffusions in random environment with ballistic behavior, beyond what was previously known.     

Bose–Einstein Condensates with Non-classical Vortex

Coupled systems of nonlinear Schrödinger equations have been used extensively to describe Bose–Einstein condensates. In this paper, we study a two-component Bose–Einstein condensate (BEC) with an external driving field in a three-dimensional space. This model gives rise to a new kind of vortex–filaments, with fractional degree and nontrivial core structure. We show that vortex–filaments is 1-rectifiable set, and calculate its mean curvature in the strong coupling (Thomas–Fermi) limit. In particular, we show that large strength of the external driving field causes vortex–filaments for a two-component BEC.     

Gated Polling Systems with Lévy Inflow and Inter-Dependent Switchover Times: A Dynamical-Systems Approach

We study asymmetric polling systems where: (i) the incoming workflow processes follow general Lévy-subordinator statistics; and, (ii) the server attends the channels according to the gated service regime, and incurs random inter-dependentswitchover times when moving from one channel to the other. The analysis follows a dynamical-systems approach: a stochastic Poincaré map, governing the one-cycle dynamics of the polling system is introduced, and its statistical characteristics are studied. Explicit formulae regarding the evolution of the mean, covariance, and Laplace transform of the Poincaré map are derived. The forward orbit of the maps transform – a nonlinear deterministic dynamical system in Laplace space – fully characterizes the stochastic dynamics of the polling system. This enables us to explore the long-term behavior of the system: we prove convergence to a (unique) steady-state equilibrium, prove the equilibrium is stationary, and compute its statistical characteristics.     

Monotone Methods for Markovian Equilibrium in Dynamic Economies

In this paper, we provide an overview of an emerging class of monotone map methods in analyzing distorted equilibrium in dynamic economies. In particular, we focus on proving the existence and characterization of competitive equilibrium in non-optimal versions of the optimal growth models. We suggest two alternative methods: an Euler equation method for a smooth, strongly concave environment, and a value function method for a non-smooth supermodular environment. We are able to extend this analysis to study models that allow for unbounded growth or a labor–leisure choice.     

The Vortex Filament Equation as a Pseudorandom Generator

In this paper, we consider the evolution of the so-called vortex filament equation (VFE),

$$\mathbf{X}_t = \mathbf{X}_s\wedge\mathbf{X}_{ss}, $$

taking a planar regular polygon of M sides as initial datum. We study VFE from a completely novel point of view: that of an evolution equation which yields a very good generator of pseudorandom numbers in a completely natural way. This essential randomness of VFE is in agreement with the randomness of the physical phenomena upon which it is based.