Birth-death master equation

RNApolymerase molecules are involved in the process. If the system is well stirred so that spatial degrees of freedom play no role, birth-death master equation approaches have been used to describe such reacting systems [33, 34]. The master equation can be simulated efficiently using Gillespie s algorithm [35]. However, if spatial degrees of freedom must be taken into account, then the construction of algorithms is still a matter of active research [36-38]. 

The function yields the stationary probability distribution of a stochastic, birth-death master equation for single variable systems... 

In this chapter we formulate the thermodynamic and stochastic theory of the simple transport phenomena diffusion, thermal conduction and viscous ffow (1) to present results parallel to those listed in points 1-7, Sect. 8.1, for chemical kinetics. We still assume local equilibrium with respect to translational and internal degrees of freedom. We do not assume conditions close to chemical or hydrodynamic equilibrium. For chemical reactions and diffusion the macroscopic equations for a given reaction mechanism provide sufficient detail, the fluxes in the forward and reverse direction, to write a birth-death master equation with a stationary solution given in terms of For thermal conduction and viscous flow we derive the excess work and then find Fokker-Planck equations with stationary solutions given in terms of that excess work. 

We assume that the fluctuations in this system are given by a birth-death master equation... 

The birth-death master equation in terms of the numbers of species X and Y is... 

Since the formal chemical kinetics operates with large numbers of particles participating in reaction, they could be considered as continuous variables. However, taking into account the atomistic nature of defects, consider hereafter these numbers N as random integer variables. The chemical reaction can be treated now as the birth-death process with individual reaction events accompanied by creation and disappearance of several particles, in a line with the actual reaction scheme [16, 21, 27, 64, 65], Describing the state of a system by a vector N = TV),..., Ns, we can use the Chapmen-Kolmogorov master equation [27] for the distribution function P(N, t)... 

It is of importance to point out that if the right-hand side is truncated after two terms (diffusion approximation), the last relation leads to an expression similar to the familiar Fokker-Planck equation (4.116). The approximation of a master equation of a birth-death process by a diffusion equation can lead to false results. Van Kampen has critically examined the Kramers-Moyal expansion and proposed a procedure based on the concept of system size expansion.135 It can be stated that any diffusion equation can be approximated by a one-step process, but the converse is not true. 

The continued fraction representation of the transition factor has been applied for solving one-variable chemical master equations (Haag Hanggi, 1979, 1980). For simple birth and death processes with birth and death rate functions / and x the nearest neighbour transition gj is ... 

In the birth and death formalism the probability p(n,t) to have n particles at time t in the homogeneous reactive system obeys the Master Equation ... 

The time evolution of an inhomogenous bistable reactive system in absence of convection is studied in the birth and death formalism. With the aid of multivariate Master Equations it is shown in simple cases that spatial homegenity can be spontaneously breaken during the passage from metastable to stable state a theory of nucleation is presented, which allows the evaluation of the nucleation rate. 

On the other hand in the chemical explosion case [8] we deal with a birth and death process for which the Master equation reads... 

We turn now to the microscopic aspects of the dynamics, in which fluctuations are incorporated in the description. Using as in Section 3 the fact that the birth and death transition probabilities are proportional to the frequencies of reactive encounters [11] we write the explicit form of the master equation as... 

In this section the master equation and the mean value equations for the general migration and birth-death process of interacting populations, proceeding along the lines of Chap. 3, will be set up. 

In this section pure birth-death processes will be discussed. Although such models have been dealt with extensively in the literature (see, for instance [4.1-8]), there still exist some seldom considered problems referring to the relation between the exact stationary or quasi-stationary solution of the master equation and the deterministic approach. This relation will be treated in Sect. 4.3.1 including an appUcation of the results obtained to observations on animal populations [4.15,17]. Further, the generalization of the stochastic standard model by including multistep birth or death processes will be investigated in Sect. 4.3.2. 

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