Fluctuation theorems master equation

Here a is the differential cross-section, and depends only on Pi Pi = l/>3 Pa and on (/U - p2) p2 Pa)-The precise number of molecules in the cell fluctuates around the value given by the Boltzmann equation, because the collisions occur at random, and only their probability is given by the Stosszahlansatz. Our aim is to compute these fluctuations. If / differs little from the equilibrium distribution one may replace the Boltzmann equation by its linearized version. It is then possible to include the fluctuations by adding a Langevin term, whose strength is determined by means of the fluctuation-dissipation theorem.510 As demonstrated in IX.4, however, the Langevin approach is unreliable outside the linear domain. We shall therefore start from the master equation and use the -expansion. The whole procedure consists of four steps. 

We have found that even in these simple cases things can go wrong. A physically reasonable Master Equation may not be of the Lindblad form. The corresponding Lindblad form may, on the other hand, violate simple rules like the fluctuation-dissipation theorem. 

We have thus seen that the requirement that the friction y and the random force 7 (Z) together act to bring the system to thennal equilibrium at long time, naturally leads to a relation between them, expressed by Eq. (8.20). This is a relation between fluctuations and dissipation in the system, which constitutes an example of the fluctuation-dissipation theorem (see also Chapter 11). In effect, the requirement that Eq. (8.20) holds is equivalent to the condition of detailed balance, imposed on transition rates in models described by master equations, in order to satisfy the requirement that thermal equilibrium is reached at long time (see Section 8.3). 

If they are constants with time, it can be shown (e.g. by using Boltzmann s famous H-theorem) that all distributions developing according to the master equation (3.14) or the Fokker-Planck equation (3.32) or (3.34) end up in the stationary distribution W (jc). After this the motion of a sample socio-configuration can only consist of stationary fluctuations around the probability peaks of the stationary distribution. While this kind of motion may describe a static society under stable psychological and situational conditions, it does not comprise the full scope of sociological phenomena. 

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