Canonical Distribution Function Research Articles

A quasi-linear response theory of statistical heavy-ion collisions: (I). Basic formulation : Transport equations and roles of Green functions

We develop a time-dependent theory of heavy-ion collisions which consistently treats the relative and the intrinsic motions by coupled equations derived from the many-body von Neumann equation. The structure of the equations determining the mean trajectory and the fluctuations of the relative motion is the same as that of the corresponding equations in the known linear response theory. The present theory differs, however, from the linear response theory, in that it presumes neither weak coupling between the relative motion and the intrinsic excitations, nor the canonical distribution function for the density operator of the intrinsic motions. We apply the theory especially to deep inelastic collisions, where the relative motion couples to intrinsic excitations through a stochastic hamiltonian. Based on the stochastic assumption, we study the properties of the Green functions that take into account the higher order effects of the coupling hamiltonian. We then discuss, in particular, the effects of the Green functions on the time evolution of the intrinsic state, which is described in terms of a coarse-grained master equation, the friction tensor and fluctuation dissipation theorems.

Semiclassical probability distribution function for finite-temperature field theory

Semiclassical methods for evaluating the canonical probability distribution function rho(phi) = in field theory for systems in thermodynamic equilibrium is studied. Field configurations which dominate the semiclassical distribution function are interpretable as finite-temperature most-probable escape paths (FTMPEP's) in field space and are related to the recently discovered caloron solutions, which are known to partially dominate the semiclassical partition function Z = ..integral..Dphi rho(phi). Presented is a semiclassical path-integral approximation for the distribution function and also discuss a Hartree self-consistent field approximation.

Nonlinear theory of the Weibel instability

A canonical distribution function is proposed to describe the instantaneous state of a single nonlinear wave–plasma system as it evolves quasi-statically in time. This function is based on two single particle constants of motion for a charged particle in a zero-frequency transverse magnetic wave and determines a wavenumber condition and two system energy constants. In the case of a onecomponent bi-Maxwellian plasma with T⊥/T‖>1, these relations are particularly simple and yield expressions for the energy in the magnetic wave field, the wavenumber, the temperatures, and the entropy of the system in terms of one unknown parameter, chosen to be the instantaneous temperature ratio, T⊥/T‖ The maximum value of the field energy is expressed in terms of only the initial temperature anisotropy, and is shown to be always less than of the system's total energy. The results are in good agreement with computer simulations of the electron Weibel instability.

On equilibrium distributions and detailed balance relations in non-isothermal systems

The significance of the detailed balance principle and equilibrium solutions of the master equation is discussed from a thermodynamic point of view for isolated and isothermal systems. Starting from a master equation for all the time dependent degrees of freedom it is shown that the uniqueness of the equilibrium distribution as a stationary solution is ensured if the detailed rate constants are balanced with the aid of the distribution which maximizes the entropy subject to the thermodynamic constraints. This procedure should precede physical assumptions which simplify the original master equation, e.g. the assumption that rapidly relaxing modes can be described by canonical distribution functions.

Wall effects in one-dimensional jellium

The one-particle canonical distribution function near a wall is investigated by different methods. Results illustrate the “crystalline” state predicted by Kunz.

Kinetic theory of dense fluids

A kinetic theory of dense fluids is presented in this series of papers. The theory is based on a kinetic equation for subsystems which represents a subset of equations structurally invariant to the sizes of the subsystem that includes the Boltzmann equation as an element at the low density limit. There exists a H-function for the kinetic equation and the equilibrium solution is uniquely given by the canonical distribution functions for the subsystems comprising the entire system. The cluster expansion is discussed for the N-body collision operator appearing in the kinetic equation. The kinetic parts of transport coefficients are obtained by means of a moment method and their density expansions are formally obtained. The Chapman-Enskog method is discussed in the subsequent paper.

Ordering of the Exponential of Quadratic Forms in Boson Operators and Some Applications

In the present paper we order operators of the general form exp [αa2 + βa+2 + γ(a+a + aa+) + δa + ϵa+], using parametric differentiation. Using this ordered form we derive the density matrices and the canonical Wigner distribution function of the harmonic oscillator and of the electromagnetic field in a simple straightforward manner.

On the Wigner distribution function and its factorization

It is proved that the most general, everywhere analytical, potential V( x) for which the one particle thermal equilibrium canonical Wigner distribution function factorizes is at most a quadratic form.

Quantum corrections to the radial distribution function of a fluid

The perturbation theory developed for liquids is used to derive an expression for the first-order quantum correction to the radial distribution function of a fluid. The result is given in terms of grand canonical ensemble distribution functions for the classical fluid. The equations giving the thermodynamic functions in terms of the radial distribution function are discussed, and differences in the quantum and classical cases emphasized. An equation relating the two- and three-body classical distribution functions, derived recently by Singh and Ram using an indirect method, is shown to be simply related to the second equation of the BGY hierarchy.

Relativistic statistical thermodynamics

The formalism of relativistic statistical mechanics, developed in previous papers, provides a very straightforward proof of the Lorentz invariance of the canonical equilibrium distribution function. This theorem automatically determines the Lorentz transformation law of the temperature and of the free energy, and hence of all the thermodynamic functions. The transformation rule for the internal energy is discussed in great detail. It is shown that for a system of finite size the energy does not transform as the fourth component of a 4-vector, as is assumed by some authors. The boundary effect is responsible for this non-vectorial character. On the contrary, local quantities such as the energy density, for which the boundaries play no role, have a tensorial character. The controversy which arose in the recent literature about these problems is discussed in detail, with an emphasis on the sources of ambiguity. It is shown that, besides the canonical distribution, there exists an infinite number of Lorentz-invariant equilibrium distributions for each value of the velocity; all these functions reduce to the usual canonical distribution in the rest frame. Each one leads to a different thermodynamic formalism, special cases of which are the formalisms suggested by Ott and by Landsberg. All these formalisms can be reduced to the same form-invariant of Planck by means of a “gauge transformation” of the temperature and of the free energy. Strict equilibrium statistical mechanics cannot determine a unique choice of the absolute gauge. If such a determination has any meaning at all, it can only come from an extension of this investigation to cover non-equilibrium processes.

Generalizations of the Virial and Wall Theorems in Classical Statistical Mechanics

A generalized virial theorem which expresses inverse compressibility in terms of integrals of virials and canonical distribution functions through the four-particle distribution is transformed to the grand canonical ensemble and becomes an expression for compressibility in terms of the same integrals formed with grand canonical distribution functions. The integrals are of a mixed (virial and fluctuation) type. While the thermodynamic functions expressed by the same integrals with canonical and grand canonical distribution functions are quite different, the two formulas agree in the thermodynamic limit because of the different asymptotic behavior of canonical and grand canonical distribution functions. We also derived an alternative form of the second virial theorem which expresses compressibility in terms of integrals over virials and grand canonical distribution functions through the three-particle distribution function only. It is shown that this form and its generalizations to higher derivatives of the density, as well as the hierarchy of fluctuation theorems and the fugacity expansions of distribution and correlation functions can all be very simply derived from a set of integro-differential equations satisfied by the grand canonical distribution functions. A generalization of the wall theorem (P/kT = ρwall) is derived and shown to be equivalent to the generalized virial theorem (canonical form).

A Perturbation Method for the Classical Time-Dependent Pair Correlation Function

A perturbation technique useful for computing many-time thermal averages of classical quantities is developed. The canonical distribution function for the system is shown to evolve isothermally from that for a free-particle system as the interaction is switched on slowly. This permits convenient use of an interaction picture in which to perform thermal averaging. The technique is applied to the calculation of the time-dependent pair correlation function in position for a uniform gas. The correlation function is shown to be a sum of two components, one the solution of a kinetic equation and essentially a generalization of the autocorrelation in equilibrium, the other a generalization of the mutual correlation function in equilibrium. The equations arise as sums over diagrams. The equations resulting from the random phase approximation, valid for the short-time behavior, are solved exactly. It is shown directly that the generalized dielectric function given in terms of the correlation function is identical with that found by solution of the kinetic equation in the random phase approximation for all frequencies.

Binary collision expansion of quantum statistical pair propagator

The binary collision expansion of the pair propagator, which is closely related to the grand canonical pair distribution function, is obtained in terms of quantum statistical Feynman-type diagrams. The result may be summarized by stating that the pair propagator can be expanded in a manner in which ( a) all interaction lines, representing potentials υ of the usual perturbation diagrams, are replaced by sets of ladder diagrams representing binary collision kernels B (2) and ( b) whenever a double ladder structure occurs in the diagrams, the statistical factor (1 ± ƒ 1)(1 ± ƒ 2) (ƒ being the Bose or the Fermi distribution function) associated with the intermediate lines with momenta p 1 and p 2 should be replaced by (1 ± ƒ 1)(1 ± ƒ 2) − 1 .

Principle of Corresponding States for Transport Properties

The principle of corresponding states can be demonstrated by use of the autocorrelation function expressions for the transport properties and the assumption that the intermolecular potential has the form u=εu*(r/σ). The result follows from the fact that both the canonical ensemble distribution function and the solution of the mechanical equations of motion may be written in reduced variables. One finds that η*=ησ2/m½ε½, κ*=κkε½/m½σ 2, and D*=Dm½/ε½σ are universal functions of T*=Tk/ε, P*=Pσ3/ε, and in the quantum mechanical case ℏ*=ℏ/σm½ε½.