Search

The “p-q” model earlier introduced by the authors to describe persistent scattering under a scalar Boltzmann equation is here examined in detail. After deriving the scattering kernel and exhibiting its properties we obtain moment and similarity solutions and show how the model effectively parametrizes all intermediate conditions between the extremes of diffusion-like “small-scattering” and the strong-collisional limit of “diffuse-scattering” characteristic of earlier, more restrictive models. Both continuous and discrete-variable versions of the model are discussed and shown to be straightforwardly interrelated. Our derivations, carried out in natural energy-like variables, parallel those given recently by Ernst and Hendriks using transform methods.

Free energies for the formation of atomic microclusters of Lennard-Jones argon are computed. All interactions are considered in the calculation of the force-constant matrix for the vibrational analysis. Introduction of these results into the early kinetic formulation of the classical theory of nucleation yielded nucleation rates for states of realistic isentropic expansions. The calculations are compared with recent Rayleigh light scattering experiments on homogeneous nucleation of argon in helium in steady supersonic nozzle flow. A steep rise of calculated nucleation rate at certain states in the expansion coincides with the observed location of the onset of condensation and remarkable agreement of experiment and theory is found. Further comparisons of the experiments with predictions obtained with the classical theory are not successful. As expected in an extreme situation of small clusters, no agreement of the liquid drop model with experiment can be obtained in a wide range of assumed material properties.

We examine the full-range initial-value problem for velocity relaxation in a one-dimensional ensemble of Rayleigh test-particles in an arbitrary heat-bath at unit mass-ratio (the special Rayleigh problem). A Laplace transformation method is used to obtain an explicit solution in two independent parity components. Of these, the even component, giving the speed relaxation is obtained exactly and agrees with our previous result obtained by the singular eigenfunction method; the odd component, explicit to within a Laplace inverse provides the time-dependent flux of particles with given velocity. By numerical inversion of the latter, we obtain both time-dependent velocity distributions and the velocity autocorrelation function for equilibrium fluctuations. In keeping with the singular character of the transport equation, the velocity ACF proves to be appreciably non-exponential in character, behaving asymptotically as t−5/2e−t independently of the heat-bath distribution. The complex admittance of a system of charged Rayleigh test-particles is also derived and shown to lead to expected behaviour for dissipation to the heat-bath and phase-lag of response to the applied field.

The concept of distributive process, recently developed as a description of systems which evolve through ‘random’ energy-transfer in binary collision complexes4) is here extended to the case of energy quanta exchanged similarly between discrete, degenerate internal energy-levels. A corresponding exact solution of the Master Equation is forthcoming, in which the eigenvectors are now orthogonal polynomials of the discrete variable, viz. the Meixner and Hahn types. The relaxation times of the discrete models prove to be identical with those of their continuous counterparts and the autocorrelation functions for equilibrium fluctuations are likewise of strictly exponential type. All results tend naturally to their continuous analogues as the quantity (hv/kBT) for the heat-bath tends to zero. A number of aspects of mathematical interest are pointed out. For example, when considering the spectral representation of the transition matrices for the discrete distributive processes we arrive inter alia at previously unknown Erdelyi-type bilinear expansion formulae for the Meixner and Hahn systems and thence a stochastic interpretation of these. Another point of interest is the occurrence of fractional sum and difference-operators, this giving a rare example of the fractional calculus in statistical physics. Apart from furnishing several exactly soluble Master equations on an infinite, discrete state-space, the ‘distributive’ transition probabilities would seem to be the first known examples, as yet, of transition probabilities which allow the consistent representation of multiple degrees of freedom in the transfer of vibrational energy between polyatomic molecules.

A model Boltzmann equation with persistence of state is described and shown to lead to both exact, similarity solutions and general solutions as moment expansions. These, unlike any previously known, are parametrized over the whole range from “small-scattering” to “diffuse-scattering” extremes.

When a medium energy neutron hits a free atom there is a finite probability that as a consequence of the recoil the atom becomes excited or ionised. For the simple case of a hydrogen atom exact calculations are reported of the transition probabilities for various initial and final states as a function of the momentum transfer.

In this paper we discuss the statics of small assemblies of soft, rare-gas type atoms (N=4 to 13) interacting under simple two-body central forces such as the Lennard-Jones and Morse type. Our main concern is to characterize the problem of isomer multiplicity in small packed structures and to devise practical algorithms for the discovery of a representative majority, if not all, stable soft-packing structures for clusters of atoms in the above size-range. We illustrate one such possible ‘Aufbau algorithm’ by demonstrating the existence of no less than 988 distinct, stable minimum configurations for 13 Lennard-Jones atoms and of correspondingly smaller numbers in the range N=7 to 12. The minimal structures obtained may be classified broadly into ‘crystallographic’ and ‘non-crystallographic’ types, the latter predominating among those of greatest binding energy. A surprising result is that, when the softer (α=3) Morse potential is used, the great majority of the Lennard-Jones minima are not support...

We present detailed tabulations of the first few eigenfunctions of the hard-sphere energy scattering kernel for a test-particle in a background heat-bath. Calculations, for a range of heat bath/test particle mass-ratios between 1 8 and 1 1024 , were carried out by a Rayleigh-Ritz method using the exact solutions of the hard-sphere Fokker-Planck equation as a basis set and supplement our previously-published results for the eigenvalues alone. The results, given as expansion coefficients in this representation thus also serve to verify the accuracy of the Fokker-Planck equation itself, the departure from this equation being reflected in the off-diagonal contributions in the Rayleigh-Ritz expansion eigenvectors. As expected, the tendency towards brownian motion behaviour with decrease in the mass-ratio parameter shows itself in a progressive convergence of a larger and larger subset of the true eigenfunctions to the corresponding Fokker-Planck set, beginning with the eigenvalue of lowest index. The class of probability distributions whose evolution is satisfactory predicted by the Fokker-Planck equation is then precisely the class that can be adequately expanded in terms of this incomplete subset. In keeping with the approximations introduced in the derivation of the Fokker-Planck equation and the qualitative nature of the hard-sphere eigenvalue spectrum, the results confirm quantitatively the considerable restrictions which the former imposes upon acceptable solution functions, excluding in particular both short-time behaviour and solutions of insufficient smoothness. A mean-square criterion for accuracy of the Fokker-Planck solutions is suggested and examined in the light of our numerical results.

We extend previous computations of mechanical stability of atomic microclusters to the realm of statistical thermodynamics, obtaining thermodynamic functions for small, solid-like Van der Waals clusters of less than some 100 atoms possessing non-crystalline structures of ‘polytetrahedral’ type. These are shown to be almost invariably at a thermodynamic advantage over alternative lattice structures of the same number of atoms, at least for the Lennard-Jones potential in the harmonic-oscillator/rigid-rotor approximation. The dependence of thermodynamic functions upon cluster size appears to be essentially monotonic in the number of internal degrees of freedom; although there are certain exceptional structures, particularly with icosahedral symmetry, there proves to be little evidence for the occurrence of ‘magic numbers’ for stability at any temperature and within the size-range considered. Particular attention is given to the heat capacity of model systems in relation to their vibrational spectra....

The status of model calculations as a tool for the elucidation of the structure of simple metallic glasses is examined with reference to both traditional geometric studies of random packings and more recent dynamical simulations using the Monte Carlo and molecular dynamics methods. A number of hypotheses relating to the occurrence of differentiated intermediate-range structure are considered and an attempt is made to assess realistically the practical limitations to the use of computers for structural synthesis.

A Fourier transform technique based on the method of Dubner and Abate is here applied to the energy-level density problem for quantum partition functions. It is shown that, for typical molecular systems at realistic energies, the integrated level count, W(E), is obtainable to near ‘spectroscopic’ accuracy and in reasonable computing time. A smooth Fourier resolution of the simple density function, g(E), is likewise possible. Though somewhat less economical, this procedure entirely removes the slight discrepancies found in the steepest-descent method and can reasonably be claimed to dispose once and for all of the problem of quantum level-density computations in molecular rate theory.

The linear relaxation process corresponding to randomly distributive energy transfer between molecules of arbitrarily-many internal degrees of freedom is reduced to an exactly-soluble initial-value problem. If system and heatbath molecules have p and q degrees of freedom respectively, the relaxation-times for the process are τk = (p+q)k/(p)k in units of the mean collision-time for interactions. The autocorrelation function for equilibrium fluctuations in sub-system energy is shown to be a simple exponential with time-constant q/(p+q) and similar exponential behaviour is found for the mean energy during relaxation from an arbitrary initial distribution.

The potential energy surfaces for clusters of some three to sixty atoms under Lennard-Jones forces have been systematically explored using numerical optimization techniques. In searching for minimum-energy configurations three particularly compact non-lattice growth schemes emerge showing tetrahedral, pentagonal (D5 h ) and icosahedral symmetry respectively. All these systems were found to be appreciably more stable than microcrystallites based on the face-centred cubic structure while certain lattice elements were shown to be metastable for small numbers of atoms. Some qualitative conclusions are then drawn concerning the occurrence of saddle-points for delocalized motion and the contribution of these to the generation of configurational entropy in small clusters. A crucial feature of the energy surfaces examined is the breakdown of strict local symmetry in compact clusters of more than two ‘shells’ of atoms and the possibility of delocalized motion of surface atoms around a solid-like core. The...

This paper explains and advocates the use of a Laplace‐transform steepest‐descent method for the computation of energy‐level densities g(E) by inversion of the partition function. The earlier Lin–Eyring approach, based on the Darwin–Fowler method, is shown to be unnecessarily obscure and in part redundant By use of a much‐simpler formulation due to Kubo, it is shown that the first‐order steepest‐descent approximation to the inversion integral follows straightforwardly from a knowledge of the relevant partition function, its energy, and specific‐heat relations. In this way several explicit equations for g(E) and its integral W(E) are derived, including two for anharmonic vibration and internal vibration–rotation. The relationship of the S–D method to Tolman's classical formula is discussed and some thermodynamic implications pointed out. Numerical results agree closely with those obtained by the direct counting of energy levels and, with certain exceptions, the indirect computations of other authors.

A study is presented of the potential energy and thermodynamic properties of assemblies of relatively small numbers of atoms (4 < N < 70) interacting through two-body central potentials. From this a primitive statistical mechanics is constructed based on the examination of compact minimum configurations and harmonic vibrational motion about these. Previous results for amorphous “polytetrahedral” and microcrystalline fcc-type clusters [M.R. Hoare and P. Pal, Advan. Phys. 20 (1971) 161; Nature 236 (1972) 35] are extended to evaluate thermodynamic as well as mechanical stability by consideration of the vibrational free-energy of various forms. Special attention is given to the thermodynamics of the “multiple twinning” process suggested by Ino, Allpress and Sanders and others to explain the production of pseudo-crystalline pentagonal and icosahedral forms in deposited films. It is concluded that these are unlikely to arise by stepwise composition of tetrahedra as previously assumed since for N < 50 the multiply-twinned structures are less favoured thermodynamically than alternative more disordered clusters, whether at zero or small finite temperatures. An alternative stepwise growth mechanism due to Fukano and Wayman is also disproved. Results indicate that growth is more likely to be haphazard with spontaneous emergence of certain compact forms such as the 13 and 55-atom icosahedra. Numerical results are presented for vibrational heat-capacity, entropy and free-energy for a variety of argon clusters in the harmonic approximation.

The classic problem of Rayleigh's piston is reconsidered in the light of modern transport theory. The velocity relaxation of a one-dimensional ensemble of test particles immersed in a similar heat bath at arbitrary mass ratio gamma is investigated, and a new reduction of the integral collision operator to an infinite-order differential expansion is obtained, which is more tractable than the conventional expression in powers of the mass ratio. In this way it is proved that the number of nonzero discrete eigenvalues of the Rayleigh collision operator is always bounded for finite mass ratio, being in fact zero for the special case gamma =1. By truncation of the collision operator expansion, a tentative bound is then obtained which suggests (subject to an unproved positive-definiteness condition) that the emptiness of the discretum actually extends over at least the mass ratio region (3( nth root 2-1))-1< gamma

We consider a class of scalar nonlinear Boltzmann equations describing the evolution of a microcanonical ensemble in which sub-systems exchange internal energy ‘randomly’ in binary interactions. In the continuous variable version these models can equally be interpreted as Boltzmann equations for Maxwell type molecules in arbitrary dimensionality. We construct general solutions in the form of a Fourier series; the expansion coefficients (Sonine or Meixner moments) satisfy the same recursive system of coupled equations as the ordinary moments, and can be solved sequentially.

Publisher Summary This chapter discusses the relationship between physics and mathematics. The relationship between physics and mathematics is in some ways one of the extraordinary intimacy, in others troubled and indefinite. The physicist tends to think of the mathematician as remote from what he, the physicist, likes to call reality, interested only in sterile abstractions and formal schemes; the mathematician tends to regard the physicist as unconcerned about the real subtleties of the equations he uses, slipshod in his proofs, and altogether too prone to draw mathematical conclusions from entirely nonmathematical insights. These criticisms arise quite naturally out of the differences in intellectual temperament of those drawn to the two subjects and are sometimes entirely justified. However, many misunderstandings are the product of the peculiar education system that so often attempts to draw a definite frontier between subjects where no such frontier exists.