Reaction-diffusion equations for interacting particle systems

We study interacting spin (particle) systems on a lattice under the combined influence of spin flip (Glauber) and simple exchange (Kawasaki) dynamics. We prove that when the particle-conserving exchanges (stirrings) occur on a fast time scale of order ^–2 the macroscopic density, defined on spatial scale ^–1, evolves according to an autonomous nonlinear diffusion-reaction equation. Microscopic fluctuations about the deterministic macroscopic evolution are found explicitly. They grow, with time, to become infinite when the deterministic solution is unstable.This work was supported by NSF Grant DMR81-14726-02.Partially supported by CNR.Partially supported by CNPq Grant No. 201682-83.     

Navier-Stokes equations for stochastic particle systems on the lattice

We introduce a class of stochastic models of particles on the cubic lattice ℤ ^d with velocities and study the hydrodynamical limit on the diffusive spacetime scale. Assuming special initial conditions corresponding to the incompressible regime, we prove that in dimensiond≧3 there is a law of large numbers for the empirical density and the rescaled empirical velocity field. Moreover the limit fields satisfy the corresponding incompressible Navier-Stokes equations, with viscosity matrices characterized by a variational formula, formally equivalent to the Green-Kubo formula. Partially supported by GNFM-CNR and MURST. Partially supported by GNFM-CNR, INFN and MURST. Partially supported by U.S. National Science Foundation grant 9403462 and David and Lucile Packard Foundation Fellowship.     

Algebraic solutions of integrodifferential CRPA equations

We formulate a version of the collisional Random-Phase-Approximation obtained from the linearization of a general kinetic equation. The resulting equations are formally solved in three different situations depending on the way the previous history is considered. An application of the formalism here developed to a two level model is made.     

变质量完整系统Gibbs-Appell方程的形式不变性

建立变质量完整系统的GibbsAppell方程,给出该方程在无限小群变换下形式不变性的定义和判据,并在确定的条件下由不变性引导出守恒量.举例说明结果的应用 关键词： 分析力学 变质量完整系统 Gibbs-Appell方程 不变性 守恒量     

Applications of dynamical algebra of invariance of integrodifferential equations

General results on the algebraic properties of integrodifferential equations are used to obtain coherent and squeezed states and Green functions for the matrixdifferential models of condensed matter theory.     

Integral equations for particle density matrices

A system of integral equations for particle density matrices is obtained. The method of expansion in one-, two-, three-particle (etc.) parameters is used. It is shown that in the classical limit the resulting equations become the system of familiar equations for particle distribution functions of classical statistical mechanics.Translated from Izvestiya VUZ. Fizika, Vol. 11, No. 8, pp. 81–86, August, 1968.     

Macroscopic Einstein equations for a system of gravitating particles with unequal masses

We derive macroscopic Einstein equations, to within terms of the second order of smallness in interaction, for a system of gravitationally interacting particles with unequal masses. We generalize the results of [1, 2], which are applicable only to a system of gravitationally interacting particles with equal masses.     

Master equations for coupled systems

A formalism to cope with the problem of dynamically coupled systems is developed. A time-dependent projection operator of the type given by Willis-Picard and Grabert-Weidlich is used to derive a time-convolutionless master equation from the Liouville equation for the total composite system. A systematic perturbational expansion formula with respect to the interaction between systems is also given. Finally, the comparison with the usual non-Markoffian master equation is discussed.     

Stochastic weighted particle methods for population balance equations

A class of coagulation weight transfer functions is constructed, each member of which leads to a stochastic particle algorithm for the numerical treatment of population balance equations. These algorithms are based on systems of weighted computational particles and the weight transfer functions are constructed such that the number of computational particles does not change during coagulation events. The algorithms also facilitate the simulation of physical processes that change single particles, such as growth, or other surface reactions.     

Coarse-grained kinetic equations for quantum systems

The nonequilibrium density matrix method is employed to derive a master equation for the averaged state populations of an open quantum system subjected to an external high frequency stochastic field. It is shown that if the characteristic time τ_stoch of the stochastic process is much lower than the characteristic time τ_steady of the establishment of the system steady state populations, then on the time scale Δt ～ τ_steady, the evolution of the system populations can be described by the coarse-grained kinetic equations with the averaged transition rates. As an example, the exact averaging is carried out for the dichotomous Markov process of the kangaroo type.     

A set of integral equations for particle distribution functions

Integral equations for particle distribution functions are obtained from Bogolyubov's integrodifferential equations. Bogolyubov's integral equations can be obtained from this set of equations. The equations can also be used to obtain new relations between particle distribution functions.     

Effective equations for disordered one-dimensional systems

Using the method of supersymmetry, effective equations are derived for the one-particle Green's function of various one-dimensional disordered models. As an example, explicit expressions for the density of states and the localization length are derived for the two-band model of a one-dimensional semiconductor.     

Hamilton's equations for constrained dynamical systems

We derive expressions for the conjugate momenta and the Hamiltonian for classical dynamical systems subject to holonomic constraints. We give an algorithm for correcting deviations of the constraints arising in numerical solution of the equations of motion. We obtain an explicit expression for the momentum integral for constrained systems.     

Two-dimensional nonlinear string-type equations and their exact integration

On the basis of a group-theoretical formulation for exactly integrable two-dimensional nonlinear dynamical systems associated with the local part of an arbitrary graded Lie algebra, we study a string-type subclass of the equations. Explicit expressions are obtained for their general solutions.