Steady-state solutions master equation

The master equation (165) leads to a closed system of 25 equations of motion for the density matrix elements. Since the laser field does not couple to the level d), the system of equations splits into two subsystems a set of 17 equations of motion directly coupled to the driving field and the other of 8 equations of motion not coupled to the driving field. It is not difficult to show that the steady-state solutions for the 8 density matrix elements are zero. Using the trace property, one of the remaining equations can be eliminated, and the system of equations reduces to the 16 coupled linear inhomogeneous equations. 

It is easy to obtain the master equation for the case Qi/Qo>l, in which the system dynamics is described by the Hamiltonian (63). By exchanging parameters A B, D changing coupling constant gate g in B, C, Di, we obtain the master equation for the case Qi/Qo>l-Correspondingly, the steady-state solutions are obtained. Especially, the stability condition (aoo)... 

In Section 2 the formalism of the Master equation, our main tool in the microscopic approach developed in this chapter, is laid down. This formalism, which constitutes a convenient intermediate between purely microscopic and macroscopic theories, accounts for microscopic dynamics through the fluctuations of the macrovariables. We review the main assumptions at the basis of this description, the formal properties of its solutions, and some results established in the early literature on this subject in connection with bifurcations leading to steady-state solutions. We subsequently focus on dynamical bifurcation phenomena and discuss, successively, thermodynamic fluctuations near Hopf bifurcation (Section 3) and in the regime of deterministic chaos (Section 4). A summary and suggestions for further study are given in the final Section 5. 

Considerable effort has been devoted to the asymptotic (in the sense of large size limit) properties of the invariant probability distribution, corresponding to the steady-state solution of the master equation. We briefly summarize some important results. 

Figure A3.13.16. Illustration of the level populations (eorresponding to the C-C oseillator states) from various treatments in the model of figure A3.13.15 for C2Hg at a total energy E = (he) 41 000 em and a tlneshold energy = (he) 31 000 em The pomts are mieroeanonieal equilibrium distributions. The erosses result from the solution of the master equation for IVR at steady state and the lines are thennal populations at the temperatures indieated (from [38] quant, is ealeulated with quantum densities of states, elass. with elassieal meehanieal densities.).

El < Eo) non-dissociative states (dissociative states are rapidly depopulated by the fast intramolecular dissociation process). As is well known, the time evolution of the populations [A(i)] is given by a series of exponentially decaying terms which ctHTiespond to an initial rovibrational relaxation, a subsequent incubation period with overlap of vibrational rriaxation of upper levels and dissociation, and the final dissociation period with steady-state of all populations [A(i)]. Explicit solutions of the master equation for the dissociation of diatomic molecules have been extensively reviewed by H. O. Pritchard in Volume 1 of this series. Such... 

Solutions of the Master Equations Rate Coefficients as a Function of Fluence, Intensity, and Time and at Steady State... 

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