non-Markovian Kinetic Equations Research Articles

A many-particle derivation of non-Markovian kinetic equations of reversible reaction A+ B⇌ C in solutions based on the effective pairs approach

The non-Markovian kinetic equations for local concentrations have been derived in the effective pairs approximation on the basis of the hierarchy for correlation patterns of the reversible reaction A+ B⇌ C. The equations are shown to agree with the kinetic and thermodynamic detailed balancing principles. As compared to the integral encounter theory equations obtained earlier, the above equations have a wider time range of applicability, and may form the basis for the development of a binary approximation for the reversible reaction A+ B⇌ C in solutions.

A many-particle derivation of the Integral Encounter Theory non-Markovian kinetic equations of the reversible reaction A+ B⇄ C in solutions

The non-Markovian binary kinetic equations for a rather realistic model of association–dissociation reaction A+ B⇄ C in liquid solutions are derived using a newly developed method based on a many-particle consideration. The method leads to an infinite hierarchy for correlation patterns that can be truncated at the first step on a two-particle level to give the Integral Encounter Theory result. It is shown that although being restricted in time, the theory predicts true equilibrium concentrations of reactants and the equilibrium constant corresponding to the thermodynamic detailed balancing principle.

General kinetic laws of dissociation and reversible reaction A+ B⇄ C in solutions

Based on the non-Markovian kinetic equations of the reversible reaction A+ B⇄ C derived earlier, we have obtained the non-Markovian kinetic equations describing the reaction under consideration on macroscopic space–time scales. They define the non-Markovian kinetics of the systems with different structure of reactants in a universal fashion, and are the direct generalization of the kinetic law of mass action of formal chemical kinetics. The analysis of their solutions has led to the establishment of the time range of the applicability of the law of mass action. Beyond these limits the law of mass action becomes invalid, and correct description of the kinetics even in the most rough approximation calls for the use of the non-Markovian theory.

Non-Markovian kinetic equations of dissociation and reversible reaction A+ B⇄ C in solutions

A consistent many-particle approach has been used to derive integro-differential kinetic equations of a bulk reaction of dissociation in spatially non-uniform solutions in a hydrodynamic approximation with a realistic description of reactants structure and translational mobility. The detailed analysis of the memory function structure of this equation has been carried out, and a microscopic detailed balancing principle of the theory of migration accelerated reactions has been formulated. This has made it possible to formulate a macroscopic kinetic detailed balancing principle for non-Markovian dissociation and recombination kinetics. The non-Markovian kinetic equation of the reversible reaction A+ B⇄ C in spatially non-uniform systems is proposed. Thermodynamic detailed balancing principle is established.

Kinetic Theory of a Weakly-Nonideal Spatially Nonuniform Polarizable Plasma

A nonmarkovian and nonlocal kinetic equation, which is a generalization of the (nonlinear) Balescu-Lenard equation, is derived for a weakly nonuniform multicomponent polarizable plasma. A specific expansion of the plasma resolvent, which allow to separate the dymamic, kinetic and hydrodynamic scales, is proposed. Explicit expressions of the collision integral and of the nonequilibrium pair correlation function are given in an approximation which take into account the effect of patial and temporal nonlocality a well as of the polarization. Balance equation for the momentum and energy densitie are calculated in the first order in nonlocality, and potential contributions to the fluxes due to polarization are obtained.

Retarded versus Time-Nonlocal Quantum Kinetic Equations

The finite duration of the collisions in Fermionic systems as expressed by the retardation time in non-Markovian Levinson-type kinetic equations is discussed in the quasiclassical limit. We separate individual contributions included in the memory effect resulting in (i) off-shell tails of the Wigner distribution, (ii) renormalization of scattering rates, (iii) renormalization of the single-particle energy, (iv) collision delay, and (v) related nonlocal corrections to the scattering integral. In this way we transform the Levinson equation into the Landau–Silin equation extended by the nonlocal corrections known from the theory of dense gases. The derived nonlocal kinetic equation unifies the Landau theory of quasiparticle transport with the classical kinetic theory of dense gases. The space-time symmetry is discussed versus particle-hole symmetry and a solution is proposed which transforms these two exclusive pictures into each other.

Short-time dynamics with initial correlations.

The short-time dynamics of correlated systems is strongly influenced by initial correlations, giving rise to an additional collision integral in the non-Markovian kinetic equation. Exact cancellation of the two integrals is found if the initial state is thermal equilibrium, which is an important consistency criterion. Analytical results are given for the time evolution of the correlation energy, which are confirmed by comparisons with molecular dynamics simulations.

Consideration of initial correlations of the same type in reactions A+ B→ B and A+ B→ C+ B

A new method is used to derive the binary non-markovian integro-differential kinetic equations of irreversible reactions with respect to initial correlations of the same type. The source of inhomogeneous integro-differential equation is determined by initial correlations which is in full accord with the properties of kinetic equations of non-equilibrium statistical mechanics. A procedure is found to reduce the derived equations into the differential form of the equations of chemical kinetics.

The effect of chemical displacement of B species in the reaction A+ B→ B

The non-Markovian binary kinetic equations of the irreversible reaction A+ B→ B taking into account B reactant displacement due to chemical conversion events have been derived for the first time. The derivation is based on the many-particle closure method recently proposed by the authors. It is shown that a conventional view of the reaction A+ B→ B as a pseudomonomolecular reaction is true only for spatially uniform systems. In the case of spatially non-uniform systems additional macroscopic flows arise. They can transform an originally uniform distribution of one of the species into a non-uniform one in the course of the reaction.

The “Mixed” Green's Function Approach to Quantum Kinetics with Initial Correlations

A method for deriving quantum kinetic equations with initial correlations is developed on the basis of the nonequilibrium Green's function formalism. The method is applicable to a wide range of correlated initial states described by nonequilibrium statistical thermodynamics. Initial correlations and the real-time evolution are treated by a unified technique employing many-component “mixed” Green's functions. The Dyson equation for the mixed Green's function leads to a set of equations for real-time Green's functions and new (cross) components linking initial correlations with dynamical processes. These equations are used to formulate a generalized Kadanoff–Baym ansatz for correlated initial states. A non-Markovian short-time kinetic equation is derived within the T-matrix approximation for the self-energies. The properties of the memory kernels in this equation are considered in detail in Born approximation for the T-matrices. The kinetic equation is demonstrated to conserve the total energy of the system. An explicit expression for the time-dependent correlation energy is obtained.

Quantum kinetics and thermalization in a particle bath model.

We study the dynamics of relaxation and thermalization in an exactly solvable model of a particle interacting with a harmonic oscillator bath. Our goal is to understand the effects of non-Markovian processes on the relaxational dynamics and to compare the exact evolution of the distribution function with approximate Markovian and non-Markovian quantum kinetics. There are two different cases that are studied in detail: (i) a quasiparticle (resonance) when the renormalized frequency of the particle is above the frequency threshold of the bath and (ii) a stable renormalized "particle" state below this threshold. The time evolution of the occupation number for the particle is evaluated exactly using different approaches that yield to complementary insights. The exact solution allows us to investigate the concept of the formation time of a quasiparticle and to study the difference between the relaxation of the distribution of bare particles and that of quasiparticles. For the case of quasiparticles, the exact occupation number asymptotically tends to a statistical equilibrium distribution that differs from a simple Bose-Einstein form as a result of off-shell processes whereas in the stable particle case, the distribution of particles does not thermalize with the bath. We derive a non-Markovian quantum kinetic equation which resums the perturbative series and includes off-shell effects. A Markovian approximation that includes off-shell contributions and the usual Boltzmann equation (energy conserving) are obtained from the quantum kinetic equation in the limit of wide separation of time scales upon different coarse-graining assumptions. The relaxational dynamics predicted by the non-Markovian, Markovian, and Boltzmann approximations are compared to the exact result. The Boltzmann approach is seen to fail in the case of wide resonances and when threshold and renormalization effects are important.

A scaling procedure in a many-particle derivation of the non-Markovian binary kinetic equations of the reaction A+ B→ B in liquid solutions

A mathematically rigorous scaling procedure for the derivation of binary non-Markovian kinetic equations has been developed. Though the procedure refers to the irreversible reaction A+ B→ B, it can serve as a reliable basis for selecting diagrams in regular many-particle derivation of non-Markovian kinetic equations for other types of elementary chemical reactions in liquid solutions.

Many-particle treatment of nonuniform reacting systems A+ B→ C and A+ B→ C+ D in liquid solutions

The recently developed method of deriving binary non-Markovian (non-stationary) kinetic equations of reactions in liquid solutions has been applied to irreversible reactions A+ B→ C and A+ B→ C+ D in spatially nonuniform systems. The method is based on the derivation and subsequent closure of hierarchies for correlation forms. Physically clear kinetic equations for local concentrations of reactants showing that the method can be applied to a wide class of the reacting systems have been obtained in the hydrodynamic approximation.

A new approach to the derivation of binary non-Markovian kinetic equations

A universal method of derivation of infinite hierarchies for partial distribution functions and correlation forms in the thermodynamic limit has been developed. It is based on the consideration of reacting systems in the Fock space. Hierarchy closure methods available in the literature are shown to give incorrect binary kinetic equations of the reaction A+ B→ B in some critical cases. A new approach to hierarchy closure has been proposed. It consists in neglecting contributions from four-particle correlations and in adapting the Faddeev method of the three-body theory to the extraction of a binary part of three-particle evolution. For the model of the reaction A+ B→ B the proposed method gives correct kinetic equations obtained earlier on the basis of diagram summation. It gives the theoretical basis for derivation of binary kinetic equations for realistic reacting systems.

Non-Markovian effects in strong-field pair creation

We analyze a quantum kinetic equation describing both boson and fermion pair production and explore analytically and numerically the solution of the non-Markovian kinetic equation. In the Markovian limit of the kinetic equation we find an analytical solution for the single particle distribution function of bosons and fermions. The numerical investigation for a homogeneous, constant electric field shows an enhancement (bosons) or a suppression (fermions) of the pair creation rate according to the symmetry character of the produced particles. For strong fields non-Markovian effects are important while they disappear for weak fields. Hence it is sufficient to apply the low density limit for weak fields but necessary to take into account memory effects for strong fields.

Quantum kinetics and thermalization in an exactly solvable model

We study the dynamics of relaxation and thermalization in an exactly solvable model with the goal of understanding the effects of off-shell processes. The focus is to compare the exact evolution of the distribution function with different approximations to the relaxational dynamics: Boltzmann, non-Markovian and Markovian quantum kinetics. The time evolution of the distribution function is evaluated exactly using two methods: time evolution of an initially prepared density matrix and by solving the Heisenberg equations of motion. There are two different cases that are studied in detail: i) no stable particle states below threshold of the bath and a quasiparticle resonance above it and ii) a stable discrete exact `particle' state below threshold. For the case of quasiparticles in the continuum (resonances) the exact quasiparticle distribution asymptotically tends to a statistical equilibrium distribution that differs from a simple Bose-Einstein form as a result of off-shell processes. In the case ii), the distribution of particles does not thermalize with the bath. We study the kinetics of thermalization and relaxation by deriving a non-Markovian quantum kinetic equation which resums the perturbative series and includes off-shell effects. A Markovian approximation that includes off-shell contributions and the usual Boltzmann equation are obtained from the quantum kinetic equation in the limit of wide separation of time scales upon different coarse-graining assumptions. The relaxational dynamics predicted by the non-Markovian, Markovian and Boltzmann approximations are compared to the exact result of the model. The Boltzmann approach is seen to fail in the case of wide resonances and when threshold and renormalization effects are important.

Damping rates of hot giant dipole resonances

The damping rate of hot giant dipole resonances (GDRs) is investigated. Besides Landau damping we consider collisions and density fluctuations as contributions to the damping of GDRs. Within the nonequilibrium Green's function method we derive a non-Markovian kinetic equation. The linearization of the latter one leads to complex dispersion relations. The complex solution provides the centroid energy and the damping width of giant resonances. The experimental damping widths are the full width half maximum and can be reproduced by the full width of the structure function. Within simple finite size scaling we give a relation between the minimal interaction strength which is required for a collective oscillation and the cluster size. We investigate the damping of giant dipole resonances within a Skyrme type of interaction. Different collision integrals are compared with each other in order to incorporate correlations. The inclusion of a conserving relaxation time approximation allows us to find the ${T}^{2}$ dependence of the damping rate with a temperature known from Fermi-liquid theory. However, memory effects turn out to be essential for a proper treatment of the damping of collective modes. We derive a Landau-like formula for the one-particle relaxation time similar to the damping of zero sound.

A many-particle approach to the derivation of binary non-Markovian kinetic equations for the reaction A+B→B

Many-particle methods of physical kinetics have been adapted to a many-particle consideration of irreversible reactions A+B→B in liquid solutions. The correctness of the diagrammatic approach developed is demonstrated using the exactly solvable problem. Appropriate scaling procedure has been devised for the extraction of a binary kinetics. On its basis, the conception of pair uncorrelated encounters of reactants in liquid solutions has been generalized, and the concept of the effective pair has been introduced. Binary non-Markovian kinetic equations of the reaction for uniform initial distribution of B species obtained earlier are reproduced. For the first time, integro-differential kinetic equations of the reaction in spatially non-uniform normal systems have been derived.

Quasihydrodynamic approximation for memory functions in non-Markovian relaxation processes in condensed matter

A quasihydrodynamic approximation is introduced for senior memory functions in the chain of non-Markovian kinetic equations for time correlation functions in many-body systems. This approximation allows to describe complicated high-frequency and short wavelength spectra for a wide range of phenomena: from the molecular scattering and the scattering of slow neutrons in liquids to the description of the decay of magnetic induction signal in crystals.

Correlation time approximation in non-Markovian kinetics

A new approximation to kinetic equations is derived which has a broad range of applicability. It allows one to take into account dominant memory and correlation effects while solving only Markovian instead of complicated non-Markovian kinetic equations. It furthermore yields a drastic simplification of kinetic equations for spatially inhomogeneous systems. The approximation is illustrated numerically on examples based on the quantum Boltzmann equation with, respectively, a static and a dynamically screened interaction potential (Lenard-Balescu equation).