Diffusion approximations of the two dimensional master equation

The effects of rotational energy transfer on unimolecular dissociation are assessed using a two-dimensional master equation approach. The collision operator for such a system can be very large, and approximations are sought that reduce the size of the matrix that has to be stored. The density of states of the vibrational energy is large for most molecules and can be treated as virtually continuous. As a consequence, motion in vibrational energy space can be treated using a diffusion equation approach. This reduces the storage requirements greatly and increases the speed at which diagonalization occurs. Results are presented for the dissociation of ethane and methane.     

Some approximate solutions to the discrete master equation

Recent mathematical developments on approximate diffusionlike solutions to the master equation are summarized. The technique is applied to two master equations of physical interest-one that describes the phenomenon of superradiance and a second that characterizes generation-recombination noise in semiconductors. For this second case, some previously obtained equilibrium results are found and the extension of these results to finite times is given.     

Some comments on modeling the linearized Boltzmann equation

Some exact solutions of the homogeneous and the inhomogeneous linearized Boltzmann equation (LBE) for rigid-sphere collisions are used to define two model equations in the general area of rarefied-gas dynamics. These equations are obtained from a systematic development of two synthetic scattering kernels that yield model equations that have as exact solutions certain known exact solutions of the homogeneous and of the inhomogeneous LBE. The first model established is defined in terms of the collisional invariants and the Chapman-Enskog integral equations for viscosity and for heat conduction. An extended model is defined also in terms of the collisional invariants and the Chapman-Enskog functions for viscosity and heat conduction, but the first and second Burnett functions are also included in the model. The variable collision frequency or generalized BGK model is also obtained as a special case. In addition, the exact mean-free paths defined, for rigid-sphere collisions and the LBE, in terms of viscosity or heat conduction are employed to define approximations of these quantities that are consistent with the use of the variable collision frequency model.     

Measurement master equation

We derive a master equation describing the evolution of a quantum system subjected to a sequence of observations. These measurements occur randomly at a given rate and can be of a very general form. As an example, we analyse the effects of these measurements on the evolution of a two-level atom driven by an electromagnetic field. For the associated quantum trajectories we find Rabi oscillations, Zeno-effect type behaviour and random telegraph evolution spawned by mini quantum jumps as we change the rates and strengths of measurement.     

Equivalence of the master equation and the Langevin equation

It is shown that the random force in the Langevin equation for a nonlinear system may be chosen in such a way that the resulting equation is mathematically equivalent to the master equation.     

The sub master equation

Three postulates are proposed concerning the probabilistic dynamics of chemically reactive systems: the occurrence of the elementary event is a random variable characterized by a Poisson process; the state of the chemical system is a multivariate random variable characterized by a Markov process; the identity of any chemical species in the system is an independent random variable. These postulates when applied to chemically reactive systems in a uniform manner lead to a hierarchy of equations describing in detail how eachk subpopulation varies with time. By summing over all permutations of the equation forf _n ⁽ⁿ⁾ (e ₁, ...,e _n ;t) we obtain the usual master equation. This paper focuses on the simple isomerization reaction.     

Some comments on the photomagnetoelectric effect

A generalization of the theory for the photomagnetoelectric effect (PME) is reported. Experimental quantities such as short circuit current, open circuit voltage and magnetoresistance are described as functions of carrier lifetime, surface velocities at the sample surfaces, electron and hole mobilities, electron and hole concentrations, absorption and quantum efficiency coefficients, intensity of light, magnetic field and sample dimensions. The results of numerical computation are also presented.     

Some comments on the many-dimensional solitons

The possibilities for the existence of truly localized soliton solutions in the realistic three-dimensional case are discussed. The gauge invariant theory of a non-linear chiral field is shown to be a good candidate for a model with solitons.     

Alternative generalized master equation approach to the dc phonon-assisted hopping

As the momentum operator has no diagonal elements between localized states, the hopping conduction theory should be formulated in terms of the linear response of the site-off-diagonal elements of the single-electron density matrix to an external field. A theory of this kind, starting from generalized master equations and yielding the dc phonon-assisted hopping conductivity and thermopower is formulated. This is an alternative to the usual approach treating the current conduction via a time-derivative of the electric dipole momentum.Stimulating discussions with Dr. B. Velický turning the present author's attention to his own old ideas about the hopping conduction via off-diagonal elements of the single-electron density matrix are appreciated.     

Quantum fluctuations,master equation and Fokker-Planck equation

In order to describe quantum fluctuations a general method is developed, which also may be applied to nonstationary systems as well as to states far from thermodynamic equilibrium. After a concise derivation of the master equation quantum mechanically determined dissipation and fluctuation coefficients are introduced, for which several theorems and relations are given. By using these coefficients there is set up a general Fokker-Planck equation for the diffusion of the statistical operator due to quantum fluctuations.     

A convenient expansion of the master equation

We present a simple and very convenient expansion of the master equation.     

A comment on the stationary distribution of the master equation

In a homogeneous Ising ferromagnet in the pair approximation, an equilibrium distribution function is directly derived from the full master equation in the limit of infinite system size.     

Chapman-Enskog development of the multivariate master equation

A systematic development for a Multivariate Master Equation (MME), describing reaction diffusion systems, is presented along the same lines as the Chapman-Enskog development of kinetic gas theory. Diffusion, which occurs at a very fast rate, brings the system near a state of diffusional equilibrium, the analogue of local equilibrium for gases. In diffusional equilibrium, the global Master Equation (ME) is shown to be an exact consequence of the MME. For finite systems, corrections to the global ME, resulting from the finiteness of the diffusion times, are calculated. The development is verified on an exactly solvable model and illustrated on the Schlögl model. The difficulties encountered in the thermodynamic limit are discussed, and possible outcomes suggested.