Gentile statistics and restricted partitions

In a recent paper (Tranet al, Ann. Phys.311, 204 (2004)), some asymptotic number theoretical results on the partitioning of an integer were derived exploiting its connection to the quantum density of states of a many-particle system. We generalise these results to obtain an asymptotic formula for therestricted or coloured partitionsp _k ^s (n), which is the number of partitions of an integern into the summand of sth powers of integers such that each power of a given integer may occur utmostk times. While the method is not rigorous, it reproduces the well-known asymptotic results fors = 1 apart from yielding more general results for arbitrary values ofs.     

Numerical study on flame spread of an n-decane droplet array in different temperature environment under microgravity

A series of numerical calculations of flame spread of an n-decane droplet array was conducted at different ambient temperatures (T_a = 300 and 573 K) for S/d₀ from 1.5 to 10, where S is the droplet interval and d₀ is the initial droplet diameter. The authors compared these numerical results with experimental results under similar conditions at different ambient temperatures for the first time in this study. Good qualitative agreement in flame spread behavior between numerical results and microgravity experiments is obtained. Flame spread mode changed with an increase in S/d₀. Also, appearance of the flame spread mode in a stepping-stone manner (Mode III in [Jpn. Soc. Mech. Eng. 68 (672) (2002) 2423]) in a normal temperature environment was verified by numerical calculations and microgravity experiments, although it was not predicted in the theoretical analysis. In addition, good qualitative agreement of flame spread rate V_f versus S/d₀ was obtained between numerical and experimental results, although numerical results were at least twice as large as experimental results. V_f had a maximum peak at a specific S/d₀ for a different ambient temperature. Employment of improved reaction model and consideration for thermal radiation heat transfer are expected to produce quantitatively better results. An increase in surface temperature of unburned droplets and the development of a flammable gas layer around the droplets were promoted in a high-temperature environment, due to an increase in heat transfer from ambient air to the droplet. As a result, V_f was increased by the higher ambient temperature, suggesting that ambient temperature plays a significant role both in the flame spread mode and the flame spread rate through promotion of a flammable gas layer around unburned droplets.     

Evaluation of thesscf andpy approximations for quadrupolar fluids

Two approximations, the single super chainf-expansion (sscf), and Percus-Yevick (py) approximation, are evaluated for a molecular fluid in which the molecules interact with a pair potential, that is the sum of Lennard-Jones and quadrupole-quadrupole parts at two values of reduced quadrupole moment. These results are compared with Monte-Carlo results. Except for the harmonic coefficienth (222;r), thesscf approximation seems to be quite accurate for the lower value of quadrupole moment but at higher valuespy approximation produces much better results except forh(220;r).     

Onsager's reaction field for the Potts model from the path integral

A new path integral formulation for theq-state Potts model is proposed. This formulation reproduces known results for the Ising model (q=2) and naturally extends these results for arbitraryq. The mean field results for both the Ising and the Potts models are obtained as a leading saddle point contribution to the corresponding functional integrals, while the systematic computation of corrections to the saddle point contribution produces the Onsager reaction field terms, which forq=2 coincide with results already known for the Ising model.     

On the Characteristic Polynomial¶ of a Random Unitary Matrix

We present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial Z of a random N×N unitary matrix, as N→∞. First we show that , evaluated at a finite set of distinct points, is asymptotically a collection of i.i.d. complex normal random variables. This leads to a refinement of a recent central limit theorem due to Keating and Snaith, and also explains the covariance structure of the eigenvalue counting function. Next we obtain a central limit theorem for ln Z in a Sobolev space of generalised functions on the unit circle. In this limiting regime, lower-order terms which reflect the global covariance structure are no longer negligible and feature in the covariance structure of the limiting Gaussian measure. Large deviations results for ln Z/A, evaluated at a finite set of distinct points, can be obtained for . For higher-order scalings we obtain large deviations results for ln Z/A evaluated at a single point. There is a phase transition at A= ln N (which only applies to negative deviations of the real part) reflecting a switch from global to local conspiracy. Received: 27 June 2000 / Accepted: 30 January 2001     

Virial expansion for the radial distribution function of a fluid using the 6 : 12 potential

The radial distribution function can be expressed in a virial expansion. Using the 6 : 12 potential the second-order density coefficient, g ₂(r), is numerically calculated for a wide range of temperatures and intermolecular separations. These results are used to calculate the second-order density coefficient, c ₂(r), in the expansion of the direct correlation function and to calculate the fourth virial coefficient, B ₄. In addition, approximate results for g ₂(r), c ₂(r), and B ₄ are calculated on the basis of the Percus-Yevick, hypernetted chain, and the self-consistent approximations of Hurst and Rowlinson. These approximate results are compared with the exact results. The Percus-Yevick theory is in good agreement with the exact results at high temperatures but is unsatisfactory at low temperatures. The hyper-netted-chain approximation is in fair agreement with the exact results at high temperatures, is in poor agreement at intermediate temperatures, but is in good agreement at low temperatures. The self-consistent approximations are in reasonably good agreement with the exact calculations at all temperatures.     

Community overlays upon real-world complex networks

An approximation for the dynamic structure factor S(q, ω) of the linear harmonic chain with periodic boundary conditions is obtained from a recurrence relation approach. It is compared with (numerically) exact results for S(q, ω). Explicit expressions for the moments of the structure factor are obtained from the short time expansion up to order n = 12. For the long-wavelength as well as the deep-inelastic limit, simple expressions are derived permitting to some extent analytic results for S(q, ω). Introducing a linear regression, an approximation scheme is defined which shows excellent agreement with the numerical results.     

Calculations of thermal conductivity of complex (dusty) plasmas using homogenous nonequilibrium molecular simulations

The heat conductivity of three-dimensional Yukawa dusty plasma liquids (YDPLs) has been investigated by employing a homogenous nonequilibrium molecular dynamics (HNEMD) technique at a low normalized force field strength (F*). The obtained results for plasma heat conductivity with suitable normalizations are measured over a wide range of various plasma states of the Coulomb coupling (Γ) and screening length (κ) in a canonical ensemble (NVT). The calculations for lattice correlations (Ψ) show that our YDPLs system remains in a nonideal strongly coupled regime for a complete range of Γ. It has been shown that the presented Yukawa system obeys a simple analytical temperature demonstration of λ₀ with a normalized Einstein frequency. The employed HNEMD algorithm is found to have a more efficient method than that of different earlier numerical methods and it gives more satisfactory results for lower to intermediate Γ with small system sizes at low F*. The obtained simulation results at nearly equilibrium F* (=?0.002) are in reasonable agreement with different earlier numerical results and with the present reference set of data showed deviations within less than ±15% for most of the present data points and generally underpredicted the λ₀ by 2–22%, depending on (Γ, κ).     

On the validity and properties of the atom-atom potential as a function of intermolecular separation,configuration, and partial wave order

The intermolecular partial wave expansion of the atom-atom potential U is reviewed briefly and developed, by using results due to Sack, so that the radial components of the expansion can be evaluated to arbitrary accuracy for all relevant partial wave orders and values of the intermolecular distance r. These results are used to study the convergence of the partial wave expansion of U as a function of partial wave order, r, intermolecular orientation, and the anisotropy of the interacting molecules. In marked contrast to previous work it is found that many of the higher order partial wave components of U are important relative to the isotropic term even for the interaction of relatively spherical molecules and that the results obtained from a truncated partial wave expansion depend significantly upon the method of summation due to the generally poor convergence of the expansion. The validity of the atom-atom potential as a representation of the correct attractive intermolecular potential is also discussed in some detail. There are basic problems associated with the representations furnished by both the isotropic and the anisotropic parts of the atom-atom potential at intermediate and large r. The different convergence properties of the r ^-1 expansions of the partial wave expansions of U and of the correct potential for these values of r is illustrated by using model interactions. While it appears that it may be possible to obtain a qualitatively reasonable representation of the attractive part of an intermolecular potential over a useful range of r from atom-atom results, this apparently cannot be achieved for wider ranges of r or for the purely anisotropic part of the potential.     

Estimation of thermal lensing effect in the high-power end-pumped direct-cut crystal lasers

At this working, analytical (with Gaussian pump beam profile) and numerical FEM (with super-Gaussian pump beam profile) methods were compared for estimation of thermal lensing effect in the diode-end-pumped solid-state laser rods. These results were applied to the design and realization of a direct-cut (i.e., cut in right-angle ) crystal Nd:YVO₄ laser. The results showed that for such crystal (direct-cut), two methods obtained almost the same results, but slightly different from experimental results. However, estimation of thermal lensing effect led to realization of a diode-end-pumped Nd:YVO₄ cw high-power laser with good optical quality.     

Long range adiabatic potentials and scattering lengths for the EF,e and h states of the hydrogen molecule

Accurate EF ¹Σ _g , e ³Σ _u and h ³Σ _g state adiabatic potentials are computed for internuclear distances up to 44 atomic units. All the energies are more accurate by a fraction of a wave number from existing results. Scattering lengths for collisions of 1S and 2S hydrogen atoms are evaluated. These differ significantly from earlier results. This is because for some states the scattering length is very sensitive to even small changes in the potential.     

Hall coefficient in solution for a simple dynamical model

A model Liouville equation is proposed for a system composed of an ion moving in a solvent fluid. Using this model, explicit results are obtained for the Ohmic conductivityL and the Hall conductivityh. These results are then used to calculate the Hall coefficientR = ehL^–2, which is a measure of the effect of non-Brownian motion, for several charge carriers of interest. Our results are in agreement with earlier findings based on a stochastic model which predictR > 1 for H⁺(aq). Our results also indicate thatR 1 for charge carriers such as Na⁺, Cl^–, and K⁺ which have a mass greater than that of a solvent molecule (here taken as 18 amu).This research was supported in part by grants from the National Science Foundation and by the Research Foundation of the State University of New York.     

Cusped gaussian wave functions

An orbital-fitting procedure has been used to fit the s- and p-orbitals expressed in terms of a cusped gaussian basis onto Clementi's accurate SCF orbitals. The results suggest that approximately double-zeta accuracy may be obtained for the s-orbitals with a basis of two 1s gaussians per orbital supplemented by a single 1s cusp function for the representation of the orbitals at and near the nucleus, and for the p-orbitals with a basis of three 2p gaussians per orbital plus the 2p cusp function. It is proposed that for the s-orbitals the cusped gaussian basis is superior to both the all-gaussian and Slater function bases for the purpose of molecular calculations. The results are less conclusive for the p-orbitals and, although the efficiency of the basis increases with increasing nuclear charge, further investigation is necessary before a definite conclusion may be reached as to the usefulness of the cusped gaussian basis for the representation of p-orbitals.     

The transparency of nuclei to nucleons and pions in a relativistic Glauber approximation

We present a selection of results obtained within the context of a relativistic eikonal model. First, results of relativistic Glauber calculations for the nuclear transparency extracted from photon-induced pion production are presented. Second, computed differential cross-sections for the ¹² C(p, 2p) are compared to data.     

The thermodynamics of symmetric two centre Lennard-Jones liquids

We have carried out molecular dynamic simulations for the thermodynamic properties of two centre Lennard-Jones fluids at lower densities and higher temperatures than have been studied previously, and have also made simulations for one additional shape. The results, together with results already given in the literature, are presented in a parameterized form which is very convenient to use in testing the simulation results against data for real liquids. Tables of second virial coefficients are also provided. We illustrate the use of our results by an analysis of pure liquid ethane, which is found to be well represented by such a model with L* = 0·67, ε/k = 137·5 K and σ = 3·506 Å. We also suggest that the experimental thermodynamic properties of suitable liquid mixtures can, with the aid of a theory for the equivalent pure liquid parameters (L _x *, ε _x , σ _x ), be satisfactorily interpreted using the general results given in this paper.     

Thermodynamic perturbation theory for simple polar fluids,I

The application of thermodynamic perturbation theory to the computation of the properties of simple polar fluids is considered. The Helmholtz free energy of a fluid of molecules interacting via a Stockmayer potential v ^S is computed through fourth order in μ, where μ is the dipole moment. Numerical results are obtained on the basis of both the ‘exact’ Monte-Carlo computations for a Lennard-Jones system and the Verlet-Weis perturbation-theoretic computations for that system. The results obtained on the latter basis are then compared with results for a fluid of molecules interacting via a central-force potential v ^ES that is conformal with the Lennard-Jones 6–12 potential and equivalent to the Stockmayer potential through order μ⁴. To facilitate the comparison the v ^ES results are computed according to the Verlet-Weis method. The v ^ES results and the fourth-order (in μ) v ^S results constitute two different approximations to the thermodynamics of the Stockmayer potential; the compressibility factors as well as the free energies of the two approximations are compared.

It is concluded that the thermodynamic contribution of the dipole term of v ^S is significant over the whole liquid region when μ²=εσ³, where ε and σ are the usual Lennard-Jones parameters. For this μ, the two approximations we consider give results in close agreement.     

Computation of parabolic cylinder functions by means of a tricomi expansion

Fast convergence expansion of the parabolic cylinder functions U(a, x), V(a, x), W(a, ± x) is obtained in terms of the Tricomi functions E_v(z). The numerical results are quite accurate for a large interval of values of “a” and for |x| 7. Tables are given for U and V in order to compare our results with other recent works on the same functions.     

Static properties and semileptonic decays of doubly heavy baryons in a nonrelativistic quark model

We evaluate static properties and semileptonic decays for the ground state of doubly heavy Ξ, Ξ', Ξ ^* and Ω, Ω', Ω ^* baryons. Working in the framework of a nonrelativistic quark model, we solve the three-body problem by means of a variational ansatz made possible by heavy-quark spin symmetry constraints. To check the dependence of our results on the inter-quark interaction we use five different quark-quark potentials that include a confining term plus Coulomb and hyperfine terms coming from one-gluon exchange. Our results for static properties (masses, charge radii and magnetic moments) are, with a few exceptions for the magnetic moments, in good agreement with a previous Faddeev calculation. Our much simpler wave functions are used to evaluate semileptonic decays of doubly heavy Ξ, Ξ'(J = 1/2) and Ω, Ω'(J = 1/2) baryons. Our results for the decay widths are in good agreement with calculations done within a relativistic quark model in the quark-diquark approximation. An erratum to this article is available at .     

Rank-Driven Markov Processes

We study a class of Markovian systems of N elements taking values in [0,1] that evolve in discrete time t via randomized replacement rules based on the ranks of the elements. These rank-driven processes are inspired by variants of the Bak–Sneppen model of evolution, in which the system represents an evolutionary ‘fitness landscape’ and which is famous as a simple model displaying self-organized criticality. Our main results are concerned with long-time large-N asymptotics for the general model in which, at each time step, K randomly chosen elements are discarded and replaced by independent U[0,1] variables, where the ranks of the elements to be replaced are chosen, independently at each time step, according to a distribution κ _N on {1,2,…,N} ^K . Our main results are that, under appropriate conditions on κ _N , the system exhibits threshold behavior at s ^∗∈[0,1], where s ^∗ is a function of κ _N , and the marginal distribution of a randomly selected element converges to U[s ^∗,1] as t→∞ and N→∞. Of this class of models, results in the literature have previously been given for special cases only, namely the ‘mean-field’ or ‘random neighbor’ Bak–Sneppen model. Our proofs avoid the heuristic arguments of some of the previous work and use Foster–Lyapunov ideas. Our results extend existing results and establish their natural, more general context. We derive some more specialized results for the particular case where K=2. One of our technical tools is a result on convergence of stationary distributions for families of uniformly ergodic Markov chains on increasing state-spaces, which may be of independent interest.     

Dynamic response and frequency dependent local field correction for an electron gas system

We present here an approximate scheme to obtain the complex frequency-dependent local field correction,G (q, ω). Using the equation of motion approach we develop here a methodology from which the results obtained by earlier workers, for density-density response function of an electron gas, can be arrived at in a simple systematic manner. We study in detail an approximation made along the similar lines as Singwi and coworkers, which leads to a compact expression for a complex frequency-dependent local field correction. We give results for damping coefficient of long wavelength plasmons which depends on ImG(q, ω). The real and imaginary parts ofG(q, ω) as functions ofω for various values ofq are also calculated. The dynamic structure factorS(q, ω) is evaluated forq=1.6q _F and compared with earlier theoretical and experimental results. An erratum to this article is available at .