Showing papers on "Master equation published in 1977"
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01 Jan 1977
TL;DR: In this article, the authors investigate the effect of random variables on the sample space of a phase transition in a 3D laser system and show that they can be used to estimate the probability of the phase transition.
Abstract: 1. Goal.- 1.1 Order and Disorder: Some Typical Phenomena.- 1.2 Some Typical Problems and Difficulties.- 1.3 How We Shall Proceed.- 2. Probability.- 2.1 Object of Our Investigations: The Sample Space.- 2.2 Random Variables.- 2.3 Probability.- 2.4 Distribution.- 2.5 Random Variables with Densities.- 2.6 Joint Probability.- 2.7 Mathematical Expectation E(X), and Moments.- 2.8 Conditional Probabilities.- 2.9 Independent and Dependent Random Variables.- 2.10 Generating Functions and Characteristic Functions.- 2.11 A Special Probability Distribution: Binomial Distribution.- 2.12 The Poisson Distribution.- 2.13 The Normal Distribution (Gaussian Distribution).- 2.14 Stirling's Formula.- 2.15 Central Limit Theorem.- 3. Information.- 3.1 Some Basic Ideas.- 3.2 Information Gain: An Illustrative Derivation.- 3.3 Information Entropy and Constraints.- 3.4 An Example from Physics: Thermodynamics.- 3.5 An Approach to Irreversible Thermodynamics.- 3.6 Entropy-Curse of Statistical Mechanics?.- 4. Chance.- 4.1 A Model of Brownian Movement.- 4.2 The Random Walk Model and Its Master Equation.- 4.3 Joint Probability and Paths. Markov Processes. The Chapman-Kolmogorov Equation. Path Integrals.- Sections with an asterisk in the heading may be omitted during a first reading..- 4.4 How to Use Joint Probabilities. Moments. Characteristic Function. Gaussian Processes.- 4.5 The Master Equation.- 4.6 Exact Stationary Solution of the Master Equation for Systems in Detailed Balance.- 4.7 The Master Equation with Detailed Balance. Symmetrization, Eigenvalues and Eigenstates.- 4.8 Kirchhoff's Method of Solution of the Master Equation.- 4.9 Theorems about Solutions of the Master Equation.- 4.10 The Meaning of Random Processes, Stationary State, Fluctuations, Recurrence Time.- 4.11 Master Equation and Limitations of Irreversible Thermodynamics.- 5. Necessity.- 5.1 Dynamic Processes.- 5.2 Critical Points and Trajectories in a Phase Plane. Once Again Limit Cycles.- 5.3 Stability.- 5.4 Examples and Exercises on Bifurcation and Stability.- 5.5 Classification of Static Instabilities, or an Elementary Approach to Thorn's Theory of Catastrophes.- 6. Chance and Necessity.- 6.1 Langevin Equations: An Example.- 6.2 Reservoirs and Random Forces.- 6.3 The Fokker-Planck Equation.- 6.4 Some Properties and Stationary Solutions of the Fokker-Planck-Equation.- 6.6 Time-Dependent Solutions of the Fokker-Planck Equation.- 6.6 Solution of the Fokker-Planck Equation by Path Integrals.- 6.7 Phase Transition Analogy.- 6.8 Phase Transition Analogy in Continuous Media: Space-Dependent Order Parameter.- 7. Self-Organization.- 7.1 Organization.- 7.2 Self-Organization.- 7.3 The Role of Fluctuations: Reliability or Adaptibility? Switching.- 7.4 Adiabatic Elimination of Fast Relaxing Variables from the Fokker-Planck Equation.- 7.5 Adiabatic Elimination of Fast Relaxing Variables from the Master Equation.- 7.6 Self-Organization in Continuously Extended Media. An Outline of the Mathematical Approach.- 7.7 Generalized Ginzburg-Landau Equations for Nonequilibrium Phase Transitions.- 7.8 Higher-Order Contributions to Generalized Ginzburg-Landau Equations.- 7.9 Scaling Theory of Continuously Extended Nonequilibrium Systems.- 7.10 Soft-Mode Instability.- 7.11 Hard-Mode Instability.- 8. Physical Systems.- 8.1 Cooperative Effects in the Laser: Self-Organization and Phase Transition.- 8.2 The Laser Equations in the Mode Picture.- 8.3 The Order Parameter Concept.- 8.4 The Single-Mode Laser.- 8.5 The Multimode Laser.- 8.6 Laser with Continuously Many Modes. Analogy with Superconductivity.- 8.7 First-Order Phase Transitions of the Single-Mode Laser.- 8.8 Hierarchy of Laser Instabilities and Ultrashort Laser Pulses.- 8.9 Instabilities in Fluid Dynamics: The Benard and Taylor Problems.- 8.10 The Basic Equations.- 8.11 The Introduction of New Variables.- 8.12 Damped and Neutral Solutions (R ? Rc).- 8.13 Solution Near R = Rc (Nonlinear Domain). Effective Langevin Equations.- 8.14 The Fokker-Planck Equation and Its Stationary Solution.- 8.15 A Model for the Statistical Dynamics of the Gunn Instability Near Threshold.- 8.16 Elastic Stability: Outline of Some Basic Ideas.- 9. Chemical and Biochemical Systems.- 9.1 Chemical and Biochemical Reactions.- 9.2 Deterministic Processes, Without Diffusion, One Variable.- 9.3 Reaction and Diffusion Equations.- 9.4 Reaction-Diffusion Model with Two or Three Variables: The Brusselator and the Oregonator.- 9.5 Stochastic Model for a Chemical Reaction Without Diffusion. Birth and Death Processes. One Variable.- 9.6 Stochastic Model for a Chemical Reaction with Diffusion. One Variable.- 9.7 Stochastic Treatment of the Brusselator Close to Its Soft-Mode Instability.- 9.8 Chemical Networks.- 10. Applications to Biology.- 10.1 Ecology, Population-Dynamics.- 10.2 Stochastic Models for a Predator-Prey System.- 10.3 A Simple Mathematical Model for Evolutionary Processes.- 10.4 A Model for Morphogenesis.- 10.5 Order Parameters and Morphogenesis.- 10.6 Some Comments on Models of Morphogenesis.- 11. Sociology and Economics.- 11.1 A Stochastic Model for the Formation of Public Opinion.- 11.2 Phase Transitions in Economics.- 12. Chaos.- 12.1 What is Chaos?.- 12.2 The Lorenz Model. Motivation and Realization.- 12.3 How Chaos Occurs.- 12.4 Chaos and the Failure of the Slaving Principle.- 12.5 Correlation Function and Frequency Distribution.- 12.6 Discrete Maps, Period Doubling, Chaos, Intermittency.- 13. Some Historical Remarks and Outlook.- References, Further Reading, and Comments.
780 citations
TL;DR: In this paper, the master equation for a thermal unimolecular reaction in gases at low pressures is formulated and steady state solutions are derived in analytical form with an exponential model of collisional transition probabilities, for vibrational energy transfer in molecules with variable densities of states.
Abstract: The master equation for a thermal unimolecular reaction in gases at low pressures is formulated. Steady‐state solutions are derived in analytical form with an exponential model of collisional transition probabilities, (i) for vibrational energy transfer in molecules with variable densities of states, and (ii) for combined rotational and vibrational energy transfer in molecules with variable heights of the centrifugal barriers. Other models of transition probabilities are treated numerically. The diffusion limit of energy transfer is discussed. In all cases, the nonequilibrium populations of excited states and the weak collision efficiency factors βc are calculated.
678 citations
TL;DR: In this paper, the interrelation between the well-known non-Markovian master equation and the new memoryless one used in the previous paper is clarified on the basis of damping theory, and the latter equation is generalized to include cases in which the Hamiltonian or the Liouvillian is a random function of time.
Abstract: The interrelation between the well-known non-Markovian master equation and the new memoryless one used in the previous paper is clarified on the basis of damping theory. The latter equation is generalized to include cases in which the Hamiltonian or the Liouvillian is a random function of time, and is written in a form feasible for perturbational analysis. Thus, the existing stochastic theory in which those cases mentioned above are discussed is equipped with a more tractable basic equation. Two problems discussed in the previous paper, i.e., the random frequency modulation of a quantal oscillator and the Brownian motion of a spin, are treated from the viewpoint of the stochastic theory without such explicit consideration of external reservoirs as was taken in the previous paper.
398 citations
TL;DR: In this article, a semiphenomenological cluster theory is developed for the dynamics of systems with a conserved (one component) order parameter, which is not limited to small deviations from equilibrium.
Abstract: A semiphenomenological cluster theory is developed for the dynamics of systems with a conserved (one component) order parameter, which is not limited to small deviations from equilibrium. Concentration fluctuations of the binary system are parametrized in terms of clusters of various "sizes" $l$; these fluctuations decay by cluster reactions and cluster diffusion. The cluster diffusivity ${D}_{l}$ is estimated using the master equation for atomic exchange processes, and is confirmed by recent computer simulations of Rao et al. Close to equilibrium the nonlinear set of kinetic equations is reduced to a Fokker-Planck equation for the concentration of large clusters, which contains an effective chemical potential produced by the small clusters. Due to the conservation law this potential slowly varies with time. From this equation, we obtain as special cases the critical behavior of the diffusion constant both in solid and liquid binary systems close to ${T}_{c}$, and the Lifshitz-Slyozov theory of grain growth (below ${T}_{c}$). Additional terms describing the coagulation of large clusters have to be included in the latter case, however. At intermediate times the Lifshitz-Slyozov mechanism may even be neglected. A dynamic scaling solution of the coagulation equation predicts that the typical linear dimension should increase $\ensuremath{\propto}{t}^{\frac{1}{(3+d)}}$ in $d$ dimensions, in agreement with our previous heuristic arguments. The results are compared to computer simulations and to experiments on real systems. For the nonlinear relaxation above ${T}_{c}$ both scaling analysis and cluster dynamics give identical predictions. We also compare our approach to other theories of spinodal decomposition, deriving them in a unified way by factorization approximations of a rigorous kinetic equation, and thus elucidate their validity.
327 citations
TL;DR: In this paper, the authors proposed a three-dimensional model of the radiation field and showed that it can be represented by three levels of level-crossing: zero magnetic field level crossing, high-field level crossing and high-frequency level crossing.
291 citations
TL;DR: In this paper, the diffusion matrix of the Fokker Planck equation is used as a contravariant metric tensor in phase space, and the covariance of the Langevin-equations and the fokker equation is demonstrated.
Abstract: The Fokker Planck equation is considered as the master equation of macroscopic fluctuation theories. The transformation properties of this equation and quantities related to it under general coordinate transformations in phase space are studied. It is argued that all relations expressing physical properties should be manifestly covariant, i.e. independent of the special system of coordinates used. The covariance of the Langevin-equations and the Fokker Planck equation is demonstrated. The diffusion matrix of the Fokker Planck equation is used as a contravariant metric tensor in phase space. Covariant drift vectors associated to the Langevin- and the Fokker Planck equation are found. It is shown that special coordinates exist in which the covariant drift vector of the Fokker Planck equation and the usual non-covariant drift vector are equal. The physical property of detailed balance and the equivalent potential conditions are given in covariant form. Finally, the covariant formulation is used to study how macroscopic forces couple to a system in a non-equilibrium steady state. A general fluctuation-dissipation theorem for the linear response to such forces is obtained.
201 citations
TL;DR: In this paper, a new memoryless expression for the equation of motion for the reduced density matrix is derived, which is equivalent to that proposed by Tokuyama and Mori, but has a more convenient form for the application of the perturbational expansion method.
Abstract: A new memoryless expression for the equation of motion for the reduced density matrix is derived. It is equivalent to that proposed by Tokuyama and Mori, but has a more convenient form for the application of the perturbational expansion method. The master equation derived from this form of equation in the first Born approximation is applied to two examples, the Brownian motion of a quantal oscillator and that of a spin. In both examples the master equation is rewritten into the coherent-state representation. A comparison is made with the stochastic theory of the spectral line shape given by Kubo, and it is shown that this theory of the line shape can be incorporated into the framework of the present theory.
194 citations
TL;DR: In this article, a new technique for handling chemical master equations, based on an expansion of the probability distribution in Poisson distributions, is introduced, which enables chemical master equation to be transformed into Fokker-Planck and stochastic differential equations and yields very simple descriptions of chemical equilibrium states.
Abstract: We introduce a new technique for handling chemical master equations, based on an expansion of the probability distribution in Poisson distributions. This enables chemical master equations to be transformed into Fokker-Planck and stochastic differential equations and yields very simple descriptions of chemical equilibrium states. Certain nonequilibrium systems are investigated and the results are compared with those obtained previously. The Gaussian approximation is investigated and is found to be valid almost always, except near critical points. The stochastic differential equations derived have a few novel features, such as the possibility of pure imaginary noise terms and the possibility of higher order noise, which do not seem to have been previously studied by physicists. These features are allowable because the transform of the probability distribution is a quasiprobability, which may be negative or even complex.
150 citations
TL;DR: In this article, the authors investigate the time evolution of stochastic non-Markov processes as they occur in the coarse-grained description of open and closed systems and show that semigroups of propagators exist for all multivariate probability distributions.
Abstract: We investigate the time evolution of stochastic non-Markov processes as they occur in the coarse-grained description of open and closed systems. We show that semigroups of propagators exist for all multivariate probability distributions, the generators of which yield a set of time-convolutionless master equations. We discuss the calculation of averages and time-correlation functions. Further, linear response theory is developed for such a system. We find that the response function cannot be expressed as an ordinary time-correlation function. Some aspects of the theory are illustrated for the two-state process and the Gauss process.
124 citations
TL;DR: In this article, the authors show how statistical fluctuations can be treated within the collective approach to heavy ion reactions and show that the equation of motion for the distribution d in the collective variables Q μ and their conjugate momenta P μ turns out to be a Fokker-Planck equation.
TL;DR: In this article, the average power in each mode as a function of range is derived from the coupled mode equations by using certain approximations, and the Monte Carlo results also establish the region of validity of these fluctuation predictions.
Abstract: Acoustic waves in a random ocean have been studied by means of coupled mode equations, a system of stochastic ordinary differential equations for the acoustic normal mode amplitudes as a function of range. The random coupling coefficients represent scattering from internal wave normal modes. Deterministic master equations, which predict the average power in each mode as a function of range, can be derived from the coupled mode equations by using certain approximations. Monte Carlo simulations, numerically solving many realizations of the coupled mode equations, have been carried out in order to establish the validity of the master equations as a function of range and acoustic frequency. In addition, fluctuations of the power about the average value are predicted by yet another system of equations, but whose coefficients are linear combinations of those appearing in the master equations. The Monte Carlo results also establish the region of validity of these fluctuation predictions. [Work supported by ONR.]
TL;DR: In this article, a comparison between the deterministic and stochastic (master equation) formulation of nonlinear chemical rate processes with multiple stationary states has been made, via two specific examples of chemical reaction schemes, that the master equations have quasi-stationary state solutions which agree with the various initial condition dependent equilibrium solutions of deterministic equations.
Abstract: A comparison has been made between the deterministic and stochastic (master equation) formulation of nonlinear chemical rate processes with multiple stationary states. We have shown, via two specific examples of chemical reaction schemes, that the master equations have quasi-stationary state solutions which agree with the various initial condition dependent equilibrium solutions of the deterministic equations. The presence of fluctuations in the stochastic formulation leads to true equilibrium solutions, i.e. solutions which are independent of initial conditions as t → ∞. We show that the stochastic formulation leads to two distinct time scales for relaxation. The mean time for the reaction system to reach the quasi-stationary states from any initial state is of O ( N 0 ) where N is a measure of the size of the reaction system. The mean time for relaxation from a quasi-stationary state to the true equilibrium state is O (e N ). The results obtained from the stochastic formulation as regards the number and location of the quasi-stationary states are in complete agreement with the deterministic results.
TL;DR: In this article, a system involving all-or-none transitions away from equilibrium is considered and an integral representation of the solution of the master equation is derived, which permits an exact evaluation of the variance in the thermodynamic limit.
Abstract: A system involving all-or-none transitions away from equilibrium is considered. Under the assumption of spatially homogeneous fluctuations an integral representation of the solution of the master equation is derived, which permits an exact evaluation of the variance in the thermodynamic limit. A systematic perturbative solution of the master equation is also developed. Both approaches yield “classical” exponents describing the divergence of the second-order variance as the instability point is approached on either side. Finally, at the instability point the second-order variance is shown to diverge as the 32 power of the volume.
TL;DR: An extension of the continuous-time random walk formalism to include internal states and to establish the connection to generalized master equations with internal states is presented.
Abstract: We present an extension of the continuous-time random walk formalism to include internal states and to establish the connection to generalized master equations with internal states. The theory allows us to calculate physical observables from which we can extract the characteristic parameters of the internal states of the system under study.
TL;DR: In this paper, it was shown that a non-linear Fokker-Planck equation is the most general continuous asymptotic representation of master equations describing internal fluctuations in the limit of large systems.
Abstract: Using the theory of Markovian diffusion processes it is established that a non-linear Fokker-Planck equation is the most general continuous asymptotic representation of master equations describing internal fluctuations in the limit of large systems. The good agreement between the results of the Fokker-Planck approximation and those of the master equation description is demonstrated on several examples. The differences with van Kampen's approach are elucidated.
TL;DR: In this article, the authors present a rigorous derivation of the stochastic Glauber dynamics of Ising models from Hamiltonian quantum mechanics and obtain an example of a nonequilibrium statistical mechanical system for which a link without mathematical gap can be established from microscopic quantum mechanics to a macroscopic irreversible thermodynamic process.
Abstract: With the help of recent results in the mathematical theory of master equations, we present a rigorous derivation of the stochastic Glauber dynamics of Ising models from Hamiltonian quantum mechanics. A thermal bath is explicitly constructed and, as an illustration, the dynamics of the Ising-Weiss model is analyzed in the thermodynamic limit. We thus obtain an example of a nonequilibrium statistical mechanical system for which a link without mathematical gap can be established from microscopic quantum mechanics to a macroscopic irreversible thermodynamic process.
TL;DR: In this article, it was shown that the transition probabilities for weakly coupled systems are equal to those that are calculated using the Fermi golden rule for a large class of systems coupled to heat baths.
Abstract: We give a proof that for a large class of systems weakly coupled to heat baths the transition probabilities per unit time obtained from the Markov approximation are equal to those that are calculated using the Fermi golden rule.
TL;DR: In this article, a stochastic analysis of the spatial and temporal structures in the Prigogine-Lefever-Nicolis model (the Brusselator) is presented, carried out through a Langevin equation derived from a multivariate master equation using the Poisson representation method.
Abstract: A stochastic analysis of the spatial and temporal structures in the Prigogine-Lefever-Nicolis model (the Brusselator) is presented. The analysis is carried out through a Langevin equation derived from a multivariate master equation using the Poisson representation method, which is used to calculate the spatial correlation functions and the fluctuation spectra in the Gaussian approximation. The case of an infinite three-dimensional system is considered in detail. The calculations for the spatial correlation functions and the fluctuation spectra for a finite system subject to different kinds of boundary conditions are also given.
TL;DR: In this paper, the authors studied the time evolution of the single event probability of macroscopic variables from a microscopic point of view, and showed that the retarded master equation can exactly be transformed into a time-convolutionless and homogeneous form.
Abstract: The time evolution of the single event probability of macroscopic variables is studied from a microscopic point of view. The explicit consideration of the preparation of the initial microdistribution leads to a unique decomposition of the macrodynamics into two parts, a local and instantaneous one and a nonlocal and retarded one. In this retarded, i.e., non-Markovian master equation no inhomogeneity occurs in contrast to previous approaches. It is shown that the retarded master equation can exactly be transformed into a time-convolutionless and homogeneous form\(\dot p(t) = \Gamma (t)p(t)\), which generates a substitutive Markov process with the same single event behaviour as the process in question.
TL;DR: A concise derivation of the now well established result that a random walk with an arbitrary pausing time distribution is describable by a time-convoluted master equation with an explicitly determined memory kernel is given in this article.
TL;DR: In this paper, the authors show that the correct statistical mechanical description of the fluctuations of the mean field theory of optical bistability depends on the experimental parameters and present a new master equation for a previously unconsidered experimentally relizable parameter range.
TL;DR: In this paper, the effect of diffusion on fluctuations is considered, and exact results are obtained for several reaction mechanisms at steady states away from equilibrium in the Ornstein-Zernike form.
Abstract: An anomalous prediction of the master equation formalism is not obtained when concentration fluctuations in chemical reactions are described by the fluctuation‐dissipation theory. The known connection between these two theories shows that the discrepancy may be eliminated by taking the properly scaled thermodynamic limit of the master equation prior to achieving steady state. The effect of diffusion on fluctuations is also considered, and exact results are obtained for several reaction mechanisms. At steady states away from equilibrium nonlocal correlations of the Ornstein–Zernike form are found. This agrees with the continuum limit of the master equation formalism used by Gardiner et al.
TL;DR: The singular perturbation method has been applied to solve the quantum mechanical Liouville equation for the relaxation phenomenon of the system in thermal contact with a heat bath.
Abstract: The singular perturbation method has been applied to solve the quantum mechanical Liouville equation for the relaxation phenomenon of the system in thermal contact with a heat bath. The master equation derived gives the proper expressions for both diagonal and off-diagonal elements of the density matrix and is capable of describing the time-dependent behavior of the system in the time range comparable with the reciprocal of the damping constants and the time range t → ∞ compared with the reciprocal of the damping constants.
01 Jan 1977
TL;DR: The generalized master equation (GME) is an entity one meets with on the wayside in one's journey from the microscopic to the macroscopic level of the dynamics of large systems in statistical mechanics as mentioned in this paper.
Abstract: The generalized master equation (GME) is an entity one meets with on the wayside in one’s journey from the microscopic to the macroscopic level of the dynamics of large systems in statistical mechanics. The prominent character in this journey is not the GME but is the Pauli master equation (PME) also known as the Master equation. The latter, with its distinctive tendencies as are evident in the H-theorem and its generally built-in irreversibility, possesses the ability to guide the weary [1] traveller safely (?) to the realm of macroscopic phenomena. The importance of the GME is therefore not always appreciated in the normal course of this journey. In fact, usually, the equation is not even allowed to live long. Almost immediately after one makes its acquaintance, a procedure known as the Markoffian approximation [2] is thrust into the GME, destroying its special characteristics and converting it into the sought-after PME. There are, However, researchers, who admire the GME for its own qualities (and not merely for its ability to give birth to the PME), who have recently studied it in its own right and have put to use its potentialities.
TL;DR: In this article, a truncated Morse oscillator model was proposed to describe the vibrational energies and the SSH transition probabilities for dissociation of diatomic molecules, and single-quantum vibrational-translational energy exchanges were shown to dominate the rate of dissociation.
Abstract: A steady‐state form of the master equation is solved to determine rate constants for dissociation of diatomic molecules. A truncated Morse oscillator model describes the vibrational energies and the (SSH) transition probabilities. Single‐quantum vibrational–translational energy exchanges are shown to dominate the rate of dissociation. The theory provides order of magnitude agreement with experimental data for H2, N2, O2, NO, F2, Cl2, Br2, and I2. Anharmonicity of the potential contributes to the lowering of the energy of activation below the bond dissociation energy. Deviations from the Boltzmann distribution cause the energy of activation to decrease at higher temperatures.
TL;DR: In this paper, the general master equations describing nonlinear birth and death processes with one variable are analyzed in terms of their eigenmodes and eigenvalues using the method of a WKB approximation.
Abstract: Nonlinear birth and death processes with one variable are considered. The general master equations describing these processes are analyzed in terms of their eigenmodes and eigenvalues using the method of a WKB approximation. Formulas for the density of eigenstates are obtained. The lower lying eigenmodes are calculated to investigate long-time relaxation, such as relaxations of metastable and unstable states. Anomalous accumulation of the lower lying eigenvalues is shown to exist when the system is infinitesimally close to a critical or marginal state. The general results obtained are applied to some instructive examples, such as the kinetic Weiss-Ising model and a stochastic model of nonlinear chemical reactions.
TL;DR: In this article, a Boltzmann equation was proposed to describe the absorption of energy by the driven mode and the transfer of energy to the rest of the molecule by intramolecular vibrational coupling.
Abstract: The molecular excitation process is represented by a system of coupled anharmonic oscillators, one of which is driven by a near-resonant applied electric field. The master equation for the density matrix of such a system is reduced to the form of a Boltzmann equation describing the absorption of energy by the driven mode and the transfer of energy to the rest of the molecule by intramolecular vibrational coupling. The solution of this Boltzmann equation is shown to lead to an energy absorption spectrum which, for sufficiently strong laser fields, is considerably broader than the anharmonic defect and shows maxima corresponding to multiphoton transitions. The magnitude which the vibrational coupling width must attain before dissociation of the molecule is possible may be estimated from the calculated energy absorption spectrum.
TL;DR: In this article, it was shown that the master equation theory is identical to the fluctuation-dissipation theory in the macroscopic limit for a variety of systems, including chemical reactions.
Abstract: Three descriptions of spontaneous fluctuations in macroscopic systems have been proposed: One uses generalized Fokker–Planck equations and treats fluctuations as a stochastic diffusion process; another uses a connection between fluctuations and dissipation and generalizes the Langevin method; the third is the master equation theory which treats fluctuations as arising from a birth and death process. For a variety of systems it is known that the master equation theory is identical to the fluctuation–dissipation theory in the macroscopic limit. For chemical reactions it is shown here that the appropriate diffusion process also becomes identical with the fluctuation–dissipation theory in the macroscopic limit.