An alternative construction of the Percus-Yevick equation based on the equilibrium BBGKY hierarchy

The existing derivations of the Percus-Yevick equation are not readily extendable into the nonequilibrium domain. In particular, the elegant Percus functional construction relies on a test particle theorem which lacks an exact nonequilibrium generalization. We propose here a new construction which utilizes some elementary ideas of functional expansions together with the equilibrium BBGKY hierarchy of equations. Also, we feel this new construction provides fresh insight into the physical basis of the equilibrium Percus-Yevick equation.This research was supported in part by a grant from the Faculty Research Award Program of the City University of New York.     

Diffusion,effusion, and chaotic scattering: An exactly solvable Liouvillian dynamics

We study diffusion and chaotic scattering in a chain of baker maps coupled together which forms an area-preserving mapping of an infinitely extended strip onto itself. This exactly solvable mapping sustains chaotic behaviors and diffusion processes. The relationship between the diffusion coefficient, the Lyapunov exponent, and the entropy per unit time is derived. The long-lived classical resonances of the Liouville evolution operator are proved to converge toward the eigenvalues of the phenomenological diffusion equation. In this sense, there is a quasi-isomorphism between the resonance spectrum of the Liouville evolution and the eigenvalue spectrum of the phenomenological diffusion equation. Furthermore, we show that a fractal repeller is associated to each non-equilibrium state in the isolated and finite multibaker chain. The nonequilibrium states are all unstable with respect to the equilibrium, validating a weak form of the second principle of thermodynamics for the present dynamical system. Consequences of nonequilibrium fractals on classical measurements are discussed. We then describe the open multibaker chain as a scattering system. Fractal properties of chaotic scattering are here shown to be related to diffusion in the chain.     

Non-Markovian Quantum Kinetics and Conservation Laws

A link between memory effects in quantum kinetic equations and nonequilibrium correlations associated with the energy conservation is investigated. In order that the energy be conserved by an approximate collision integral, the one-particle distribution function and the mean interaction energy are treated as independent nonequilibrium state parameters. The density operator method is used to derive a kinetic equation in second-order non-Markovian Born approximation and an evolution equation for the nonequilibrium quasi-temperature which is thermodynamically conjugated to the mean interaction energy. The kinetic equation contains a correlation contribution which exactly cancels the collision term in thermal equilibrium and ensures the energy conservation in nonequilibrium states. Explicit expressions for the entropy production in the non-Markovian regime and the time-dependent correlation energy are obtained.     

Statistical approach to the kinetics of nonuniform fluids

We present a new formalism in Fourier space for the study of spatially nonuniform fluids in nonequilibrium states which generalizes previous work on uniform fluids. Starting from the Liouville equation we obtain a hierarchy of equations for the reduced distribution functions which gives their rate of change at any given order of the system mean density as a sum of a finite number of terms. Using a finite-ranged repulsive interaction potential we derive, as a first application of the formalism, the Boltzmann integrodifferential equation for an infinite system which is initially in a “weakly” inhomogeneous state. This is accomplished introducing an initial statistical assumption, namely initial molecular chaos; this condition is seen to hold during the time evolution described by the resulting kinetic equation.     

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.     

A new,accurate, statistical mechanical theory for simple fluids

A new integral equation in which the hypernetted chain and Percus-Yevick approximations are “mixed” as a function of interparticle separation is described. Calculations with fluids varying from hard spheres to the one-component plasma show good agreement with Monte Carlo calculations.     

A unified quantum theory of mechanics and thermodynamics. Part I. Postulates

A unified axiomatic theory that embraces both mechanics and thermodynamics is presented in three parts. It is based on four postulates; three are taken from quantum mechanics, and the fourth is the new disclosure of the existence of quantum states that are stable (Part I). For nonequilibrium and equilibrium states, the theory provides general original results, such as the relation between irreducible density operators and the maximum work that can be extracted adiabatically (Part IIa). For stable equilibrium states, it shows for the first time that the canonical and grand canonical distributions are the only stable distributions (Part IIb). The theory discloses the incompleteness of the equation of motion of quantum mechanics not only for irreversible processes but, more significantly, for reversible processes (Part IIb). It establishes the operational meaning of an irreducible density operator and irreducible dispersions associated with any state, and reveals the relationship between such dispersions and the second law (Part III).     

Quantum kinetic theory of trapped particles in a strong electromagnetic field

The idea of treating quantum systems by semiclassical representations using effective quantum potentials (forces) has been successfully applied in equilibrium by many authors, see e.g. [D. Bohm, Phys. Rev. 85 (1986) 166 and 180; D.K. Ferry, J.R. Zhou, Phys. Rev. B 48 (1993) 7944; A.V. Filinov, M. Bonitz, W. Ebeling, J. Phys. A 36 (2003) 5957 and references cited therein]. Here, this idea is extended to nonequilibrium quantum systems in an external field. A gauge-invariant quantum kinetic theory for weakly inhomogeneous charged particle systems in a strong electromagnetic field is developed within the framework of nonequilibrium Green’s functions. The equation for the spectral density is simplified by introducing a classical (local) form for the kinetics. Nonlocal quantum effects are accounted for in this way by replacing the bare external confinement potential with an effective quantum potential. The equation for this effective potential is identified and solved for weak inhomogeneity in the collisionless limit. The resulting nonequilibrium spectral function is used to determine the density of states and the modification of the Born collision operator in the kinetic equation for the Wigner function due to quantum confinement effects.     

Memory function for dressed particles

This paper is concerned with the calculation of the memory function and derivation of a kinetic equation for one-body phase space correlation functions. The theory uses a one-body additive projection operator and a division of the Liouville operator with an unperturbed part that describes dressed particles. Binary collisions are neglected, for the theory aims at describing the screening and backflow effects of a type contained in the plasma kinetic theory of Balescu and Lenard. We obtain an explicit kinetic equation which is an improvement of these theories for the plasma case, and involves the exact equilibrium pair and triplet distributions. The equation also describes systems with strong short-range forces and shows how the screening effects occur in this case as well. The unifying function is the direct correlation function. The theory is meant to provide understanding for a more complete theory of fluids where a proper account is given of close collisions.Work supported by National Science Foundation, Grant No. GH 35691.     

A density-corrected quantum Boltzmann equation

Binary correlations are a recognized part of the pair density operator, but the influence of binary correlations on the singlet density operator is usually not emphasized. Here free motion and binary correlations are taken as independent building blocks for the structure of the nonequilibrium singlet and pair density operators. Binary correlations are assumed to arise from the collision of twofree particles. Together with the first BBGKY equation and a retention of all terms that are second order in gas density, a generalization of the Boltzmann equation is obtained. This is an equation for thefree particle density operator rather than for the (full) singlet density operator. The form for the pressure tensor calculated from this equation reduces at equilibrium to give the correct (Beth-Uhlenbeck) second virial coefficient, in contrast to a previous quantum Boltzmann equation, which gave only part of the quantum second virial coefficient. Generalizations to include higher-order correlations and collision types are indicated.     

Linear quantum Enskog equation. I. Homogeneous quantum fluids

The quantum-statistical generalization of the well-known classical, linear revised Enskog equation is derived for spatially uniform systems. This new quantum kinetic equation allows the study of equilibrium time correlation functions and their associated transport coefficients of normal quantum fluids where static correlations and degeneracy effects due to particle statistics (both are treated exactly) are important. Furthermore, we derive the quantum-statistical analog of the classical ring operator. These microscopic and systematic derivations are based on a recently developed superoperator formalism (including cluster expansion techniques) that, as a main feature, allows a clear distinction between static and dynamic correlations, which is crucial in the discussion of the Enskog approximation.     

Solution of the master equation for quantum systems weakly coupled to reservoirs and far from thermal equilibrium

The Liouville operator of the master equation is decomposed into a part describing the proper system and another operator describing the interaction of the proper system with a set of reservoirs. If the motion of the proper system is described by a Hamiltonian, the corresponding solutions of the density matrix equation are spanned by all conserved quantities and are thus highly degenerate. We demonstrate how this degeneracy may be lifted by applying some sort of perturbation theory for degenerate systems. If there is only one relevant conserved quantity and if the heatbaths connect the quantum numbers of this conserved quantity by nearest-neighbour steps, the solution of the problem can be found explicitly. Otherwise the solution of the original master equation is reduced to a much simpler master equation of orderN, whereN is the number of conserved quantities. An application to a new type of parametric processes in nonlinear optics is given.     

On the calculation of the classical Liouville operator for the step-type interparticle interaction

A new method of the determination of the Liouville operator for the step-type interparticle interaction is presented. The interaction part of the Liouville operator is calculated for the particles interacting via a spherical repulsive and spherical attractive barrier. Limiting cases of an infinitely high (hard core) and infinitely deep barrier are discussed. These results can be used for the determination of the interaction part of the Liouville operator for an arbitrary step-type interaction which can be expressed as a sum of several attractive and repulsive barriers.     

Response function theory for far-from-equilibrium statistical systems

A formalism to determine the response function of a sample in conditions far from thermal equilibrium is presented. It consists in a generalization of scattering theory coupled to the statistical theory of irreversible processes, the nonequilibrium statistical operator method, developed by Zubarev. The scattering cross section is expressed in terms of double-time correlation functions, which are related to appropriate nonequilibrium thermodynamic Green's functions. The latter are also used to treat generalized transport equations, and, as an illustration, the method is applied to the study of the time-resolved Raman spectroscopy of a photoexcited semiconductor plasma.     

Formal structure of kinetic theory

We study the Liouville equation in the domain of small deviations from absolute equilibrium. The solution is expressed in terms of amplitudes ofn-body additive functions which are orthogonal with respect to the Gibbs weight factor. In the memory operator approach the memory operators are formally exact continued fractions. We show that with the isolation in the Liouville operator of a one-body additive operatorL _o, any memory operator can be written alternatively as an exact infinite series, each term of which can be calculated exactly. This yields improvements of the dressed particle approximation. We discuss the choice ofL _o, which is in general time dependent. The theory is developed both for smooth potentials and for hard spheres, where we use pseudo-Liouville operators. The theory can be formulated in an equivalent manner by introducing modified cumulant distributions, which are closely related to the amplitudes. The modified cumulants have the same spatial asymptotic properties as ordinary cumulants, but have superior short-time and small-distance behavior.Work supported by the National Science Foundation.     

Covariant Wigner function approach to the relativistic quantum electron gas in a strong magnetic field

This paper is concerned with a unified approach to some equilibrium properties of the relativistic quantum electron plasma embedded in a strong external magnetic field. This unified approach rests on the systematic use of a covariant Wigner function. The equilibrium Wigner function of the noninteracting gas is derived and its main properties are studied. In particular, it satisfies equations that are the complete analog of the usual Liouville equation and thus can be termed “relativistic quantum Liouville equation” whose properties are considered. The equations of state are rederived in this formalism and the results obtained earlier by Canuto and Chiu are found anew. Also, the covariant Wigner funetion of the magnetized vacuum is derived: it is needed, in this formalism, in order to obtain, e.g., the vacuum polarization tensor. Since we are also interested in the plasma modes, the fluctuations of one-particle quantities—and their spectrum—(in particular, of the four current) are calculated in view of their use in the fluctuation-dissipation theorem. We also outline a microscopic proof of this theorem, on the basis of a BBGKY hierarchy for the covariant Wigner functions, and point out the existence of an effective plasma frequency.     

The construction of kinetic equations for nonequilibrium quantum systems

A method of construction of a Markoffian solution of the Liouville equation for nonequilibrium systems, leading to closed kinetic equations, is described. The method is illustrated on the basis of an electron-phonon system; higher orders in the interaction are taken into account.     

Nonperturbative weak-coupling analysis of the quantum Liouville field theory

Two independent weak-coupling expansions are developed for the Liouville quantum field theory on a circle. In the first, the coupling of the nonzero modes is treated as a perturbation on the exact solution to the zero-mode problem (quantum mechanics with an exponential potential). The second approach is a weak-coupling approximation to an explicit operator solution which expresses various Liouville operators as functions of a free massless field using a Bäcklund transformation. It is shown that the free state space associated with the latter solution must be restricted to the sector which is odd with respect to a type of “parity.” Various matrix elements are computed to order g¹⁰ using both approaches, yielding identical results.