The Langevin Equation for x t

A Langevin equation for the motion of x (t) has the following general form  

Here the mean value ( ) has to be taken over the possible random forces occurring in the sample society via the stochastic transitions of its members. The stochastic nature of the random forces on the rhs of (2.35) in turn leads to non-deterministic, stochastic paths of the variable x(t) Paths x(t) develop differently under the influence of the stochastic forces (t) in (2.35) even if they start out from the same initial value jk (0). The mean deviation and mean square deviation of x (t) from the initial value after a short time interval At can, however, be calculated by integrating (2.35) iteratively over with the initial 

2 Opinion Formation - an Elementary Example of Semi-Quantitative Sociology 

This result should be compared with (2.33 b, 34 b) where the short time mean deviation and mean square deviation of x (divided by At) are calculated on the basis of the Fokker-Planck equation for P (x 0 and where it turns out that they agree with the drift coefficient and fluctuation coefficient, respectively, of the Fokker-Planck equation. Equating the rhs of (2.33 b, 37) as well as those of (2.34b, 38), the relations between the coefficients of the Fokker-Planck equation (2.26) and the Langevin equation (2.35) 

As discussed in the introduction it is quite often sufficient for the solution -except in the case of bifurcation - to estabhsh the paths of the mean values of relevant variables. It is easy to derive equations for (x), and (x ), from the Fokker-Planck equation (2.26) using the definition (2.32). Multiplying (2.26) with X and x respectively, after partial integration over the interval [-1,1] and taking into account the boundary condition (2.29) the exact equations are obtained as 

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