Integral equations for particle density matrices

A system of integral equations for particle density matrices is obtained. The method of expansion in one-, two-, three-particle (etc.) parameters is used. It is shown that in the classical limit the resulting equations become the system of familiar equations for particle distribution functions of classical statistical mechanics.Translated from Izvestiya VUZ. Fizika, Vol. 11, No. 8, pp. 81–86, August, 1968.     

Lower bounds for Wiener integrals

Techniques are developed to sharpen lower bounds for density matrices occurring in statistical mechanics. The Wiener integrals are treated by insertion of trial functionals and parametric representations of unity that involve functionals of the path. Jensen's inequality is applied to suitable parameter-dependent path measures. These yield stronger forms than the basic Feynman bound. We also introduce trajectory insertions, and use coupling constant integration and the hierarchy for correlation functions to improve the bounds.     

Second quantized reduced Bloch equations and the exact solutions for pairing Hamiltonian

In this article, we present a set of hierarchy Bloch equations for the reduced density operators in either canonical or grand canonical ensembles in the occupation number representation. They provide a convenient tool for studying the equilibrium quantum statistical mechanics for some model systems. As an example of their applications, we solve the equations for the model system with a pairing Hamiltonian. With the aid of its symplectic group symmetry, we obtain the statistical reduced density matrices with different orders. As a special instance for the solutions, we also get the reduced density matrices of the ground state for a superconductor.     

Statistical mechanics of nonlinear hydrodynamic fluctuations

The paper studies nonlinear hydrodynamic fluctuations by the methods of nonequilibrium statistical mechanics. The generalized Fokker-Planck equation for the distribution function of coarse-grained densities of conserved quantities is derived from the Liouville equation and then is investigated by using the gradient expansions in the flux correlation matrix. We have obtained the functional-differential Fokker-Planck equation describing the nonlinear hydrodynamic fluctuations in spatially nonuniform systems to second order in gradients of coarse-grained fluctuating fields. An outline of the derivation of Fokker-Planck equations containing the Burnett terms is also given. The explicit coordinate representation for the hydrodynamic Fokker-Planck equation is discussed in the case of one-component simple fluid. The general scheme of a change of coarse-grained functional variables is developed for hydrodynamic Fokker-Planck equations. The corresponding transformation rules are found for “drift” terms, “diffusion coefficients” and thermodynamic forces. The dynamical equations and stationary conditions for averages of functions (functionals) of hydrodynamic fields are discussed by using the Fokker-Planck operators acting on such functions. The explicit form of these operators are found for various sets of fluctuating fields. As an application of the formalism the calculation of the stationary correlation functions is presented for a simple nonequilibrium steady state.     

A Hopf-like equation and perturbation theory for differential delay equations

We extend techniques developed for the study of turbulent fluid flows to the statistical study of the dynamics of differential delay equations. Because the phase spaces of differential delay equations are infinite dimensional, phase-space densities for these systems are functionals. We derive a Hopf-like functional differential equation governing the evolution of these densities. The functional differential equation is reduced to an infinite chain of linear partial differential equations using perturbation theory. A necessary condition for a measure to be invariant under the action of a nonlinear differential delay equation is given. Finally, we show that the evolution equation for the density functional is the Fourier transform of the infinite-dimensional version of the Kramers-Moyal expansion.     

Properties of the functional formalism and their application to the theory of classical fluids

A general formalism is derived relating any generating functional of a hierarchy of functions to some other functionals yieldingUrsell, Husimi, and similar expansions of the original hierarchy and vice versa. There are two expansions starting with an equation of the O.-Z. type. This formalism is applied to the grand partition function with an external potential which is a generating functional for the molecular distribution functions. When the external potential is induced by adding particles to the system we obtain several hierarchies of integral equations related to each other in a simple fashion. As the Kirkwood-Salsburg, Mayer-Montroll, Green equations, the P. Y., HNC and a HNC similar approximation with their extensions are special cases of these hierarchies the relations between them become transparent. At the same time the heuristic feature in the choice of functionals and independent functions in earlier derivations of some of these equations is removed.     

Method of molecular distribution functions in the description of adsorption processes in nanoscale structures

The short-range order during gas adsorption by micropores can be described based on integral equations of statistical mechanics for partial distribution functions. The assumptions necessary for correct analysis of physical sorption are considered, and adsorption characteristics of carbon nanofibers in relation to hydrogen are calculated.     

Diagrammatic analysis of correlations in polymer fluids: Cluster diagrams via Edwards’ field theory

Edwards’ functional integral approach to the statistical mechanics of polymer liquids is amenable to a diagrammatic analysis in which free energies and correlation functions are expanded as infinite sums of Feynman diagrams. This analysis is shown to lead naturally to a perturbative cluster expansion that is closely related to the Mayer cluster expansion developed for molecular liquids by Chandler and co-workers. Expansion of the functional integral representation of the grand-canonical partition function yields a perturbation theory in which all quantities of interest are expressed as functionals of a monomer-monomer pair potential, as functionals of intramolecular correlation functions of non-interacting molecules, and as functions of molecular activities. In different variants of the theory, the pair potential may be either a bare or a screened potential. A series of topological reductions yields a renormalized diagrammatic expansion in which collective correlation functions are instead expressed diagrammatically as functionals of the true single-molecule correlation functions in the interacting fluid, and as functions of molecular number density. Similar renormalized expansions are also obtained for a collective Ornstein-Zernicke direct correlation function, and for intramolecular correlation functions. A concise discussion is given of the corresponding Mayer cluster expansion, and of the relationship between the Mayer and perturbative cluster expansions for liquids of flexible molecules. The application of the perturbative cluster expansion to coarse-grained models of dense multi-component polymer liquids is discussed, and a justification is given for the use of a loop expansion. As an example, the formalism is used to derive a new expression for the wave-number dependent direct correlation function and recover known expressions for the intramolecular two-point correlation function to first-order in a renormalized loop expansion for coarse-grained models of binary homopolymer blends and diblock copolymer melts.     

Functional equations for path integrals

We consider the density matrices that arise in the statistical mechanics of the electron-phonon systems. In the path integral representation the phonon coordinates can be eliminated. This leads to an action that depends on pairs of points on a path, that depends explicitly on time differences, and that contains the phonon occupation numbers. The integral is reduced to a standard form by scaling to the thermal length. We use the technique of integration by parts and add specially chosen generating functionals to the action. We set down functional derivative equations for the source-dependent density matrix and for the mass operator. This allows us to develop a series of approximations for the operator in terms of exact propagators. The crudest approximation is a coherent potential approximation applicable at a general temperature.     

Statistical properties of chaotic wavefunctions in two and more dimensions

Starting with Berry's hypothesis for fixed energy waves in a classically chaotic system, and casting it in a Green function form, we derive wavefunction correlations and density matrices for few or many particles. Universal features of fixed energy (microcanonical) random wavefunction correlation functions appear which reflect the emergence of the canonical ensemble as N↦∞. This arises through a little known asymptotic limit of Bessel functions. The Berry random wave hypothesis in many dimensions may be viewed as an alternative approach to quantum statistical mechanics, when extended to include constraints and potentials.     

Approximate solutions of the Liouville equation. II. Stationary variational principles

Stationary variational functionals for the Laplace transform of the Liouville distribution are constructed. The value of the functional is the autocorrelation function that one wishes to compute. It is shown that the functionals may be transformed to a renormalized form. Trial functions not involving the potential explicitly give rise to time-dependent autocorrelation functions determined only by equilibrium spatial correlation functions. Another class of functionals is constructed by independently varying the parity symmetric and antisymmetric parts of the distribution function. Trial functions need only be assumed for one of these—the optimum value of the other one is given exactly. This procedure is used to improve the simplest known theories for velocity and density autocorrelation functions.Work supported by a grant from the National Science Foundation.     

Hierarchy of equations for reduced density matrices in the case of thermodynamic equilibrium

A hierarchy of equations for s-particle density matrices at thermodynamic equilibrium is obtained, with the equation for the nonequilibrium density matrix as the starting point. When deducing the hierarchy the hypothesis of maximum statistical independence for the density matrices is used. The hierarchy obtained is an analogue of the classical equilibrium BBGKY hierarchy and goes over into it when . It is shown that thermodynamic quantities can be expressed in terms of functions that enter only into the first hierarchy equations. The hierarchy is analysed in detail in the case of a uniform fluid. As an example in which the equations can be solved easily enough, a hard-sphere system wherein triplet correlations are neglected is considered. Different approximations that can be used when solving the equations derived are discussed. Comparisons are made with the results of other theoretical treatments.     

A statistical view of exclusive and inclusive cross sections

The interrelations between exclusive and inclusive cross sections are investigated by means of efficiency matrices. A partition function Z is used as a generating functional for these cross sections which yields similarly also topological and other mixed cross sections. Regarding the partition function itself as a cross section, the general properties of Z and of statistical quantities derived from ln Z are investigated as functions of the chemical potentials. These statistical methods are used to analyze three simple examples of data for multiplicity distributions, in particular for the third distribution from the Echo Lake experiment a behaviour of Z is found which has similarities with a phase transition in statistical mechanics.     

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.     

Dissipative evolution of quantum statistical ensembles and nonlinear response to a time-periodic perturbation

We present a detailed discussion of the evolution of a statistical ensemble of quantum mechanical systems coupled weakly to a bath. The Hilbert space of the full system is given by the tensor product between the Hilbert spaces associated with the bath and the bathed system. The statistical states of the ensemble are described in terms of density matrices. Supposing the bath to be held at some - not necessarily thermal - statistical equilibrium and tracing over the bath degrees of freedom, we obtain reduced density matrices defining the statistical states of the bathed system. The master equations describing the evolution of these reduced density matrices are derived under the most general conditions. On time scales that are large with respect to the bath correlation time and with respect to the reciprocal transition frequencies of the bathed system, the resulting evolution of the reduced density matrix of the bathed system is of Markovian type. The detailed balance relations valid for a thermal equilibrium of the bath are derived and the conditions for the validity of the fluctuation-dissipation theorem are given. Based on the general approach, we investigate the non-linear response of the bathed subsystem to a time-periodic perturbation. Summing the perturbation series we obtain the coherences and the populations for arbitrary strengths of the perturbation.Received: 26 November 2003, Published online: 30 January 2004PACS: 05.30.-d Quantum statistical mechanics - 33.35. + r Electron resonance and relaxation - 33.25. + k Nuclear resonance and relaxation     

Kinetic equations for a quantum gas with bound states

The kinetic theory for dense gases is modified to take into account the existence of bound states. A molecular chaos condition is used which corresponds to the division of the two- and three-particle Hubert spaces into scattering and boundstate subspaces. A kinetic stage results from a long-time limit which converges to yield time-independent functionals for the two- and three-particle density matrices, as functionals of the density matrices for atoms and molecules. Coupled kinetic equations are obtained which describe the gas as a reacting mixture of atoms and diatomic molecules. These include the effects of scattering and rearrangement collisions between the atom and the molecule, and of molecular formation and dissociation.     

Canonical solutions of variational problems and canonical equations of mechanics

The canonical (non-parametric) solutions of the variational problems for integral functionals are considered and the canonical solutions of variational problems of mechanics in Minkowski spaces are derived. By combining the variational principles of least action, flow, and hyperflow canonically invariant equations for the energy-momentum variable are obtained. From these equations the equations for the action and wave functions as a general solution of the combined variational problems of mechanics are derived. These equations are applicable for describing different types of particles and interactions and are summarized within the approach of general relativity.     

Functional methods in thermofield dynamics: A real-time perturbation theory for quantum statistical mechanics

A generating functional is constructed for real-time Green functions in quantum statistical mechanics in the context of thermofield dynamics. The KMS condition is taken as an axiom which together with field equations fixes the generating functional for causal Green functions in an interacting quantum field theory. This leads to Feynman rules for diagrammatic real-time perturbation theory.     

Quantum theory of pure liquid metals as two-component systems

With the ultimate aims of clarifying the interpretation and the utility of effective ion-ion interactions in liquid metals, and of understanding the unusual isotopic mass dependence of the shear viscosity of liquid metal Li, a fully quantum statistical mechanical theory is developed from the many-body Hamiltonian of the conduction electron-positive ion assembly.We have set up quantum equations of motion which are analogs of classical continuity and conservation equations by expanding the equation for the Wigner distribution function about its diagonal. The most important of these equations for our present purposes relates the time derivative of the current density j(r, t) to the flux of current and to density-density correlation functions for electrons, electron-ions, and ions.This theory is then applied to neutron scattering by liquid metals. While the theory is sufficiently general in principle to treat electron-ion interaction of arbitrary strength, it is shown that when the interacion is weak, the usual results are recovered, along with the effective ion-ion interaction. In this latter connection, it is also demonstrated how the effective Ornstein-Zernike function C(q) in a liquid metal is related to bare ion and bare electron direct correlation functions and to the bare electron partial structure factor. Combining C(q) with one of the classical equations of liquid structure such as Born-Green or Percus-Yevick then relates the effective ion-ion interaction to the partial correlation functions of the bare ions and electrons.It is further shown how gradient expansions of the correlation functions lead to equations of motion for the density, current, and energy density which are simply the hydrodynamic equations of the present quantum theory of two-component systems. It is pointed out that the analog of the Navier-Stokes equation for the two-component system may be used to identify the quantity $\text{4}\text{3}\text{η + ζ}$ for the liquid metal, η and ζ being respectively the shear and bulk viscosities. Finally, it is demonstrated that $\text{4}\text{3}\text{η + ζ}$ depends explicitly on functional derivatives of the nonequilibrium pair distribution functions of ion-ion, electron-ion, and electron-electron correlations.     

Playing with fidelities

The notion of partial fidelities as invented by A. Uhlmann for pairs of finite-dimensional density matrices is extended to the νN-algebraic context and is considered and thoroughly discussed in detail from a mathematical point of view. Especially, in the case of semifinite νN-algebras, formulae and estimates for the partial fidelity between the functionals of a dense cone of inner derived normal positive linear forms are obtained. Also, some generalities on the notion of fidelity in quantum physics are collected in Appendix, and another system of mathematical axioms for fidelity over density operators, which is based on the concept of relative majorization and which is intimately related to complete positivity, is proposed.