non-Markovian Kinetic Equations Research Articles

Dynamic Correlations in Open Quantum Systems: The Dephasing Model

We study the dynamical correlations in open quantum systems on an example of an exactly solvable dephasing model. The system of non-Markovian kinetic equations for the generalized coherence and quasi-temperature is derived up to the 2-nd order in the coupling constant using the generalized quantum master equation [20]. Numerical and analytical solutions of the kinetic equations are obtained. The analytical result shows a redistribution of the real and imaginary parts of the coherence as compared to the exact one. We give a physical interpretation to the quasi-temperature, relating it to the nonequilibrium kinetic energy of the environment.

Немарковське кінетичне рівняння Фоккера–Планка для системи твердих куль

Немарковське кінетичне рівняння Фоккера–Планка для системи твердих куль

Analysis of the methods for the derivation of binary kinetic equations in the theory of fluorescence concentration quenching

In the framework of unified many-particle approach the familiar problem of fluorescence concentration quenching in the presence of pumping (light pulse) of arbitrary intensity is considered. This process is a vivid and the simplest example of multistage bulk reaction including bimolecular irreversible quenching reaction and reversible monomolecular transformation as elementary stages. General relation between the kinetics of multistage bulk reaction and that of the elementary stage of quenching has been established. This allows one to derive general kinetic equations (of two types) for the multistage reaction in question on the basis of general kinetic equations (differential and integro-differential) of elementary stage of quenching. Relying on the same unified many-particle approach we have developed binary approximations with the use of two (frequently employed in the literature) many-particle methods (such as simple superposition approximation and the method of extracting pair channels in three-particle correlation evolution) to the derivation of non-Markovian binary kinetic equations. The possibility of reducing the obtained binary equations to the Markovian equations of formal chemical kinetics has been considered. As an example the exact solution of the problem (for the specific case) is examined, and the applicability of two many particle methods of derivation of binary equations is analyzed.

On the non-Markovian Enskog equation for granular gases

We develop a rigorous formalism for the description of the kinetic evolution of many-particle systems with dissipative interaction. The links of the evolution of a hard sphere system with inelastic collisions described within the framework of marginal observables governed by the dual Bogolyubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy and the evolution of states described by the Cauchy problem of the Enskog kinetic equation for granular gases are established. Moreover, we consider the Boltzmann–Grad asymptotic behavior of the constructed non-Markovian Enskog kinetic equation for granular gases in a one-dimensional space.

Influence of the force interaction on accumulation of macroscopic correlations in elementary 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 taking account of initial microscopic correlations and force interactions between reactants has been treated theoretically with modern many-particle method for the derivation of non-Markovian binary kinetic equations. The method shows that taking into consideration initial correlations and force interactions leads to the redefinition of the Markovian rate constant only in the expressions derived earlier. Thus, just as in the reaction A + A →C and the reaction A + B →C neglecting force and initial correlations, the Modified Encounter Theory (MET), when reduced to equations of a Regular Form, both extends the time applicability range of ET homogeneous rate equation, and yields the inhomogeneous equation of the Generalized Encounter Theory (GET). It reveals macroscopic correlations induced by the encounters in the reservoir of free walks in full agreement with physical considerations. Time accumulation of macroscopic correlations obeys the same time law as in the previously considered case neglecting force interactions. Just the rate of the process will change, according to traditional redefinition of the steady-state constant of the reaction.

Proton-driven spin diffusion in rotating solids via reversible and irreversible quantum dynamics

Proton-driven spin diffusion (PDSD) experiments in rotating solids have received a great deal of attention as a potential source of distance constraints in large biomolecules. However, the quantitative relationship between the molecular structure and observed spin diffusion has remained obscure due to the lack of an accurate theoretical description of the spin dynamics in these experiments. We start with presenting a detailed relaxation theory of PDSD in rotating solids that provides such a description. The theory applies to both conventional and radio-frequency-assisted PDSD experiments and extends to the non-Markovian regime to include such phenomena as rotational resonance (R(2)). The basic kinetic equation of the theory in the non-Markovian regime has the form of a memory function equation, with the role of the memory function played by the correlation function. The key assumption used in the derivation of this equation expresses the intuitive notion of the irreversible dissipation of coherences in macroscopic systems. Accurate expressions for the correlation functions and for the spin diffusion constants are given. The theory predicts that the spin diffusion constants governing the multi-site PDSD can be approximated by the constants observed in the two-site diffusion. Direct numerical simulations of PDSD dynamics via reversible Liouville-von Neumann equation are presented to support and compliment the theory. Remarkably, an exponential decay of the difference magnetization can be observed in such simulations in systems consisting of only 12 spins. This is a unique example of a real physical system whose typically macroscopic and apparently irreversible behavior can be traced via reversible microscopic dynamics. An accurate value for the spin diffusion constant can be usually obtained through direct simulations of PDSD in systems consisting of two (13)C nuclei and about ten (1)H nuclei from their nearest environment. Spin diffusion constants computed by this method are in excellent agreement with the spin diffusion constants obtained through equations given by the relaxation theory of PDSD. The constants resulting from these two approaches were also in excellent agreement with the results of 2D rotary resonance recoupling proton-driven spin diffusion (R(3)-PDSD) experiments performed in three model compounds, where magnetization exchange occurred over distances up to 4.9 Å. With the methodology presented, highly accurate internuclear distances can be extracted from such data. Relayed transfer of magnetization between distant nuclei appears to be the main (and apparently resolvable) source of uncertainty in such measurements. The non-Markovian kinetic equation was applied to the analysis of the R(2) spin dynamics. The conventional semi-phenomenological treatment of relxation in R(2) has been shown to be equivalent to the assumption of the Lorentzian spectral density function in the relaxatoin theory of PDSD. As this assumption is a poor approximation in real physical systems, the conventional R(2) treatment is likely to carry a significant model error that has not been recognized previously. The relaxation theory of PDSD appears to provide an accurate, parameter-free alternative. Predictions of this theory agreed well with the full quantum mechanical simulations of the R(2) dynamics in the few simple model systems we considered.

Non-Markovian relaxation of atomic systems and a calculation of the shape of spectral lines

A non-Markovian kinetic equation for a system of two identical interacting two-level atoms has been derived. The solution to this equation has been used for calculating the shape of spectral lines of this system.

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.

Kinetic Equations for Local Concentrations of Reactants in Spatially Inhomogeneous Solutions

Spatially inhomogeneous dilute solutions of reactants placed in a continual solvent (Waite approach) are examined. Based on the familiar non-Markovian kinetic equations of the Integral Encounter Theory, the Markovian kinetic equations were first obtained that allowed for the force action of reactants on each other due to binary encounters in solution during reactant macroscopic displacement. It has been shown that the value of “collisional integral” of reactants determined by their stochastic motion is small as compared to the “collisional integral” with the solvent molecules in the frame of the binary approach to the description of the solvent evolution. Nevertheless, taking account of the encounters of reactants with each other may turn out to be of primary importance, since it results in new physical effects.

Kinetics of dense matter: Correlations and memory

A link between correlation contributions to kinetic equations for dense quantum systems and the energy conservation is considered. In order that the energy be conserved by an approximate collision integral, the one-particle density matrix and the mean interaction energy are treated as independent state parameters. It is shown how the density operator method can be used to derive non-Markovian kinetic equations including the correlation effect associated with the energy conservation. A quantum generalization of the Enskog theory is discussed.

Inertial mechanism: Dynamical mass as a source of particle creation

A kinetic theory of vacuum particle creation under the action of an inertial mechanism is constructed within a nonperturbative dynamical approach. At the semi-phenomenological level, the inertial mechanism corresponds to quantum field theory with a time-dependent mass. At the microscopic level, such a dependence may be caused for different reasons: the non-stationary Higgs mechanism, the influence of a mean field or condensate, the presence of a conformal multiplier in the scalar-tensor gravitation theory, etc. In what follows, a kinetic theory in the collisionless approximation is developed for scalar, spinor, and massive vector fields in the framework of the oscillator representation, which is an effective tool for transition to the quasiparticle description and for derivation of non-Markovian kinetic equations. Properties of these equations and relevant observables (particle number and energy densities, pressure) are studied. The developed theory is applied here to describe the vacuum matter creation in conformal cosmological models and explain the observed number density of photons in the cosmic microwave background radiation. As another example, the self-consistent evolution of scalar fields with non-monotonic self-interaction potentials (the W-potential and Witten-Di Vecchia-Veneziano model) is considered. In particular, conditions for the appearance of tachyonic modes and the problem of the relevant definition of a vacuum state are discussed.

Pair production beyond the Schwinger formula in time-dependent electric fields

We investigate electron-positron pair production in pulse-shaped electric background fields using a non-Markovian quantum kinetic equation. We identify a pulse length range for subcritical fields still in the nonperturbative regime where the number of produced pairs significantly exceeds that of a naive expectation based on the Schwinger formula. From a conceptual viewpoint, we find a remarkable quantitative agreement between the (real time) quantum kinetic approach and the (imaginary time) effective action approach.

Hamiltonian and Brownian systems with long-range interactions: IV. General kinetic equations from the quasilinear theory

We develop the kinetic theory of Hamiltonian systems with weak long-range interactions. Starting from the Klimontovich equation and using a quasilinear theory, we obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. This equation is valid at order 1 / N in a proper thermodynamic limit and it coincides with the kinetic equation obtained from the BBGKY hierarchy. For N → + ∞ , it reduces to the Vlasov equation governing collisionless systems. We describe the process of phase mixing and violent relaxation leading to the formation of a quasistationary state (QSS) on the coarse-grained scale. We interpret the physical nature of the QSS in relation to Lynden-Bell’s statistical theory and discuss the problem of incomplete relaxation. In the second part of the paper, we consider the relaxation of a test particle in a thermal bath. We derive a Fokker–Planck equation by directly calculating the diffusion tensor and the friction force from the Klimontovich equation. We give general expressions of these quantities that are valid for possibly spatially inhomogeneous systems with long correlation time. We show that the diffusion and friction terms have a very similar structure given by a sort of generalized Kubo formula. We also obtain non-Markovian kinetic equations that can be relevant when the auto-correlation function of the force decreases slowly with time. An interesting factor in our approach is the development of a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems.

A hard sphere model of chemical reactions theory from the point of view of many-particle approach

On the basis of the previously obtained non-Markovian kinetic equation for irreversible chemical reaction A+B→C+B, we have examined a model of contact reaction between hard (impenetrable) spheres (singular force potential). The nature of the additional contribution to the rate constant is explained. As is found, for reactants with a hard core it is necessary to allow for the presence of initial correlations between them. A class of initial conditions for arbitrary form of the pair potential energy is established.

Consistent account of force correlations in a many-particle derivation of non-Markovian kinetic equations for chemical reactions in solutions

The previously proposed many-particle method for the derivation of kinetic equations of chemical reactions in liquid solutions is applied to the consideration of irreversible reaction A + B → C + B . Both force and chemical space correlations resulting from the force interactions between all reactants and chemical reaction between A and B species are taken into account on the same basis. Examination of chemical and force interactions in the system of reactants shows that their contributions to the time evolution of correlation patterns differ essentially because of different analytical properties of the above interactions. It is shown that proper account of the force interaction between reactants corresponds to the change in the character of a relative mobility in the ultimate evolution of a reaction pair (ultimate propagator), not only in the process of correlation accumulation in effective pairs, as is sometimes assumed.

Deviation from the kinetic law of mass action for reactions induced by binary encounters inliquid solutions

In considering the irreversible chemical reaction in liquid solutions two many-particle approaches to the derivation of binarynon-Markovian kinetic equations are compared: simple superposition decoupling and amethod of extracting ‘pair’ channels from three-particle correlation evolution. It isshown that both methods provide an almost identical description of this reaction.However, in studies of reversible reactions in liquid solutions only the channel extractionmethod gives a correct physically clear description of the reaction though it consists of asequence of steps: the development of integral encounter theory (IET), effectivepairs approximation (EPA), modified encounter theory (MET), and the finalregular form (RF) of kinetic equations. It is shown that the rate equations oftenencountered in the literature correspond to the independence of transient channels of‘scattering’ in the bimolecular reversible reaction (), while the independent transient channel of ‘decay’ in the reversible reaction is defined solely by time integral convolution. In the general case transient channels innon-Markovian theory are not independent, and their interference manifests itself as anon-Markovian inhomogeneous source in binary non-Markovian kinetic equations in regularform.Based on the derived equations new universal kinetics (independent of models) of chemicalequilibrium attainment have been obtained. It is shown that these kinetics can differessentially from the kinetics corresponding to the kinetic law of mass action of formalchemical kinetics.

General kinetic laws of monomolecular-bimolecular reaction A+B C in solutions.

Non-Markovian kinetic equations of the reversible monomolecular-bimolecular reactions of the type A+B right arrow over left arrow C (at arbitrary ratio between A and B concentrations) derived earlier are used in the calculation of kinetics on macroscopic space-time scales. It is found that the kinetics of the systems with different structure of reactants is universal, and it is the direct generalization of the kinetic law of mass action of formal chemical kinetics. The analysis of the kinetics allows one to establish the time range of the applicability of the law of mass action. It is shown that beyond these limits the usual kinetic law of mass action becomes invalid, and correct description of the kinetics even in the most rough approximation calls for the non-Markovian corrections to usual kinetic laws.

Ultrafast dynamics of laser-pulse excited semiconductors: non-Markovian quantum kinetic equations with nonequilibrium correlations

Ultrafast dynamics of laser-pulse excited semiconductors: non-Markovian quantum kinetic equations with nonequilibrium correlations

Magnetotransport in the two-dimensional Lorentz model: non-markovian Grad limit of the BBGKY hierarchy

We consider the classical two-dimensional Lorentz model of non-interacting charged point particles acted upon by a perpendicular uniform magnetic field. Particles are elastically scattered by hard disks randomly distributed in the motion plane. Trajectories of gas particles between collisions are arcs of cyclotron circles. We consider the Grad limit of the corresponding BBGKY hierarchy. The Lorentz model in magnetic field provides an example where the Grad limit of the solution for the one-particle density cannot be found by formally applying the limit to evolution equations. The resulting non-markovian kinetic equation is shown to coincide with the generalization of the Boltzmann equation originally proposed upon intuitive arguments.