Main equation

In Physical, a main equation is a differential equation describing the temporal evolution of a system. It is a equation of rate for the states of the system.

The evolution of the probability $P\_k$ of being in the discrete state K follows an equation of the type:

$\backslash \; frac\; \{dP\_k\}\; \{dt\}\; =\; \backslash \; sum\_\; \backslash \; ell\; \backslash \; Gamma\_\; \{K\; \backslash \; ell\}\; P\_\; \backslash \; ell,$

The matrix $\backslash \; Gamma\_\; \{\backslash \; ell\; K\}$ is sometimes called matrix of the rates of transitions.

This equation is found in mathematics during the probabilistic treatment of the Chaîne of Markov.

As the total probability preserves $\backslash \; sum\_\; \{\backslash \; ell\}\; \backslash \; Gamma\_\; \{\backslash \; ell\; K\}\; =\; 0$

and the main equation can thus be rewritten

$\backslash \; frac\; \{dP\_k\}\; \{dt\}\; =\; \backslash \; sum\_\; \backslash \; ell\; (\backslash \; Gamma\_\; \{K\; \backslash \; ell\}\; P\_\; \backslash \; ell\; -\; \backslash \; Gamma\_\; \{\backslash \; ell\; K\}\; P\_k).$

This form directly makes it possible to see the starting rates $\backslash \; Gamma\_\; \{\backslash \; ell\; K\}$ of the state K and the rates of arrived $\backslash \; Gamma\_\; \{K\; \backslash \; ell\}$ towards this state.

A generalization of this equation is the equation of Fokker-Planck for the evolution of an infinite number of state K.

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