The cauchy problem for BBGKY hierarchy of quantum kinetic equations with coulomb potential

The existence of a unique solution, in terms of initial data of the BBGKY hierarchy of quantum kinetic equations with coulomb potential, is proved. This is based on non-relativistic quantum mechanics and utilizing a semigroup method.     

Uniqueness for the BBGKY hierarchy for hard spheres in one dimension

We prove that the stationary BBGKY hierarchy for an infinite system of hard spheres in one dimension has a unique solution for all densities, within a symmetry class that pertains to either a fluid array or to a perfect crystalline array. The solution is shown to correspond to the uniform fluid, which is the only equilibrium state of the infinite system. The proof is subject to the recursion relation for the correlation functions found by Salsburg, Zwanzig, and Kirkwood, which we show exactly reduces the infinite hierarchy to a pair of coupled equations. A brief discussion is given of the existence of multiple solutions of an approximate BBGKY equation.     

A discrete velocity model of a gas: Global in time solutions of the BBGKY hierarchy

In the present paper we study the evolution of a system of hard disks moving in the plane with a finite number of velocities as in the framework of a discrete velocity model of the Enskog equation, proposed in previous papers. Starting from the BBGKY hierarchy of such a system we give existence and uniqueness results for the initial value problem in suitable Banach spaces. In particular, the main result presented is the global in time weak solution to the BBGKY hierarchy for local equilibrium initial data, in the thermodynamic limit.     

Two-body collisions and mean-field theory: The Brueckner-Boltzmann equation

Starting from the BBGKY hierarchy of density matrices, a quantum mechanical Boltzmann equation, including a mean field, is derived. Both the mean field, which is of the well-known Brueckner-Hartree-Fock form, as well as the collision term are expressed in terms of the self-consistent Brueckner G-matrix. The relation with the generalized Boltzmann equation of Kadanoff and Baym is discussed. It is shown that the usual quantum mechanical theories like TDHF and Uehling-Uhlenbeck appear as limiting cases.     

Stationary solutions of the bogoliubov hierarchy equations in classical statistical mechanics. 2

In the preceding paper under the same title we have formulated a theorem which describes the set of states (i.e., probability measures on phase space of an infinite system of particles inR ^v) corresponding to stationary solutions of the BBGKY hierarchy. We have proved the following statement: ifG is a Gibbs measure (Gibbs random point field) corresponding to a stationary solution of the BBGKY hierarchy, then its generating function satisfies a differential equation which is conjugated to the BBGKY hierarchy. The present paper deals with the investigation of the conjugated equation for the generating function in particular cases.     

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.     

The BBGKY hierarchy in quantum statistical mechanics

It is shown that the infinite volume limit of the equilibrium reduced density matrices, shown by Ginibre to exist at low densities, satisfy the quantum time independent BBGKY hierarchy of equations. This extends analogous results for classical systems due to Gallavotti.Supported by AFOSR Contract Number F44620-71-C-0013.     

On the Stationary BBGKY Hierarchy for Equilibrium States

We consider infinite classical systems of particles interacting via a smooth, stable and regular two-body potential. We establish a new direct integration method to construct the solutions of the stationary BBGKY hierarchy, assuming the usual Gaussian distribution of momenta. We prove equivalence between the corresponding infinite hierarchy and the Kirkwood–Salsburg equations. A problem of existence and uniqueness of the solutions of the hierarchy with appropriate boundary conditions is thus solved for low densities. The result is extended in a milder sense to systems with a hard core interaction.     

Not to Normal Order—Notes on the Kinetic Limit for Weakly Interacting Quantum Fluids

The derivation of the Nordheim-Boltzmann transport equation for weakly interacting quantum fluids is a longstanding problem in mathematical physics. Inspired by the method developed to handle classical dilute gases, a conventional approach is the use of the BBGKY hierarchy for the time-dependent reduced density matrices. In contrast, our contribution is motivated by the kinetic theory of the weakly nonlinear Schrödinger equation. The main observation is that the results obtained in the latter context carry over directly to weakly interacting quantum fluids provided one does not insist on normal order in the Duhamel expansion. We discuss the term by term convergence of the expansion and the equilibrium time correlation 〈a(t)^* a(0)〉.     

Eigenmodes in the long-time behavior of a coupled spin system measured with nuclear magnetic resonance

The many-body quantum dynamics of dipolar coupled nuclear spins I=1/2 on an otherwise isolated cubic lattice are studied with nuclear magnetic resonance. By increasing the signal-to-noise ratio by 2 orders of magnitude compared with previous reports for the free induction decay (FID) of (19)F in CaF(2) we obtain new insight into its long-time behavior. We confirm that the tail of the FID is an exponentially decaying cosine, but our measurements reveal a second decay mode with comparable frequency but twice the decay constant. This result is in agreement with a recent theoretical prediction for the FID in terms of eigenvalues for the time evolution of chaotic many-body quantum systems.     

Relativistic quantum kinetic analysis of a pion-nucleon system

A relativistic plasma of nucleons interacting through pions via the usual isospin-invariant Yukawa coupling is analyzed in the framework of the covariant Wigner function technique. The method is manifestly covariant and the temperature effects are considered. The relativistic quantum BBGKY hierarchy for the pion-nucleon system is derived. By generalizing the Bogolioubov analysis of the classical BBGKY hierarchy a non-perturbative renormalizable method is elaborated which allows the solution of the kinetic problem in form of power series of two cluster parameters which measure the importance of correlations. In the lowest order of the cluster expansion (Hartree approximation or zero-order approximation) the quasi-nucleon Fock space is introduced, the fermion Wigner function in the thermodynamic equilibrium is obtained and the vacuum effects are renormalized. In this approximation the plasma behaves as a perfect Fermi gas of nucleons and antinucleons, but there exists an abnormal configuration with a uniform pion condensate which is unstable. In the next approximation (quadratic in the small parameters) the quasi-pion dispersion relation is obtained and the vacuum polarization tensor is renormalized. The quasi-pion rest-mass spectra (“plasma frequency”) and the effective-coupling behaviour as functions of the thermodynamic state are given. By estimating the size of the cluster parameters the self-consistency of the approximation scheme is proved. The quasi-pion Fock space is introduced and the quasi-pion equilibrium Wigner function is obtained. From these results the problem of the higher-order corrections to the Hartree thermodynamics is outlined.     

Duals for Non-Abelian Lattice Gauge Theories by Categorical Methods

We introduce duals for non-Abelian lattice gauge theories in dimension at least three by using a categorical approach to the notion of duality in lattice theories. We first discuss the general concepts for the case of a dual-triangular lattice (i.e., the dual lattice is triangular) and find that the commutative tetrahedron condition of category theory can directly be used to define a gauge-invariant action for the dual theory. We then consider the cubic lattice (where the dual is cubic again). The case of the gauge group SU(2) is discussed in detail. We will find that in this case gauge connections of the dual theory correspond to SU(2) spin networks, suggesting that the dual is a discrete version of a quantum field theory of quantum simplicial complexes (i.e. the dual theory lives already on a quantized level in its classical form). We conclude by showing that our notion of duality leads to a hierarchy of extended lattice gauge theories closely resembling the one of extended topological quantum field theories. The appearance of this hierarchy can be understood by the quantum von Neumann hierarchy introduced by one of the authors in previous work.     

Truncation of time-dependent many-body theories

We investigate truncation schemes for the many-body dynamics of finite fermion systems beyond the time-dependent Hartree-Fock (TDHF) approximation. One approach starts from the quantum Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY-)hierarchy of equations of motion for density matrices. It is shown that simple truncation of higher correlations within this scheme is inconsistent because essential exchange correlations are lost. As an alternative, we study the (extended) exp(S) or coupled-cluster formalism. It provides a wider range of applicability. But it also runs into problems with non-unitary propagation after truncation.     

Continuum Approximations to Systems of Correlated Interacting Particles

Journal of Statistical Physics - We consider a system of interacting particles with random initial conditions. Continuum approximations of the system, based on truncations of the BBGKY hierarchy,...     

Second approximation of the temperature expansion of the BBGKY hierarchy

A solution is found to the BBGKY hierarchy in the first and second approximations of the temperature expansion by representing the distribution function as a sum of derivatives of functions with respect to the coordinates to the fourth order inclusively. The Grüneison parameter for a van der Waals crystal is found as a function of the temperature and density.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 14–18, January, 1976.     

Expansion of the kinetic hierarchy for a massive particle: The repeated ring and Fokker-Planck equations

The tagged particle BBGKY hierarchy is systematically expanded in inverse powers of the square root of the particle mass. In the Brownian limit, for fixed Knudsen number, the hierarchy reduces to the Brownian limit of the repeated ring equation which itself reduces to the Fokker-Planck equation. The friction coefficient of the Fokker-Planck equation is found to be a functional of the solution of Dorfman, van Beijeren, and McClure's extended Boltzmann equation for a fixed object in a flowing gas. As a consequence, the tagged particle diffusion coefficient calculated in the Brownian limit of the repeated ring equation is valid for all particle sizes.     

Many-body localization: An introduction and selected topics

What happens in an isolated quantum system when both disorder and interactions are present? Over the recent years, the picture of a non-thermalizing phase of matter, the many-localized phase, has emerged as a stable solution. We present a basic introduction to the topic of many-body localization, using the simple example of a quantum spin chain that allows us to illustrate several of the properties of this phase. We then briefly review the current experimental research efforts probing this physics. The largest part of this review is a selection of more specialized questions, some of which are currently under active investigation. We conclude by summarizing the connections between many-body localization and quantum simulations.     

The Enskog repeated ring equations for tagged molecule motion and variational solutions

The Enskog repeated ring equations (ERRE) for tagged molecule motion are derived from the BBGKY hierarchy. The hierarchical deviation demonstrates the problem associated with using naive truncations for complex fluid systems. A moment-variational solution is suggested for the ERRE. The moment-variational method is applied to the ERRE for the two- and three-dimensional overlapping Lorentz gas (LG). Both self-consistent and non-self-consistent equations for the diffusion constant are solved. The results compare favorably with previous calculations and molecular dynamics (MD).