Exact solutions of the nonlinear Boltzmann equation

A review is given of research activities since 1976 on the nonlinear Boltzmann equation and related equations of Boltzmann type, in which several rediscoveries have been made and several conjectures have been disproved. Subjects are (i) the BKW solution of the Boltzmann equation for Maxwell molecules, first discovered by Krupp in 1967, and the Krook-Wu conjecture concerning the universal significance of the BKW solution for the large(v, t) behavior of the velocity distribution functionf (v, t); (ii) moment equations and polynomial expansions off (v, t); (iii) model Boltzmann equation for a spatially uniform system of very hard particles, that can be solved in closed form for general initial conditions; (iv) for Maxwell and non-Maxwell-type molecules there exist solutions of the nonlinear Boltzmann equation with algebraic decrease at ; connections with nonuniqueness and violation of conservation laws; (v) conjectured super-H-theorem and the BKW solution; (vi) exactly soluble one-dimensional Boltzmann equation with spatial dependence.Reference due to C. Cercignani.     

Coordinate-dependent 3+1 formulation of the general relativity equations for a perfect fluid

This paper defines mass, momentum, and energy densities for a perfect fluid, and derives a coordinate-dependent 3+1 decomposition of the equation of motion in terms of a scalar potential c ² [(–g _g44) 1/2 –1] and a vector potentialA _i cg _4i /(–g ₄₄)^1/2. The momentum equation has the form of the Euler equation except there is an additional force proportional to the vector potential and the rate of change of kinetic energy per unit volume. The momentum and energy equations are integrated to obtain the equations previously derived for a particle. The momentum equation is solved for the total acceleration of a fluid element. The equations are exact and do not depend on the choice of coordinate system.     

Analyticity in Hubbard Models

The Hubbard model describes a lattice system of quantum particles with local (on-site) interactions. Its free energy is analytic when t is small, or t ²/U is small; here, is the inverse temperature, U the on-site repulsion, and t the hopping coefficient. For more general models with Hamiltonian H=V+T where V involves local terms only, the free energy is analytic when T is small, irrespective of V. There exists a unique Gibbs state showing exponential decay of spatial correlations. These properties are rigorously established in this paper.     

Dynamics of the generalized Glauber-Ising chain in a magnetic field

A one-dimensional kinetic Ising model with nearest neighbor interactionJ and magnetic fieldH 0 is treated in both linear and nonlinear response, using the most general single spin-flip transition probabilities that depend on nearest neighbor states only. The dynamics is reformulated in terms of kinetic equations for the concentration n_l ⁺(t) [@#@ n_l(t) of clusters containingl up- [or down-] spins, which is exact in the homogeneous case. The initial relaxation time ^* of the magnetization is obtained rigorously for arbitraryJ, H, and temperatureT. The relaxation function is found by numerical integration forJ/T < 2. It is shown that coagulation of minus-clusters becomes negligible for bothJ/T andH/T large, and the resulting set of equations is solved exactly in terms of an eigenvalue problem. A perturbation theory is developed to take into account the neglected coagulation terms. The relaxation function is found to be non-Lorentzian in general, in contrast to the Glauber results atH = 0, which are recovered as a special case. In addition, nonlinear and linear relaxation functions differ forH 0. Consequences for the application to biopolymers are briefly mentioned.Supported in part by the Deutsche Forschungsgemeinschaft (SFB 130).     

The dispersionless Lax equations and topological minimal models

It is shown that perturbed rings of the primary chiral fields of the topological minimal models coincide with some particular solutions of the dispersionless Lax equations. The exact formulae for the tree level partition functions ofA _n topological minimal models are found. The Virasoro constraints for the analogue of the -function of the dispersionless Lax equation corresponding to these models are proved.     

Row transfer matrix spectra of cyclic solid-on-solid lattice models

The eigenvalue spectra of cyclic solid-on-solid (CSOS) row transfer matrices are studied. An equivalence is established between the inversion identity and the Bethe ansatz functional equations and these equations are solved in the thermodynamic limit by a Wiener-Hopf perturbation technique for the bands of leading excitations. TheL-state CSOS model, with crossing parameter=s/L, possesses a 2(L – s)-fold degenerate largest eigenvalue corresponding to the 2(L – s) coexisting phases. The expressions for the largest eigenvalue and free energy coincide with those of the eight-vertex model. The string excitations for 2s < L and 2s > L admit different classifications and are treated separately. The correlation length is calculated in both regimes, yielding the critical exponentv=L/2s, in agreement with the scaling relations.     

On Average Properties of Inhomogeneous Fluids in General Relativity: Dust Cosmologies

For general relativistic spacetimes filled with irrotational dust a generalized form of Friedmann's equations for an effective expansion factor a _D of inhomogeneous cosmologies is derived. Contrary to the standard Friedmann equations, which hold for homogeneous-isotropic cosmologies, the new equations include the backreaction effect of inhomogeneities on the average expansion of the model. A universal relation between backreaction and average scalar curvature is also given. For cosmologies whose averaged spatial scalar curvature is proportional to a _D ^-2, the expansion law governing a generic domain can be found. However, as the general equations show, backreaction acts as to produce average curvature in the course of structure formation, even when starting with space sections that are spatially flat on average.     

Letter: A Model of Dark Energy for the Accelerating Universe

Recent observations of large scale structure of the Universe, especially that of Type Ia supernovae, indicate that the Universe is flat and is accelerating, and that the dominant energy density in the Universe is the cosmic dark energy. We propose a model in which the cosmic effective Yang-Mills condensate familiar in particle physics plays the role of the dark energy that causes the acceleration of the Universe. Since the quantum effective Yang-Mills field in certain states has the equation of state p _y = – _y , when employed as the cosmic matter source, it naturally results in an accelerating expansion of the Universe. With the matter components ( _m 1/3) being added into the model, the composition of YM condensate and matter components can give rise to the desired equation of state w –2/3 for the Universe.     

Kinetic theory of adsorption and desorption

In this paper we use stochastic methods to discuss adsorption and desorption. The paper derives generalized coefficients of sticking and accomodation depending on surface temperatureT _s and gas temperatureT _g and shows, that for additive Markov processes, these kinetic coefficients are identical. Furthermore, exact solutions of the kinetic equations for certain simple transition probabilitiesP(, ) are found and an approximation method for more complicatedP(, ) is given. The comparison of the theory with experimental results for noble gas-metal systems indicate a quadratic relationship between the first moment of the transition probabilityP(, ) and the well depth of the physisorption system.     

Nearly separable behavior of Fermi-Pasta-Ulam chains through the stochasticity threshold

For the periodic Fermi-Pasta-Ulam chain with quartic potential we prove the relation p _k ² _T (1+) _k ² q _k ² , i.e., the proportionality, already at early timesT, between averaged kinetic and harmonic energies of each mode. The factor depends on the parameters of the model, but not on the mode and the number of degrees of freedom. It grows with the anharmonic strength from the value =0 of the harmonic limit (virial theorem) up to few units for the system much above the threshold. In the stochastic regime the time necessary to reduce the fluctuations ink to a fixed percentage remains at least one order of magnitude smaller than the time necessary to reach a similar level of equipartition. The persistence of such a behavior even above the stochasticity threshold clarifies a number of previous numerical results on the relaxation to equilibrium: e.g., the existence of several time scales and the relevance of the harmonic frequency spectrum. The difficulties in the numerical simulation of the thermodynamic limit are also discussed.CNR-INFM.     

Processes of Electron Excitation Energy Conversion in Europium Tris‐Naphthoyltrifluoroacetonate

The absolute quantum yield and duration of ⁵ D ⁷ F _j luminescence of the Eu³⁺ ion in europium trisnaphthoyltrifluoroacetonate has been measured on excitation of luminescence in the UV S ₀ S ₁ absorption band of the ligand in the temperature range 77–365 K. Based on the analysis of the experimental dependences obtained and solution of the balance equations for stationary and nonstationary cases, we have developed a new approach to the determination of the scheme and rate constants of actual transitions. The rate constants of all processes of electron excitation energy conversion have been determined. The role of the state with charge transfer in these processes has been revealed.     

Energy level crossings in quantum mechanics

From the eigenvalue equationH \ _n () =E _n ()\ _n () withH H ₀ +V one can derive an autonomous system of first order differential equations for the eigenvaluesE _n () and the matrix elementsV _mn () where is the independent variable. To solve the dynamical system we need the initial valuesE _n ( = 0) and \ _n ( = 0). Thus one finds the motion of the energy levelsE _n (). We discuss the question of energy level crossing. Furthermore we describe the connection with the stationary state perturbation theory. The dependence of the survival probability as well as some thermodynamic quantities on is derived. This means we calculate the differential equations which these quantities obey. Finally we derive the equations of motion for the extended caseH =H ₀ +V ₁ + ² V ₂ and give an application to a supersymmetric Hamiltonian.     

Shear flow in the two-body Boltzmann gas. II. Small and large γ expansion of the shear viscosity

In two and three dimensions, the relaxation time Boltzmann equation can be solved analytically for the distribution function for a system of two hard particles subject to isothermal shear. The previous solutions of Morriss, and Ladd and Hoover are shown to be formally equivalent. The integral representation for the average of each of the elements of the pressure tensor in the steady state is obtained for both sllod and dolls tensor equations of motion. Rigorous equations are derived which relate the viscosity and the normal stress differences in these two methods. We obtain asymptotic expansions for each element of the pressure tensor for both small and large. For high shear rates, the viscosity is found to vanish as ^–2 log in both two and three dimensions.     

Quantized Knizhnik-Zamolodchikov equations,quantum Yang-Baxter equation,and difference equations forq-hypergeometric functions

Thes₂ quantized Knizhnik-Zamolodchikov equations are solved inq-hypergeometric functions. New difference equations are derived for generalq-hypergeometric functions. The equations are given in terms of quantum Yang-Baxter matrices and have the form similar to quantum Knizhnik-Zamolodchikov equations for quantum affine algebras introduced by Frenkel and Reshetikhin.This work was supported by NSF grant DMS-9203929.     

General Volume-Preserving Mechanical Systems

We present the general form of equations that generate a volume-preserving flow on a symplectic manifold M, ) via the highest Euler–Lagrange cohomology. It is shown that for every volume-preserving flow there are some 2-forms that play a similar role to the Hamiltonian in Hamilton mechanics. The ordinary canonical equations are included as a special case with a 2-form 1/(n - 1)H, where H is the corresponding Hamiltonian.     

Two-Loop Improved Truncation of the Ghost-Gluon Dyson-Schwinger Equations: Multiplicatively Renormalizable Propagators and Nonperturbative Running Coupling

The coupled Dyson-Schwinger equations for the gluon and ghost propagators are investigated in the Landau gauge using a two-loop improved truncation that preserves the multiplicative renormalizability of the propagators. In this truncation all diagrams contribute to the leading-order infrared analysis. The infrared contributions of the nonperturbative two-loop diagrams to the gluon vacuum polarization are computed analytically, and this reveals that infrared power-behaved propagator solutions only exist when the squint-diagram contribution is taken into account. For small momenta the gluon and ghost dressing functions behave like (p ²)² and (p ²)^–, respectively, and the running coupling exhibits a fixed point. The values of the infrared exponent and fixed point depend on the precise details of the truncation. The coupled ghost-gluon system is solved numerically for all momenta, and the solutions have infrared behaviors consistent with the predictions of the infrared analysis. For truncation parameters chosen such that = 0.5, the two-loop improved truncation is able to produce solutions for the propagators and running coupling which are in very good agreement with recent lattice simulations.Received March 17, 2003; accepted May 9, 2003 Published online September 24, 2003     

Multicomponent composites,electrical networks and new types of continued fraction I

The development of bounds on the complex effective conductivity tensor * (that relates the average current to the average electric field in a multicomponent composite) has been hindered by lack of a suitable continued-fraction representation for *. Here a new field equation recursion method is developed which gives an expression for * as a continued fraction of a novel form incorporating as coefficients the component conductivities and a set of fundamental geometric parameters reflecting the composite geometry. A hierarchy of field equations is set up such that the solutions of the (j+1)th-order equation generate the solutions of thejth-order equation. Consequently the effective tensor ^(j) associated with thejth-order field equation is expressible as a fractional linear matrix transformation of ^(j+1). These transformations combine to form the continued fraction expansion for *=⁽⁰⁾ which is exploited in the following paper, Part II, to obtain bounds: crude bounds on ^(j), forj1, give narrow bounds on *. The continued fraction is a generalization to multivariate functions of the continued fraction expansion of single variable Stieltjes functions that proved important in the development of the theory of Páde approximants, asymptotic analysis, and the theory of orthogonal polynomials in the last century. The results extend to other transport problems, including conduction in polycrystalline media, the viscoelasticity of composites, and the response of multicomponent, multiterminal linear electrical networks.     

Metastable states of an asymmetric spin-glass with one stored pattern

The number of metastable states in a spin-glass with asymmetric Gaussian couplings (J _ij J _ji ) is studied. In order to store one pattern a generalized stability condition on the local energies is considered for a part of the system (=n/N) only. The average number of metastable states is calculated analytically as a function of the overlapm in the thermodynamic limit. For the contribution of the storage restriction an approximation is introduced to simplify the calculations. The corresponding saddlepoint equations are solved numerically for two distribution functions of the local energies. For several values of the parameters we find two bands with an exponential number of metastable states. At a critical value _crit the second band disappears. The critical value for the storage parameter is calculated numerically. As expected we find _crit1 for 0.Work performed within the research program of the Sonderforschungsbereich 341, Köln-Aachen-Jülich     

Absolute parallelism: Spherical symmetry and singularities

A one-parameter class of compatible field equations of n-frames h ^a is considered, characterized by the fact that after the substitution h ^a = HPH ^a (H = detH ^a , p is the constant depending on the dimensionality n and on the equation parameter ), the equations can be written in a trilinear form which contains the matrix H_a and its derivatives, but not H ^a . The equations remain regular for degenerated matrices H_a, if r = rank H_a ^< 2, the points with r < n being singular (if n > 1 + 1/). It is shown that there exist spherically symmetric solutions with singularities appearing for r < n; as the singularity develops, there appears a domain bounded by singular points in which, for certain values of and n, the field h ^a becomes complex.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 47–52, September, 1991.The author thanks I. L. Bukhbinder for his interest to this paper and useful discussions during the seminar.