Derivation Of Kinetic Equations Research Articles

New approach to derivation of quantum kinetic equations with initial correlations

We propose a new approach to the derivation of kinetic equations from dynamics of large particle quantum systems, involving correlations of particle states at initial time. The developed approach is based on the description of the evolution within the framework of marginal observables in scaling limits. As a result the quantum Vlasov-type kinetic equation with initial correlations is constructed and the statement relating to the property of a propagation of initial correlations is proved in a mean field limit.

Derivation of kinetic equations from non-Wiener stochastic differential equations

Kinetic differential-difference equations containing terms with fractional derivatives and describing α -stable Levy processes with 0 < α < 1 have been derived in a unified manner in terms of one-dimensional stochastic differential equations controlled merely by the Poisson processes.

Mechanochemical synthesis of nanoparticles by a dilution method: derivation of kinetic equations

A kinetic equation was derived for the mechanochemical preparation of nanosystems by a dilution method; the equation was used to calculate mechanical activation times necessary for the completion of exchange reactions and mass-transfer coefficients in ball milling.

Manifestation of macroscopic correlations in elementary reaction kinetics. II. Irreversible reaction A+B→C

The applicability of the Encounter Theory (ET) (the prototype of the Collision Theory) concepts for widely occurring diffusion assisted irreversible bulk reaction A+B→C (for example, radical reaction) in dilute solutions with arbitrary ratio of initial concentrations of reactants has been treated theoretically with modern many-particle method for the derivation of non-Markovian binary kinetic equations. The method shows that, just as in the reaction A+A→C considered earlier, the agreement with the Encounter Theory is observed when the familiar Integral Encounter Theory is used which is just a step in the derivation of kinetic equations in the framework of the method employed. It allows for two-particle correlations only, and fails to consider the correlation of reactant simultaneously with a partner and with a reactant in the bulk. However, the next step leading to the Modified Encounter Theory under reduction of equations to a regular form both extends the time applicability interval of ET homogeneous rate equation (as for reactions proceeding in excess of one of the reactants), and yields the inhomogeneous equation of the Generalized Encounter Theory (GET) that reveals macroscopic correlations induced by the encounters in a reservoir of free walks in full agreement with physical considerations. This means that the encounters of reactants in solution are correlated at rather large time interval of the reaction course. However, unlike the reaction A+A→C of identical reactants, the reaction A+B→C accumulation of the above macroscopic correlations (even with the initial concentrations of reactants being equal) proceeds much slower. Another distinction is that for the reaction A+A→C the long-term behavior of ET and GET kinetics is the same, while in the reaction A+B→C these kinetics behave differently. It is of interest that just taking account of the above macroscopic correlations in the reaction A+B→C (in GET) results in the universal character of the long-term behavior of the kinetics for the case of equal initial concentrations of reactants and that where one of the reactants is in excess. This is more natural from the point of view of the reaction course on the encounters of reactants in solutions.

Manifestation of macroscopic correlations in elementary reaction kinetics. I. Irreversible reaction A+A→product

Using an modern many-particle method for the derivation of non-Markovian binary kinetic equations, we have treated theoretically the applicability of the encounter theory (ET) (the prototype of the collision theory) concepts to the widely known diffusion assisted irreversible bulk reaction A+A-->product (for example, radical reaction) in dilute solutions. The method shows that the agreement with the ET is observed when the familiar integral ET is employed which in this method is just a step in the derivation of kinetic equations. It allows for two-particle correlations only, but fails to take account of correlation of reactant simultaneously with the partner of the encounter and the reactant in the bulk. However, the next step leading to the modified ET under transformation of equations to the regular form both extends the time range of the applicability of ET rate equation (as it was for reactions proceeding with one of the reactants in excess), and gives the equation of the generalized ET. In full agreement with physical considerations, this theory reveals macroscopic correlations induced by the encounters in the reservoir of free walks. This means that the encounters of reactants in solution are correlated on a rather large time interval of the reaction. Though any nonstationary (non-Markovian) effects manifest themselves rather weakly in the kinetics of the bimolecular reaction in question, just the existence of the revealed macroscopic correlations in the binary theory is of primary importance. In particular, it means that the well-known phenomena which are generally considered to be associated solely with correlation of particles on the encounter (for example, chemically induced dynamic nuclear polarization) may be induced by correlation in the reservoir of free random walks of radicals in solution.

Derivation of an evolution equation for two-time correlation function

An expansion is developed of the two-point correlation function in the two-time correlation functions up to the expansion terms of third order. This expansion is necessary for the rigorous derivation of kinetic equations that objectively describe physical phenomena in a ‘single’ plasma with Langmuir turbulence. The correspondence of the expansion terms with various physical processes of the traditional weak plasma turbulence theory is discussed. Perspectives are outlined of application of the results to studies in the physics of non-ideal plasmas.

Renormalization Group Method Applied to Kinetic Equations: Roles of Initial Values and Time

The so-called renormalization group (RG) method is applied to derive kinetic and transport equations from the respective microscopic equations. The derived equations include the Boltzmann equation in classical mechanics, the Fokker–Planck equation, and a rate equation in a quantum field theoretical model. Utilizing the formulation of the RG method which elucidates the important role played by the choice of the initial conditions, the general structure and the underlying assumptions in the derivation of kinetic equations in the RG method are clarified. It is shown that the present formulation naturally leads to the choice for the initial value of the microscopic distribution function at arbitrary time t0 to be on the averaged distribution function to be determined. The averaged distribution function may be thought of as an integral constant of the solution of the microscopic evolution equation; the RG equation gives the slow dynamics of the would-be initial constant, which is actually the kinetic equation governing the averaged distribution function. It is further shown that the averaging as given above gives rise to a coarse-graining of the time-derivative which is expressed with the initial time t0, and thereby leads to time-irreversible equations even from a time-reversible equation. It is shown that a further reduction of the Boltzmann equation to fluid dynamical equations and the adiabatic elimination of fast variables in the Fokker–Planck equation are also performed in a unified way in the present method.

Microscopic derivation of kinetic equations for interacting optical two-level systems

Kinetic equations are derived for optical two-level atoms interacting with a molecular subsystem treated as a thermostat. It is assumed that the kinetics is determined by the electric dipole interaction perturbed by the thermal motion. Dynamic parts of the kinetic equations coincide with the corresponding terms of optical Bloch equations, whereas nonlinear relaxation and shift terms have the specific form and are absent in the phenomenological generalized Bloch equations. It is shown that the relaxation kinetics can be substantially different from the exponential one and depends on the initial state of the system. In particular, the inversion relaxation is frozen at small deviations from the equilibrium. The possibility of observation of the optical bistability is discussed.

The Boltzmann equation for dissipative disparate mixtures

This paper deals with the modelling and analysis of the nonlinear kinetic equations for a disparate mixture of particles undergoing dissipative binary collisions and coagulation fragmentation phenomena. The first part of the paper deals with collision dynamics, the second part with derivation of kinetic equations.

Irreversibility and the probabilistic treatment of the dynamics of classical particles

It is shown that B B Kadomtsev's idea of small 'external noise' securing a time-irreversible evolution reduces the justification of statistical physics to a fundamentally new formulation which requires that the dynamics of a multiparticle system be treated by neglecting small acceleration particles. It turns out that this formulation not only leads naturally to irreversible evolution but also suggests a new way of constructing kinetic equations capable of correctly accounting for fluctuations. At the early, small-time, stage of evolution, the original correlations are forgotten and 'post-collisional' ones form. It is the final portion of the first stage and the formation of the 'post-collisional' correlations which can be described by a closed form kinetic equation. A relation between Kadomtsev's external noise idea and Bogolyubov's derivation of kinetic equations is established, leading to a new physical interpretation of the 'molecular chaos' hypothesis.

Inelastic granular gas: Computer simulations and kinetic theory of the cooling state

We developed computer simulations of an inelastic granular gas which show that the energy decay proposed by Goldhirsch and Zanetti [Phys. Rev. Lett. 70, 1619 (1993)] has a limited validity. Moreover, we give an exact solution of the Liouville equation for the moments of the two-body homogeneous cooling distribution. The latter includes velocity correlations which raises questions about the derivation of kinetic equations for inelastic gases.

General linear compartment model with zero input: I. Kinetic equations

The derivation of kinetic equations is described for n-compartment linear models, in which the substance may be simultaneously introduced into one or more compartments at t = 0 and eliminated from any compartment. For a given zero-input, general formulas are derived which describe the amount of tracer in any of the compartments as a function of time and the model parameters. New algorithms have been developed which allow the expression of the kinetic equations.

On the validity of the Brussels formalism in statistical mechanics

The so-called 'Brussels' approach to the derivation of kinetic equations usually proceeds by representing the Liouville operator L in the form L0+ lambda L1 where lambda is a perturbation parameter. It can be formulated in terms of an operator P (or a set of such operators) commuting with L0 and projecting from a Hilbert space H, spanned by all square-integrable phase-space densities or density matrices rho , into a subspace H1 in which the reduced or kinetic description is to apply. For the case where H1 is finite-dimensional, the authors prove two main results: (i) if the time-domain collision operator, defined by psi (t)=PL1Q exp(QLQt)QL1P where Q=1-P, is bounded above in norm by a decreasing exponential function of mod t mod and satisfies the condition that the Hermitian part of integral 0infinity eiyt psi (t)dt be invertible for all real y, then for sufficiently small positive lambda the long-time asymptotic approach to equilibrium in the subspace H1 is an exponential decay or exponentially decaying oscillation and is correctly given by the Brussels perturbation method. (ii) If the norm of the collision operator decays to zero at t to infinity more slowly than any exponential then, regardless of the value of lambda , the asymptotic behaviour does not have the exponential form predicted by the Brussels method.

On the Lippmann-Schwinger equation in Liouville space. Π̊ subdynamics

We study some aspects of the Lippmann-Schwinger equation in Liouville space: The relation between the Møller approach and Π̊ subdynamics, the obtention of H-invariants from the corresponding H 0-invariants and the derivation of kinetic equations.

Nonunitary transformations and hydrodynamic limit in the one-dimensional hard-point gas

The ergodic-theoretical formalism of nonunitary “Λ” transformations is compared with the derivation of kinetic equations by the hydrodynamic scaling limit. The one-dimensional hard-point system is used as an illustration. Both methods rest on the K-mixing property of the physical systems considered. We show that the specific form of the operator Λ is important only on the microscopic time scale: on the macroscopic scale, it acts like a projector (with broken time-reversal symmetry). We also discuss the connection between the Kolmogorov-Sinai (K-S) entropy of the model dynamical system, the relative entropy production rate in the nonunitary formalism and the macroscopic entropy production rate.

Allowance for intermolecular forces of attraction in the derivation of kinetic equations

This paper proposes a method for solving the Bogolyubov hierarchy for dense gases with real interaction potentials. The idea of the method is to decouple each equation of the hierarchy with respect to scales that characterize the strength of the intermolecular repulsion and attraction. Kinetic equations for moderately dense and dense gases are derived and some model equations of the kinetic theory of dense gases with real interaction potentials are constructed.

Derivation of kinetic equation in Fröhlich anharmonic model

This work is based on a microscopic approach and the conclusion is made that there is no phonon condensation in the Fröhlich model.

Derivation of kinetic equation for model system with discrete spectrum without the hypothesis of correlation damping

Derivation of kinetic equation for model system with discrete spectrum without the hypothesis of correlation damping

Derivation of kinetic equations in the statistical T-matrix approximation

Derivation of kinetic equations in the statistical T-matrix approximation

Eine verallgemeinerung des stratonovich-verfahrens für anwendungen in der statistischen mechanik

Zusammenfassung R.L. Stratonovich has given a derivation of kinetic equations, if a stochastic process is determined by a system of fluctuation equations. In this paper it is examined how the equations of motion of a particle of a system of mass points can be interpreted as a system of fluctuation equations. These equations are more general than those which are treated by Stratonovich. A derivation of kinetic equations is given which is applicable to the more general case and which contains the results of Stratonovich. The kinetic equation can be interpreted as a generalization of the well-known Fokker-Planck equation. The stochastic force field which is generated by the particles of the system of mass points is briefly discussed.