Master equation multivariate

It was demonstrated in X.5 that the O-expansion applies to multivariate master equations, provided that the macroscopic equations possess a single stationary solution, and that it is globally stable. The difference with the one-variable case was that no general method exists to solve the macroscopic equations. With respect to unstable situations, however, there is the added difference that the variety of possible instabilities is much larger than for a single variable. 

The treatment of our example was facilitated by the fact that the master equation (3.1) for the reaction part was linear. Now consider again the reaction (X.1.1), studied in X.l for the case of perfect stirring. In order to obtain equations for the first and second moments one has to apply the Q-expansion to each separate cell, i.e., one has to expand the multivariate master equation (1.1) in powers of A 1/2. This imposes an additional condition on the cell size that A must be large enough to contain many particles. 

So far we only considered transport of particles by diffusion. As mentioned in 1 the continuous description was not strictly necessary, because diffusion can be described as jumps between cells and therefore incorporated in the multivariate master equation. Now consider particles that move freely and should therefore be described by their velocity v as well as by their position r. The cells A are six-dimensional cells in the one-particle phase space. As long as no reaction occurs v is constant but r changes continuously. As a result the probability distribution varies in a way which cannot be described as a succession of jumps but only in terms of a differential operator. Hence the continuous description is indispensable, but the method of compounding moments can again be used. 

We start from the multivariate master equation for the probability distribution of number of particles in local elements. The local concentration of a species X in a volume element at the position r is denoted by x(f) and its statistical average is denoted by x which is assumed to be independent of the position due to the assumption of a homogeneous steady state. Then, the covariance of the fluctuations... 

Nicolis, G. Malek-Mansour, M. (1980). Systematic analysis of the multivariate master equation for a reaction-diffusion system. J. Stat. Phys., 22, 495-512. 

To go beyond these qualitative arguments, we turn to the Multivariate Master Equation, eq. (1). We may easily derive an equation for the spatial correlation function, eq. (6) by neglecting third and higher order moments. This Gaussian type of approximation is expected to be valid well before bifurcation, as pointed out in the previous section. One obtains ... 

As for the stochastic kinetics of the reaction, it is described by a multivariate Master Equation which cannot be solved exactly even its stationary solution is unknown. The continuous approximations only yield local information on the probability distribution, which are not sufficient to study the relaxation between the macroscopic stationary states, at least far from the threshold of bistability. 

According to the multivariate Master Equation formalism the probability p(n-, n2 t) of having n molecules in cell 1 and n2 molecules in cell 2 at time t satisfies the 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. 

The system according to the general formalism of 2.3 is divided into N cells and the probability p(n. ..nj t) to find n particles X in cell i at time t 1 = 1. ..N obeys the Multivariate Master Equation (9). In order to proceed on we have to forget the discrete nature of n we describe the state of cell i by its concentration x = n /Q =en Q = 1/e being the volume of a cell cund we consider x. as a continuous variable. Expressing Eq. (9) with the x and expanding it in powers of we get an approximate continuous Multivariate Master Equation ... 

The stochastic description of a compartmental system is usually performed in terms of a multivariate master equation. For a chemical system this equation has the form... 

The numerical investigation of this equation shows that in the bistable regime the minimum in the diffusional dependence of the transition time between stationary states also occurs Cas in the case of the multivariate master equation ). Fig. 2 shows the results for the Schlogl model Cthe system size is 100 and 200D. 

We turn now to the stochastic simulation of the multivariate master equation along the lines of Section 3.2. Figure 21 depicts the time evolution of the local variance of X at the middle of the system. As in Section 4.1 we... 

The problem of fluctuation lissipation relations in multivariable systems is analyzed in [15] the mathematics needed for that task goes beyond the level chosen for this book, and hence only a brief verbal smnmary is presented. A statistical ensemble is chosen, which consists of a large number of replicas of the system, such as for example the Selkov model, each characterized by different composition vectors. There exists a master equation for this probability distribution of this ensemble, which serves as a basis for this approach an analytical solution of this master equation is given in [15]. 

The theoretical method developed here provides a rigorous approach to the description of the internal dynamics of flexible aliphatic tails. The treatment is able to link the master equations used in connection with the RIS approximation to the multivariate Fokker Planck or diffusive equations, avoiding loosely defined phenomenological parameters. 

Standardizing the spectral response is mathematically more complex than standardizing the calibration models but provides better results as it allows slight spectral differences - the most common between very similar instruments - to be corrected via simple calculations. More marked differences can be accommodated with more complex and specific algorithms. This approach compares spectra recorded on different instruments, which are used to derive a mathematical equation, allowing their spectral response to be mutually correlated. The equation is then used to correct the new spectra recorded on the slave, which are thus made more similar to those obtained with the master. The simplest methods used in this context are of the univariate type, which correlate each wavelength in two spectra in a direct, simple manner. These methods, however, are only effective with very simple spectral differences. On the other hand, multivariate methods allow the construction of matrices correlating bodies of spectra recorded on different instruments for the above-described purpose. The most frequent choice in this context is piecewise direct standardization... 

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