Generalized enskog theory for homogeneous systems

We propose a generalization of the Enskog equation for homogeneous dense systems including the complete three-particle dynamics. To this end the time derivative of the one-particle distribution is represented in the thermodynamic limit as the sum of three terms describing the effect of the initials-particle correlations, collisions withins-particle clusters, and coupling ofs-particle clusters to the surrounding gaseous medium, respectively. The analysis of casess=2 ands=3 is performed both for hard spheres and for a smooth, repulsive interaction. On assuming the equilibrium structure and spatial dependence of terms reflecting the effect of the medium, we obtain fors=2 the Enskog equation, and fors=3 a new equation, going beyond the Enskog theory. Apart from the Enskog collision term it contains additional contributions, and can be shown to reduce to the Choh-Uhlenbeck equation in the long-time, low-density limit.     

Non-equilibrium statistical mechanics in the general theory of relativity: II. Linear fields in a kinetic approximation

The first paper in this series introduced a new, manifestly covariant approach to non-equilibrium statistical mechanics in classical general relativity. The object of this second paper is to apply that formalism to the evolution of a collection of particles that interact via linear fields in a fixed curved background spacetime. Given the viewpoint adopted here, the fundamental objects of the theory are a many-particle distribution function, which lives in a many-particle phase space, and a many-particle conservation equation which this distribution satisfies. By viewing a composite N-particle system as interacting one- and (N ? 1)-particle subsystems, one can derive exact coupled equations for appropriately defined reduced one- and (N ? 1)-particle distribution functions. Alternatively, by treating all the particles on an identical footing, one can extract an exact closed equation involving only the one-particle distribution. The implementation of plausible assumptions, which constitute straightforward generalizations of standard non-relativistic “kinetic approximations”, then permits the formulation of an approximate kinetic equation for the one-particle distribution function. In the obvious non-relativistic limit, one recovers the well-known Vlasov-Landau equation. The explicit form for the relativistic expression is obtained for three concrete examples, namely, interactions via an electromagnetic field, a massive scalar field, and a symmetric second rank tensor field. For a large class of interactions, of which these three examples are representative, the kinetic equation will admit a relativistic Maxwellian distribution as an exact stationary solution; and, for these interactions, an H-theorem may be proved.     

Nonequilibrium statistical mechanics in the general theory of relativity I. A general formalism

This is the first in a series of papers, the overall objective of which is the formulation of a new covariant approach to nonequilibrium statistical mechanics in classical general relativity. The object here is the development of a tractable theory for self-gravitating systems. It is argued that the “state” of an N-particle system may be characterized by an N-particle distribution function, defined in an 8N-dimensional phase space, which satisfies a collection of N conservation equations. by mapping the true physics onto a fictitious “background” spacetime, which may be chosen to satisfy some “average” field equations, one then obtains a useful covariant notion of “evolution” in response to a fluctuating “gravitational force.” For many cases of practical interest, one may suppose (i) that these fluctuating forces satisfy linear field equations and (ii) that they may be modeled by a direct interaction. In this case, one can use a relativistic projection operator formalism to derive exact closed equations for the evolution of such objects as an appropriately defined reduced one-particle distribution function. By capturing, in a natural way, the notion of a dilute gas, or impulse, approximation, one is then led to a comparatively simple equation for the one-particle distribution. If, furthermore, one treats the effects of the fluctuating forces as “localized” in space and time, one obtains a tractable kinetic equation which reduces, in the newtonian limit, to the standard Landau equation.     

Quantum Dynamics with Mean Field Interactions: a New Approach

We propose a new approach for the study of the time evolution of a factorized N-particle bosonic wave function with respect to a mean-field dynamics with a bounded interaction potential. The new technique, which is based on the control of the growth of the correlations among the particles, leads to quantitative bounds on the difference between the many-particle Schrödinger dynamics and the one-particle nonlinear Hartree dynamics. In particular the one-particle density matrix associated with the solution to the N-particle Schrödinger equation is shown to converge to the projection onto the one-dimensional subspace spanned by the solution to the Hartree equation with a speed of convergence of order 1/N for all fixed times.     

Decay of correlations. IV. Necessary and sufficient conditions for a rapid decay of correlations

We considerN-particle systems whose probability distributions obey the master equation. For these systems, we derive the necessary and sufficient conditions under which the reducedn-particle (n) probabilities also obey master equations and under which the Ursell functions decay to their equilibrium values faster than the probability distributions. These conditions impose restrictions on the form of the transition rate matrix and thus on the form of its eigenfunctions. We first consider systems in which the eigenfunctions of theN-particle transition rate matrix are completely factorized and demonstrate that for such systems, the reduced probabilities obey master equations and the Ursell functions decay rapidly if certain additional conditions are imposed. As an example of such a system, we discuss a random walk ofN pairwise interacting walkers. We then demonstrate that for systems whoseN-particle transition matrix can be written as a sum of one-particle, two-particle, etc. contributions, and for which the reduced probabilities obey master equations, the reduced master equations become, in the thermodynamic limit, those for independent particles, which have been discussed by us previously. As an example of suchN-particle systems, we discuss the relaxation of a gas of interacting harmonic oscillators.Supported in part (grants to D.B. and K.E.S.) by the Advanced Research Projects Agency of the Department of Defense as monitored by the U.S. Office of Naval Research under Contract N00014-69-A-0200-6018, and in part (grant to I.O.) by the National Science Foundation.     

Mouvement Brownien dans une assemblee de batonnets minces dans la phase nematique

Using the kinetic methods of the Brussels school, we establish the equation (to the second order in the perturbation) for the return to equilibrium of the one-particle energy distribution function in the nematic phase of a fluid made of thin slabs interacting through a P₂-type potential. On the basis of the mean field equilibrium theory developed by Maier and Saupe for such a fluid, we show that for a very heavy brownian particle, this equation reduces to a Fokker-Planck type equation; the friction coefficient thus obtained is compared with the friction coefficient obtained for the isotropic phase and we show that they are equal for the transition temperature.     

Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum

We consider a system ofN hard disks in ?² in the Boltzmann-Grad limit (i.e.N → ∞,d ↘ 0,N·d → λ^?1>0, whered is the diameter of the disks). If λ is sufficiently small and if the joint distribution densities factorize at time zero, we prove that the time-evolved one-particle distribution converges for all times to the solution of the Boltzmann equation with the same initial datum.     

N-particle correlations in the McKean model

In the McKean model the BBGKY hierarchy is equivalent to a simple hierarchy of coupled equations for thep-particle correlation functions. Approximate solutions are obtained by truncating the hierarchy. The convergence of the truncation method is studied by comparison with the exact solution for the model, which can be given in closed form. In the long-time limit the exact solution is linearized around the equilibrium value, showing the decay of the correlations. It turns out thatp-particle correlations decayp times faster than the nonequilibrium one-particle distribution.     

Polarization properties of fluctuations of the field created by randomly located dipoles

The vector properties of the strength of a field created by an ensemble of N parallel dipoles located, on average, uniformly are considered. For N → ∞, the problem is reduced to the Poisson problem. The direction of the dipoles specifies the symmetry axis of fluctuations. The moments of probability distributions for the Cartesian components of the strength are calculated. The anisotropy of the fluctuations is approximately 15–20%. Under conditions that are of interest for spectroscopic applications, the N-particle probability distribution contains a weak broad negative background, the shape of which replicates the one-particle distribution. The analogy between the problem considered and the theory of spectral line broadening in the impact approximation, in particular, the Dicke narrowing of the Doppler profile, is analyzed. Conditions of applicability of the theory developed are discussed.     

Generalized density expansion in the kinetic theory and the resistivity of liquid metals

For the system of electrons and immovable interacting centers an exact equation for averaged electron Green's function is formulated. The expansion of self-energy part over the one-particle t-matrices and explicit Green's functions is derived. It represents a kind of a generalized density series containing the correlation functions of the centres. In the low approximation over t-matrix, the transition probability (t)²S in the kinetic equation is obtained (S = the structure factor of centers).     

Calculation of Coefficients of Fractional Parentage for Large-Basis Harmonic-Oscillator Shell Model

A new procedure for large-scale calculations of the coefficients of fractional parentage (CFPs) for a single j-orbit with isospin is presented. The approach is based on a simple enumeration scheme for antisymmetric A-particle states and an efficient method for constructing the eigenvectors of an idempotent matrix. We investigate the characteristics of the introduced CFP basis and the application of this procedure to the ab initio harmonic-oscillator shell-model approach. The results of CFP calculations for the j=1/2,…,41/2 orbits are presented (the full sets of one-particle and two-particle CFPs up to the j=9/2 orbit are obtained). The new computer code for calculation of the CFPs proves to be very quick, efficient, and numerically stable and produces results possessing only small numerical uncertainties.     

New properties of a class of generalized kinetic equations

We give some properties of a new class of hard-sphere kinetic equations of great generality, introduced earlier by Polewczak. The assumptions used to obtain the general class are very weak, and the equations include not only the standard and revised Enskog equations, but also generalizations thereof that can be expected to yield essentially exact transport coefficients. In particular, there is a natural two-particle realization that is obtained from maximizing the information entropy subject to prescribed two-particle and one-particle probability distribution functions;k-particle analogs fork > 2 also naturally follow. We obtain Liapunov functionals for the whole class of equations under consideration and discuss the question of which of these functionals can be expected to play the role ofH-functions. We also obtain several more special results that include new lower bounds on the potential part of theH-function for the revised Enskog equation. The bounds are instrumental in obtaining global existence theorems and also imply that the necessary condition for invertibility of the nonequilibrium extension of local activity as a functional of local density is satisfied.     

Initial state dependence of nonlinear kinetic equations: The classical electron gas

The method of nonequilibrium cluster expansion is used to stydy the decay to equilibrium of a weakly coupled inhomogeneous electron gas prepared in a local equilibrium state at the initial time,t=0. A nonlinear kinetic equation describing the long time behavior of the one-particle distribution function is obtained. For consistency, initial correlations have to be taken into account. The resulting kinetic equation-differs from that obtained when the initial state of the system is assumed to be factorized in a product of one-particle functions. The question of to what extent correlations in the initial state play an essential role in determining the form of the kinetic equation at long times is discussed. To that end, the present calculations are compared with results obtained before for hard sphere gases and in general gases with strong short-range forces. A partial answer is proposed and some open questions are indicated.     

Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent

The quantal system of Bose particles described by the non-linear Schrödinger equation i?φ/?t = -$\text{1}\text{2}$?²φ/?x² + cφ1φ², with c= cxf∞ and via the ground state with finite particle density, is the 1- dimensional gas of impenetrable bosons studied by M. Girardeau, T.D. Schultz, A. Lenard, H.G. Vaidya and C.A. Tracy. We show that the 2-point (resp. 2n-point) function, or the 1-particle (resp. n-particle) reduced density matrix, of this system satisfies a non-linear differential equation (resp. a system of non-linear partial differential equations) of Painlevé type. Derivation of these equations is based on the link between field operators in a Clifford group and monodromy preserving deformation theory, which was previously established and applied to the 2-dimensional Ising model and other problems. Several related topics are also discussed.     

The exact solution of an elimination problem in kinetic theory: I. The one-dimensional hard sphere gas

A class of initial value problems for a one-dimensional hard sphere gas is considered where a specified particle has a given distribution f⁽¹⁾(z₁; 0) and the rest are in equilibrium at t=0. An exact expansion is obtained for a certain n-particle reduced distribution function f⁽ⁿ⁾(z₁;…;z_n; t) in terms of the 1-particle reduced distribution function f⁽¹⁾(z₁; t) for the specified particle by starting with separate expressions for these functions in terms of f⁽¹⁾(z₁; 0). Expansions for the corresponding cluster functions are first obtained and then graph theoretic methods applied to obtain a solution.     

Comment on symmetry properties of the linear Enskog kinetic operators

The one-particle average consistent with the structure of the revised Enskog theory is introduced. Symmetry properties of the linear kinetic operators reflecting those of theN-particle pseudo-Liouville operators are derived, implying a recently proved symmetry of kinetic expressions for equilibrium time correlation functions.     

Metastable states for the Becker-Döring cluster equations

The Becker-Döring equations, in whichc _l (t) can represent the concentration ofl-particle clusters or droplets in (say) a condensing vapour at timet, are $$\begin{array}{*{20}c} {{{dc_l (t)} \mathord{\left/ {\vphantom {{dc_l (t)} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = J_{l - 1} (t) - J_l (t)} & {(l = 2,3,...)} \\ \end{array} $$ with $$J_l (t): = a_l c_1 (t)c_l (t) - b_{l + 1} c_{l + 1} (t)$$ and eitherc ₁=const. (‘case A’) or \(\rho : = \sum\limits_1^\infty {lc_l } \) =const. (‘case B’). The equilibrium solutions arec _l =Q _l z ^l , where \(Q_l : = \prod\limits_2^l {({{a_{r - 1} } \mathord{\left/ {\vphantom {{a_{r - 1} } {b_r }}} \right. \kern-0em} {b_r }})} \) . The density of the saturated vapour, defined as \(\rho _s : = \sum\limits_1^\infty {lQ_l z_s ^l } \) , wherez _s is the radius of convergence of the series, is assumed finite. It is proved here that, subject to some further plausible conditions on the kinetic coefficientsa _l andb _l , there is a class of “metastable” solutions of the equations, withc ₁?z _s small and positive, which take an exponentially long time to decay to their asymptotic steady states. (An “exponentially long time” means one that increases more rapidly than any negative power of the given value ofc ₁?z _s (or, in caseB,ρ?ρ _s ) as the latter tends to zero). The main ingredients in the proof are (i) a time-independent upper bound on the solution of the kinetic equations (this upper bound is a steady-state solution of case A of the equations, of the type used in the Becker-Döring theory of nucleation), and (ii) an upper bound on the total concentration of particles in clusters greater than a certain critical size, which (with suitable initial conditions) remains exponentially small until the time becomes exponentially large.     

Ab-initio calculations of pair-correlation effects on the hyperfine structure of the 4s 4p and 4s 3d configurations of Ca

The hyperfine structure of the 4s 4p and 4s 3d configurations of Ca has been evaluated using many-body perturbation theory. Single excitations and pair-correlation effects have been included by solving coupled one-particle and two-particle differential equations numerically. The correlation between the valence electrons has been treated selfconsistently by solving these equations iteratively for the valence pair. Core-polarisation effects have been evaluated by hyperfine-induced single-particle functions. From the calculated and experimental results of the two configurations,Q=?49(5) mb has been evaluated for the nuclear quadrupole moment of⁴³Ca.     

Pairing in multiparticle amplitudes: Inclusive distributions

A model for multi-pion production in the central region in high-energy collisions is studied which describes factorizable emission of pion pairs. A mathematical identification between the exclusive cross section for pion emission in our model (with all interference terms) and the configurational probability distribution function for a classical system of interacting molecules in equilibrium is exploited to obtain an expansion for the asymptotic single-particle inclusive distribution, the two-particle inclusive correlation function, and the exponent of s in the total cross section by means of cluster diagrams. An integral equation is exhibited for summing the terms corresponding to the cluster diagrams.A specific model is then considered, which we call “s-channel pole dominance”. In this model the amplitude is assumed to be large only when the subenergies of pairs of pions are near the mass of a low-lying two-pion resonance, and the transverse momentum of each resonance is small. The dependence of the amplitude on other variables is ignored, so that we effectively have independent emission of two-pion resonances with non-zero width. It is seen that an I = 0 or I = 1 resonance results in a positive two-particle inclusive I = 2 correlation function at small rapidity separations, as s → ∞, and that the correlation function can have an exponential “tail” in rapidity of qualitatively longer range than the resonance. A crude numerical simulation of a broad I = 0 spinless resonance is discussed, and the resulting I = 2 inclusive correlation function is seen to be quite large at small rapidity separations, and to have the same exponential “tail” as the I = 0 correlation function.     

Disordered system withn orbitals per site: Lagrange formulation without replica trick,and scaling law for the density of states

A Lagrange formulation of the gauge invariantn orbital model of disordered electronic systems is given for the one-particle Green's function. The replica trick is avoided by starting from a formulation on a Grassmann algebra. A vector model of a real, two component vector is derived. Fluctuations around the saddle point solution of the model are studied. A non-linear transformation allows the consideration of all the important fluctuations. In contrast to the 1/n-expansion of Oppermann and Wegner it is possible to take then=∞ band edges into account. In a vicinity of these band edges a scaling law for the density of states is found: $$\rho ({\rm E}) = n^{ - \user2{\xi }} \bar \rho (n^{2\user2{\xi }} (|E| - E_0 ))$$ with an exponent ξ=2/(6?d) ford<2 and large values ofn.