Coulomb pair density matrix. II

The Coulomb pair density matrixG (r, r) for attractive and repulsive potentials and for all values of parameters is determined in the form of simple series or integrals. These results are useful in both theoretical and numerical studies.     

Transport properties of the continuous-time random walk with a long-tailed waiting-time density

We derive asymptotic properties of the propagatorp(r,t) of a continuous-time random walk (CTRW) in which the waiting time density has the asymptotic form(t)T /t ⁺¹ whentT and 0<<1. Several cases are considered; the main ones are those that assume that the variance of the displacement in a single step of the walk is finite. Under this assumption we consider both random walks with and without a bias. The principal results of our analysis is that one needs two forms to characterizep(r,t), depending on whetherr is large or small, and that the small-r expansion cannot be characterized by a scaling form, although it is possible to find such a form for larger. Several results can be demonstrated that contrast with the case in which t= ₀ ()d is finite. One is that the asymptotic behavior ofp(0,t) is dominated by the waiting time at the origin rather than by the dimension. The second difference is that in the presence of a fieldp(r,t) no longer remains symmetric around a moving peak. Rather, it is shown that the peak of this probability always occurs atr=0, and the effect of the field is to break the symmetry that occurs when t. Finally, we calculate similar properties, although in not such great detail, for the case in which the single-step jump probabilities themselves have an infinite mean.     

On the isothermal density derivative ofG(r) and a new theory of the pair correlation function of hard spheres

A new representation is obtained for the isothermal density derivative ofg(r). It explicitly exhibits the contributions of potential energy terms that are not pairwise additive. Consideration of a previously known result shows that one has to be rather cautious when using it to obtain information on the triplet correlation function from the well-known relation between this function andg/, due to large cancellations which take place at high density. By integrating with respect to density the new representation forg/, after a suitable closure has been introduced, we obtain an augmented Percus-Yevick equation for hard spheres which has full thermodynamic consistency. The equation of state and the cavity functiony(r) are very accurate at low density and considerably improve PY at medium density, so that this appears to be a useful new approach to the theory of fluids, but it is necessary to improve the closure in order to treat a dense fluid.This paper is dedicated to Jerry Percus on the occasion of his 65th birthday.     

On singularities of electronic state density in liquid metals and disordered metallic alloys

We have performed a qualitative consideration of the problem of structurally dependent singularities of the density of electronic states,N(E), in disordered metallic systems: liquids, glasses and alloys. The appearance of Kohn anomalies atq ₀2K _F (q ₀ being the maximum position of the structure factorS(q);K _F , the Fermi momentum) in thermodynamic properties of liquid and amorphous metals is analyzed. It has been shown that the appearance of these contributions is due to the presence of a diffuse (inelastic) part ofS(q) and that they are no analogs of the Van Hove singularities (VHS) in crystals. The corollary of the hypothesis about the quasicrystalline structure of liquids, i.e. the appearance of blurred VHS's inN(E), is treated. A strict solution of the problem of the Gaussian static disorder effect on the VHS in crystalline alloys is presented. A new energy scale in the relaxation time and, consequently, in kinetic coefficients has been revealed.     

The number process as low density limit of Hamiltonian models

We study a nonrelativistic quantum system coupled, via a quadratic interaction [cf. formula (1.10) below], to a free Boson gas in the Fock state. We prove that, in the low density limit (z ²=fugacity0), the matrix elements of the wave operator of the system at timet/z ² in the collective coherent vectors converge to the matrix elements, in suitable coherent vectors of the quantum Brownian motion process, of a unitary Markovian cocycle satisfying a quantum stochastic differential equation driven by some pure number process (i.e. no quantum diffusion part and only the quantum analogue of the purely discontinuous, or jump, processes). This proves that the number (or quantum Poisson) processes, introduced by Hudson and Parthasarathy and studied by Frigerio and Maassen, arise effectively as conjectured by the latter two authors as low density limits of Hamiltonian models.     

Critical temperature and energy gap in superconductors with singular density of states

The influence of-functional (symmetric as well as asymmetric) singularities in density of states (DOS) on the critical temperature and zero temperature energy gap is calculated. Surprisingly, we have obtained the same function for the off-symmetry of the peak position in DOS on the corresponding critical temperature as for the temperature dependence of the energy gap in the strong-coupling limit. The enhancement of the critical temperature due to the singularity (compared with the constant DOS near the Fermi surface) is much lower for strong-coupling superconductors than in the weak-coupling limit. Hence, the singularity in DOS cannot be the exclusive reason for large values of critical temperatures in highTc superconductors.This work was supported by the grant GA SAV 188/1991.     

The role of the distribution of nucleons in finite nuclei

The total energy of many-nucleon system is expressed as a functional E[ _p(r), _n(r)] of the proton and neutron densities _p(r) and _n(r), respectively. The distribution(r) of nucleons in the nucleus, which is essential to determine the energy functional, is chosen. The energy density formalism is applied to finite nuclei, and then the binding energies per nucleon together with the mean square radii, for some medium and heavy nuclei, are obtained. Finally the achieved results are compared with the corresponding experimental values.     

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     

Modular properties of the hard hexagon model

The physical quantities (or powers thereof) in the hard-hexagon model that were computed exactly by Baxter are shown to be modular functions with respect to the number-theoretic group ₁[N]. This allows us to determine the analytic structure of, the partition function per site in the thermodynamic limit, and, the density, as functions of the activityz.     

Correlation tensors of the blackbody radiation in rectangular cavities

Electromagnetic equilibrium fluctuations in finite cavities filled with a dissipative medium (dielectric function ()=+i) and bounded by walls of infinite conductivity are considered. Expanding the fields in terms of a complete and orthonormal set of functions and solving the Maxwell equations the response of the EM field to external forces (polarization and magnetization) is obtained. With the aid of the fluctuation dissipation theorem and the linear response functions the 2nd order correlation tensors of the EM field are derived.For rectangular cavities explicit considerations are made. In the case of transparent media (=0) the spectral energy density of the EM radiation is calculated.     

Propagation of self-gravitating density waves in the deDonder gauge on a gravitational background field

Arguments are given for using the deDonder instead of the synchronous gauge in describing the propagation of density perturbations in a preexisting gravitational field. Since in the deDonder gauge the corresponding reference frame is fixed on the background, the physical interpretation of results is obvious, while in the synchronous gauge it is at least very difficult to extract the physical consequences from the results. For the propagation of density perturbations, with large spatial extension, a decisive difference is found between the two gauges. While in the synchronous gauge there is a growing mode in the density contrast (at least for adiabatic perturbations on a background matter substratum withp as equation of state), in the deDonder gauge there is not. The calculation in deDonder gauge leads to upper boundaries for the spatial extension of unstable density perturbations, and thus may give a hint for upper boundaries of galaxy masses.     

Propagation of density perturbations in the deDonder gauge on a gravitational background field II

For the problem of propagation of density waves in a preexisting gravitational field, the advantages of the deDonder gauge over the commonly used synchronous gauge are outlined. In a background matter substratum withp as equation of state there are in the deDonder gauge only decaying modes of the perturbation density contrast with arbitrary large spatial extension, whereas in synchronous gauge there is one growing mode (calculated for vanishing spatial divergency of the perturbation in the 4-velocity, i.e.,usk(1),j/j0). The calculations are extended to the case of finite spatial extensions of the density perturbations. This is done by expanding all perturbations in a power series of the inverse square of the speed of light with the result of getting a recursive set of differential equations in both gauges for the equation of motion of the density perturbations. The lowest orders of this equation are the same in both gauges, but only in the deDonder gauge is the correct Newtonian limit of propagation of density waves in an expanding universe obtained. The correction by the next higher orders in the deDonder gauge are dependent explicitly on the spatial extension of the perturbations; whereas in synchronous gauge this is not the case. For attaining the Newtonian limit this dependence is a necessary condition. At appropriately large spatial extensions, however exact, this dependence in deDonder gauge leads ultimately to a decaying of density contrast modes growing in zeroth order (at least forp=0 andp/3 as equations of state for the background matter substratum). Hence, there are upper boundaries in the spatial extensions of instable growing modes of density contrast.     

Diffusion in a one-dimensional gas of hard point particles

We simulate the classical diffusion of a particle of massM in an infinite one-dimensional system of hard point particles of massm in equilibrium. Each computer run corresponds to about 10⁸ collisions of the diffusive particle. We find that (t) 1/t fort large enough, and a crossover from an M m regime where=2 to=3 forM=m. The diffusion constant has a sharp maximum atM=m. We study moments x(t)² and x(t)⁴, and examine the behavior ofq ² (t)=x(t)⁴/3x(t)²². We find thatq(t)1 (consistent with a normal distribution) in theM limit (for all timest) and in the t limit for allM. On sabbatical leave from IVIC-Instituto Venezolano de Investigaciones Cientificas.     

Photon-photon angular correlation ine +e− collision in QED

The angular correlation of two monochromatic photons produced ine ⁺e^– collision (annihilation), defined as the average of the cosine of the angle of separation between their outgoing momenta, is studied completely in momentum space to lowest order in the fine structure constant in QED. This is done from the expression of thee ⁺e^–-momentum-spin,-polarization-averaged conditional probability density for the angular distributions of the photons, given that the process has occurred. Although the density is in general nonvanishing for less than as well as greater than/2, the angular correlation is found to be strictly negative for all energies, indicating that the photons tend to travel in opposing directions in a statistical sense. The back-to-back motion, however, is very quickly unfavored as we move to higher energies.     

Impact of the side-path model on the muon cycling rate density dependence in D-T mixtures

The side-path model previously suggested by us allows for muon transferfrom tritiumto deuterium via intermediate dtµ^* resonances formed in tµ(2s)-D₂ collisions. Taking this effect into account, the density dependence of the muon cycling rate in D-T mixtures at low temperatures is analysed. Compared with cascade models where only one-sided transfer is present, the predicted density dependence of _c is in better agreement with measurements.     

Singular behavior of the density of states and the Lyapunov coefficient in binary random harmonic chains

We study the integrated density of statesH( ²) of a chain of harmonic oscillators with a binary random distribution of the masses. We show in particular that there is a dense set of values of the squared frequency for which the differenceH( ²+)-H( ²) has a singularity of the type ¦¦², multiplied by a periodic function of ln ¦¦, where the exponent and the period depend continuously on ². In the region where < 1/2,H is not differentiate on a dense set of points. The same type of singularities is also present in the Lyapunov coefficient.     

Phase separation of symmetrical polymer mixtures in thin-film geometry

Monte Carlo simulations of the bond fluctuation model of symmetrical polymer blends confined between two neutral repulsive walls are presented for chain lengthN _A=N _B=32 and a wide range of film thicknessD (fromD=8 toD=48 in units of the lattice spacing). The critical temperaturesT _c (D) of unmixing are located by finite-size scaling methods, and it is shown that , wherev ₃0.63 is the correlation length exponent of the three-dimensional Ising model universality class. Contrary to this result, it is argued that the critical behavior of the films is ruled by two-dimensional exponents, e.g., the coexistence curve (difference in volume fraction of A-rich and A-poor phases) scales as , where ₂ is the critical exponent of the two-dimensional Ising universality class ( ₂=1/8). Since for largeD this asymptotic critical behavior is confined to an extremely narrow vicinity ofT _c (D), one observes in practice effective exponents which gradually cross over from ₂ to ₃ with increasing film thickness. This anomalous flattening of the coexistence curve should be observable experimentally.     

Effect of the target density on the cross section of charge exchange between fast ions and atoms

When fast X^q+ X^{q^ + } ions collide with atomic or molecular targets, the total charge exchange cross section decreases with increasing target density. This is because the excitation levels of resulting X^{(q - 1)+} X^{(q - 1)^ + } ions are suppressed because of ionization by target atoms. The effect of target density on the total charge exchange cross section may amount to one order of magnitude or more depending on the charge and energy of an incident ion, as well as on the density and inner shell configuration of target atoms. Numerical calculations are performed for partial (in the principal quantum number n) cross sections σ(n) and total cross sections σ_tot=Σ_nσ(n) of charge exchange in the case of collisions of fast multiply charged ions having an energy E in the range 100 keV/u-10 MeV/u with gas or solid targets.     

Localization of a high-power microwaves absorption in the cylindrical plasma column at ωLH ≪ ω ≪ ωce

This paper describes experiments which make it possible to determine the place of the absorption of electromagnetic waves in a cold (T _e=5–10eV) not fully ionized plasma column at _ce/=1 and _ce/> 1.The authors wish to thank L.Bárdo for helping with the experiments.     

On the thermodynamicV-representability of one-particle density matrices

We consider thermodynamicallyV-representable one-matrices, i. e., one-particle density matrices that are obtained by reducing the Gibbs grand canonical density matrix of a quantum mechanical many-particle system subject to a suitable external potential, and show them to obey an inequality lower bounding their eigenvalues in terms of those of the one-particle kinetic energy operator. The result imposes a severe constraint on the asymptotic behavior of the eigenvalues of any one-matrix to beV-representable. For noninteracting particles, the corresponding upper bound is also proven, implying that a one-matrix can be interactionlesslyV-representable for at most one temperature. We expect the upper bound to be valid more generally, as is illustrated by a model of coupled harmonic oscillators where theV-representable one-matrices can be explicitly calculated, and discuss its implications for certain aspects of density-matrix functional theory.