Master equation

About: Master equation is a research topic. Over the lifetime, 10541 publications have been published within this topic receiving 276095 citations.

Transport Phenomena in Einstein-Bose and Fermi-Dirac Gases. I

Abstract: In this paper the general theory of transport phenomena in simple gases is concluded, and numerical values of the gas coefficients for heat conductivity and viscosity are obtained as a function of the temperature and density for the particular case of molecules acting as rigid elastic spheres. In the Introduction will be found a qualitative discussion of the principal results obtained and an interpretation of these results according to elementary considerations. In Section 1 the method of solution of the integral equations with which the formal theory of Part I was concluded is given together with the resulting general equations for the heat conductivity and viscosity coefficients. Since the integral equations can be solved only by a method of successive approximations, the expressions for the gas coefficients are in the form of infinite series the rapidity of convergence of which depends on a suitable choice of a complete set of auxiliary functions. In Section 2 all of the integrals appearing in the first two terms of the infinite series are evaluated. That only two terms are required is due to the fact that one is able to make an excellent choice of functions to represent the auxiliary set. In the evaluation of these integrals restriction is made in the application of the theory to small values of the degeneracy parameter $A$, since only terms in the zeroth and first power of $A$ are retained. This restriction is only slightly greater than that imposed by the fundamental postulates of the general theory which restrict its applicability to moderately rare gases. In Section 3 the final equations for the gas coefficients are applied to gases consisting of molecules which interact quantum-mechanically as rigid elastic spheres of diameters and masses associated with the gases helium and hydrogen.

Ballistic-Diffusive Heat-Conduction Equations

Abstract: We present new heat-conduction equations, named ballistic-diffusive equations, which are derived from the Boltzmann equation. We show that the new equations are a better approximation than the Fourier law and the Cattaneo equation for heat conduction at the scales when the device characteristic length, such as film thickness, is comparable to the heat-carrier mean free path and/or the characteristic time, such as laser-pulse width, is comparable to the heat-carrier relaxation time.

Monte Carlo simulation of the atomic master equation for spontaneous emission.

Abstract: A Monte Carlo simulation of the atomic master equation for spontaneous emission in terms of atomic wave functions is developed. Realizations of the time evolution of atomic wave functions are constructed that correspond to an ensemble of atoms driven by laser light undergoing a sequence of spontaneous emissions. The atomic decay times are drawn according to the photon count distribution of the driven atom. Each quantum jump of the atomic electron projects the atomic wave function to the ground state of the atom. Our theory is based on a stochastic interpretation and generalization of Mollow's pure-state analysis of resonant light scattering, and the Srinivas-Davies theory of continuous measurements in photodetection. An extension of the theory to include mechanical light effects and a generalization to atomic systems with Zeeman substructure are given. We illustrate the method by simulating the solutions of the optical Bloch equations for two-level systems, and laser cooling of a two-level atom in an ion trap where the center-of-mass motion of the atom is described quantum mechanically.

Generalised P-representations in quantum optics

Abstract: A class of normal ordering representations of quantum operators is introduced, that generalises the Glauber-Sudarshan P-representation by using nondiagonal coherent state projection operators. These are shown to have practical application to the solution of quantum mechanical master equations. Different representations have different domains of integration, on a complex extension of the usual canonical phase-space. The 'complex P-representation' is the case in which analytic P-functions are defined and normalised on contours in the complex plane. In this case, exact steady-state solutions can often be obtained, even when this is not possible using the Glauber-Sudarshan P-representation. The 'positive P-representation' is the case in which the domain is the whole complex phase-space. In this case the P-function may always be chosen positive, and any Fokker-Planck equation arising can be chosen to have a positive-semidefinite diffusion array. Thus the 'positive P-representation' is a genuine probability distribution. The new representations are especially useful in cases of nonclassical statistics.

Coarse master equations for peptide folding dynamics

Abstract: We construct coarse master equations for peptide folding dynamics from atomistic molecular dynamics simulations. A maximum-likelihood propagator-based method allows us to extract accurate rates for the transitions between the different conformational states of the small helix-forming peptide Ala5. Assigning the conformational states by using transition paths instead of instantaneous molecular coordinates suppresses the effects of fast non-Markovian dynamics. The resulting master equations are validated by comparing their analytical correlation functions with those obtained directly from the molecular dynamics simulations. We find that the master equations properly capture the character and relaxation times of the entire spectrum of conformational relaxation processes. By using the eigenvectors of the transition rate matrix, we are able to systematically coarse-grain the system. We find that a two-state description, with a folded and an unfolded state, roughly captures the slow conformational dynamics. A f...