Rate equations in the hopping transport

The usual kinetic equations for the site occupation probabilities in an external field are solved exactly in a simple one-dimensional periodic model with two kinds of atoms using a) free boundary conditions and order of limitsN, 0 needed for a proper treatment of the dc conductivity here b) boundary conditions with metallic contacts and order of limitsN, 0 and c) the same boundary conditions but reversed order of limiting processes 0,N typical of e.g. numerical and percolation treatments. (N and are the number of sites and frequency.) It is demonstrated that though the bulk dc conductivity is the same in all three cases, local bulk properties of the material are strongly dependent on the régime used. The role of the order of all three limiting processes 0,N+ andn+ (N–n+) for local shifts of the chemical potential _n in the dc limit is examined (n is the number of the relevant site calculated from a boundary of the chain). It is shown especially that the rate equation treatment (régime a) on the one hand and numerical or percolation treatments (régime c) on the other hand never yield the same bulk values of _r.     

Variance calculations and the Bessel kernel

In the Laguerre ensembleof n xN Hermitian matrices, it is of interest both theoretically and for applications to quantum transport problems to compute the variance of a linear statistic, denoted var_N f, asN . Furthermore, this statistic often contains an additional parameter a for which the limit is most interesting and most difficult to compute numerically. We derive exact expressions for both lim_N var_N f and lim , lim_N var_N f.     

Characterization of the Weyl solutions

For the Weyl solutions(z, x) of the Schrödinger and Dirac equations, asymptotics for |z| are obtained. This gives a possibility of selecting Weyl solutions by their behaviour when |z| . Some applications are given.     

Absence of asymptotic-time symmetry breaking in zero-bias spin-boson model with partial relaxation

The symmetric spin-boson model without external field is treated for any type of coupling to the boson bath and any initial bath density matrix. With initially fully aligned spin (_z (0)= =1), the proof is given that a partial relaxation (_z (+) t1<) implies that there is no asymptotic-time (up-and-down) symmetry breaking (i.e. that _z (+)=0). For the problem of a particle (interacting with free bosons) in a symmetric double well without spatial symmetry breaking before the infinite time limit, this means that att + the particle distribution becomes symmetric (irrespective of the full initial asymmetry) unless the particle fully remains (att + ) in Ihe starting well.     

The Percolation Transition for the Zero-Temperature Stochastic Ising Model on the Hexagonal Lattice

On the planar hexagonal lattice , we analyze the Markov process whose state (t), in , updates each site v asynchronously in continuous time t0, so that _v (t) agrees with a majority of its (three) neighbors. The initial _v (0)'s are i.i.d. with P[ _v (0)=+1]=p[0,1]. We study, both rigorously and by Monte Carlo simulation, the existence and nature of the percolation transition as t and p1/2. Denoting by ⁺(t,p) the expected size of the plus cluster containing the origin, we (1) prove that ⁺(,1/2)= and (2) study numerically critical exponents associated with the divergence of ⁺(,p) as p1/2. A detailed finite-size scaling analysis suggests that the exponents and of this t= (dependent) percolation model have the same values, 4/3 and 43/18, as standard two-dimensional independent percolation. We also present numerical evidence that the rate at which (t)() as t is exponential.     

Trapping and cascading of eigenvalues in the large coupling limit

We consider eigenvaluesE of the HamiltonianH =–+V+W,W compactly supported, in the limit. ForW0 we find monotonic convergence ofE to the eigenvalues of a limiting operatorH (associated with an exterior Dirichlet problem), and we estimate the rate of convergence for 1-dimensional systems. In 1-dimensional systems withW0, or withW changing sign, we do not find convergence. Instead, we find a cascade phenomenon, in which, as , each eigenvalueE stays near a Dirichlet eigenvalue for a long interval (of lengthO( )) of the scaling range, quickly drops to the next lower Dirichlet eigenvalue, stays there for a long interval, drops again, and so on. As a result, for most large values of the discrete spectrum ofH is close to that ofE , but when reaches a transition region, the entire spectrum quickly shifts down by one. We also explore the behavior of several explicit models, as .Max Kade Foundation FellowPartially supported by USNSF under Grant DMS-8416049On leave of absence from Department of Mathematics and Statistics, Case Western Reserve University, Cleveland, OH 44106, USA. Partially supported by USNSF under Grant DMS-8620231 and the Case Institute of Technology, RIG     

On the asymptotic behavior of the solutions of the Caldirola-Kanai equation

We study in this Letter the asymptotic behavior, as t+, of the solutions of the one-dimensional Caldirola-Kanai equation for a large class of potentials satisfying the condition V(x)+ as |x|. We show, first of all, that if I is a closed interval containing no critical points of V, then the probability P _t (t) of finding the particle inside I tends to zero as t+. On the other hand, when I contains critical points of V in its interior, we prove that P _t (t) does not oscillate indefinitely, but tends to a limit as t+. In particular, when the potential has only isolated critical points x ₁, ..., x _N our results imply that the probability density of the particle tends to in the sense of distributions.Supported by Fulbright-MEC grant 85-07391.     

The equilibrium thermodynamics of a spin-boson model

We consider the equilibrium thermodynamics of a Dicke-type model forN identical spins of arbitrary magnitude interacting linearly and homogeneously with a boson field in a volumeV _N, in the limitN,V _N, withN/V _N=const. The system exhibits a second-order phase transition; complete information on the spin polarizations and their correlations is obtained. The proofs use a general result on the free energy of quantum spin systems based on the large deviation principle and the Berezin-Lieb inequalities.     

The limits of Brans-Dicke spacetimes: a coordinate-free approach

We investigate the limit of Brans-Dicke spacetimes applying a coordinate-free technique. We obtain the limits of some known exact solutions. It is shown that these limits may not correspond to similar solutions in the general relativity theory.     

Exact results for a generalized classicalO(n) matrix spin model

A generalizedO(n) matrix version of the classical Heisenberg model, introduced by Fuller and Lenard as a classical limit of a quantum model, is solved exactly in one dimension. The free energy is analytic and the pair correlation functions decay exponentially for all finite temperatures. It is shown, however, that even for a finite number of spins the model has a phase transition in then limit. The transition features a specific heat jump, zero long-range order at all temperatures, and zero correlation length at the critical point. The Curie-Weiss version of the model is also solved exactly and shown to have standard mean-field type behavior for all finiten and to differ from the one-dimensional results in then limit.     

Notes on the classical and the nonrelativistic limits in quantum mechanics

A method is presented which can be used to discuss both the classical and also the nonrelativistic limit of quantum mechanics. A one-to-one correspondence may be established between the asymptotic convergence of the resolvent and that of the timedependent solution. In so far as the question of dynamics is concerned we investigate the relation between families of nonrelativistic Hamiltonians and the corresponding Dirac-Hamiltonians when c± or when c±0. The nonrelativistic free theory formally shows the same pattern when ±0 (the classical limit) or when ±. The investigation finally shows how the asymptotic convergence of the relativistic theory can take place under some fairly general conditions of the radiation field.     

Multiparticle fractal aggregation

Kinetic fractal aggregation in a particle bath where a fractionf of the sites are initially occupied is studied withd=2 computer simulations. Independent particles diffusing to a fixed cluster produce an aggregate with fractal dimensionD 1.7 up to a correlation length(f). At larger lengthsD2.(f) asf 0. When the particles remain fixed but the cluster undergoes a rigid random walkD appears constant at larger scales but varies withf. D 1.95 at largef andD 1.7 asf 0. In both cases, the aggregate sizeN(t) grows with timet ^(f) . Aggregation on a surface by independently diffusing particles produces shapes reminiscent of electrochemical dendritic growth. The dependence of growth rate and geometry is studied as a function of particle concentration and sticking probability.     

Isospectral transformations in relativistic and nonrelativistic mechanics

The modified Korteweg de-Vries hierarchy of partial differential equations generating transformations of the one-dimensional Dirac equation, is shown to reduce in the limitc to the Korteweg de-Vries hierarchy, generating isospectral transformations of the Schrödinger equation. The former hierarchy reduces into relativistic and the latter into nonrelativistic isoperiodic transformation in the limit0.     

End patterns of self-avoiding walks

Consider a fixed end pattern (a short self-avoiding walk) that can occur as the first few steps of an arbitrarily long self-avoiding walk on ^d. It is a difficult open problem to show that asN , the fraction ofN-step self-avoiding walks beginning with this pattern converges. It is shown that asN , this fraction is bounded away from zero, and that the ratio of the fractions forN andN+2 converges to one. Similar results are obtained when patterns are specified at both ends, and also when the endpoints are fixed.     

A two-dimensional Fokker-Planck equation degenerating on a straight line

By an example of a two-dimensional hydrodynamic system, second-order Langevin equations with two correlated noise sources are investigated. It is shown that the asymptotic expression (t) for the stationary distribution functionP depends on the order in which the limiting transitions;t andN ₂₂0 (N ₂₂ is the power of one of the noises) are made. Using the method of local expansions in trigonometric form, approximate expressions are written for the distribution functionP at small but finiteN ₂₂ tending atN ₂₂0 to the known exact solution.     

Thermodynamics of a spin 1/2 Anderson impurity in theU→∞ limit

The Bethe-ansatz equations describing the thermodynamics of the non-degenerate Anderson model are derived in theU limit (double occupation of the localized level is excluded). The set of Bethe-ansatz equations for theU limit is considerably different from the one for the finiteU case. The Kondo limit, the Fermi liquid behavior at lowT and the highT perturbation expansion for the thermodynamic potential are extracted from these equations.Heisenberg-fellow of the Deutsche Forschungsgemeinschaft     

Kinematics of a Spacetime with an Infinite Cosmological Constant

A solution of the sourceless Einstein's equation with an infinite value for the cosmological constant is discussed by using Inönü–Wigner contractions of the de Sitter groups and spaces. When , spacetime becomes a four-dimensional cone, dual to Minkowski space by a spacetime inversion. This inversion relates the four-cone vertex to the infinity of Minkowski space, and the four-cone infinity to the Minkowski light-cone. The non-relativistic limit c is further considered, the kinematical group in this case being a modified Galilei group in which the space and time translations are replaced by the non-relativistic limits of the corresponding proper conformal transformations. This group presents the same abstract Lie algebra as the Galilei group and can be named the conformal Galilei group. The results may be of interest to the early Universe Cosmology.     

Dispersion in momentum space and the existence ofU(t,t')

The operatorU(t,t') giving transition probabilities between finite times or connecting free and interacting fields does not exist (apart from the ultraviolet divergence problem) because of the 3-translation invariance of current quantum field theory. To remedy this, the idealization that one has an infinite timeT = to prepare initial, or measure final,n-particle momentum eigenstates is discarded here. It is shown that random space-time (which itself eliminates ultraviolet divergences from field theory) implies and fixes uniquely a random momentum space if free particle momentaK are determined by time-of-flight measurements withT < . In particular, the dispersion ofK m/T, where is the space-time dispersion andm is the particle mass. Stochastic momentum space is incorporated into field theory in a preliminary way; because 3-translation form-invariance is slightly violated, the unitaryU-operator expressed as the usualT-exponential exists and the limitU S ast ,t' – is welldefined withoutad hoc tricks like the adiabatic cut-off. A frame-dependence is necessarily introduced into fields andU-operator, and the transformation properties expressing Lorentz covariance are of the same more general type encountered in previous work on quantum field theory over stochastic spacetime.     

A derivation of the Broadwell equation

We consider a stochastic system of particles in a two dimensional lattice and prove that, under a suitable limit (i.e.N, 0,N²const, whereN is the number of particles and is the mesh of the lattice) the one-particle distribution function converges to a solution of the two-dimensional Broadwell equation for all times for which the solution (of this equation) exists. Propagation of chaos is also proven.Research partially supported by CNR-PS-MMAIT     

Two results concerning asymptotic behavior of solutions of the Burgers equation with force

We consider the Burgers equation with an external force. For the case of the force periodic in space and time we prove the existence of a solution periodic in space and time which is the limit of a wide class of solutions ast . If the force is the product of a periodic function ofx and white noise in time, we prove the existence of an invariant distribution concentrated on the space of space-periodic functions which is the limit of a wide class of distributions ast .