A dynamical monte carlo algorithm for master equations with time-dependent transition rates

A Monte Carlo algorithm for simulating master equations with time-dependent transition rates is described. It is based on a waiting time image, and takes into account that the system can become frozen when the transition rates tend to zero fast enough in time. An analytical justification is provided. The algorithm reduces to the Bortz-Kalos-Lebowitz one when the transition rates are constant. Since the exact evaluation of waiting times is rather involved in general, a simple and efficient iterative method for accurately calculating them is introduced. As an example, the algorithm is applied to a one-dimensional Ising system with Glauber dynamics. It is shown that it reproduces the exact analytical results, being more efficient than the direct implementation of the Metropolis algorithm     

On the master equation with time-dependent transition probabilities

The asymptotic behaviour of solutions of the master equation with time-dependent transition probabilities by means of the Kullback information is investigated. We assume that there are no absorbing states.     

A phase-consistent derivation of the electromagnetic multipole operators

The aim of this work is to draw attention to the precise formulation of theelectromagnetic-multipole transition operators and moments, especially the electric ones. Their precise structure fixes their transformations under some operations, as time reversal and hermitian conjugation, and in this way they play an important role as far as properties of their matrix elements are concerned.Another point of discussion is the consistent use of a well-defined phase convention for the angular-momentum eigenfunctions and their role in the development of the BCS formalism. A numerical example proves the importance of the use of well-defined phase-consistent eigenfunctions, operators, occupation probabilities, etc.     

On the derivation of thermohydrodynamic equations from the Boltzmann equations

The Boltzmann equation deals with a distributionf(x, ), wherex denotes the space variable and is the momentum. The hydrodynamic equations deal with-moments of the distribution. The paper deals with the derivation of the hydrodynamic equations in the case that the collision kernel is Maxwellian, i.e., independent of the velocity. For such a kernel, a computational tool, based on the theory of representations of the orthogonal group, is developed. With this tool it is possible to derive systems of equations for any number of moments. The construction of closed systems is based on asymptotic estimates for solutions of Boltzmann equations. These show that, in some definite sense, an approximating system involving moments of high order is more accurate than a system of lower order.     

Renormalization of composite operators in time-dependent backgrounds

We study the phenomenon of composite operator renormalization and mixing in systems where time-translational invariance is broken and the evolution is out-of-equilibrium. We show that composite operators mix also through non-local memory terms which persist for periods whose duration is set by the mass scales in the problem.     

Remarks concerning the derivation and the expansion of the master equation

The present work consist of two parts: In the first part we apply the method of quasilinearization to the differential equation describing the time development of the quantum-mechanical probability density. In this way we derive the master equation without resorting to perturbation theory. In the second part of the paper, for a general form of the master equation which is an integro-differential equation, we test the accuracy of the Fokker-Planck approximation with the help of a solvable model. Then we study an alternative way of reducing the integro-differential equation to a partial differential equation. By expanding the transition probability W(q, q′), and the distribution function in terms of a complete set of functions, we show that for certain forms of W(q, q′), the master equation can be transformed exactly to partial differential equations of finite order.     

On the relativistic spin projection operators

In the Rarita-Schwinger formalism, the relativistic spin projection operators are discussed with the help of the Pauli-Lubanski four-vector. It is shown that this approach is equivalent to the conventional one, but moreover, it enables one to derive recurrence relations for the spin projection operators. Such relations can be useful in practical applications.     

The Markovian master equations for unstable particles

The theory of Markovian master equations is applied to a certain model of unstable particles. The exponential decay law is obtained in the weak coupling limit. The connection to the method of one-parameter contracting semigroups on a single particle Hilbert space is given.     

Linearized master equations in the correlation-patterns representation

The evolution operator in the correlation-patterns representation is studied in the linear approximation. The linearized master evolution equations satisfied by the vacuum and the correlations are derived. The equations are applicable to inhomogeneous systems. The structure of the introduced operators is discussed.     

The Markov master equations and the Fermi golden rule

We give a proof that for a large class of systems weakly coupled to heat baths the transition probabilities per unit time obtained from the Markov approximation are equal to those that are calculated using the Fermi golden rule.     

On a simple derivation of master equations for diffusion processes driven by white noise and dichotomic Markov noise

A very simple way is presented of deriving the partial differential equations (the master equations) satisfied by the probability density for certain kinds of diffusion processes in one dimension, in which the driving term is a Gaussian white noise, or a dichotomic noise, or a combination of the two. The method involves the use of certain ‘formulas of differentiation’ to derive the equations obeyed by the characteristic functions of the processes concerned, and thence the corresponding master equations. The examples presented cover a substantial number of diffusion processes that occur in physical modelling, including some master equations derived recently in the literature for generalizations of persistent diffusion.     

The initial value problem of birth-death master equations

The initial value problem of birth-death master equations with a bistable potential is solved in terms of a new matching technique which takes two time regions instead of the three time regions of the scaling theory.     

Physically transparent derivation of the Hartree-Fock equations

A simple and physically transparent derivation of the Hartree-Fock equation is given. This derivation reveals the physical essence of this approximation.     

Solutions of the exact time-dependent microbending equations

Bandwidth improvement factor and losses of two-dimensional fibers with power-law index profiles, submitted to microbends with power-law curvature spectra, are given here in a graphical form. The domain of validity of the analytical solutions is discussed.     

Geometrical derivation of collective operators and paths in TDHF

It is first found that the intrinsic parity of an operator under time reversal and the interpretation of the operator as coordinate- or momentum-like in a TDHF calculation are not simply related. This is because the TDHF particle-hole basis is, in general, complex. The TDHF equation is then reformulated in the plane tangent to the Slater determinant manifold. This plane is spanned by the particle-hole basis. The particle-hole matrix elements of the Hartree-Fock Hamiltonian define the energy gradient vector in this tangent plane. This gradient is real when the Slater determinant is real. A TDHF calculation initiated from a real determinant induces, during the first infinitesimal time step, a purely imaginary variation of this determinant along the gradient. The gradient is thus identified with the matrix elements of a boost operator. The next infinitesimal time step defines, in turn, a displacement operator. These operators are retained as collective if the TDHF path is stable under changes of velocities. Various criteria are found for this stability condition. The theory cannot be applied straightforwardly to translations and rotations for there is no energy gradient to generate coordinate operators. Particle-hole matrix elements of boost operators can be obtained, however, by a multiplication by i of the matrix elements of displacement operators, since the latter are known explicitly. It is finally found that the rotation of a wavefunction is contradictory with angular momentum conservation in general. Conservation can be ensured by a rotation of the density only and a more elaborate evolution of the velocity field.