Homogeneous anisotropic cosmological models with variable gravitational and cosmological “Constants”

The Einstein field equations with perfect fluid source and variable andG for Bianchi-type universes are studied under the assumption of a power-law time variation of the expansion factor, achieved via a suitable power-law assumption for the Hubble parameter suggested by M. S. Berman. All the models have a power-law variation of pressure and density and are singular at the epocht=0. The variation ofG(t) as 1/t and (t) as 1/t ² is consistent with these models.     

Compressible Flow: Turbulence at the Surface

In a study of compressible flow, we have tracked the motion of particles that float on a turbulent body of water. The second moment of longitudinal velocity differences scales as in incompressible flow. However the separation R ²(t) of particle pairs does not vary in time according to the Richardson–Kolmogorov prediction R ²(t)t ³. As expected, the self diffusion d ²(t) shows a crossover between ballistic motion d ²(t)t ² at small t and uncorrelated motion d ²(t)t in the longtime limit.     

A new class of long-tailed pausing time densities for the CTRW

We present some asymptotic results for the family of pausing time densities having the asymptotic (t) property(t) [t ln¹⁺(t/T)]^–1. In particular, we show that for this class of pausing time densities the mean-squared displacement r ²(t) is asymptotically proportional to ln(t/T), and the asymptotic distribution of the displacement has a negative exponential form.     

Arbitrarily slow decay of correlations in quasiperiodic systems

For diffusive motion in random media it is widely believed that the velocity autocorrelation functionc(t) exhibits power law decay as time;t. We demonstrate that the decay ofc(t) in quasiperiodic media can be arbitrarily slow within the class of integrable functions. For example, ind=1 with a potentialV(x)=cosx+coskx, there is a dense set of irrationalk's such that the decay ofc(k, t) is slower than 1/t ⁽¹⁺⁾ for any>0. The irrationals producing such a slow decay ofc(k, t) arevery well approximated by rationals.     

Effect of disorder on relaxation in the one-dimensional Glauber model

We study the long-time relaxation of magnetization in a disordered linear chain of Ising spins from an initially aligned state. The coupling constants are ferromagnetic and nearest-neighbor only, taking valuesJ ₀ andJ ₁ with probabilitiesp and 1–p, respectively. The time evolution of the system is governed by the Glauber master equation. It is shown that for large timest, the magnetizationM(t) varies as [exp(–₀ t](t), where ₀ is a function of the stronger bond strengthJ ₀ only, and (t) decreases slower than an exponential. For very long times, we find that ln (t) varies as –t ^1/3. For low enough temperatures, there is an intermediate time regime when ln (t) varies as –t ^1/2. The results can be extended to more general probability distributions of ferromagnetic coupling constants, assuming thatM(t) can only increase if any bond in the chain is strengthened. If the coupling constants have a continuous distribution in which the probability density varies as a power law near some maximum valueJ ₀, we find that ln (t) varies as –t ^1/3(lnt)^2/3 for large times.     

The inverses-wave scattering problem for a class of potentials depending on energy

The inverse scattering problem is considered for the radials-wave Schrödinger equation with the energy-dependent potentialV ⁺(E,x)=U(x)+2 Q(x). (Note that this problem is closely related to the inverse problem for the radials-wave Klein-Gordon equation of zero mass with a static potential.) Some authors have already studied it by extending the method given by Gel'fand and Levitan in the caseQ=0. Here, a more direct approach generalizing the Marchenko method is used. First, the Jost solutionf ⁺(E,x) is shown to be generated by two functionsF ⁺(x) andA ⁺(x,t). After introducing the potentialV ^–(E,x)=U(x)–2 Q(x) and the corresponding functionsF ^–(x) andA ^–(x,t), fundamental integral equations are derived connectingF ⁺(x),F ^–(x),A ⁺(x,t) andA ^–(x,t) with two functionsz ⁺(x) andz ^–(x);z ⁺(x) andz ^–(x) are themselves easily connected with the binding energiesE _n ⁺ and the scattering matrixS ⁺(E),E>0 (the input data of the inverse problem). The inverse problem is then reduced to the solution of these fundamental integral equations. Some specific examples are given. Derivation of more elaborate results in the case of real potentials, and applications of this work to other inverse problems in physics will be the object of further studies.Physique Mathématique et Théorique, Equipe de recherche associée au C.N.R.S.     

t1/3 Superdiffusivi ty of Fini te-Range Asymme tric Exclusion Processes on $${\ma thbb{Z}}$$

We consider finite-range asymmetric exclusion processes on with non-zero drift. The diffusivity D(t) is expected to be of . We prove that D(t) ≥ Ct ^1/3 in the weak (Tauberian) sense that as . The proof employs the resolvent method to make a direct comparison with the totally asymmetric simple exclusion process, for which the result is a consequence of the scaling limit for the two-point function recently obtained by Ferrari and Spohn. In the nearest neighbor case, we show further that tD(t) is monotone, and hence we can conclude that D(t) ≥ Ct ^1/3(log t)^−7/3 in the usual sense. Supported by the Natural Sciences and Engineering Research Council of Canada. Partially supported by the Hungarian Scientific Research Fund grants T37685 and K60708.     

Macroscopic Determinism in Interacting Systems Using Large Deviation Theory

We consider the quasi-deterministic behavior of systems with a large number, n, of deterministically interacting constituents. This work extends the results of a previous paper [J. Statist. Phys. 99:1225–1249 (2000)] to include vector-valued observables on interacting systems. The approach used here, however, differs markedly in that a level-1 large deviation principle (LDP) on joint observables, rather than a level-2 LDP on empirical distributions, is employed. As before, we seek a mapping _t on the set of (possibly vector-valued) macrostates such that, when the macrostate is given to be a ₀ at time zero, the macrostate at time t is _t (a ₀) with a probability approaching one as n tends to infinity. We show that such a map exists and derives from a generalized dynamic free energy function, provided the latter is everywhere well defined, finite, and differentiable. We discuss some general properties of _t relevant to issues of irreversibility and end with an example of a simple interacting lattice, for which an exact macroscopic solution is obtained.     

General formulae for polarization observables in two-body break-up of deuteron photodisintegration

The formal expressions of all possible polarization observables ind(,N)N with polarized photons and oriented deuterons are derived in terms of thet-matrix elements. Furthermore, using the multipole expansion of thet-matrix, all observables are expanded in terms of Legendre polynomials or associated functions, the coefficients of which are given as bilinear forms of the multipole moments and allow a model independent analysis of experimental data.Supported by the Deutsche Forschungsgemeinschaft (SFB 201)     

On the Superdiffusive Behavior of Passive Tracer with a Gaussian Drift

In the present article we consider a motion of a passive tracer particle, whose trajectory satisfies the Itô stochastic differential equation d x(t) = V(t, x(t)) dt + d w(t), where w(·) is a Brownian motion, V is a stationary Gaussian random field with incompressible realizations independent of w(·) and >0. We prove the superdiffusive character of the motion under certain conditions on the energy spectrum of the velocity field. The result is shown both for steady (time independent) and time dependent and Markovian velocity fields. In addition, we provide explicit upper and lower bounds for the Hurst exponent of the trajectory. All previous rigorous results concerned explicitely solvable shear flows cases.     

Imaginary-time path integral for a relativistic spinless particle in an electromagnetic field

A rigorous path integral representation of the solution of the Cauchy problem for the pure-imaginary-time Schrödinger equation _t (t, x)=–[H–mc ²](t,x) is established.H is the quantum Hamiltonian associated, via the Weyl correspondence, with the classical Hamiltonian [(cp–eA(x))²+m ² c ⁴]^1/2+e(x) of a relativistic spinless particle in an electromagnetic field. The problem is connected with a time homogeneous Lévy process.     

A limit theorem for stochastic acceleration

We consider the motion of a particle in a weak mean zero random force fieldF, which depends on the position,x(t), and the velocity,v(t)= (t). The equation of motion is (t)=F(x(t),v(t), ), wherex(·) andv(·) take values in ^d ,d3, and ranges over some probability space. We show, under suitable mixing and moment conditions onF, that as 0,v (t)v(t/²) converges weakly to a diffusion Markov processv(t), and ² x (t) converges weakly to , wherex=lim ² x (0).     

Diffusion in a periodic potential with a local perturbation

We consider the diffusion of a particle at X_t in a drift field derived from a smooth potential of the formV+B, whereV is periodic andB is a bump of compact support. With no bump,B=0, the mean squared displacementE(t) E |X _t – X₀|² =D(V)t +C +O(e ^–t ),>0, in any dimension. WhenB0, we establish in one dimension the asymptotic expansion , 0, ast. Our analysis relies on the Nash estimates developed in previous work for the transition density of the process and their consequences for the analytic structure,of the Laplace transform ofE(t).     

Quantization and instability of the damped harmonic oscillator subject to a time-dependent force

We consider the one-dimensional motion of a particle immersed in a potential field U(x) under the influence of a frictional (dissipative) force linear in velocity () and a time-dependent external force (K(t)). The dissipative system subject to these forces is discussed by introducing the extended Bateman’s system, which is described by the Lagrangian: which leads to the familiar classical equations of motion for the dissipative (open) system. The equation for a variable y is the time-reversed of the x motion. We discuss the extended Bateman dual Lagrangian and Hamiltonian by setting specifically for a dual extended damped–amplified harmonic oscillator subject to the time-dependent external force. We show the method of quantizing such dissipative systems, namely the canonical quantization of the extended Bateman’s Hamiltonian ?. The Heisenberg equations of motion utilizing the quantized Hamiltonian surely lead to the equations of motion for the dissipative dynamical quantum systems, which are the quantum analog of the corresponding classical systems. To discuss the stability of the quantum dissipative system due to the influence of an external force K(t) and the dissipative force, we derived a formula for transition amplitudes of the dissipative system with the help of the perturbation analysis. The formula is specifically applied for a damped–amplified harmonic oscillator subject to the impulsive force. This formula is used to study the influence of dissipation such as the instability due to the dissipative force and/or the applied impulsive force.     

Optimal Gaussian solutions of nonlinear stochastic partial differential equations

We present a linearization procedure of a stochastic partial differential equation for a vector field (X _i (t,x)) (t[0, ),xR ^d ,i=l,...,n): _t X _i (t,x)=b _i (X(t, x)) +D, X _i (t, x) + _i f _i (t, x). Here is the Laplace-Beltrami operator inR ^d , and (f _i (t,x)) is a Gaussian random field with f _i (t,x)f _j (t,x) = _ij (t – t)(x – x). The procedure is a natural extension of the equivalent linearization for stochastic ordinary differential equations. The linearized solution is optimal in the sense that the distance between true and approximate solutions is minimal when it is measured by the Kullback-Leibler entropy. The procedure is applied to the scalar-valued Ginzburg-Landau model in R¹ withb ₁(z) =z - vz ³. Stationary values of mean, variance, and correlation length are calculated. They almost agree with exact ones if 1.24 ( ² ₁ ⁴ /D ₁ ^1/3:= _c . When _c , there appear quasistationary states fluctuating around one of the bottoms of the potentialU(z) = b ₁(z)dz. The second moment at the quasistationary states almost agrees with the exact one. Transient phenomena are also discussed. Half-width at half-maximum of a structure function decays liket ^–1/2 for small t. The diffusion term _x ² X accelerates the relaxation from the neighborhood of an unstable initial stateX(0,x) 0.     

Exactly solvable models through the generalized empty interval method,for multi-species interactions

Multi-species reaction-diffusion systems, with nearest-neighbor interaction on a one-dimensional lattice are considered. Necessary and sufficient constraints on the interaction rates are obtained, that guarantee the closedness of the time evolution equation for E _n(t)'s, the expectation value of the product of certain linear combination of the number operators on n consecutive sites at time t. The constraints are solved for the single-species left-right-symmetric systems. Also, examples of multi-species system for which the evolution equations of E _n(t)'s are closed, are given. Received 25 September 2002 / Received in final form 3 December 2002 Published online 14 February 2003 RID="a" ID="a"e-mail: mamwad@iasbs.ac.ir     

Existence of a homoclinic point for the Hénon map

We prove analytically that for the Hénon map of the plane into itself (s, t)(t+1–1.4a ², 0.3s), there exists a transversal homoclinic point.     

Kinetic processes of muonium formation by thermalized electrons in the spur model

Diffusion of a thermalized electron in the field of a Coulomb centre is considered. The time dependence of the probability for the electron not to be captured by the Coulomb centre ( ⁺) at the momentt,P _D (t), is obtained. The quantityP _D (t) coincides with the precession amplitude at the muon frequency and is determined by the parameter/l _CT ( is an averaged distance from an electron to a muon att=0 andl _CT =e ²/(2T) is the Onsager length) that reflects the relation between electron potential and kinetic energies at t=0. When/l _CT –1 the precession amplitudeP _D (t) decreases up to 30% during the timel _CT ² /D-(D-is the diffusion coefficient) and then it becomes almost constant. The dependence ofP _D () on/l _CT is presented.     

A supersymmetric transfer matrix and differentiability of the density of states in the one-dimensional Anderson model

LetH=–+V onl ²(), whereV(x),x, are i.i.d.r.v.'s with common probability distributionv. Leth(t)=e ^–itv dv(v) and letk(E) be the integrated density of states. It is proven: (i) Ifh isn-times differentiable withh ^(j)(t)=O((1+|t|)^–) for some >0,j=0, 1, ...,n, thenk(E) is aC ⁿ function. In particular, ifv has compact support andh(t)=O((1+|t|)^–) with >0, thenk(E) isC . This allowsv to be singular continuous. (ii) Ifh(t)=O(e ^–|t|) for some >0 thenk(E) is analytic in a strip about the real axis.The proof uses the supersymmetric replica trick to rewrite the averaged Green's function as a two-point function of a one-dimensional supersymmetric field theory which is studied by the transfer matrix method.Research partially supported by the NSF under grant MC-8301889     

Expectation Values of Observables in Time-Dependent Quantum Mechanics

Let U(t) be the evolution operator of the Schrödinger equation generated by a Hamiltonian of the form H ₀(t) + W(t), where H ₀(t) commutes for all twith a complete set of time-independent projectors . Consider the observable A=_j P _jjwhere _j j , >0, for jlarge. Assuming that the matrix elements of W(t) behave as for p>0 large enough, we prove estimates on the expectation value for large times of the type where >0 depends on pand . Typical applications concern the energy expectation H₀(t) in case H ₀(t) H ₀or the expectation of the position operator x²(t) on the lattice where W(t) is the discrete Laplacian or a variant of it and H ₀(t) is a time-dependent multiplicative potential.