Linear quantum enskog equation II. Inhomogeneous quantum fluids

This is the second part of a work concerned with the quantum-statistical generalization of classical Enskog theory, whereby the first part is extended to spatially inhomogeneous fluids. In particular, working with Liouville operators and using cluster expansions and projection operators, we derive the inhomogeneous linear quantum Enskog equation and express the dynamic structure factor and the nonlocal mobility tensor in terms of the corresponding quantum Enskog collision operator. Thereby static correlations due to excluded volume effects and quantum-statistical correlations due to the fermionic (bosonic) character of the pairwise strongly interacting particles are treated exactly. When static correlations are neglected, this Enskog equation reduces to the inhomogeneous linear quantum Boltzmann equation (containing an exchange-modifiedt-matrix). In the classical limit, the well-known linear revised Enskog theory is recovered for hard spheres.     

Der zweite quantenmechanische Virialkoeffizient eines Boltzmann-Gases

The second quantum mechanical virial coefficient for a gas is given in the same approximation, in which the masterequation and the Boltzmann equation hold. The close connection between the macroscopic equations of motion and the equation of state of a real gas in equilibrium is shown. The virial coefficient of the Boltzmanngas is given in compact form by the real part of the forward scattering amplitude, weighed with the Boltzmann distribution at the given temperature.     

Non equilibrium density expansion in the dilute one component plasma

In plasmas the 3-body collision operator diverges even after its most diverging part has been included in the Balescu Lénard collision operator. The virial expansion of transport coefficient after the Boltzmann order includes fractional powers and logarithms.     

A BBGKY-based density gradient approximation of interparticle forces: Application for discrete Boltzmann methods

The force term accounting for interparticle interactions, with application to discrete Boltzmann simulation methods, is derived using a density gradient expansion of the BBGKY collision operator. It is shown that previous calculations, based on essentially the same mean-field-theory philosophy, do not apply the density gradient approximation in a self consistent fashion. Thus these previous models have errors in the second virial coefficient as well as coefficients associated with gradient terms in the pressure tensor. This new treatment corrects these shortcomings.     

Quantenstatistische Zustandsgleichung von stark ionisierten Wasserstoffplasmen mittlerer Dichte

A quantum statistical formula for the equation of state of fully ionized plasmas in the mean temperature and mean density region is given. The complete second virial coefficient for the case of Debye screening is taken into account. The binary Slater sums are represented by interpolation between the analytical expressions for small and for large radii respectively. Using the interpolation formulae the equation of state is calculated numerically for the temperatures T = 20000 K and T = 80000 K. The resulting curves are in good agreement with the analytical equation of state for very small densities.     

Quantum Statistical Second Virial Coefficient for Real Gases and Plasmas

Using the resolvent representation of the binary Boltzmann operator the second virial coefficient is expressed by a contour integral over the Jost function of the two-particle scattering problem. This new formula is useful for many general investigations and also for practical calculations if the Jost functions for the scattering problem are known. As special examples real gases with hard core potentials and fully ionized plasmas with Coulomb interactions are treated. For the plasma case the second virial coefficient is derived without approximations with respect to the interaction parameter. The contribution of the bound states to the free energy is discussed and compared with the effect of ionic association to the classical free energy and the conductivity.     

On the quantum corrections to the virial coefficients of the equation of state of a fluid

The first quantum correction to the virial coefficients of the equation of state of a fluid is derived in the presence of a weak three-body potential ?(i, j, k). Results for the third and fourth virial coefficients are given. Representing the potential energy of interaction of a pair and a triplet, by the Lennard-Jones (12-6) model and the triple dipole dispersion potential model of Axilrod and Teller, the first quantum correction to the third virial coefficient is calculated for many values of T*. The theoretical result is compared with the experimental data of helium.     

Critical temperature of infinitely long chains from Wertheim's perturbation theory

Wertheim's theory is used to determine the critical properties of chains formed by m tangent spheres interacting through the pair potential u(r). It is shown that within Wertheim's theory the critical temperature and compressibility factor reach a finite non-zero value for infinitely long chains, whereas the critical density and pressure vanish as m ^-1.5. Analysing the zero density limit of Wertheim's equation or state for chains it is found that the critical temperature of the infinitely long chain can be obtained by solving a simple equation which involves the second virial coefficient of the reference monomer fluid and the second virial coefficient between a monomer and a dimer. According to Wertheim's theory, the critical temperature of an infinitely long chain (i.e. the Θ temperature) corresponds to the temperature where the second virial coefficient of the monomer is equal to 2/3 of the second virial coefficient between a monomer and dimer. This is a simple and useful result. By computing the second virial coefficient of the monomer and that between a monomer and a dimer, we have determined the Θ temperature that follows from Wertheim's theory for several kinds of chains. In particular, we have evaluated Θ for chains made up of monomer units interacting through the Lennard-Jones potential, the square well potential and the Yukawa potential. For the square well potential, the Θ temperature that follows from Wertheim's theory is given by a simple analytical expression. It is found that the ratio of Θ to the Boyle and critical temperatures of the monomer decreases with the range of the potential.     

Kinetic derivation of the second virial coefficient for a quantum gas

A kinetic equation has been proposed earlier for an inhomogeneous quantum dilute gas obeying Boltzmann statistics. We show here that it leads to the correct equilibrium properties of the gas by computing the second virial coefficient as it can be deduced from the trace of the pressure tensor and by checking the result with the value deduced from various approaches.     

Quantum-statistical kinetic equations

Considering a homogeneous normal quantum fluid consisting of identical interacting fermions or bosons, we derive an exact quantum-statistical generalized kinetic equation with a collision operator given as explicit cluster series where exchange effects are included through renormalized Liouville operators. This new result is obtained by applying a recently developed superoperator formalism (Liouville operators, cluster expansions, symmetrized projectors,P _q rule, etc.) to nonequilibrium systems described by a density operator(t) which obeys the von Neumann equation. By means of this formalism a factorization theorem is proven (being essential for obtaining closed equations), and partial resummations (leading to renormalized quantities) are performed. As an illustrative application, the quantum-statistical versions (including exchange effects due to Fermi-Dirac or Bose-Einstein statistics) of the homogeneous Boltzmann (binary collisions) and Choh-Uhlenbeck (triple collisions) equations are derived.     

Ursell operators in statistical physics of dense systems: the role of high order operators and of exchange cycles

The purpose of this article is to discuss cluster expansions in dense quantum systems, as well as their interconnection with exchange cycles. We show in general how the Ursell operators of order l≥ 3 contribute to an exponential which corresponds to a mean-field energy involving the second operator U₂, instead of the potential itself as usual - in other words, the mean-field correction is expressed in terms of a modification of a local Boltzmann equilibrium. In a first part, we consider classical statistical mechanics and recall the relation between the reducible part of the classical cluster integrals and the mean-field; we introduce an alternative method to obtain the linear density contribution to the mean-field, which is based on the notion of tree-diagrams and provides a preview of the subsequent quantum calculations. We then proceed to study quantum particles with Boltzmann statistics (distinguishable particles) and show that each Ursell operator U_n with n≥ 3 contains a “tree-reducible part”, which groups naturally with U₂ through a linear chain of binary interactions; this part contributes to the associated mean-field experienced by particles in the fluid. The irreducible part, on the other hand, corresponds to the effects associated with three (or more) particles interacting all together at the same time. We then show that the same algebra holds in the case of Fermi or Bose particles, and discuss physically the role of the exchange cycles, combined with interactions. Bose condensed systems are not considered at this stage. The similarities and differences between Boltzmann and quantum statistics are illustrated by this approach, in contrast with field theoretical or Green's functions methods, which do not allow a separate study of the role of quantum statistics and dynamics. Received 18 October 2001     

Theory and simulations of rigid polyelectrolytes

Theoretical and numerical studies are reported on stiff, linear polyelectrolytes within the framework of the cell model, first reviewing analytical results obtained on a mean-field Poisson—Boltzmann level, and then using molecular dynamics simulations to show the circumstances under which these fail quantitatively and qualitatively. For the hexagonally packed nematic phase of the polyelectrolytes the osmotic coefficient is computed as a function of density. In the presence of multivalent counterions it can become negative, leading to effective attractions. This is shown to result from a reduced contribution of the virial part to the pressure. The osmotic coefficient and ionic distribution functions are computed from Poisson—Boltzmann theory with and without a recently proposed correlation correction. Simulation results for the case of poly(p-phenylene) are presented and compared with recently obtained experimental data on this stiff polyelectrolyte. Ion—ion correlations in the strong coupling regime are studied and compared with the predictions of the recently advocated Wigner crystal theories.     

Triple-collision operator for a quantum gas with bound states

The triple-collision operator of dense gas theory is analyzed for a quantum-mechanical gas obeying Boltzmann statistics. The contribution of two-body bound states is extracted by using Faddeev's representation of the three-body resolvent. The result is a binary atom-molecule collision operator which includes the effects of molecular formation and breakup, and inelastic and rearrangement collisions. An additional contribution is a modification of the Boltzmann collision operator due to the binding of one member of the colliding pair to a third particle. The analysis is carried out in the framework of the Green-Kubo formulas so the operators considered are linear and the results are in a form suitable for the evaluation of the transport coefficients.     

A quantum theory of higher virial coefficients

Recently a theory of time-delay phenomena in few particle scattering has been developed. The results of this theory are used to investigate the quantum virial coefficient problem in the case of Boltzmann statistics. Working within the frame work of Faddeev's time-dependent scattering theory we find explicit formulas for the higher virial coefficients. One aim of statistical mechanics is to derive all the equilibrium properties of a macroscopic system from the dynamical laws of the constituent particles. Our solutions for the higher virial coefficients prove that the macroscopic properties of a quantum gas are sensitive only to the time-delay aspect of the collision process. The analysis does not restrict the type of interactions between the constituents with the one exception of long-range Coulomb forces. In particular, the interaction may be attractive and strong enough to form stable few-particle clusters. Thus our solution describes the equilibrium in a quantum gas where the interactions are responsible for creating different species types.     

Quantum kinetic equation for nonequilibrium dense systems

Using the density matrix method in the form developed by Zubarev, equations of motion for nonequilibrium quantum systems with continuous short range interactions are derived which describe kinetic and hydrodynamic processes in a consistent way. The T-matrix as well as the two-particle density matrix determining the nonequilibrium collision integral are obtained in the ladder approximation including the Hartree-Fock corrections and the Pauli blocking for intermediate states. It is shown that in this approximation the total energy is conserved. The developed approach to the kinetic theory of dense quantum systems is able to reproduce the virial corrections consistent with the generalized Beth-Uhlenbeck approximation in equilibrium. The contribution of many-particle correlations to the drift term in the quantum kinetic equation for dense systems is discussed.     

Density dependence of electric properties of binary mixtures of inert gases

A computational study of the density dependence of the refractivity and dielectric constants, the electric-field induced second harmonic generation (ESHG) hyperpolarizabilities and of the Kerr constants of binary mixtures of helium, neon and argon is presented. Potentials and interaction properties of the homonuclear A₂ and heteronuclear AB dimers (A,?B=He, Ne, Ar) are taken from a previous study [J. López Cacheiro, B. Fernández, D. Marchesan, S. Coriani, C. Hättig, A. Rizzo. Molec. Phys., 102, 101 (2004)]. Dispersion coefficients for the second virial coefficients allow for the determination of the density dependence at any frequency far from the lowest resonance. Fully quantum mechanical results are presented and a comparison with the corresponding classical estimates is discussed. Deep minima are predicted to occur in the ESHG second virial coefficient curve drawn as a function of the molar fraction of one of the components in binary mixtures of He/Ar and Ne/Ar. This phenomenon, observed over a wide range of temperatures, should be easily verifiable experimentally.     

Locality,factorizability, and the Maxwell Boltzmann distribution

A classical gas at equilibrium satisfies the locality conditionif the correlations between local fluctuations at a pair of remote small regions diminish in the thermodynamic limit. The gas satisfies a strong locality conditionif the local fluctuations at any number of remote locations have no (pair, triple, quadruple....) correlations among them in the thermodynamic limit. We prove that locality is equivalent to a certain factorizability condition on the distribution function. The analogous quantum condition fails in the case of a freeBose gas. Next we prove that strong locality is equivalent to the total factorizability of the distribution function, and thus (given Liourilles theorem) to the Maxwell Boltzmann distribution for an ideal gas.Dedicated to Professor Max Jammer on the occasion of his eightieth birthday. April 13. 1995.