Master equations oscillators

Figure A3.13.15. Master equation model for IVR in highly excited The left-hand side shows the quantum levels of the reactive CC oscillator. The right-hand side shows the levels with a high density of states from the remaining 17 vibrational (and torsional) degrees of freedom (from [38]).

In Eqs. (II. 1)—(II.4) we have assumed that there is only one system oscillator. In the case where there exists more than one oscillator mode, in addition to the processes of vibrational relaxation directly into the heat bath, there are the so-called cascade processes in which the highest-frequency system mode relaxes into the lower-frequency system modes with the excess energy relaxed into the heat bath. These cascade processes can often be very fast. The master equations of these complicated vibrational relaxation processes can be derived in a straightforward manner. 

The parameter is the damping constant, and (n) is the mean number of reservoir photons. The quantum theory of damping assumes that the reservoir spectrum is flat, so the mean number of reservoir oscillators (n) = ( (O)bj(O j) = ( (1 / ) — 1) 1 in the yth mode is independent of j. Thus the reservoir oscillators form a thermal system. The case ( ) = 0 corresponds to vacuum fluctuations (zero-temperature heat bath). It is convenient to consider the quantum dynamics of the system (56)-(59) in the interaction picture. Then the master equation for the density operator p is given by... 

Two special cases of the theory illustrate the important features. The first is the relaxation of an ensemble of noninteracting harmonic oscillators in contact with a heat bath, and subject to nearest neighbor transitions in the discrete translational energy space. The master equations which describe the evolution of the ensemble can be written13... 

In 1958 Montroll and Shuler1 analyzed in great detail the master equation for a harmonic oscillator interacting with a gas of particles which only have translational energy and which are in equilibrium. The interaction was assumed to lead to transition between neighboring levels only, and in this approximation the master equation is... 

In general the relaxation to equilibrium of E(t) is nonexponential, since the rate matrix in the master equation has an infinite number of (in principle) nondegenerate eigenvalues if there are an infinite number of states n). There are, however, two instances where the relaxation is approximately exponential. In the first instance one assumes that the initial nonequilibrium state has appreciable population only in the first two oscillator eigenstates, and further that k,. 0 k,, m and k0. t k0 m for m > 2. If one neglects terms involving these small rate constants, the master equation reduces to a pair of coupled rate equations for a two-level system ... 

A solvable model which we have not investigated is the one of coupled harmonic oscillators. This was introduced by Ford and his collaborators [Ford 1965] and also by Ref. [Ullersma 1966], This model provides a formally exact derivation of the Master Equation. Many features of irreversible evolution can be investigated exactly within this model for example see Ref. [Haake 1985 Strunz 2003]. The result is also equivalent with the approaches in Refs. [Cal-deira 1983 Unruh 1989],... 

In this example the master equation formalism is appliedto the process of vibrational relaxation of a diatomic molecule represented by a quantum harmonic oscillator In a reduced approach we focus on the dynamics of just this oscillator, and in fact only on its energy. The relaxation described on this level is therefore a particular kind of random walk in the space of the energy levels of this oscillator. It should again be emphasized that this description is constructed in a phenomenological way, and should be regarded as a model. In the construction of such models one tries to build in all available information. In the present case the model relies on quantum mechanics in the weak interaction limit that yields the relevant transition matrix elements between harmonic oscillator levels, and on input from statistical mechanics that imposes a certain condition (detailed balance) on the transition rates. 

Consider now the overall relaxation process. As was done in Section 8.3.3, this process can be represented by a master equation for the probability Pn to be in quantum state n of the oscillator. 

We find the density matrix elements from the master equation of the system. In the frame rotating with the laser frequency a>L and within a secular approximation, in which we ignore all terms oscillating with (colrf - a>L) and ((]>2d — the master equation for the density operator of the system is given by... 

Under the assumption of weak system-bath interactions, going into the Markovian limit, the probabilities P to occupy the n state of the molecular oscillator satisfy the master equation [31] ... 

The master equation formalism can be extended to include more than a single molecular oscillator by considering several localized vibrational sets [20]. Furthermore, it can be utilized for describing heat flow in a multiterminal junction [33], and for investigating electronic energy transfer between metals [34]. 

Now considering = X. as a set of concentrations for chemical species /, the master equations of the system evolution are written as dXJdt = K J V, Xj) - DpC, which is a set of ordinary differential equations. Since J(V, X) is nonlinear in X the system may produce in some cases (close to some complex oscillations, explaining qualitatively the existence of a prepitting noise. 

In this chapter we shall examine some of the effects of molecular fluctuations on chemical oscillations, waves and patterns. There are many ways one can attempt to study fluctuation dynamics in reacting systems, the most familiar of which are master equation models [ 1 ]. Here we present results obtained using a specific class of cellular automaton models, termed lattice-gas cellular automata [2-4]. These cellular automaton models provide a mesoscopic description of the spatially-distributed reacting system and are constructed to model the microscopic collision dynamics. The modeling strategy and rule construction are different from those for traditional cellular automata and are based on lattice-gas cellular automaton models for hydrodynamics [5]. However, reactive lattice-gas models differ from the corresponding hydrodynamics models in a number of important respects and are closely related to master equation descriptions of the reactive dynamics. 

In this example the master equation formalism is applied to the process of vibrational relaxation of a diatomic molecule represented by a quantum harmonic oscillator. 

These rates can be used in the master equation (8.70) for the probability P(n, t) (denoted below Pn t)) to find the oscillator in its nth level at time V. 

---