Stability of Equilibria with a Condensate

A quantum system composed of a spatially infinitely extended free Bose gas with a condensate, interacting with a quantum dot, which can trap finitely many Bosons, has multiple equilibria at fixed temperature. We extend the notion of return to equilibrium to systems possessing a multitude of equilibrium states and show that the above system returns to equilibrium in a weak coupling sense: any local perturbation of an equilibrium state converges in the long time limit to an asymptotic state. The latter is, modulo an error term, an equilibrium state which depends, in an explicit way, on the initial local perturbation. The error term vanishes in the small coupling limit.We deduce this stability result from properties of structure and regularity of eigenvectors of the Liouville operator, the generator of the dynamics. Among our technical results is a virial theorem for Liouville type operators which has new applications to systems with and without a condensate.Supported by a CRM-ISM postdoctoral fellowship and by McGill University     

Correlation Decay in Certain Soft Billiards

Motivated by the 2D finite horizon periodic Lorentz gas, soft planar billiard systems with axis-symmetric potentials are studied in this paper. Since Sinais celebrated discovery that elastic collisions of a point particle with strictly convex scatterers give rise to hyperbolic, and consequently, nice ergodic behaviour, several authors (most notably Sinai, Kubo, Knauf) have found potentials with analogous properties. These investigations concluded in the work of V. Donnay and C. Liverani who obtained general conditions for a 2-D rotationally symmetric potential to provide ergodic dynamics. Our main aim here is to understand when these potentials lead to stronger stochastic properties, in particular to exponential decay of correlations and the central limit theorem. In the main argument we work with systems in general for which the rotation function satisfies certain conditions. One of these conditions has already been used by Donnay and Liverani to obtain hyperbolicity and ergodicity. What we prove is that if, in addition, the rotation function is regular in a reasonable sense, the rate of mixing is exponential, and, consequently, the central limit theorem applies. Finally, we give examples of specific potentials that fit our assumptions. This way we give a full discussion in the case of constant potentials and show potentials with any kind of power law behaviour at the origin for which the correlations decay exponentially.     

Closed-orbit theory of spatial density oscillations in finite fermion systems

We investigate the particle and kinetic-energy densities for N noninteracting fermions confined in a local potential. Using Gutzwiller's semiclassical Green function, we describe the oscillating parts of the densities in terms of closed nonperiodic classical orbits. We derive universal relations between the oscillating parts of the densities for potentials with spherical symmetry in arbitrary dimensions and a "local virial theorem" valid also for arbitrary nonintegrable potentials. We give simple analytical formulas for the density oscillations in a one-dimensional potential.     

Virial scaling corrections to Koopmans' ionization potentials in atoms

Relaxation corrections to Koopmans' approximation for atomic ionization potentials are derived on the basis of the virial theorem. They require no information in addition to that available in the Hartree-Fock computation for the neutral system, and considerably improve the agreement with more accurate ionization potentials.     

Second and third virial coefficient of a quantum gas from two-particle scattering amplitude

Starting from the cluster expansion of the partition function the second and third virial coefficient of a quantum gas is expressed in terms of the two-particle scattering amplitude. In the case of spherically symmetric interaction the result forB(T) agrees with the well known expression ofBeth andUhlenbeck, but the method given here is also valid for non-spherically symmetric and even for non-local potentials. For the third virial coefficientC(T) an expression is derived in lowest order in the two-particle scattering amplitude which is suitable for numerical calculations.     

Quantum and relativistic virial inequalities

The generalization of the virial theorem is discussed. The case where the potential energy is a sum of homogeneous functions of various degree is investigated. If the potential energy U is composed of a gravitational (or Coulomb) energy and an energy of the short-range repulsion of particles, then virial inequalities of the form 2¯K + < 0 are valid, where K is the kinetic energy. For classical systems of this type, but with a Hamiltonian relativistic in the momenta, the inequality 3N < ¦¦ holds, where N is the number of particles in the system, = kT, T is the temperature, and k is Boltzmann's constant.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 76–79, June, 1979.     

Partial virial theorems for atomic-molecular systems

New virial relations for three-and four-particle atomic-molecular systems are proposed. Using operators of extension or squeezing of interparticle distances, it is shown that, for all pairs of j and k particles in S states of these systems, the following partial virial relations are valid: 〈2T _jk 〉+〈 V _jk 〉=0, where 〈V _jk 〉 is the average Coulomb interaction energy for a pair of particles and 〈T _jk 〉 is a part of the average kinetic energy of the system. There are three and six such relations for three-and four-particle systems, respectively. The conventional virial theorem (〈 2T〉+〈V〉=0) for the average total kinetic and potential energies of the system (〈 T〉 and 〈V〉, respectively) corresponds to the summation of partial virial relations over all pairs of particles. It is shown by an example of variational calculations of the helium atom ⁴He²⁺ e ^? e ^? and the helium muon-electron mesoatom ⁴He²⁺μ^? e ^? that partial virial relations are a highly sensitive indicator of the accuracy of wave functions.     

The Microscopic Foundations of Vlasov Theory for Jellium-Like Newtonian N-Body Systems

The kinetic equations of Vlasov theory, in the weak formulation, are rigorously shown to govern the $N\rightarrow \infty $ limit of the Newtonian dynamics of $D\ge 2$ -dimensional $N$ -body systems with attractive harmonic pair interactions and locally integrable repulsive inverse power law pair interactions, provided a mild higher moment hypothesis on the forces (which is shown to propagate globally in time for each $N$ ) will hold uniformly in $N$ at later times if it holds uniformly in $N$ initially (the uniformity in $N$ of this moment condition is demonstrated to hold for an open set of initial data). Logarithmic interactions are included as a limiting case. The proof is based on the Liouville equation, more precisely the first member of the pertinent BBGKY hierarchy, and does not invoke the Hewitt–Savage theorem, nor any regularization of the interactions. In addition, a rigorous proof of the virial theorem and of some of its interesting ramifications is given.     

Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions

We evaluate the virial coefficients B_k for for hard spheres in dimensions Virial coefficients with k even are found to be negative when This provides strong evidence that the leading singularity for the virial series lies away from the positive real axis when . Further analysis provides evidence that negative virial coefficients will be seen for some k > 10 for D = 4, and there is a distinct possibility that negative virial coefficients will also eventually occur for D = 3.     

On the Virial Theorem in Quantum Mechanics

We review the various assumptions under which abstract versions of the quantum mechanical virial theorem have been proved. We point out a relationship between the virial theorem for a pair of operators H, A and the regularity properties of the map . We give an example showing that the statement of the virial theorem in [CFKS] is incorrect. Received: 7 January 1999 / Accepted: 2 June 1999     

PROOF OF THE VALIDITY OF THE VIRIAL THEOREM IN THE EQUATION OF STATE BY THE THOMAS-FERMI-DIRAC MODEL

The present paper is to demonstrate the validity of the virial theorem for electrons in the revised Thomas-Fermi-Dirac model, which has not yet received a rigorous proof. The theorem has actually a wider sense in application both in classical and quantum-mechanical dynamics.     

On the Virial Theorem for the Relativistic Operator of Brown and Ravenhall,and the Absence of Embedded Eigenvalues

A virial theorem is established for the operator proposed by Brown and Ravenhall as a model for relativistic one-electron atoms. As a consequence, it is proved that the operator has no eigenvalues greater than max(2Z - )mc², where is the fine structure constant, for all values of the nuclear charge Z below the critical value Z_c: in particular, there are no eigenvalues embedded in the essential spectrum when Z 3/4 . Implications for the operators in the partial wave decomposition are also described.     

Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics

A convergence theorem of the fractional step Lax-Friedrichs scheme and Godunov scheme for an inhomogeneous system of isentropic gas dynamics (1<5/3) is established by using the framework of compensated compactness. Meanwhile, a corresponding existence theorem of global solutions with large data containing the vacuum is obtained.Partially supported by U.S. NSF Grant # DMS-850403     

Some generalizations of the virial theorem

Generalizations of the virial theorem are derived: In atomic physics, in systems including electromagnetic radiation, in Newtonian gravitation, and in general relativity and also some types of nuclear forces. The cases discussed are limited to potentials which can be produced by the exchange of one particle, which include potentials of the form1/r. The method used is to set equal a change in energy produced by an infinitesimal similarity transformation to a change of energy obtained by a first-order perturbation.Work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract # W-7405-Eng-48.     

Approximating Multi-Dimensional Hamiltonian Flows by Billiards

The behavior of a point particle traveling with a constant speed in a region , undergoing elastic collisions at the regions’s boundary, is known as the billiard problem. Various billiard models serve as approximation to the classical and semi-classical motion in systems with steep potentials (e.g. for studying classical molecular dynamics, cold atom’s motion in dark optical traps and microwave dynamics). Here we develop methodologies for examining the validity and accuracy of this approximation. We consider families of smooth potentials , that, in the limit , become singular hard-wall potentials of multi-dimensional billiards. We define auxiliary billiard domains that asymptote, as to the original billiards, and provide, for regular trajectories, asymptotic expansion of the smooth Hamiltonian solution in terms of these billiard approximations. The asymptotic expansion includes error estimates in the C ^r norm and an iteration scheme for improving this approximation. Applying this theory to smooth potentials that limit to the multi-dimensional close to ellipsoidal billiards, we predict when the billiard’s separatrix splitting (which appears, for example, in the nearly flat and nearly oblate ellipsoids) persists for various types of potentials.     

Standard Potentials for Non-Even Dynamics

The notion of standard potentials is introduced for a general dynamics. This is a generalization of earlier works of Araki and Moriya which is restricted to even dynamics. Most formulae in the present analysis are the same as the case of even dynamics: The time derivative of a local observable is times the sum of commutators with all potentials, and application of the conditional expectation to the local algebra for a region I to a potential for a region J leave the potential unchanged if and annihilate it otherwise (the standardness of the potential). However, the convergence condition for the potential takes a different form for the odd part of the potential. The equivalence of various characterizations of equilibrium states remain valid, except that the variational principle is out of the game for non-even dynamics because the translation invariance and non-evenness of dynamics are incompatible as is already known.     

New accurate binary hard sphere mixture radial distribution functions at contact and a new equation of state

New and very accurate formulae for additive binary hard sphere (HS) mixture radial distribution functions (RDFs) at contact are proposed in a simple analytical form. Using the virial theorem, the formulae also provide a new HS mixture equation of state (EOS). The new RDF formulae are the most accurate currently available. The new EOS is of comparable accuracy with that of Malijevsky, A., and Veverka, J. (1999, Phys. Chem. chem. Phys., 1, 4267), which is the most accurate HS mixture EOS currently available. However, the new EOS proposed here is of much simpler analytical form.     

Methodology of Analytic and Computational Studies on Quantum Systems

The concept of separation of procedures and the ST-transformation are briefly reviewed together with the equivalence theorem that a d-dimensional quantum system with finite-range interactions is equivalent to the corresponding (d+1)-dimentional classical system with finite-range interactions. This theorem yields the introduction of the quantum transfer-matrix method. Thermo quantum dynamics is formulated using the quantum transfer-matrix method. This new formulation has the great merit that the thermal average Q for any observable Q in the thermodynamic limit is expressed as an expectation value over a temperature-dependent state vector in the single (conjugate) Hilbert space in the contrast to the usage of the double Hilbert space in thermo field dynamics.     

Generalized virial theorem for the Klein-Gordon equation

The virial theorem for the Klein-Gordon equation has been generalized with respect to non-integrable scale-invariant probe functions. Ground-state energies as well as certain upper bounds on the coupling constants have been established. For definiteness several kinds of attractive power potentials have been considered.