Nonequilibrium statistical mechanics of systems interacting with nonadditive forces. III. Properties to first order in the density

We obtain the transport coefficients up to first order in the density for a system of particles interacting with nonadditive forces. To zeroth order in the density we recover the usual results obtained from the Boltzmann equation. To first order in the density one finds that the transport coefficients consist of two parts: the value of the corresponding quantities for a system with only additive forces, plus a contribution due to nonadditive forces.     

微扰Burgers-KdV方程的孤子解

对于一个其耗散项可看作微扰的Burgers-KdV(B-KdV)方程u_t+uu_x+βu_xxx=εu_xx,|ε|?1,考虑一级近似和行波情形,建立一套求通解的直接扰动方法,利用零级方程的单孤子解,获得一级方程的孤子型通解,它包含任意多个不同的孤子解,每个孤子解分别描述一个位于半无限空间的孤子阵列,分析表明,耗散使得“亮孤子”变矮变窄,“暗孤子”变浅变窄. 关键词：     

Heterogeneous Photocatalytic Oxidation of Pollutants in Air on TiO<Subscript>2</Subscript> Particles

A semiempirical method has been developed for analyzing the mechanism of heterogeneous reactions based on the Langmuir–Hinshelwood kinetic model modified using the first- order double-exponential decay approach. The method proved useful for describing the kinetics of photocatalytic oxidation (PCO) on TiO₂ particles in air for a wide range of substances: ketones, organophosphorus compounds, alkyl sulfides, and chlorinated hydrocarbons. The range of substances can certainly be considerably expanded. An equation of implicit function was derived that describes the kinetics of heterogeneous PCO of the zeroth, first, and intermediate (between the zeroth and first) orders. Approximation of the experimental time dependence of concentration using this equation makes it possible to determine the reaction order including the intermediate one, the characteristic decay time of the substance, and the fraction of the exponential components in the kinetic equation. This semiempirical method was used for processing both the original experimental data obtained in the present study and the literature data. The time dependences of trichloroethylene (TCE) concentrations in a closed space during the heterogeneous PCO on TiO₂ aerosol catalyst particles were studied using a specially designed unit. The catalytic activity increased with the aerosol concentration C _as : at C _as = 10.23, 14.17, and 19.85 g/m³, 90% purification of air from TCE was reached in 8.5, 5.0, and 1.5 min, respectively.     

Perturbation theory for QED bound states

A new method is presented for deriving a systematic perturbative expansion for QED bound states, which does not rely upon solving any new or old equation. The starting point is a given nonperturbative zeroth order Green's function, obtained by a suitable “relativistic dressing” of the nonrelativistic Green's function for the Schrödinger equation with Coulomb potential, which embodies the Coulombic bound states and is known. The comparison with the complete Green's function as given by standard perturbative QED gives a perturbative kernel which is then used for the expansion of the QED Green's function in terms of the given non-perturbative zeroth order Green's function.     

Liquid metal pair potentials from structure data

In linear screening theory, the zeroth order result is the equality of the potential of mean force and the pair potential. The Kirkwood superposition approximation holds to the same order.The next order yields an equation for the pair potential identical with the first order iteration of Johnson and March from the Born-Green equation.     

A theory of Doppler and binary collision broadening by foreign gases—II.: Isolated spectral line structure

The kinetic equation, derived in the preceeding paper I and describing combined Doppler and binary collision broadening is examined. A series solution is developed for the case of an isolated spectral line which, to first order in the density, is Lorentzian under appropriate conditions. The general solution retains the effects of non-instantaneous collisions and binary collision correlations. An approximation for the collision terms of the kinetic equation is also proposed.     

On a level shift occurring in the treatment of a discrete state coupled to two interacting continua

It is shown how the partitioning method of Löwdin may be used to obtain approximate solutions to the Dirac equation. By using a novel separation of the partitioned wave equation perturbation theory may be employed with the solutions of the Schrödinger equation as the zeroth order functions. The method is demonstrated for the 1s, 2s and 2p states of the hydrogen atom and in particular the energies correct to order mc ² α⁶ are obtained. The first and second order contributions to the energy are both finite so the problem of divergences is avoided.     

Equilibrium time correlation functions in the low-density limit

We consider a system of hard spheres in thermal equilibrium. Using Lanford's result about the convergence of the solutions of the BBGKY hierarchy to the solutions of the Boltzmann hierarchy, we show that in the low-density limit (Boltzmann-Grad limit): (i) the total time correlation function is governed by the linearized Boltzmann equation (proved to be valid for short times), (ii) the self time correlation function, equivalently the distribution of a tagged particle in an equilibrium fluid, is governed by the Rayleigh-Boltzmann equation (proved to be valid for all times). In the latter case the fluid (not including the tagged particle) is to zeroth order in thermal equilibrium and to first order its distribution is governed by a combination of the Rayleigh-Boltzmann equation and the linearized Boltzmann equation (proved to be valid for short times).Supported in part by NSF Grant PHY 78-22302.     

Generalized Boltzmann equation for lattice gas automata

In this paper a theory is formulated that predicts velocity and spatial correlations between occupation numbers that occur in lattice gas automata violating semi-detailed balance. Starting from a coupled BBGKY hierarchy for then-particle distribution functions, cluster expansion techniques are used to derive approximate kinetic equations. In zeroth approximation the standard nonlnear Boltzmann equation is obtained; the next approximation yields the ring kinetic equation, similar to that for hard-sphere systems, describing the time evolution of pair correlations. The ring equation is solved to determine the (nonvanishing) pair correlation functions in equilibrium for two models that violate semidetailed balance. One is a model of interacting random walkers on a line, the other one is a two-dimensional fluid-type model on a triangular lattice. The numerical predictions agree very well with computer simulations.     

Scaling of a Slope: The Erosion of Tilted Landscapes

We formulate a stochastic equation to model the erosion of a surface with fixed inclination. Because the inclination imposes a preferred direction for material transport, the problem is intrinsically anisotropic. At zeroth order, the anisotropy manifests itself in a linear equation that predicts that the prefactor of the surface height–height correlations depends on direction. The first higher order nonlinear contribution from the anisotropy is studied by applying the dynamic renormalization group. Assuming an inhomogeneous distribution of soil substrate that is modeled by a source of static noise, we estimate the scaling exponents at first order in an ε-expansion. These exponents also depend on direction. We compare these predictions with empirical measurements made from real landscapes and find good agreement. We propose that our anisotropic theory applies principally to small scales and that a previously proposed isotropic theory applies principally to larger scales. Lastly, by considering our model as a transport equation for a driven diffusive system, we construct scaling arguments for the size distribution of erosion “events” or “avalanches.” We derive a relationship between the exponents characterizing the surface anisotropy and the avalanche size distribution, and indicate how this result may be used to interpret previous findings of power-law size distributions in real submarine avalanches.     

Approximate solutions of the Liouville equation. IV. The two-body additive approximation

The two-body additive approximation on the time-dependent Liouville distribution, first introduced in part I of this series, is put into the conventional form of a self-contained kinetic equation for the doublet distribution. From this point of view the approximation consists in truncating the BBGKY chain by expressing the triplet distribution as a functional of lower distributions at the same value of the time variable. To accomplish this, it is necessary to study two associated purely spatial integral equations. The doublet kinetic equation can then be written in terms of solutions of these integral equations and comparison with conventional methods of truncating the BBGKY chain can then be made. For the purpose of comparison a method of truncating the chain based on the Kirkwood superposition approximation is introduced and discussed briefly. The momentum structure of the resulting doublet kinetic equation is similar, but the nonlocality in space of our truncation introduces distinct differences in the spatial structure. The inconsistency between conventional truncations and the exact initial conditions used for the calculation of time-dependent correlation functions is pointed out. This inconsistency is not shared by the two-body additive approximation.     

Fractional statistical mechanics

The Liouville and first Bogoliubov hierarchy equations with derivatives of noninteger order are derived. The fractional Liouville equation is obtained from the conservation of probability to find a system in a fractional volume element. This equation is used to obtain Bogoliubov hierarchy and fractional kinetic equations with fractional derivatives. Statistical mechanics of fractional generalization of the Hamiltonian systems is discussed. Liouville and Bogoliubov equations with fractional coordinate and momenta derivatives are considered as a basis to derive fractional kinetic equations. The Fokker-Planck-Zaslavsky equation that has fractional phase-space derivatives is obtained from the fractional Bogoliubov equation. The linear fractional kinetic equation for distribution of the charged particles is considered.     

Diffusive corrections to asymptotics of a strong-field quantum transport equation

The asymptotic analysis of a linear high-field Wigner-BGK equation is developed by a modified Chapman-Enskog procedure. By an expansion of the unknown Wigner function in powers of the Knudsen number ?, evolution equations are derived for the terms of zeroth and first order in ?. In particular, a quantum drift-diffusion equation for the position density of electrons, with an ?-order correction on the field terms, is obtained. Well-posedness and regularity of the approximate problems are established, and a rigorous proof that the difference between exact and asymptotic solutions is of order ?², uniformly in time and for arbitrary initial data is given.     

Kinetic equation for a weakly interacting classical electron gas

A kinetic equation is derived for the two-time phase space correlation function in a dilute classical electron gas in equilibrium. The derivation is based on a density expansion of the correlation function and the resummation of the most divergent terms in each order in the density. It is formally analogous to the ring summation used in the kinetic theory of neutral fluids. The kinetic equation obtained is consistent to first order in the plasma parameter and is the generalization of the linearized Balescu-Guersey-Lenard operator to describe spatially inhomogeneous equilibrium fluctuations. The importance of consistently treating static correlations when deriving a kinetic equation for an electron gas is stressed. A systematic derivation as described here is needed for a further generalization to a kinetic equation that includes mode-coupling effects. This will be presented in a future paper.     

Projection method of solving the Bogolyubov equation for the generating functional in classical statistical physics

An approximate method is considered for the solution of the Bogolyubov equation, which is characterized from the physical viewpoint by successively taking account of corrections of ever higher order. In the zeroth approximation the known Vlasov equation is obtained, in the first approximation a system of equations for the unary distribution and second-order correlation functions, and in the second approximation, a system of three equations for the appropriate correlation functions. The properties of the first approximation equations are investigated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 95–99, April, 1984.The authors are grateful to É. A. Arinshtein and N. M. Placid for useful discussions.     

The kinetic equation and the equation of momentum transfer for a plasma

An attempt is made to derive a simple form of the collision integral of the kinetic equation for a plasma, by using Rostoker’s equation which expresses the pair correlation function in terms of the distribution functions of the particles, and the conditional probability of one particle shielding the other. The conditional probability function is assumed to be given by its equilibrium value. By taking first order velocity-moment of the resulting kinetic equation, the equation of momentum transfer has been obtained.     

The Melting Curve of Argon by Using Lindemann's Criterion

In this work we applied Lindemann's criterion to build the melting curve of argon between 83.8 and 260 K, taking into account the strong anharmonicity. For this purpose we used the Correlative Method of Unsymmetrized Self-Consistent Field (CUSF), including anharmonic terms up to the fourth order in the zeroth approximation, deriving the equation of state of the crystal.     

Transient acoustic radiation from impulsively accelerated bodies by the finite element method

Bodies under impulsive motion, immersed in an infinite acoustic fluid, severely put to test any numerical method for the transient exterior acoustic problem. Such problems, in the context of the finite element method (FEM), are not well studied. FE modeling of such problems requires truncation of the infinite fluid domain at a certain distance from the structure. The volume of computation depends upon the extent of this domain as well as the mesh density. The modeling of the fluid truncation boundary is crucial to the economy and accuracy of solution and various boundary dampers have been proposed in the literature for this purpose. The second order damper leads to unsymmetric boundary matrices and this necessitates the use of an unsymmetric equation solver for large problems. The present paper demonstrates the use of FEM with zeroth, first and second order boundary dampers in conjunction with an unsymmetric, out of core, banded equation solver for impulsive motion problems of rigid bodies in an acoustic fluid. The results compare well with those obtained from analytical methods.