System Of Independent Particles Research Articles

Level I theory of large deviations in the ideal gas

LetX1,X2,... be i.i.d. random elements (the states of the particles 1,2,...). Letf be an ℝd-valued, measurable function (an observable) and letB ⊂Rdbe a convex Borel set. DenoteSn=f(X1)+f(X2)+...+f(Xn). Using large-deviation theory, it may be shown that, under certain regularity conditions, there exists a point υB (the dominating point of B) so that, givenSn/ne B, actually Sn/n→υ B in probability as n→∞. Having this conditional weak law of large numbers as our starting point, we consider physical systems of independent particles, especially the ideal gas. Given an observed energy level, we derive convergence results for empirical means, empirical distributions, and microcanonical distributions. Results are obtained for a closed system with a fixed number of particles as well as for an open particle system in the space (a Poisson random field). Our approach is elementary in the sense that we need not refer to the abstract “level II” theory of large deviations. However, the treatment is not restricted to the so-called discrete ideal gas, but we consider the continuous ideal gas.

A First Order Generalized Thomas‐Fermi Approximation for the Electron Density in Inversion Layers

AbstractAn operator evaluation of the one‐particle density matrix of a degenerate system of independent particles in first order with respect to the gradient of the potential developed by Macke and Rennert yields an analytic expression for the particle density. This method is extended here to potentials with an infinite step and to finite temperatures – a situation which is characteristic for inversion electrons in MIS‐systems. The resulting density can be expressed as the Airy transform of the zeroth order (local density approximation). The first order yields both the tunneling into the classically forbidden region and oscillations of the density near the step of the potential. The operator evaluation of the density matrix is shown to be equivalent to solving a Schrödinger like equation. The first order density yields results for the subband structure of (100)Si inversion and accumulation layers at OK in remarkable agreement to density functional calculations of Ando.

Occupation time large deviations of the voter model

This paper is a sequel to [5] and [6]. We continue our study of occupation time large deviation probabilities for some simple infinite particle systems by analysing the so-called voter model ζt (see e.g., [11] or [8]). In keeping with our previous results, we show that the large deviations are “classical” in high dimensions (d≧5 for ζt) but “fat” in low dimensions (d≦4). Interaction distinguishes the voter model from the independent particle systems of [5] and [6], and consequently exact computations no longer seem feasible. Instead, we derive upper and lower bounds which capture the asymptotic decay rate of the large deviation tails.

Electron correlation in alkaline-earth atoms.

Conditional probability distributions for the valence electrons of alkaline-earth atoms in their ground and valence excited states are constructed using large-basis Sturmlan configuration-interaction wave functions for two electrons in frozen, effective-core potentials. The distributions indicate that the ground states of Be, Mg, Ca, Sr, and Ba, and the valence excited states of Be, are much more like linear triatomic molecules, akin to doubly excited He, than like independent-particle systems such as the ground states of He and He-like ions.

Symmetric Exclusion Processes: A Comparison Inequality and a Large Deviation Result

We consider an infinite particle system, the simple exclusion process, which was introduced in the 1970 paper "Interaction of Markov Processes," by Spitzer. In this system, particles attempt to move independently according to a Markov kernel on a countable set of sites, but any jump which would take a particle to an already occupied site is suppressed. In the case that the Markov kernel is symmetric, an inequality by Liggett gives a comparison, for expectations of positive definite functions, between the exclusion process and a system of independent particles. We apply a special case of this inequality to an auxiliary process, to prove another comparison inequality, and to derive a large deviation result for the symmetric exclusion system. In the special case of simple random walks on $Z$, this result can be transformed into a large deviation result for an infinite network of queues.

Systematic treatment of the anderson localization on the basis of the projection operator formalism

A derivation within the projection operator technique is given for the density and current response functions of a system of independent particles moving in a random potential. The essential point is the derivation of kinetic equations for the current relaxation kernel instead of for the density propagator as in a previous treatment on the basis of the projection operator formalism. In these equations the divergent contributions from the 2k F -scattering mechanism can be systematically separated from those of diffusional scattering. Especially, both the self-consistent current relaxation theory, developed by Gotze and the self-consistent treatment of Vollhardt and Wolfle are rederived from simple approximations of the kinetic equations. The outlined method represents a systematic approach to the Anderson localization. It may be applicable also to more realistic models for the Anderson localization as well as extended to the evaluation of general transport coefficients on a level far beyond the usual perturbation theories.

On the extended Thomas-Fermi approximation to the cranked harmonic oscillator

The extended Thomas-Fermi model is used to calculate the average energy of a system of independent particles moving in a cranked harmonic oscillator. We minimize the smoothed energy with respect to the harmonic frequencies and find a great similarity between our results and the liquid drop results. Moreover, the semiclassical stationary solution violates the isotropy of the velocities distribution in the intrinsic frame. As in the quantal case, the equilibrium of the nucleus is governed by the well-known self-consistent relation of Mottelson: ω 1 2< x 2 2 > = ω 2 2< x 2 2> = ω 3 2< x 3 2>. A comparison is made with the classical rotation of a gravitational mass.

Anderson Localization ind&lt;~2Dimensions: A Self-Consistent Diagrammatic Theory

A diagrammatic theory is presented for the density response function of a system of independent particles moving in a random potential in terms of a current relaxation kernel $M(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})$ (essentially the inverse of the diffusion coefficient). In the presence of time-reversal invariance, $M(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})$ is shown to have infrared divergencies in $dl~2$ dimensions. A self-consistent treatment of the divergent terms yields a finite static electric polarizability $\ensuremath{\alpha}$, a dynamical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{2}(\ensuremath{\omega}\ensuremath{\rightarrow}0)$, and a finite localization length in $dl~2$ for arbitrarily weak disorder.

Anderson Localization ind&lt;~2Dimensions: A Self-Consistent Diagrammatic Theory

A diagrammatic theory is presented for the density response function of a system of independent particles moving in a random potential in terms of a current relaxation kernel $M(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})$ (essentially the inverse of the diffusion coefficient). In the presence of time-reversal invariance, $M(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})$ is shown to have infrared divergencies in $d&lt;~2$ dimensions. A self-consistent treatment of the divergent terms yields a finite static electric polarizability $\ensuremath{\alpha}$, a dynamical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{2}(\ensuremath{\omega}\ensuremath{\rightarrow}0)$, and a finite localization length in $d&lt;~2$ for arbitrarily weak disorder.

Functional integral approach to the static susceptibility of the Anderson model

We consider the Anderson model for dilute magnetic alloys and use the functional integral approach to calculate the static susceptibility. The Coulomb interaction at the magnetic impurity is represented by the random fields coupled to the charge and the spin fluctuations. On treating the charge fluctuations within the extremal approximation, the free-energy functional is approximated by its potential part plus the harmonic terms in its kinetic part. Within this approximation, the functional integral is evaluated over all the paths by using the transfer matrix technique. The static susceptibility reduces to an expression of a system of independent particles moving in a one-dimensional potential. In the weak-coupling limit (U/πΔ<1,U is the Coulomb interaction at the impurity and Δ is the width of its virtually boundd-state), it behaves like the Pauli susceptibility and coincides with perturbation theory results, up to first order inU/πΔ. In strong-coupling limits (U/πΔ>1), at low temperatures, it shows the tendency to saturation (Kondo effect) and, at high temperatures, behaves like a Curie susceptibility and coincides with the perturbation results, up to first order in πΔ/U.

Probabilistic methods for stationary problems of linear transport theory

The steady state of a system of independent particles which undergo elastic collisions can be expressed in terms of the absorption probabilities of the associated Markov process. For the slab albedo problem, this representation enables the application of probabilistic methods to obtain explicit upper and lower bounds on the steady-state density. In particular, the bounds prove the 1/L, decrease of the steady-state flux as a function of the slab widthL (Fick's law).

Quantum mechanics in rotating frames. I. The impossibility of rigid flow

The Hamiltonian describing a system of particles in a rotating coordinate system is derived and it is shown that the simple classical solution of rigid flow is forbidden in quantum mechanics, even at very low angular velocities. This effect is closely parallel to the Aharonov–Bohm effect, which likewise has its origin in the single-valuedness requirement of the wave function. An analytical approach to perturbation theory is used to include the effects of the Coriolis and centrifugal forces and to derive the current flows for some independent-particle systems, that is, for the Inglis cranking model. It is shown, by explicit construction, that the currents are not rigid even when the moment of inertia assumes the rigid-flow value, as it does for the harmonic oscillator single-particle potential under conditions of self-consistency. Furthermore, it is shown that, for a more general potential, even the moment of inertia is not rigid.

Random invariant measures for Markov chains, and independent particle systems

Let P be the transition operator for a discrete time Markov chain on a space S. The object of the paper is to study the class of random measures on S which have the property that MP=M in distribution. These will be called random invariant measures for P. In particular, it is shown that MP=M in distribution implies MP=M a.s. for various classes of chains, including aperiodic Harris recurrent chains and aperiodic irreducible random walks. Some of this is done by exploiting the relationship between random invariant measures and entrance laws. These results are then applied to study the invariant probability measures for particle systems in which particles move independently in discrete time according to P. Finally, it is conjectured that every Markov chain which has a random invariant measure also has a deterministic invariant measure.

On the extended Thomas-Fermi approximation to the kinetic energy density

The extend Thomas-Fermi (ETF) model is used to express the average kinetic energy density τ ̃ (r) of a system of independent particles in terms of the average nucleon density ϱ ̃ (r) . Numerical tests made using Strutinsky-averaged densities from a local Woods-Saxon potential indicated that using terms up to and including the Weizsäcker term is not sufficiently accurate. However, if the next order terms are included, the average kinetic energy of a large nucleus is reproduced to within 2–5 MeV. The expansion is not very useful for the exact densities as it leaves out all shell effects. We have also expanded the ETF approximation to include the effects of nonlocalities in the potential; in particular the effective mass and spin-orbit contributions to the kinetic energy are given.

Rotating nuclei in a semiclassical description

A recently proposed semiclassical method for extracting the smoothly varying part of the total energy of an independent particle system is applied to a rotating system. Expressions for the average density distribution, angular momentum, total energy and effective moment of inertia are given.

Ergodic properties of an infinite system of particles moving independently in a periodic field

We investigate the ergodic properties of a general class of infinite systems of independent particles which undergo nontrivial “collisions” with an external field, e.g. fixed convex barriers (the Lorentz gas). We relate the ergodic properties of these systems to the ergodic properties for a single particle moving in a finite box (with periodic boundary conditions) with the same dynamics. We prove that when the one particle system is mixing or aK-system for a sequence of boxes approaching infinity so is the infinite particle system with an equilibrium measure obtained as a Poisson construction over the one particle phase space.

The spatial coherence factor in light scattering from a system of independent particles

We derive the spatial coherence factor associated with the intensity correlation function of light scattered from N independent particles. The expression agrees with that derived by assuming a gaussian light field. We show also that the so-called cross-spectral purity condition is valid in this case. The applicability of this expression to the case of clipped intensity correlation function and to the case of strongly interacting particles are also discussed.

A Note on the Partition Function for Systems of Independent Particles

The N-particle Bose-Einstein and Fermi-Dirac canonical partition functions are expressed explicitly in terms of the single-particle partition function. The nature of the approximations made in using corrected Maxwell-Boltzmann statistics is thereby made more obvious. A recurrence relation is developed to facilitate calculations using the exact partition function.

Separation of the center-of-mass in independent particle systems

It is shown that if an independent-particle Hamiltonian can be separated into a center-of-mass and internal part, it must be a second degree polynomial in the momentum and position operators. In particular the spin can enter only through an additive term which is independent of the orbital quantities. If the masses of the particles are the same then the single particle Hamiltonians must be the same; this rules out the possibility of treating protons and neutrons differently in a separable, independent particle Hamiltonian. Invariance requirements further restrict the form of the Hamiltonian.

Calculated level density of 24Na

The level density of 24Na as a function of the angular momentum is calculated by employing two different models, an independent-particle system and an SU 3 model with the Elliott quadrupole-quadrupole interaction. The level densities are compared with experimental results. Also the effective moment of inertia and the distribution of the deformation parameter among the nuclear levels are calculated.