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.     

Small ħ asymptotics for quantum partition functions associated to particles in external Yang-Mills potentials

To a gauge field on a principalG-bundlePM is associated a sequence of quantum mechanical Hamiltonians, as Planck's constant 0 and a sequence of representations _n ofG is taken. This paper studies the associated quantum partition functions, trace exp (–tH _n ), and produces a complete asymptotic expansion, as 0, =1/n, of which the principal term, proportional to the classical partition function, is the familiar classical limit.Research supported in part by Deutsche Forschungsgemeinschaft and NSF grant NoPhy 81-09011A-01On leave of absence from Freie Universität, BerlinResearch supported by NSF grant MCS 820176A01     

Correlation function formulas for some infrared asymptotic-free scalar field lattice models via the block renormalization group

We consider a class of scalar field lattice models with action 1/2()+V(), V small. After n block renormalization group transformations, new formulas are obtained for the finite lattice generating and correlation functions. For some infrared asymptotic-free models in the thermodynamic and n limits, the formulas for correlation functions are especially simple, isolate the correct dominant long-distance behavior, and can be used to control the subdominant contributions.     

Some Questions of the Nevanlinna Theory for the Complex Half-Plane

In this paper we give variants of the logarithmic derivative lemma and the second main theorem for the complex half-plane. Here the second main theorem is a special case of a theorem concerning polynomial functions of the coordinates of a holomorphic curve f : C ₊ P ⁿ . This theorem is a variant for C ₊ of a theorem of A. Eremenko and M. Sodin, who considered an entire curve f : C ₊ P ⁿ . The proofs are realized because of the introduction of natural classes of subharmonic, -subharmonic, holomorphic, meromorphic functions in C ₊ . Such functions we shall call just functions.     

Typical cluster size for two-dimensional percolation processes

The typical cluster size for two-dimensional percolation models is discussed. It is shown that, forW ₀={xZ ²0x}, [–lim _n(1/n) logP _p (W ₀=n)]^–1p–p _c ^– aspp _c , provided thatE _p (W ₀²)/E _p (W ₀)p–P _c ^– aspp _c . Furthermore, we introduce a new quantityf _s (p), which may be thought of as the singular part of the free energy, and show thatf _s (p)¦p–p _c ¦^2v provided that the correlation length ¦p–p _c ¦^–v aspp _c .     

On nonlinear stationary half-space problems in discrete kinetic theory

We show that every steady discrete velocity model of the Boltzmann equation on the real line, _i·(d/dx)f ⁱ=C ⁱ(f), which satisfies anH-theorem and for which all _i0, has solutions on the half-line (0, ) which take prescribed non-negativef ⁱ(O) if _i>0 and approach a certain manifold of Maxwellians asx. Such solutions give the density distribution in a Knudsen boundary layer in the discrete velocity case.     

Comment on a paper by G. H. Weiss,S. Havlin,and A. Bunde

A simple proof is pointed out for the asymptotic exponential decay of then-step survival probability of a random walk on a finite lattice with traps in the limit asn . Some bounds are mentioned, which are valid for finiten and for symmetric random walks.     

Dynamical chaos in the highly excited states of a quantum Daffing oscillator in an external harmonic field

The energy spectrum of the highly excited states of a quantum Daffing oscillator is considered. It is shown that the energy levels satisfy E_n>E_n+1 and E_n -F²/8², when n , where F² and are the intensity and frequency of the external field. It is suggested that this property of the spectrum is the quantum manifestation of a stochastic attractor in the classical nonlinear limit. This property also takes place for large F².Omsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 90–94, March. 1993.     

Non-commutative spheres

LetA be the irrational rotation algebra, i.e. theC ^*-algebra generated by two unitariesU, V satisfyingVU=e ²ⁱ UV, with irrational, and consider the fixed point subalgebraB under the flip automorphismUU ^–1,VV ^–1. We prove thatB is an AF-algebra.Dedicated to Professor Huzihiro Araki on the occasion of his 60'th birthday     

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.     

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     

Measure and dimension of solenoidal attractors of one dimensional dynamical systems

Letf:MM be aC -map of the interval or the circle with non-flat critical points. A closed invariant subsetAM is called a solenoidal attractor off if it has the following structure: , where{I _k ⁽ⁿ⁾ is the cycle of intervals of periodp _n. We prove that the Lebesgue measure ofA is equal to zero and if sup(p _n+1/p_n)< then the Hausdorff dimension ofA is strictly less than 1.     

On the Fokker-Planck equation with time-dependent drift and diffusion coefficients and its exponential solutions

The static critical behavior of a mixed hydrogen-bonded ferro-antiferroelectric system is investigated with use of a pseudo-spin Ising model in a transverse field, which describes the tunneling of protons. Assuming an infinite-ranged random interbond coupling, then0 replica method is applied to evaluate the averaged free energy, from which the replica-symmetric solution of the Sherrington-Kirkpatrick type is obtained. It is shown that a proton pseudo-spin glass (PG) phase exists below a freezing temperatureT _f , whereT _f and the order parameters for the PG and ferroelectric phase are parametrized by the tunneling frequency . For _c, where _c = and is the random distribution width, no ordering is possible even forT0. Numerical solutions for the PG order parameter and the dielectric susceptibility atT0 are obtained, and the phase diagram for a simple model of a mixed system atT0 is presented.Dedicated to Prof. Dr. H.E. Müser on the occasion of his 60th birthday     

Phase Transformations α → β → γ → δ → ε in Titanium Hydride TiH x with Increase in Hydrogen Concentration

Structural phase transformations of the hcp bcc fct fcc fct ( ) type in titanium hydrides TiH _x experimentally found to occur with increase in the hydrogen concentration x are studied theoretically. Free energy phases are statistically calculated, and their concentration dependence is determined. Phase transformation temperatures are found from intersection points of free energy curves and from points of common tangents to the curves. The energy parameters of the system are estimated. The calculation results are compared with the experimental data.     

The number of bound states of singular oscillating potentials

For a spherically symmetric potential such that rVL ¹(a, ), a>0, and is such that, if we define W=– _r V(t) d(t), W belongs to L ¹ (0, ) and rW0 as r0, we show that the number of bound states in any partial-wave satisfies the bound n2 ₀ r W ² dr. It was shown in a previous paper [1] that this class of potentials is regular from the point of view of abstract scattering theory as well as from the time-independent theory and the Jost function approach. We show also that, for large values of the coupling constant, n(gV) has the asymptotic behaviour C _±g ₀ W(r) dr as g±.     

Phase diagram and scattering intensity of binary amphiphilic systems

A Ginzburg-Landau model with a scalar and a vector order parameter, which describe the concentration and orientation of the amphiphile, respectively, is used to study the phase diagram and the scattering intensity of binary amphiphilic systems. With increasing amphiphile concentration, the calculated phase diagram shows the typical sequence of ordered phases observed experimentally, that is micellar liquid cubic micellar hexagonal lamellar cubic bicontinuous invers hexagonal. The scattering intensity in the homogeneous phase is calculated in the oneloop approximation. In the vicinity of a phase transition to an ordered phase, the intensity is found to show a 1/q behavior for not too small wave vectorsq, followed by a small peak, and a 1/q ² decay for large wave vectors, in agreement with experimental observations in theL ₃-(or sponge-)phase.Dedicated to Prof. H. Wagner on the occasion of his 60th birthday     

Neutron diffraction studies of the premartensitic transformation B2 → B19′ in Ti49Ni51 single crystals

We have studied structural changes in the high-temperature B2-phase in a large single crystal at temperatures near the premartensitic transformation B2 B19. We are the first to observe an extra 1/2 (110) reflection in neutron diffraction patterns taken along the [110]_B2 direction as the sample is cooled below 420 K, but still far from the martensite start temperature (M_s=180 K). This extra reflection heralds the formation of long-range order in atomic displacements with wave vectorq=(1/2±)[110]2/a. Premartensitic diffraction effects (caused by the development and correlation of lattice waves of atomic displacements with wave vectorsq ₁2/a[1/3, 1/3, 0] andq ₁2/1[1/3, 1/3] that were clearly visible in this same single crystal before the martensitic transformation B2 R, appeared at even lower temperatures with substantially lower intensities.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 56–61, January, 1995.     

Bose–Einstein Condensation in a Harmonic Trap: Effect of Interactions on T c

For a harmonically trapped dilute Bose gas with uniformly repulsive interactions which is assumed to satisfy a certain condition on the extensivity of fluctuations, I find on upper bound on the condensate fraction f. If BEC is defined by the condition that f>const.N ^– , <1/2, I argue that in the limit N, V _o 0, NV _o const. where V _o is the space integral of the potential, the interactions cannot increase the critical temperature over that of the noninteracting gas.     

Level I theory of large deviations in the ideal gas

LetX ₁,X ₂,... be i.i.d. random elements (the states of the particles 1,2,...). Letf be an ^d-valued, measurable function (an observable) and letB R^dbe a convex Borel set. DenoteS _n=f(X₁)+f(X₂)+...+f(X_n). Using large-deviation theory, it may be shown that, under certain regularity conditions, there exists a point _B (the dominating point of B) so that, givenS _n/n B, actually S_n/n _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.