Solutions of the Equations in Relevant Cases

35) are incorporated into the master equation (4.18) and by using (4.22, 23, 28) in the mean value and variance equations (4.29, 31) as well as in the Fokker-Planck equation (4.32), the explicit form for all kinds of equations of motion is obtained. Relevant cases for which solutions can be obtained either analytically or by numerical methods will now be considered. 

Firstly the stationary solution p i (m, n) to the master equation has to be sought. This solution takes into account the most probable configurations (m, n) and the fluctuations around them in a stationary situation established by the migration process after the relaxation of a possible initial imbalance. 

Haken [4.12] has derived a form for the solution pst (0 to the general stationary master equation, i.e. to the equation 

By using (4.38) this result can be proved to be unique and therefore independent of the choice of the chain The normalization of the solution (4.39) finally 

Simultaneously conditions for w (j, i) equivalent to (4.38) but not containing the stationary solution can be derived from (4.39) For any closed loop 

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