Kinetics of reversible polymerization

This paper extends the kinetic theory of irreversible polymerization (Smoluchowski's equation) by including fragmentation effects in such a way, that the most probable (equilibrium) size distribution from the classical polymerization theories is contained in our theory as the stationary distribution. The time-dependent cluster size distributionc _k (a(t)) in Flory's polymerization modelsRA _f andA _f RB _g , expressed in terms of the extent of reaction, has the same canonical form as in equilibrium, and the time dependence of(t) is determined from a macroscopic rate equation. We show that a gelation transition may or may not occur, depending on the value of the fragmentation strength, and, in case a phase transition takes place, we give Flory- and Stockmayer-type postgel distributions.Inspec numbers: 0550, 6460, 8235, 6475.     

Algebraic spatial correlations in lattice gas automata violating detailed balance

In this paper we discuss the existence of generic long-range correlations in spatially homogeneous and stable equilibrium states of closed lattice gas automata whose stochastic collision rules violate the symmetry conditions of detailed balance and in addition satisfy local conservation laws. Such correlations occur even though the collision rules are strictly local and invariant under all symmetries of the lattice. First a phenomenological (Langevin equation) approach is discussed. Next we present a theoretical analysis on the basis of an approximate microscopic (ring kinetic) theory. This theory is used to calculate the amplitude ofr ^– tails in the spatial correlations, and the result is compared with computer simulations.     

Correlation functions for quasi-linear response theory

A quasi-linear regression formula is derived by an expansion around quasi-static equilibrium. It relates the relaxation of thermodynamic forces to the regression of correlations of thermodynamic coordinates in quasi-static equilibrium. Correlation functions and memory kernels can be introduced in almost complete analogy to linear response theory. A non-linear, non-Markovian kinetic equation is derived. The kinetic coefficients are given in terms of correlation functions of stochastic forces in quasi-static equilibrium similar to the linear theory.     

LocalH-theorem for a kinetic variational theory

A localH-theorem is derived for a recently proposed extension of Enskog kinetic theory to a dense model fluid composed of particles with interactions extending beyond a hard core.On leave from: Katedra Fizyki, Uniwersytetu Szczecinskiego, 70-451 Szczecin, Poland.     

Nonlinear desingularization in certain free-boundary problems

We consider a nonlinear, elliptic, free-boundary problem involving an initially unknown setA that represents, for example, the cross-section of a steady vortex ring or of a confined plasma in equilibrium. The solutions are characterized by a variational principle which allows us to describe their behaviour under a limiting process such that the diameter ofA tends to zero, while the solutions degenerate to the solution of a related linear problem. This limiting solution is the sum of the Green function of the linear operator and of a smooth function satisfying the boundary conditions. Mathematically speaking, this limiting process, that we call nonlinear desingularization, is a novel kind of bifurcation phenomenon since the nonlinear effect here involves smoothing the singularity of the associated linear problem.Research partially supported by A FOSR and NSF grants     

On the nonergodicity of the transverse magnetization in the transverse Ising model

The nonergodic behavior exhibited by the transverse spin correlation function _q=0 ^xx (t) the transverse Ising model obtained as the solution of approximate kinetic equations (derived on the basis of Résibois and De Leener's method), is shown to be an intrinsic property of the model and not the result of the approximations made in the derivation of the kinetic equations.Chargé de Recherches au Fonds National Belge de la Recherche Scientifique.     

Vibration-rotation wavefunctions and energies for any molecule obtained by a variational method

This paper presents an alternative method to the usual approach via perturbation theory for the determination of vibrational-rotational energy levels of a molecule in a given electronic state. It is assumed that the electronic Born-Oppenheimer equation has been solved, by an ab initio method, to give a potential function which is used in the nuclear Born-Oppenheimer equation. But the method can also be used with any potential obtained by any method. An approximate solution to the nuclear equation is derived in the form of a linear combination of expansion functions, the coefficients being determined by the standard linear variational method. Angular momentum theory is used to show that the nuclear wavefunction for m = 0 can be represented by a linear combination of functions of the form

where qi are variables which are closely related to the vibrational normal coordinates, and β, γ are two of the Euler angles. m is the eigenvalue of the Z-component of angular momentum operator in space-fixed axes OXYZ. The H_vi (qi) denote Hermite polynomials while Y^J8 (β, γ) are spherical harmonics. It is explicitly shown how all the matrix elements can be evaluated using a (3N–6) dimensional numerical integration technique. The theory in its present form is not suitable for molecules which are linear in the equilibrium configuration. In the following paper the method is used in a calculation on the water molecule.     

Contractive Metrics for a Boltzmann Equation for Granular Gases: Diffusive Equilibria

We quantify the long-time behavior of a system of (partially) inelastic particles in a stochastic thermostat by means of the contractivity of a suitable metric in the set of probability measures. Existence, uniqueness, boundedness of moments and regularity of a steady state are derived from this basic property. The solutions of the kinetic model are proved to converge exponentially as t to this diffusive equilibrium in this distance metrizing the weak convergence of measures. Then, we prove a uniform bound in time on Sobolev norms of the solution, provided the initial datum has a finite norm in the corresponding Sobolev space. These results are then combined, using interpolation inequalities, to obtain exponential convergence to the diffusive equilibrium in the strong L¹-norm, as well as various Sobolev norms.     

On weak and strong convergence to equilibrium for solutions to the linear Boltzmann equation

This paper considers the linear space-inhomogeneous Boltzmann equation for a distribution function in a bounded domain with general boundary conditions together with an external potential force. The paper gives results on strong convergence to equilibrium, whent, for general initial data; first in the cutoff case, and then for infinite-range collision forces. The proofs are based on the properties of translation continuity and weak convergence to equilibrium. To handle these problems generalH-theorems (concerning monotonicity in time of convex entropy functionals) are presented. Furthermore, the paper gives general results on collision invariants, i.e., on functions satisfying detailed balance relations in a binary collision.     

H-theorem for the (modified) nonlinear Enskog equation

We construct anH-function suitable for a system of dense hard spheres satisfying the (modified) nonlinear Enskog equation and we show that _t H 0. The equality sign holds only when the system has reached absolute equilibrium, in which caseS=– k_B H becomes the exact equilibrium entropy of the hard-sphere fluid.     

Equilibrium polymerization in a solvent: Solution on the Bethe lattice

The lattice model for equilibrium polymerization in a solvent proposed by Wheeler and Pfeuty is solved exactly on a Bethe lattice (core of a Caylay tree) with general coordination numberq. Earlier mean-field results are reobtained in the limitq, but the phase diagrams show deviations from them for finiteq. Whenq=2, our results turn into the solution of the one-dimensional problem. Although the model is solved directly, without the use of the correspondence between the equilibrium polymerization model and the diluten0 model, we verified that the latter model may also be solved on the Bethe lattice, its solution being identical to the direct solution in all parameter space. As observed in earlier studies of the puren0 vector model, the free energy is not always convex. We obtain the region of negative susceptibility for our solution and compare this result with mean field and renormalization group (-expansion) calculations.     

Exact solution of the discrete (1+1)-dimensional SOS model with field and surface interactions

We present the solution of a linear solid-on-solid (SOS) model. Configurations are partially directed walks on a two-dimensional square lattice and we include a linear surface tension, a magnetic field, and surface interaction terms in the Hamiltonian. There is a wetting transition at zero field and, as expected, the behavior is similar to a continuous model solved previously. The solution is in terms ofq-series most closely related to theq-hypergeometric functions_{1 1}.     

Response function of the second kind in many-body systems

We analyze a new type of response function which portrays the properties of a system perturbed by an external field in terms of the perturbed two-point correlations of density fluctuations rather than in terms of perturbed averages of physical quantities. This response function of the second kind satisfies both fluctuation-dissipation-like theorems, relating it to three-point equilibrium functions, and hierarchical relationship linking it to conventional quadratic (rather than linear) response functions. In the equal-time limit, when the two density fluctuations are observed at the same time, the response function of the second kind is intimately connected to the two-particle correlation function of kinetic theory. This linkage opens an avenue for developing novel approximation techniques for correlated many-body systems.     

Kinetic processes of muonium formation by non-thermalized electrons in the spur model

An approximate analytical solution is obtained for the kinetic equation for electrons in the field of a Coulomb centre formed by a muon in the end of its track. It is shown that thermalization times and 2> could differ by several orders of magnitude, depending on the behaviour of transport cross sections. The fraction of non-thermalized electrons returning to the muon and forming muonium (P _Mu ) depends on pressure and temperature. Therefore,SR experiments give us an opportunity to obtain information on the distribution function of track electrons.     

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).     

Static spherically symmetric solutions in general projective relativity

Static spherically symmetric solutions have been obtained for general projective relativity withn=0 andn0 both in isotropic and curvature coordinates. In curvature coordinates, only a restricted exact solution is possible. However, an approximate solution can always be obtained following a method similar to Vanden Bergh. In these spacetimes there is no horizon, but only a naked singularity atr=0. Thus there are no black holes. It is shown that there is no solution in static, spherically symmetric, conformally flat spacetime.     

Probability Metrics and Uniqueness of the Solution to the Boltzmann Equation for a Maxwell Gas

We consider a metric for probability densities with finite variance on ^d , and compare it with other metrics. We use it for several applications both in probability and in kinetic theory. The main application in kinetic theory is a uniqueness result for the solution of the spatially homogeneous Boltzmann equation for a gas of true Maxwell molecules.     

Energetics of growing epilayer-substrate combinations of comparable bond strength and in Kurdjumov-Sachs orientation

In this paper we address the problems related to critical misfit and thickness in epilayer-substrate combinations of comparable bond strengths; specifically the case in which a pseudomorphic monolayer (ML) is stable and the critical thickness is about three MLs or less. Of particular interest are the average energies related to misfit strain f _KS and misfit dislocations (MDs)—in the latter case the individual contributions of the oscillatory strains V and the epilayer-substrate disregistry V_MD. The individual energies are of interest because they may play different roles in the realization of specific growth modes. The analytical approach involves the following assumptions: (a) a rigid substrate as source of a periodic epilayer atom-substrate interaction potential which we model in terms of a low order truncated Fourier series; and (b) an epilayer which (i) deforms harmonically with zero strain gradient normal to the film plane, (ii) grows in Kurdjumov-Sachs (KS) orientation due to small misfit. f _KS and in the layer-by-layer growth mode.Arguments are presented claiming that this interfacial situation may be approximated by a one-dimensional problem in which epilayer stiffness constants and equilibrium structure, as well as epilayer-substrate interaction depend on epilayer thickness; which poses a complex problem. An approximate solution could be obtained by assuming these quantities to be independent of thickness and proximities of the vacuum and the substrate. The most prominent conclusions are that the equilibrium density of MDs and hence the transition from misfit accommodation by MS to one containing MDs is a catastrophic process and that sustained minimum energy may require the overcoming of an energy barrier. While elementary implementation of the results to equilibrium growth mode theory suggests—independently of the catastrophic nature—that energetically favored misfit strain relief by misfit dislocations may, or may not, effect a transition to Stranski-Krastanov growth, a crude numerical calculation favors the transition. A proper implementation of the results require extensive numerical calculations and is planned for the near future.     

On the Form of Parametrized Gravitation in Flat Spacetime

In a framework describing manifestly covariant relativistic evolution using a scalar time , consistency demands that -dependent fields be used. In recent work by the authors, general features of a classical parametrized theory of gravitation, paralleling general relativity where possible, were outlined. The existence of a preferred time coordinate changes the theory significantly. In particular, the Hamiltonian constraint for is removed From the Euler-Lagrange equations. Instead of the 5-dimensional stress-energy tensor, a tensor comprised of 4-momentum density mid flux density only serves as the source. Building on that foundation, in this paper we develop a linear approximate theory of parametrized gravitation in the spirit of the flat spacetime approach to general relativity. Using a modified form of Kraichnan's flat spacetime derivation of general relativity, we extend the linear theory to a family of nonlinear theories in which the flat metric and the gravitational field coalesce into a single effective curved metric.