Showing papers on "Master equation published in 1979"
TL;DR: In this paper, the authors examined the transport of electronic excitations between randomly distributed sites and showed that transport is nondiffusive at short times and diffusive at long times, and that the time regime in which diffusive transport occurs is dependent on density.
Abstract: The transport of electronic excitations between randomly distributed sites is examined. The Green function solution to the master equation is expanded as a diagrammatic series. Topological reduction of the series results in an expression for the Green function which is equivalent in form to the Green function solution of a generalized diffusion equation. The diagrammatic technique used suggests an interesting class of self‐consistent approximations. This self‐consistent method of approximation is applied to the specific case of the Forster transfer rate. The solutions obtained are well‐behaved for all times and all site densities and indicate that transport is nondiffusive at short times and diffusive at long times. The mean squared displacement of the excitation and the time derivative of the mean squared displacement are calculated. These calculations illustrate that the time regime in which diffusive transport occurs is dependent on density. For low density systems transport becomes diffusive only at v...
311 citations
TL;DR: It is shown that under conditions of high excitation intensities, the fluorescence decays approximately according to the (time)1/2 law.
162 citations
15 Feb 1979
TL;DR: In this article, the cumulant expansion is used to derive two formally different master equations for a two-level molecular system interacting with a bath, starting with the two master equations reducing to the same form in the markovian limit for the bath, where its correlation time is much shorter than the relaxation time.
Abstract: The cumulant expansion is used to derive two formally different master equations for a two-level molecular system interacting with a bath, starting wit The two master equations reduce to the same form in the markovian limit for the bath (where its correlation time is much shorter than the relaxation pr A detailed comparison is made between the predictions of the two approaches which enables us to understand their range of validity and limitations. We apply the formalism to the vibrational relaxation and dephasing of a molecular impurity in a solid matrix and obtain a closed expression for the vib In contrast to the simple stochastic approaches we predict that the line shape in the non-markovian limit contains information regarding the interactio However, the fluctuations in the mean interaction energy of the two-level system with the bath, if correlated with the frequency modulation, result in
111 citations
TL;DR: In this article, a comparison of the Fokker-Planck equations obtained by the Ito prescription and by the Stratonovich prescription for physical systems described by a Langevin equation with non-additive fluctuations is presented.
Abstract: We present a comparison of the Fokker-Planck equations obtained by the Ito prescription and by the Stratonovich prescription for physical systems described by a Langevin equation with non-additive fluctuations. Our main conclusion is that the Stratonovich prescription is the one that should always be used to describe physical systems. This conclusion is shown to be consistent with results obtained from path integral and Master equation approaches.
68 citations
TL;DR: In this paper, the authors proposed to treat the charge equilibration process as a collective high frequency mode and discuss the implications for the first stages of a heavy ion collision, and showed how its dynamics can be treated by means of a quantal master equation.
Abstract: We propose to treat the charge equilibration process as a collective high frequency mode and discuss the implications for the first stages of a heavy ion collision. We show how its dynamics can be treated by means of a quantal master equation. We solve numerically the dynamical equations for a two-dimensional model, using charge excess and mass asymmetry as dynamical degrees of freedom.
63 citations
TL;DR: In this article, the authors considered a system perturbed by an external field and subject to dissipative processes and derived an inhomogeneous master equation, i.e., a master equation with dissipative terms and streaming terms, using Zwanzig projection operator technique in Liouville space.
Abstract: We consider a system perturbed by an external field and subject to dissipative processes. From the von Neumann equation for such a system in the weak coupling limit we derive an inhomogeneous master equation, i.e., a master equation with dissipative terms and streaming terms, using Zwanzig’s projection operator technique in Liouville space. From this equation the response function, as well as expressions for the generalized conductivity and susceptibility, is obtained. It is shown that for large times only the diagonal part of the density operator is required. The various expressions are found to be in complete harmony with previous results (Part I) obtained via the van Hove limit of the Kubo–Green linear response formulas. In order to account for the properties at quantum frequencies, the evolution of the nondiagonal part in the weak coupling limit is also established. The complete time dependent behavior of the dynamic variables in the van Hove limit is expressed by B (t) =exp[−(Λd−iL0) t] B, where Λd is the master operator and L0 the Liouville operator in the interaction picture. The cause of irreversibility is discussed. Finally, the inhomogeneous master equation is employed to obtain as first moment equation a Boltzmann equation with streaming terms, applicable to quantum systems.
62 citations
TL;DR: In this paper, the quantal form of the classical Liouville equation is investigated on the basis of a recently developed generalized Hamiltonian theory, which comprises the use of complex classical coordinates and momenta.
Abstract: The quantal form of the classical Liouville equation will be investigated on the basis of a recently developed generalized Hamiltonian theory. The essential novelty inthat theory comprises the use of complex classical coordinates and momenta. We first show how for the nondissipative harmonic oscillator driven by an external classical force, the theory leads to the correct well-known quantum analogue of the classical Liouville equation. We then generalize this procedure to include frictional phenomena for which the novel theory has been observed to be particularly suited. The resulting quantal master equation for the simple linearly damped harmonic oscillator demonstrates that one cannot expect to find a proper quantum mechanical description of dissipative systems in terms of a single Schrodinger wave function. The master equation will then be transformed into its Wigner representation, providing a convinient form for discussion. The diffusion coefficients occuring in the resultant Fokker-Planck equation will be seen to be intimately connected with the survival of Heisenberg's uncertainty principle for dissipative systems. Apart from conceptual elegance, the present approach has superiority to a previous one in at least three aspects: i) there is no need to introduce ad-hoc quantal noise operators, ii) the above mentioned diffusion coefficients are specified and emerge in a natural way, and iii) the present approach has the important advantage of easy extension to more general systems.
50 citations
TL;DR: In this paper, a non-Markovian process can be approximated on a course time scale by a Markov description and the conditions for this approximation to be valid are discussed.
Abstract: Certain problems in physics and chemistry lead to the definition of a class of stochastic processes. Although they are not Markovian they can be treated explicitly to some extent. In particular, the probability distribution for large times can be found. It is shown to obey a master equation. This demonstrates how a non-Markovian process can be approximated on a course time scale by a Markov description. The conditions for this approximation to be valid are discussed.
50 citations
01 Aug 1979
TL;DR: In this article, a model for parametrization of the rate coefficient matrix in the master equation for unimolecular reactions induced by monochromatic infrared radiation (URIMIR) is put forward, which uses only easily measurable molecular parameters.
Abstract: Four master equations are formulated for unimolecular reactions induced by collisions, by blackbody radiation, by intense, incoherent, quasimonochromatic infrared radiation and by coherent, monochromatic infrared radiation. The relationship between these equations is discussed and the fundamental differences which are due to the different structures of the rate coefficient matrix are pointed out. The properties of the solutions of these equations are investigated in some detail for both time dependent and time independent rate coefficient matrices. A systematic method for determining all the characteristic parameters (eigenvalues, characteristic populations and characteristic times) of the master equations from experimental data or numerical calculations is proposed. A model for the parametrization of the rate coefficient matrix in the master equation for unimolecular reactions induced by monochromatic infrared radiation (URIMIR) is put forward, which uses only easily measurable molecular parameters. The implications of this model for spontaneous infrared emission of highly excited polyatomic molecules are considered. The limitations of the use of intensity proportional rate coefficients (phenomenological cross sections) for optical excitation in URIMIR are pointed out. The determination of the steady state characteristics in URIMIR from experimental data is discussed. In particular, a simple, new method for obtaining the steady state rate constant from measurements of the product yield as a function of laser energy fluence is presented and illustrated with the first evaluation of rate constants for the IR-photodissociation of several molecules from published experimental data of other authors. The intensity dependence of the rate constant in URIMIR is predicted to be nonlinear and nontrivial in the limit of both low and high radiation intensities. The reducibility of the rate coefficient matrix for URIMIR is considered and the consequences for the interpretation of experimental data are illustrated.
47 citations
TL;DR: In this paper, the authors show that the uniqueness of the equilibrium distribution as a stationary solution is ensured if the detailed rate constants are balanced with the aid of the distribution which maximizes the entropy subject to the thermodynamic constraints.
45 citations
TL;DR: In this article, an extended time-dependent Hartree-fock approximation with collision-free limit is proposed, which is obtained by a truncation of the Martin-Schwinger hierarchy at the second level and by using a simple representation of the one-body Green's function in terms of timedependent occupation numbers.
Abstract: In the time-dependent Hartree-Fock approximation, the fermions are assumed to interact only through the mean field and the collisions between particles are neglected. We formulate an extended time-dependent Hartree-Fock approximation which incorporates particle collisions due to the residual interaction, with the usual time-dependent Hartree-Fock approximation as the collisionless limit. It is obtained by a truncation of the Martin-Schwinger hierarchy at the second level and by using a simple representation of the one-body Green's function in terms of time-dependent occupation numbers. The final set of coupled equations consists of a modified time-dependent Hartree-Fock equation and a master equation for the occupation numbers. These results are physically transparent and describe properly the physics of the collision process. They may also be simple enough to be of practical use to study heavy-ion collisions or the dynamics of other fermion systems. Furthermore, as the configuration-space analog of the quantum Boltzmann equation, many important results concerning statistical dynamics are obtained. Concepts such as entropy, temperature, and local and thermal equilibrium can be quantitatively introduced. The well-known $H$ theorem that entropy never decreases can be readily recovered. With the collision term explicitly exhibited, the macroscopic equations (equations of continuity, momentum flux, and energy) and their associated conservation theorems can also be derived. Analytic solutions for the master equation for simple cases lead to new "level crossing" formulas having characteristics distinctly different from the Landau-Zener level-crossing formula and illuminate the salient features as to how a nonequilibrium fermion system approaches thermal equilibrium.NUCLEAR REACTIONS Extension of time-dependent Hartree-Fock approximation. Collisions between particles. Master equation for occupation probabilities. Entropy, temperature, and thermal equilibrium. $H$ theorem. Analytic solution of the master equation. New types of level crossing formula.
TL;DR: In this article, a new approximation scheme for solving the multivariate master equation is presented, and the results are compared with those of computer experiments, showing that a very good agreement is obtained, even in the neighbourhood of the critical point where a nonequilibrium phase transition occurs.
TL;DR: In this paper, an energy-grained master equation was used to model the multiphoton pumping process in SF6 and an energydependent absorption cross section that was approx. inversely dependent on the level of excitation was needed to reproduce exptl. data on the fraction decompd.
TL;DR: In this paper, the non-markovian master equations for one time and multi-time correlation functions of an open quantum system are derived to all orders in the interaction with the stochastic perturbations.
Abstract: The stochastic and quantum dynamics of open quantum systems interacting with stochastic perturbations in considered. The master equations for one time and multi-time correlation functions of such a system are derived to all orders in the interaction with the stochastic perturbations. The importance of the non-markovian character of such equations in the study of various problems in optical resonance is discussed. The simplified form of the non-markovian master equations in Born approximation is also given. It is shown that such non-markovian master equations in Born approximation are exact if there is only one random perturbation, of the telegraphic signal type, acting on the system. The master equations for the linear response functions of an open system interacting with stochastic perturbations are also derived. The non-markovian master equations for multitime correlations are used to study the behaviour of two level atoms interacting with fluctuating laser fields. Both amplitude and phase fluctuations are taken into account. Explicit results are presented for the spectrum of resonance fluorescence, absorption spectrum, photon antibunching effects etc. The calculations are done for arbitrary values of the relaxation parameters and intial conditions. In general the fluorescence spectrum is found to be asymmetric for off resonant fields.
TL;DR: In this article, Monte Carlo quasiclassical trajectory calculations were used to calculate the rate matrix for vibrational state changes and dissociation from individual vibrational levels in H 2 -Ar collissions at 4500 K.
TL;DR: In this paper, the authors report the results of a generalization of the continuous-time random walk theory of Montroll and Weiss to include correlations over two jumps and discuss the ease of tetrahedral interstitial sites in body centered cubic lattices.
TL;DR: In this article, a dynamical real-space renormalization group transformation for the master equation of a kinetic Ising model in one dimension is proposed, where the transformation can be expressed as a group transformation.
Abstract: We define and solve a dynamical real-space renormalization group transformation for the master equation of a kinetic Ising model in one dimension.
TL;DR: In this paper, a spin-1 Ising system with dipolar and quadrupolar interactions is placed in contact with a phonon heat bath, and a master equation on the time scale of one microscopic event is derived from the Liouville equation.
Abstract: A spin‐1 Ising system with dipolar and quadrupolar interactions is placed in contact with a phonon heat bath. Using the projection operator formalism of Zwanzig and Nakajima, a master equation on the time scale of one microscopic event is derived from the Liouville equation. The transition rates thus obtained depend upon the detailed dynamics of the system, bath, and interactions between the two. Coarse‐graining removes these details and yields a phenomenological transition rate similar to the form proposed by Langer. The kinetic equations which describe the most and least likely temporal evolution of the system are then derived. Finally, the time dependence of the two order parameters which characterize the system is examined numerically.
TL;DR: In this article, the authors explore the dynamics of triplet electronic energy transfer in an impurity band of a substitutionally-disordered material, and derive approximate results for the average density of excitation within the framework of the pair-approximation.
TL;DR: In this article, the master kinetic equation without a priori random phase assumptions is derived in the quasiclassical approximation for the quantum K-systems, and it is shown how the noniagonal elements of density matrix decay and the memory about initial conditions vanishes.
Abstract: Quantum K -systems can usually be regarded as the systems which are conventional K -systems at ℏ = 0, i.e. they have the property of mixing trajectories in a phase space. The master kinetic equation without a priori random phase assumptions is derived in the quasiclassical approximation for the quantum K -systems. It is shown how the nondiagonal elements of density matrix decay and the memory about initial conditions vanishes. A quantum nonlinear oscilator perturbed by a periodically time-dependent field is considered as an example.
TL;DR: In this article, the authors extended the concept of distributive process to the case of energy quanta exchanged similarly between discrete, degenerate internal energy levels, and showed that the relaxation times of the discrete models prove to be identical with those of their continuous counterparts and the autocorrelation functions for equilibrium fluctuations are likewise of strictly exponential type.
Abstract: The concept of distributive process, recently developed as a description of systems which evolve through ‘random’ energy-transfer in binary collision complexes4) is here extended to the case of energy quanta exchanged similarly between discrete, degenerate internal energy-levels. A corresponding exact solution of the Master Equation is forthcoming, in which the eigenvectors are now orthogonal polynomials of the discrete variable, viz. the Meixner and Hahn types. The relaxation times of the discrete models prove to be identical with those of their continuous counterparts and the autocorrelation functions for equilibrium fluctuations are likewise of strictly exponential type. All results tend naturally to their continuous analogues as the quantity (hv/kBT) for the heat-bath tends to zero.
A number of aspects of mathematical interest are pointed out. For example, when considering the spectral representation of the transition matrices for the discrete distributive processes we arrive inter alia at previously unknown Erdelyi-type bilinear expansion formulae for the Meixner and Hahn systems and thence a stochastic interpretation of these. Another point of interest is the occurrence of fractional sum and difference-operators, this giving a rare example of the fractional calculus in statistical physics.
Apart from furnishing several exactly soluble Master equations on an infinite, discrete state-space, the ‘distributive’ transition probabilities would seem to be the first known examples, as yet, of transition probabilities which allow the consistent representation of multiple degrees of freedom in the transfer of vibrational energy between polyatomic molecules.
TL;DR: In this paper, a molecular dynamics technique is presented which shows that in nonequilibrium chemical systems short range fluctuations are Poisson type and long range fluctuations can be predicted by a global master equation.
TL;DR: In this article, the authors considered the problem of two-photon transitions between atomic levels having the same parity and derived a Markovian master equation for the probability function, which may be solved exactly in the steady state.
Abstract: We consider the problem of two-photon transitions between atomic levels having the same parity. The semiclassical theory is developed from Maxwell's equations and the resulting transcendental equations for the pulse envelope solved exactly. They lead to an unstable pulse envelope if the higher-order processes are included. We then treat the laser as a birth/death process, deriving a Markovian (macroscopic) master equation for the probability function, which may be solved exactly in the steady state. By a suitable truncation procedure, this equation predicts a stable steady-state envelope, and is consistent with the microscopic quantum theory when virtual processes are neglected. A comparison is made between the macroscopic (master equation) approach and the microscopic quantum theory at the level of the moment equations. We find that, in the high-gain limit, the fluctuations predicted by the two theories are the same.
TL;DR: In this article, a ladder-climbing model of the dissociation of a diatomic molecule diluted in a heat bath is formulated and solved analytically for a pseudo-second-order dissociation rate in terms of the energy level structure and collisional transition probabilities.
Abstract: The master equation is formulated and solved analytically for a ladder-climbing model of the dissociation of a diatomic molecule diluted in a heat bath. The dissociation rate is explicitly related to the normal modes of relaxation of the internal degrees of freedom of the molecule giving, in closed form, an expression for the pseudo-second-order dissociation rate in terms of the energy-level structure and collisional transition probabilities. This form of the rate constant shows clearly the criteria for the occurrence of ''bottleneck'' and ''network'' effects. The dissociation of H/sub 2/ in Ar is studied in detail between 2000 and 6000/sup 0/K. The principal qualitative conclusions in respect of the rate of dissociation are that (a) nonequilibrium effects are significant even at 2000/sup 0/K; (b) rotation enhances the reaction rate, although it is only at very high temperatures (above 6000/sup 0/K) that rotational energy contributes equally with vibrational energy in causing dissociation; (c) the presence of tunneling gives only a very modest enhancement in the rate. The relative contributions to the Arrhenius temperature coefficient of the rate of the following theoretical constructs are delineated. A parallel set of calculations on the dissociation of D/sub 2/ in Ar also gives results in goodmore » agreement with experiment. The principal determinants of the kinetic isotope effect are the differences between H/sub 2/ and D/sub 2/ in the internal relaxation rates and in the energy-level densities. 7 figures, 83 references, 7 tables.« less
TL;DR: In this article, the authors presented the continued fraction solution for the stationary probability of discrete master equations of one-variable processes with at least two-particle jumps and showed that these processes do in general not obey a detailed balance condition.
Abstract: We present the continued fraction solution for the stationary probability of discrete master equations of one-variable processes. After we elucidate the method for simple birth and death processes we focus the study on processes which introduce at least two-particle jumps. Consequently, these processes do in general not obey a detailed balance condition. The outlined method applies as well to solutions of eigenmodes of the stochastic operator. Further we derive explicit continued fraction solutions for the Laplace transform of conditional probabilities. All the various continued fraction coefficients are given directly in terms of the transition rates and they obey recursion relations. The method is illustrated for the stationary solution of a simple nonlinear chemical reaction scheme originated by Nicolis.
TL;DR: In this paper, a method is presented for identifying the quasi-stable states of a simple class of spatially homogeneous, nonlinear, nonequilibrium chemical systems, and numerically calculating the associated mean transition times, mean fluctuation periods and effective fluctuation ranges.
Abstract: A method is presented for identifying the quasi-stable states of a simple class of spatially homogeneous, nonlinear, nonequilibrium chemical systems, and for numerically calculating the associated mean transition times, mean fluctuation periods and effective fluctuation ranges. The method of analysis is based on a “stochastic simulation” approach instead of a “master equation” approach, and it therefore focuses on the behavior of a typical individual system instead of on the collective behavior of a statistical ensemble of systems. Results of explicit calculations are presented for a model set of reactions proposed by Schlogl, and some clarification is achieved regarding hysteresis effects and the effects of an absorbing null state.
TL;DR: In this article, the Mori-Zwanzig formalism is applied to analyze the dynamics of the homogeneous (zero-dimensional) Brusselator and a comprehensive description of particle number correlations is given below, near and above the hard instability.
Abstract: The Mori-Zwanzig formalism is applied to analyze the dynamics of the homogeneous (zero-dimensional) Brusselator. A comprehensive description of particle number correlations is given below, near and above the hard instability. The static correlations which must be calculated beforehand, are obtained a) by explicitly solving the master equation (small system size), b) from Monte Carlo simulations (medium system size), c) from a Fokker-Planck-Landau approximation (large system size). The dynamic correlations are characterized by a bare relaxation matrix and a memory kernel. Below the transition the memory effects are small, and the relaxation behaviour can be expressed in terms of second order static correlations. In the limit cycle regime, the bare oscillation period increases with the relative excess of the staticY- over theX-autocorrelations while the decay constant decreases both with the distance from the transition and the system size, indicating the existence of order. The memory effects are generally inversely proportional to the system size. Their main effect for large systems is therefore to modify the decay rate above the transition (the ensemble dephasing).
21 Mar 1979
TL;DR: In the special case of ionic movement fast compared with the channel open-closed kinetics the results agree with those derived from the usual Master equation approach to electric fluctuations in nerve membrane channels.
Abstract: A theoretical approach to transport noise in kinetic systems, which has recently been developed, is applied to electric fluctuations around steady-states in membrane channels with different conductance states. The channel kinetics may be simple two state (open-closed) kinetics or more complicated. The membrane channel is considered as a sequence of binding sites separated by energy barriers over which the ions have to jump according to the usual single-file diffusion model. For simplicity the channels are assumed to act independently. In the special case of ionic movement fast compared with the channel open-closed kinetics the results agree with those derived from the usual Master equation approach to electric fluctuations in nerve membrane channels.
TL;DR: A general master equation for the linearly damped harmonic oscillator is investigated in this article, and an extended version of Hasse's existence condition for a pure state representation by means of a nonlinear frictional Schrodinger equation with a nonhermitian, but norm conserving hamiltonian is presented.
TL;DR: In this paper, the main ideas and methods of calculations within the framework of the generating functional technique are considered in a systematical way, which result in an explicit expression for the nonequilibrium generating functionals in terms of the coarse-grained generating functional being the functional mapping of the quasiequilibrium statistical operator.
Abstract: The main ideas and methods of calculations within the framework of the generating functional technique are considered in a systematical way. The nonequilibrium generating functionals are defined as functional mappings of the nonequilibrium statistical operator and so appear to be dependent on a certain set of macroscopic variables describing the nonequilibrium state of the system. The boundary conditions and the differential equation of motion for the generating functionals are considered which result in an explicit expression for the nonequilibrium generating functionals in terms of the so-called coarse-grained generating functional being the functional mapping of the quasiequilibrium statistical operator. Various types of integral equations are derived for the generating functionals which are convenient to develop the perturbation theories with respect to either small interaction or small density of particles. The master equation for the coarse-grained generating functionals is obtained and its connection with the generalized kinetic equations for a set of macrovariables is shown. The derivation of the generalized kinetic equations for some physical systems (classical and quantum systems of interacting particles, the Kondo system) is treated in detail, with due regard for the polarization effects as well as the energy and momentum exchange between the colliding particles and the surrounding media.