Showing papers on "Master equation published in 1976"

Network theory of microscopic and macroscopic behavior of master equation systems

Abstract: A general microscopic and macroscopic theory is developed for systems which are governed by a (linear) master equation. The theory is based on a network representation of the master equation, and the results are obtained mostly by application of some basic theorems of mathematical graph theory. In the microscopic part of the theory, the construction of a steady state solution of the master equation in terms of graph theoretical elements is described (Kirchhoff's theorem), and it is shown that the master equation satisfies a global asymptotic Liapunov stability criterion with respect to this state. The Glansdorff-Prigogine criterion turns out to be the differential version and thus a special case of the global criterion. In the macroscopic part of the theory, a general prescription is given describing macrostates of the systems arbitrarily far from equilibrium in the language of generalized forces and fluxes of nonlinear irreversible thermodynamics. As a particular result, Onsager's reciprocity relations for the phenomenological coefficients are obtained as coinciding with the reciprocity relations of a near-to-equilibrium network.

A quantum-mechanical master equation treatment of the dynamical Stark effect

Abstract: The description begins with an operator master equation for the atom plus incident field. Reduced atomic matrix elements are derived for arbitrary field strengths. First- and second-order correlation functions in the scattered field are also obtained and discussed in relation to the scattered spectrum and intensity-fluctuation measurements. This formalism has the appealing feature that all information is readily available from the one set of four coupled equations. The deficiencies in both the one-photon approximation and the semiclassical perspective are established in a natural and transparent fashion.

Markovian master equations. II

Abstract: We prove some theorems on the behaviour of solutions of master equations in the weak coupling limit, obtaining an exponential decay law under more general conditions than in an earlier paper. As well as applying the theory to a new type of example, we analyse some previously unstudied aspects of the dissipative behaviour.

Correlations in stochastic theories of chemical reactions

Abstract: A comprehensive study of correlations in linear and nonlinear chemical reactions is presented using coupled chemical and diffusion master equations. As a consequence of including correlations in linear reactions the approach to the steady-state Poisson distribution from an initial non-Poissonian distribution is given by a power law rather than the exponential predicted by neglecting correlations. In nonlinear reactions we show that a steadystate Poisson distribution is achieved in small volumes, whereas in large volumes a non-Poissonian distribution is built up via the correlation. The spatial correlation function is calculated for two examples, one which exhibits an instability, the other which exhibits a second-order phase transition, and correlation length and correlation time are calculated and shown to become infinite as the critical point is approached. The critical exponents are found to be classical.

The principle of increasing mixing character and some of its consequences

Abstract: The “Principle of Increasing Mixing Character”, previously postulated by one of us (and derived for the case of an ensemble of isolated systems obeying a “master equation”) as a stronger version of the second law of thermodynamics, is re-derived using von Neumann's density matrix formulation of statistical mechanics. To make the principle more convenient for applications, it is reformulated in terms of “Mixing Homomorphic Functions”, a set of state functions all of which must increase in an allowed irreversible process in an-isolated system. The entropy is one such function, but no one function, and no finite set of functions, suffices to determine the increase of mixing character. The principle is extended to the case of a system which is not isolated, but in contact with a heat bath, for which it takes a form which we name the “Principle of Decreasing Mixing Distance” from the equilibrium distribution. As examples, applications are made to two simple cases: diffusion in an ideal solution, and chemical reactions in ideal gas mixtures.

Stochastic theory of the kinetics of phase transitions

Abstract: We present a stochastic theory of the kinetics of phase transitions in univariant, nonuniform systems. We assume a master equation and a relation of the transition probability to the free energy [J. S. Langer, Ann. Phys. (N.Y) 65, 53 (1971)]. The free energy is taken to be of the Cahn–Hilliard form. By means of path integral methods we obtain a formal solution from which we derive a deterministic differential equation for the most probable variation of the density distribution in time. This equation is of the Landau–Ginzburg type. We show the existence of a Liapunoff function for this kinetic equation, which is then used to derive simply the classical result of nucleation theory for the critical radius of a droplet. Next we show from the kinetic equation that if the structure of the free energy is such that a phase transition occurs and metastable states are possible, then the kinetics of the decay to the stable and metastable states is nonlinear, of the cubic type. Finally we derive from the kinetic equa...

Generalized Onsager-Machlup function and classes of path integral solutions of the Fokker-Planck equation and the master equation

Abstract: We determine the path integral solution of a stochastic process described by a generalized Langevin equation with coordinate-dependent fluctuating forces and white spectrum. Since such equations do not permit a unique determination of the distribution function but require the Ito or Stratonovich prescription, we first pass over to the corresponding Fokker-Planck equation adopting such a prescription. By means of the one-dimensional case we show that the path integral solutions are not uniquely determined in form but allow for a class of equivalent representations. Adopting an especially suitable representation we then present the path integral solution of the multi-dimensional Fokker-Planck equation. The exponent of the exponential function occurring in that solution can be interpreted as generalization of the Onsager-Machlup function. Finally we give a path-integral solution of the master equation. In the “Fokker-Planck limit” again a generalized Onsager-Machlup function is obtained.

Transient nucleation in H2O–H2SO4 mixtures: A stochastic approach

Abstract: Transient phenomena in the rate of nucleation are of importance in the interpretation of experimental data on nucleation in the H2SO4–H2O vapor system. Furthermore, fluctuations in the rate of nucleation can actually be measured. In view of these facts, a stochastic approach to the theory of nucleation is developed with eventual application to the H2SO4–H2O system. The central goal of the theory is the evaluation of the distribution of ’’first passage times,’’ for a nucleus, over the relevant free energy barrier. This distribution is calculated from the distribution of a ’’walker’’ on a lattice. However, the distribution of the walker is derived from a ’’master equation,’’ and for this purpose it is demonstrated that walks on lattices and master equations are, in this sense, ’’equivalent’’ even for a lattice which is not translationally invariant. Given the distribution of first passage times, it is possible to compute the transient rate of nucleation, as well as the distribution of fluctuations of the rate about some instantaneous mean. This theory is then applied to the H2SO4–H2O vapor system, and the results indicate that, under many typical experimental conditions, the time to reach the steady state may be quite long (hundreds or thousands of seconds), but that nevertheless, some nucleation should be observed in the laboratory time frame (in agreement with experiment). The character of nucleation as an ’’extreme event’’ is demonstrated by the appearance of a large band gap in the spectrum of relaxation times for the process.

Effect of temperature and quencher concentration on vibrational relaxation in condensed media

Abstract: In this paper, we are concerned with the temperature effect on vibrational relaxation and the vibrational energy transfer from the vibrationally excited donor to the acceptor. For the temperature effect, we present numerical results to show the temperature dependence of the rate constant of vibrational relaxation and to discuss the validity of the rate constants obtained from the use of the weak coupling approximation and the strong coupling approximation. It is shown that although the temperature effect is extremely large over the temperature range T=0 to T=ϑE, the Einstein temperature of the medium, for the temperature range T=0 to T=0.3 ϑE, the rate constant varies slowly with temperature. For the vibrational energy transfer, we derive the master equation to describe the time dependent behavior of the excited donor, and the expression for the rate constant of vibrational energy transfer. The master equation is solved to study the temporal behavior of the excited donor as a function of the acceptor conc...

Computer molecular dynamics studies of a chemical instability

Abstract: Computer molecular dynamics studies are carried out on a binary mixture of hard disks in which one species can convert to the other by photoinduced transition or by collision‐induced reactions. Phenomenological rate laws are postulated for a simple model system which is then shown to possess multiple steady‐state concentrations under suitable conditions. The computer results for a 450‐disk system show that such solutions do exist, and that in general the postulated rate equations adequately describe the system behavior. Concentration fluctuations are examined and found to agree with the predictions of a master equation. Fluctuations are found to be enhanced near a point of marginal stability.

A derivation and comparison of two equations (Landau–Ginzburg and Cahn) for the kinetics of phase transitions

Abstract: We postulate a master equation and derive by path integral methods the Cahn equation for the most probable evolution of a nonuniform system. To do so we need to impose a constraint of local conservation of the variable of interest. The same derivation without this constraint has been shown previously to lead to a Landau–Ginzburg equation. Thus the two equations have a common origin. The Cahn equation is applicable to conserved variables in systems in which spatial inhomogeneities occur over distances much larger than a characteristic distance λ of the system (like mean free path, or nearest neighbor separation). The Landau–Ginzburg equation is applicable to nonconserved variables, and to conserved variables when spatial inhomogeneities occur over distances much smaller than the characteristic distances for those variables. For conserved variables a precursor of the Cahn equation in the derivation reduces to the Landau–Ginzburg equation as λ is increased.

Entropy and macroscopic disequilibrium. I. Isothermal time evolution with applications to vibrational relaxation

Abstract: A practical method for using bulk averaged values of observables for the characterization and prediction of the molecular population time evolution during isothermal relaxation is presented. In practical applications to realistic examples of vibrational relaxation very few bulk averages were required to accurately predict the population distribution even when the initial population was very strongly inverted. The time dependence of the macroscopic observables which are employed as input is conveniently formulated in terms of sum rules. The bulk average values are used as constraints in a maximal entropy procedure for the determination of the population distribution. It is shown that the procedure is of a variational type. Monotonic convergence of the information theoretic predicted distribution to the exact one is guaranteed upon inclusion of additional macroscopic input. The concept of ’’independent moments’’ is introduced for this purpose. Only independent observables are informative, i.e., provide independent data which are required for convergence. The number of informative observables decreases with time and is typically very much smaller than the number of significantly populated molecular energy states. The method is illustrated by comparing its predictions to the results of a numerical solution of the master equation, with a realistic set of rate constants and for different initial conditions. The application of the surprisal analysis to the interpretation, characterization, and compaction of the population distribution is demonstrated. Turning to predictions (’’surprisal synthesis’’), only strongly inverted initial populations required three independent moments, during the initial stages. Over much of the relaxation a single moment (’’vibrational temperature’’) sufficed for an accurate prediction. The limits where the characteristic vibrational temperature is high or low compared to the temperature of the buffer gas are discussed. Special reference is made to rotational relaxation. The rate of internal entropy production due to the irreversible relaxation and the rate of increase of the global entropy are discussed and shown to be positive.

A unified quantum theory of mechanics and thermodynamics. Part IIb. Stable equilibrium states

Abstract: Part IIb presents some of the most important theorems for stable equilibrium states that can be deduced from the four postulates of the unified theory presented in Part I. It is shown for the first time that the canonical and grand canonical distributions are the only distributions that are stable. Moreover, it is shown that reversible adiabatic processes exist which cannot be described by the dynamical equation of quantum mechanics. A number of conditions are discussed that must be satisfied by the general equation of motion which is yet to be discovered.

Fluctuations and the Boltzmann equation. I

Abstract: The evolution of a homogeneous dilute gas is treated as a Markov process in the complete set of $K$ coarsegrained velocity states of all $N$ particles. From the Siegert master equation for the process a Fokker-Planck equation is derived which describes, in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$, the fluctuations in the occupation numbers ${n}_{i}(t)$, whose average behavior is governed by the (appropriately discretized) Boltzmann equation: The continuum limit $K\ensuremath{\rightarrow}\ensuremath{\infty}$ corresponds to fluctuations in the usual molecular distribution function $f(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}};t)$. On similar reasoning, a Fokker-Planck equation is obtained for the fluctuation process near equilibrium, where the average is governed by the linearized Boltzmann equation. The theory of linear irreversible processes, which offers a statistical description of fluctuations on a thermodynamical basis, is applied to the linearized Boltzmann equation---treated as a linear phenomenological equation---following the development given recently by Fox and Uhlenbeck: The resulting stochastic equation is seen to be equivalent to the Fokker-Planck equation obtained from the master equation, yielding a multidimensional Ornstein-Uhlenbeck process which describes the fluctuations in molecular phase space.

Structure and dynamics of methylene iodide in solution from nuclear spin–lattice relaxation studies

Abstract: The nuclear spin–lattice relaxation of carbon‐13 enriched methylene iodide (diiodomethane) dissolved in benzene‐d6 has been studied both with and without proton decoupling and using various pulse techniques to perturb the AX2(13CH2) spin system from thermal equilibrium. The return of the spin system to steady state was monitored using carbon‐13 Fourier transform nuclear magnetic resonance techniques. It is shown that the equation of motion of the spin density matrix reduces in general to the master equation for populations in terms of interlevel transition rates, and that a linear transformation based on the complete set of irreducible spherical tensor operators which span the spin space further simplifies the equation of motion. The relaxation was modeled as intramolecular dipole–dipole interactions modulated by rotational reorientation of the molecule plus other mechanisms which can be treated collectively as external random magnetic fields interacting with the nuclear spins of interest. Extreme narrowi...

Stochastic theory of molecular collisions

Abstract: This paper investigates treating inelastic collisions as a stochastic process for the diffusion of probability between quantum states. It is shown that when the interaction is strong, the time‐dependent Schrodinger equation may be approximated by a Pauli master equation. Under certain circumstances this can be further approximated by a Fokker–Planck partial differential equation. These equations can be used to model various collision phenomena. Inelastic and dissociative collision processes are discussed, and it is indicated how the theory may be extended to systems with several degrees of freedom. The theory extends smoothly between (a) the perturbation and (b) statistical limits, although it should be better near limit (b). Nevertheless, comparisons even near limit (a) with a similar exact quantum mechanical collinear vibrational problem show that the diffusion model gives qualitative agreement and predicts certain trends correctly. For example, the usual sudden and adiabatic regimes are readily achieve...

Steady state solutions of master equations without detailed balance

Abstract: Steady state solutions of master equations with one variable are constructed. The method of solution is based on a transformation of the original equation for the probability into one for a slowly varying function. The method is of general applicability and is particularly useful in obtaining solutions in the case where detailed balance does not hold. Examples of such systems in chemical reaction models and the two photon laser are discussed.

Detailed balance in open quantum Markoffian systems

Abstract: Agarwal's definition of detailed balance for open quantum Markoffian systems is shown to arise from microreversibility in an analogous fashion to the familiar classical concept. It is therefore presented as the appropriate formal generalisation of the classical result to the quantum-mechanical regime. This fully quantum-mechanical approach is discussed in relation to the Fokker-Planck equations of the phase-space calculus and the Pauli master equation; two contexts in which a pseudo-classical form of detailed balance is well known. Our discussion is illustrated through the examples of the damped harmonic oscillator and the single mode laser.

A Comment on Chemical Langevin Equations

Abstract: It is noted that the diffusion Langevin stochastic sources in chemical reaction-diffusion theories should really arise from a stochastic source term added to the deterministic form of Fick's law. This gives rise to results for correlation functions which agree with those from stochastic master equations provided parameters are appropriately chosen.

Quantum-mechanical theory of the organic-dye laser

Abstract: We develop a fully quantum-mechanical theory for the organic-dye-solution laser, obtain density-matrix equations of motion for the single-mode radiation-density operator and the matter-density operator, and solve and investigate the steady-state case. We generalize the usual Born-Markoff approximation master equation for two matter states to include four matter states, each one of which interacts with the laser radiation field. This allows us to treat exactly the organic-dye molecular triplet-state levels which participate in the laser operation in an essential way. For experimentally realizable conditions the steady-state solution contains features which are qualitatively different from nondye lasers. These effects are directly attributable to intensity-dependent triplet-state radiation absorption losses. At threshold the diagonal matrix elements of the radiation photon distribution (R/subn/) can be a decreasing function of the photon number n with an inflection point rather than the usual truncated Gaussian. This necessitates redefinition of threshold. For pumping just above threshold there may be both a maximum and a minimum in R/subn/ rather than just a maximum as in usual laser theories. We also specify the effect of triplets on the narrowing of R/subn/ for pumping above threshold and the subsequent widening of R/subn/ for pumping well above threshold. Many of our results more » require photon-counting experiments for verification. Also, our equations are easily reducible to a semiclassical theory, where our treatment of the triplets variables is an improvement over the usual rate equations. (AIP) « less

The onset of instabilities in nonequilibrium systems

Abstract: The nonlinear master equation previously proposed by Malek-Mansour and Nicolis is applied to the analysis of unstable transitions leading to temporally or spatially organized patterns. Thecorrelation length of the destabilizing fluctuations is determined, and a number of striking analogies with equilibrium phase transitions are pointed out.

Transfer of vibrational-rotational energy in thermal reactions: application to polyatomics

Abstract: A simple model is used to deal with vibrational-rotational energy transfer in a thermal dissociation of a polyatomic molecule under non-equilibrium conditions when collisional energy transfer is rate-determining. Transition probabilities are assumed to be factorizable into purely vibrational and purely rotational terms, both given by an exponential model; a gaussian model is also used for the vibrational term. The master equation leads to a two-dimensional integral equation which is solved by means of the so-called “fixed-υ” approximation developed previously, in which no vibrational energy transfer takes place while rotational energy is being transferred. Two sets of calculations are performed, one with rotational energy transfer much faster than vibrational energy transfer, and the other with both of equal importance. Relative to purely vibrational energy transfer, rotational energy transfer is shown to lead to increased rate of dissociation and increased depopulation of levels near threshold, especially in the first set of calculation, and also to a further decrease of activation energy below the bond-dissociation energy. The model system used is the dissociation of H2O2 and the calculated data are in good agreement with experiment where available.

The Problem of Constraints in General Relativity: Solution of the Lichnerowicz Equation

Abstract: This paper is dedicated to Andre Lichnerowicz. The splitting property through conformal methods has been discovered by him, as well as the master equation whose solutions on a manifold give admissible initial data sets for Einstein’s equations. In his fundamental paper of 1944 (‘Journal de Mathematiques pures et appliquees) he uses the master equation, which will now be called the Lichnerowicz equation, to construct the first rigorous general solutions of the n-body problem in general relativity.

On the Construction of a Time-Reversed Markoff Process

Abstract: For a given Markoff process characterized by a set of transition probability densities there exists another process with time reversed (the retrodictive vs predictive process in the theory of measurements) such that any one of them multiplied by a single-event density may be symmetric with respect to an interchange of the events expressed as space-time variables, yielding a joint probability density. It is shown how this time-reversed process can be constructed by means of the generating operator of the associated evolution equation, and the basic properties with explicit applications to master equations and Fokker-Planck equations. Onsager's microscopic reversibility is reformulated on this basis. Possible sym. metries concerning time-correlation functions under the Markoffian law is summarized in comparion with the Kubo formula. In the application to Fokker-Planck equations, the Onsager-Machlup most probable paths are extended to general type of diffusion processes, and it is shown that the present method corresponds to a gauge transformation in dynamics of a charged particle which leaves its paths invariant. Reciprocity has been a fundamental subject in the theory of irreversible proces­ ses, since Onsager initiated the approach to the problem based on the consideration of microscopic reversibility.!) The Onsager relations for a linear dissipative sys­ tem in an external magnetic field @, given by L~,( -8) =L,~(@), have since been discussed in a number of papers. Kubo's general linear response theory•> among them provided an accurate statistical-mechanical foundation of these relations, ex­ pressing the microscopic reversibility in the time correlation functions between Hamiltonian-driven dynamical variables. Recently, the interests have been revised in connection with the statistical mechanics for non-equilibrium or open systems far from equilibrium conditions. Van Kampen discussed a possibility of extending the relations straightforwardly to the nonlinear regime of flux-force equations. 3> Another systematic approach has been developed by using equations of evolutions for probability densities based on the theory of Markoff processes. 4> An interesting finding in the latter approach has been that the microscopic reversibility as represented by a form of detailed balance condition (or its equivalent) is a situation rather restrictive for such non­ equilibrium states: There exist a number of important examples of violation such as complex optical systems and chemical reactions. A typical non-trivial system, a single-mode laser, is a special example for which the potential condition equiva­ lent to the reversibility is well satisfied, as discussed by Graham and Haken.

The spatial distribution of moving macromolecules undergoing isomerization

Abstract: In this paper we use probabilistic arguments to derive and discuss the spatial distribution of molecules which are undergoing electrophoresis or centrifugation while at the same time they are switching back and forth between two configurational states. An exact solution is obtained for arbitrary values of the diffusion coefficients of the two isomerizing states. The traditional analytic method of solving this problem, namely, by use of “master equations,” is not completely satisfactory because one can find only the Fourier transform of the solution rather than the solution itself. Our treatment yields the answer one would get if it were feasible to perform the inversion of the Fourier transform.