Approximate Master Equations Research Articles

Calculation of the micromaser spectrum. I. Green's-function approach and approximate analytical techniques.

We first calculate the ``exact'' micromaser spectrum by solving numerically for the Green's function of the approximate master equation for the one-atom (micro-) maser. We then proceed to calculate approximate analytical expressions for the maser linewidth using (i) a phase-operator approach, (ii) an ansatz similar to the standard quantum theory of the laser, and (iii) a linear expansion of the correlation function. Novel features, quite distinct from the familiar Schawlow-Townes linewidth, are found, e.g., the linewidth decreases as the number of thermal photons increases.

An approximate master equation for systems driven by linear Ornstein-Uhlenbeck noise

By use of an operator method, we construct a novel approximate evolution equation for a one-dimensional probability distribution of a single-degree-of-freedom system driven by Ornstein-Uhlenbeck noise. This equation is an integro-differential equation of the time-convolutionless type and its steady-state solution is presented. A class of non-linear equations with multiplicative noise is considered.

Approximate master equations for an exactly solvable quantum system

Approximate master equations for the Friedrichs model are discussed. The approximate Zwanzig equation predicts behaviour very close to the exact one, but presents only minor simplifications of necessary calculations, while an approximate convolutionless equation usually gives a reasonably good account of the behaviour of exact solutions and simplifies the calculations significantly. It is also shown that the van Hove and Markov limits give incorrect asymptotics in a physically interesting case.

Master equations for photochemistry with intense infrared light (IV). A unified treatment of case B and case C including nonlinear effects

AbstractA unified master equation for unimolecular reactions induced by monochromatic infrared radiation (URIMIR) is presented. Its effective rate coefficient matrix covers both case B (Pauli equation) and case C, properly including the nonlinearity of the latter. Exact quantum mechanical model solutions are compared with results from the approximate unified master equation. The exact analytical solutions of the master equation are presented for the URIMIR of some realistic molecular models. The important new properties of the transition range between case B and case C are quantitatively discussed with respect to time dependent and steady state level populations, time dependent and steady state rate coefficients and their nonlinear intensity dependence, and with respect to the influence of molecular properties. The role of case C for the interpretation of static field effects and its importance for efficient isotope separation are pointed out.

Foundations of the Callaway Theory of Thermal Conductivity

The equation derived by Peierls for the occupation numbers ${N}_{q}$ of phonon states with wave vector q and appropriate polarization is linearized to give a maser equation for the deviations ${n}_{q}\ensuremath{\equiv}{N}_{q}\ensuremath{-}{{N}_{q}}^{0}$ from equilibrium. This master equation, with transition probabilities calculated from first-order time-dependent perturbation theory, is then compared term by term with the linearized form of an approximate master equation proposed by Callaway. The latter assumes that the effects of normal and umklapp processes can be represented by two relaxation frequencies, ${{\ensuremath{\tau}}_{N}}^{\ensuremath{-}1}$ and ${{\ensuremath{\tau}}_{r}}^{\ensuremath{-}1}$, respectively, which are both proportional to ${q}^{2}$. This assumption is contradicted by the present analysis, which shows that the condition that the solution of the Callaway equation should also satisfy the Peierls equation in the case of steady-state heat flow leads to ${{\ensuremath{\tau}}_{N}}^{\ensuremath{-}1}$ proportional to $q$ in first approximation. Furthermore, the Callaway equation contains nonzero transition probabilities for a great many phonon processes which are absent from the Peierls equation. Thus, while Callaway's equation, with a modified ${\ensuremath{\tau}}_{N}$, can indeed give a good approximation to the phonon occupation numbers in the steady state, it does not describe the approach to that state from an arbitrary initial distribution. These conclusions are found to remain true when boundary and point-defect scattering are included, in addition to three-phonon processes. A first approximation to the wave number and temperature dependence of the corresponding contribution to the Callaway relaxation times is given for each scattering mechanism.