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.     

Periodic nonlinear waves on a half-line

Nontrivial solutions of the equationu _tt=u _xx–g(u) which are 2-periodic int and which decay asx are shown to exist ifg(a)=0 andg(0)>1. Breather-like solutions, which also decay asx –, can be interpreted as homoclinic solutions in thex-dynamics; their existence is still in question for generalg.     

The free energy of a class of Hopfield models

We show that in the limitp ,N 0,=p/N 0 the limit free energy of the Hopfield model equals in probability the Curie-Weiss free energy. We prove also that the free energy of the Hopfield model is self-averaging for any finite .     

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.     

Fluctuation of the free energy in the Hopfield model

We prove that in the ergodic region [T>J ²(1 + r)] the deviation of the total free energy of the Hopfield neural network converges in distribution asN to a (shifted) Gaussian variable. Moreover, the free energy per site converges in probability to lim(1/N)ln _N .     

On the invariant measures for the two-dimensional euler flow

It is proven that the canonical Gibbs measure associated with a gas of vortices of intensity ± converges, in the limitN, 0,Nconst, to a Gaussian measure, which is invariant for the two-dimensional Euler equation.On leave from Dipartimento di Matematica Università di Roma Tor Vergata Roma, Italy.On leave from Dipartimento di Matematica Università di Roma La Sapienza, Roma, Italy.     

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     

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 non-relativistic limit ofP(ϕ)2 quantum field theories: Two-particle phenomena

It is proved that for two-particle phenomena theP()₂ quantum field theories with speed of lightc converge to non-relativistic quantum mechanics with a function potential in the limitc.Supported by NSF Grant No. PHY 7506746     

Retrieval and chaos in extremely dilutedQ-Ising neural networks

Using a probabilistic approach, the deterministic and the stochastic parallel dynamics of aQ-Ising neural network are studied at finiteQ and in the limitQ. Exact evolution equations are presented for the first time-step. These formulas constitute recursion relations for the parallel dynamics of the extremely diluted asymmetric versions of these networks. An explicit analysis of the retrieval properties is carried out in terms of the gain parameter, the loading capacity, and the temperature. The results for theQ network are compared with those for theQ=3 andQ=4 models. Possible chaotic microscopic behavior is studied using the time evolution of the distance between two network configurations. For arbitrary finiteQ the retrieval regime is always chaotic. In the limitQ the network exhibits a dynamical transition toward chaos.     

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     

Radiation laws in the vicinity of metallic boundaries

Corrections to Planck's radiation law and to the Stefan Boltzmann law in the vicinity of a dissipative halfspacez<0 are studied. The dissipation is described by a frequency independent conductivity . The halfspacez0 is empty.For a perfectly reflecting wall (=) the proximity corrections of the thermal electric and magnetic energy mutually cancel out. Therefore the space-dependent corrections are only due to the finite conductivity of the wall.The dissipative properties of the system lead to divergencies in the limitz0. In the limitz all corrections vanish. In properly scaledz>0 ranges analytical expressions for the corrections to the radiation laws are calculated.As a by-product the density of states of surface polaritons in the passive medium (z>0) are derived.     

The spherical-model limit in a random field

The spherical-model limitn of then-vector model in a random field, with either a statistically independent distribution or with long-range correlated random fields, is studied to demonstrate the correctness of the replica method in which then and replica limits limits are interchanged, provided the replica and thermodynamic limits are taken in the right order, in the case of long-range correlated random fields. A scaling form for the two-point correlation function relevant to the first-order phase transition below the lower critical dimensionality of the random system is also obtained.     

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.     

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.     

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.     

Positive resonances and the limiting amplitude principle

We define the positive resonance points of self-adjoint operators without using the analytical continuation of corresponding resolvents and show that the limiting amplitude principle for the abstract wave equation does not take place in general, if ² = , where is the disturbing frequency and is the resonance point. The asymptotics of corresponding solutions as t are obtained, which imply the growth of the oscillation amplitude as t , 0<<1, or as ln t, t .     

A nonequilibrium entropy for dynamical systems

It is proposed to define entropy for nonequilibrium ensembles using a method of coarse graining which partitions phase space into sets which typically have zero measure. These are chosen by considering the totality of future possibilities for observation on the system. It is shown that this entropy is necessarily a nondecreasing function of the timet. There is no contradiction with the reversibility of the laws of motion because this method of coarse graining is asymmetric under time reversal. Under suitable conditions (which are stated explicitly) this entropy approaches the equilibrium entropy ast+ and the fine-grained entropy ast–. In particular, the conditions can always be satisfied if the system is aK-system, as in the Sinai billiard models. Some theorems are given which give information about whether it is possible to generate the partition used here for coarse graining from time translates of a finite partition, and at the same time elucidate the connection between our concept of entropy and the entropy invariant of Kolmogorov and Sinai.Research supported in part by NSF grants PHY78-03816 and PHY78-15920.     

Nonstationary Markov chains and convergence of the annealing algorithm

We study the asymptotic behavior as timet + of certain nonstationary Markov chains, and prove the convergence of the annealing algorithm in Monte Carlo simulations. We find that in the limitt + , a nonstationary Markov chain may exhibit phase transitions. Nonstationary Markov chains in general, and the annealing algorithm in particular, lead to biased estimators for the expectation values of the process. We compute the leading terms in the bias and the variance of the sample-means estimator. We find that the annealing algorithm converges if the temperatureT(t) goes to zero no faster thanC/log(t/t ₀) ast+, with a computable constantC andt ₀ the initial time. The bias and the variance of the sample-means estimator in the annealing algorithm go to zero likeO(t^–1+) for some 0<1, with =0 only in very special circumstances. Our results concerning the convergence of the annealing algorithm, and the rate of convergence to zero of the bias and the variance of the sample-means estimator, provide a rigorous procedure for choosing the optimal annealing schedule. This optimal choice reflects the competition between two physical effects: (a) The adiabatic effect, whereby if the temperature is loweredtoo abruptly the system may end up not in a ground state but in a nearby metastable state, and (b) the super-cooling effect, whereby if the temperature is loweredtoo slowly the system will indeed approach the ground state(s) but may do so extremely slowly.