Equations for the Moments of Co-ordinates

In this section we refer to the stochastic equation (2.29) to calculate the mode moments, that is, the averaged values of the products of the normal co-ordinates and their velocities. It is convenient in this section to omit the label of mode and to rewrite the dynamic equation for the relaxation mode in the form of two linear equations 

To calculate second-order moments of co-ordinates and velocities, one can start with the rates of change of quantities that can be written as follows 

one can use equations (2.33) to obtain equations for the moments. After one has determined the averaged values of the products of the variables and the random force, the equations for the moments take the form 

It is easy to see that, at zeroth velocity gradients, the right-hand sides of the above equations are identically equal to zero. 

The set of equations (2.34)-(2.36) for the second-order moments of coordinates and velocities can be simplified, if we consider the situation when the distribution of velocities corresponds to equilibrium, that is, we put m — 0. In this case, equation (2.35) is followed by relation 

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