Nonlinear transport in the Boltzmann limit

Formal expressions for the irreversible fluxes of a simple fluid are obtained as functionals of the thermodynamic forces and local equilibrium time correlation functions. The Boltzmann limit of the correlation functions is shown to yield expressions for the irreversible fluxes equivalent to those obtained from the nonlinear Boltzmann kinetic equation. Specifically, for states near equilibrium, the fluxes may be formally expanded in powers of the thermodynamic gradients and the associated transport coefficients identified as integrals of time correlation functions. It is proved explicitly through nonlinear Burnett order that the time correlation function expressions for these transport coefficients agree with those of the Chapman-Enskog expansion of the nonlinear Boltzmann equation. For states far from equilibrium the local equilibrium time correlation functions are determined in the Boltzmann limit and a similar equivalence to the Boltzmann equation solution is established. Other formal representations of the fluxes are indicated; in particular, a projection operator form and its Boltzmann limit are discussed. As an example, the nonequilibrium correlation functions for steady shear flow are calculated exactly in the Boltzmann limit for Maxwell molecules.Research supported in part by NSF grant PHY 76-21453.     

Two-Dimensional Coulomb Systems on a Surface of Constant Negative Curvature

We study the equilibrium statistical mechanics of classical two-dimensional Coulomb systems living on a pseudosphere (an infinite surface of constant negative curvature). The Coulomb potential created by one point charge exists and goes to zero at infinity. The pressure can be expanded as a series in integer powers of the density (the virial expansion). The correlation functions have a thermodynamic limit, and remarkably that limit is the same one for the Coulomb interaction and some other interaction law. However, special care is needed for defining a thermodynamic limit of the free energy density. There are sum rules expressing the property of perfect screening. These generic properties can be checked on the Debye–Hückel approximation, and on two exactly solvable models, the one-component plasma and the two-component plasma, at some special temperature.     

Microscopic theory of the long-wavelength modes of two-component plasmas and ionic liquids: III. The space-time correlation functions

The identity between the exact screening length obtained from the static charge density correlation function and the one which appears in the Einstein relation between the transport coefficients of electrical conductivity and mass diffusion is demonstrated from first principles. For the space-time correlation functions of the number densities we show that their long-wavelength behaviour is completely determined by the four hydrodynamical modes of the two-component system of neutral particles. For charged particle systems there are only three hydrodynamical modes while we have moreover to add the two charge relaxation modes in order to exhaust the long-wavelength limit of the first sum-rule. The strengths with which the various modes appear in the space-time correlation functions have been computed exactly in the limit of long wavelengths.     

Correlations in the plasma broadening of ion spectral lines

The results of a kinetic theory of spectral line broadening are applied to Stark broadening of ion lines in a unified classical path theory, paralleling existing results for neutral radiators. In the small plasma parameter limit, the radiator-electron interactions are screened by equilibrium electron correlation functions and a frequency-dependent dielectric function.     

Charge distribution and correlation functions near the boundary of a weakly nonideal plasma

We consider the electric double layer formed near the boundary of a weakly nonideal multicomponent classical plasma as a result of interparticle correlations. On the basis of the generalized plasma uncoupling of the equilibrium chain of BBGKY equations, which correctly takes into account correlations of particles at small distances, we find the two-particle and one-particle correlation functions near the surface of the plasma. The asymptotic form of the two-particle correlation function along the boundary of the plasma falls off as the power law 1/r³. The solution obtained here is qualitatively consistent with earlier results, but it takes into account particle correlations at small distances and hence does not have a singularity at the boundary of the plasma. Therefore in a charge-asymmetric plasma there are two correlation radii, the Debye radius D and the radius /tr2D. The thermodynamic functions of the surface layer are calculated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 72–77, March, 1989.     

Derivation of reference distribution functions for Tokamak-plasmas by statistical thermodynamics

A general approach for deriving the expression of reference distribution functions by statistical thermodynamics is illustrated, and applied to the case of a magnetically confined plasma. The local equilibrium is defined by imposing the minimum entropy production, which applies only to the linear regime near a stationary thermodynamically non-equilibrium state and the maximum entropy principle under the scale invariance restrictions. This procedure may be adopted for a system subject to an arbitrary number of thermodynamic forces, however, for concreteness, we analyze, afterwords, a magnetically confined plasma subject to three thermodynamic forces, and three energy sources: (i) the total Ohmic heat, supplied by the transformer coil; (ii) the energy supplied by neutral beam injection (NBI); and (iii) the RF energy supplied by ion cyclotron resonant heating (ICRH) system which heats the minority population. In this limit case, we show that the derived expression of the distribution function is more general than that one, which is currently used for fitting the numerical steady-state solutions obtained by simulating the plasma by gyro-kinetic codes. An application to a simple model of fully ionized plasmas submitted to an external source is discussed. Through kinetic theory, we fixed the values of the free parameters linking them with the external power supplies. The singularity at low energy in the proposed distribution function is related to the intermittency in the turbulent plasma.     

Correlation functions of infinite system of interacting Brownian particles; Local in time evolution close to equilibrium

The time-dependent correlation functions of infinite nonequilibrium systems of interacting diffusing particles are obtained in the thermodynamic limit in the case when the initial correlation functions coincide with the equilibrium correlation functions of the Gibbs system in an external field.     

Distribution functions of a slightly nonideal multicomponent plasma

A regular perturbation theory technique is constructed using the plasma parameters of the equilibrium Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) equation chain for a multicomponent plasma in which higher order terms are small at all admissable interparticle distances, in contrast to the Debye equations. As a result, two correlation radii appear in a charge-asymmetric plasma: the Debye radius D and 2 D. The divergence of correlation functions characteristic of Debye splitting is eliminated in any order of the perturbation theory. For the proposed theory to be applicable the plasma parameter must be limited more severely, the deeper the depth of the potential well which appears upon approach of oppositely charged ions.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 53–58, May, 1988.     

The thermodynamic limit for jellium

The thermodynamic limit of the free energy, energy, pressure, and entropy is established for a neutral system of charged particles interacting with a fixed, uniformly charged background (jellium).     

Pressures for a One-Component Plasma on a Pseudosphere

The classical (i.e., non-quantum) equilibrium statistical mechanics of a two-dimensional one-component plasma (a system of charged point-particles embedded in a neutralizing background) living on a pseudosphere (an infinite surface of constant negative curvature) is considered. In the case of a flat space, it is known that, for a one-component plasma, there are several reasonable definitions of the pressure, and that some of them are not equivalent to each other. In the present paper, this problem is revisited in the case of a pseudosphere. General relations between the different pressures are given. At one special temperature, the model is exactly solvable in the grand canonical ensemble. The grand potential and the one-body density are calculated in a disk, and the thermodynamic limit is investigated. The general relations between the different pressures are checked on the solvable model.     

Correlation inequalities and the thermodynamic limit for classical and quantum continuous systems

We use Ginibre's general formulation of Griffiths' inequalities to derive new correlation inequalities for two-component classical and quantum mechanical systems of distinguishable particles interacting via two body potentials of positive type. As a consequence we obtain existence of the thermodynamic limit of the thermodynamic and correlation functions in the grand canonical ensemble at arbitrary temperatures and chemical potentials. For a large class of systems we show that the limiting correlation functions are clustering. (In a subsequent article these results are extended to the correlation functions of two-component quantum mechanical gases with Bose-Einstein statistics). Finally, a general construction of the thermodynamic limit of the pressure for gases which are not H-stable, above collapse temperature, is presented.Research supported in part by the U.S. National Science Foundation under grant MPS 75-11864A Sloan Foundation Fellow     

Statistical mechanics of quasi-one-dimensional systems

A definition of a quasi-one-dimensional system as a generalized Cayley or Husimi tree with a nonzero surface to bulk ratio in the thermodynamic limit is given. Sufficient conditions for the existence of the thermodynamic limit of the free energy for such a system are derived and a thorough discussion of the thermodynamic limit properties of the one-particle distribution functions is given. These results are made more precise for the case of systems with Hamiltonians which are invariant under a special type of measure-preserving group of transformations, in particular for the d-dimensional rotation group. For this latter case, the phase transitions which can occur in quasi-one-dimensional systems upon application of small external fields are studied in some detail. A number of completely solved examples is given to illustrate the general theory. These include the classical Heisenberg model on a Cayley tree and generalizations thereof.     

The poisson representation. II Two-time correlation functions

Basic formulas for the two-time correlation functions are derived using the Poisson representation method. The formulas for the chemical system in thermodynamic equilibrium are shown to relate directly to the fluctuationdissipation theorems, which may be derived from equilibrium statistical mechanical considerations. For nonequilibrium systems, the formulas are shown to be generalizations of these fluctuation-dissipation theorems, but containing an extra term which arises entirely from the nonequilibrium nature of the system. These formulas are applied to two representative examples of equilibrium reactions (without spatial diffusion) and to a nonequilibrium chemical reaction model (including the process of spatial diffusion) for which the first two terms in a systematic expansion for the two-time correlation functions are calculated. The relation between the Poisson representation method and Glauber-SudarshanP-representation used in quantum optics is discussed.     

Mayer coefficients in two-dimensional coulomb systems

We show that, for neutral systems of particles of arbitrary charges in two dimensions, with hard cores, coefficients of the Mayer series for the pressure exist in the thermodynamic limit below certain thresholds in the temperature. Our methods apply also to correlation functions and yield bounds on the asymptotic behavior of their Mayer coefficients.     

Bank of optical and physical characteristics for solving problems of radiative plasma dynamics

A bank of data on the thermophysical and optical properties of the plasma of a number of elements and their mixtures is presented, which is the basis for simulating the problems of radiative plasma dynamics. For a substance in the state of thermodynamic equilibrium, this bank contains information on the equations of state, thermodynamic functions, shock adiabatics, and also spectral, group, and mean absorption coefficients. In the state of shock-radiative equilibrium, using a radiational-collisional model, the steady states of an optically transparent plasma and its spectral and integral radiative losses are determined, and evaluations of the role of self-radiation reabsorption are given. The rates of the processes of ionization and recombination in an unsteady plasma are discussed. Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 67, No. 2, pp. 238–243, March–April, 2000. The work was carried out with financial support of the International Scientific-Technical Center (project ISTC V23-96) and of the Republican Fundamental Research Foundation (project T96-132).     

Spin waves,vortices, and the structure of equilibrium states in the classicalXY model

We prove that, for spin systems with a continuous symmetry group on lattices of arbitrary dimension, the surface tension vanishes at all temperatures. For the classicalXY model in zero magnetic field, this result is shown to imply absence of interfaces in the thermodynamic limit, at arbitrary temperature. We show that, at values of the temperature at which the free energy of that model is continuously differentiable, i.e. at all except possibly countably many temperatures, there iseither aunique translation-invariant equilibrium state, or all such states are labelled by the elements of the symmetry group, SO(2). Moreover, there areno non-translation-invariant, but periodic equilibrium states. We also reconsider the representation of theXY model as a gas of spin waves and vortices and discuss the possibility that, in four or more dimensions, translation invariance may be broken by imposing boundary conditions which force an (open) vortex sheet through the system. Among our main tools are new correlation inequalities.     

Equilibrium properties of two-component classical plasmas

Two-component overall neutral classical Coulomb Gas is considered in the canonical ensemble for any value of the space dimensionality ν. The equilibrium properties, i.e. pair correlations and thermodynamic functions are investigated in two complementary ways. The first one is adequate in considering the low temperature range and uses the “molecular” interaction within a pair of unlike charges as a zero order starting point. On the other hand, the high-temperature fully ionized and translation-invariant plasma is considered within the nodal expression with respect to the classical plasma parameter. These two ways are possible through the use of effective temperature-dependent classical interaction for ν > 2. As a by-product, we obtain a unified treatment of the Coulomb Gas thermal properties with respect to dimensionality (integer or real). We also obtain a contrasting comparison with corresponding properties of the one component plasma model which are already known. In this analysis the ν = 2 two-component Coulomb Gas seems to be a landmark for the other TCP'₈. I do not consider degeneracy effects. I consider diffraction corrections in a first order expansion with respect to the Coulomb interaction, in the high-temperature range. The “Hydrogen atom” spectrum is explained for all ν. The long-range hypernetted chain resummation of the pair correlation functions asymptotic behavior does not hold for symmetrical (Z₁ = ?Z₂) plasmas; the corresponding onset of short-range order disappears when the plasma parameter increases. The modified long- and short-range behaviors of the pair correlation functions are then displayed with the canonical thermodynamics.     

On the linear chain with arbitrary masses and force constants. II. Ergodic properties

The ergodic properties of linear and quadratic phase functions of the classical linear chain are studied for the uniform statistical distributions on the energy surface and in the manifold belonging to fixed values of the energy and the total momentum. This is done for the finite chain by using the time dependent correlation functions studied in a previous paper¹). The thermodynamic limit is also discussed. As an example, sufficient conditions on the masses and force constants are given to ensure that the kinetic energy of a certain particle remains nonergodic in the thermodynamic limit, the conditions defining a non-exceptional set of chains.     

Thermodynamics of relativistic rotating perfect fluids

A new formulation of thermodynamics for special and general relativistic rotating perfect fluids is developed. Both isolated systems and portions of isolated systems electrically uncharged or charged are treated. Exploiting the symmetry of motion of stationary axisymmetric fluids, the global thermodynamic functions, including total energy and spin, are defined as free scalars, represented by hypersurface integrals of conserved vectors. Local equilibrium parameters such as local temperature and chemical potential are scalar functions. There also exist global equilibrium parameters, global temperature and global chemical potential, which are free scalars. The connection between local and global conditions of thermodynamic equilibrium is made clear and explicit. Thermodynamic potentials are introduced in the context of treating open systems in a relativistically invariant way.     

Ionization Equilibrium in Nonideal Alkali and Noble Gas Plasmas

Using the method of thermodynamic Green functions, a generalized Saha equation is derived which accounts for different non-ideality effects, as, e. g., interaction between charged particles and neutral composites (atoms). The degree of ionization was calculated for the alkali metals Li—Cs and for Xe in a wide n-T range. Furthermore, the chemical potential μ and the pressure p were determined for these elements.