One-Point Non-Equilibrium Correlation Functions

We turn to the non-equilibrium moments of co-ordinates and velocities of linear macromolecules. As a first step, we shall consider one-point second-order moments. The expressions for co-ordinates and velocities (4.7) with the same arguments can be used to make up proper combinations, and by averaging over the ensemble of realisation of random forces, we find the moments with accuracy to the first-order terms in velocity gradients. Then, by taking into account the properties of equilibrium moments and the antisymmetry of tensor cun, we find that 

The last expression can be simplified in the case, when the inertial forces acting on the Brownian particles are unimportant, that is m = 0. This is the only case that is of interest for application. In this case, by taking expressions (4.4) and (4.14) into account, we find the auxiliary relation 

1 To keep the correct expression for correlation functions when limit rn 0 is approached, it is convenient to use the function of a non-negative argument 

So as the moments (4.17) and (4.18) are expressed in terms of the functions M(s) and /t(.s), it is convenient to express the third moment in these functions as well. After calculating, we obtain 

We see that the non-equilibrium moments are expressed in terms of the equilibrium moment of co-ordinate M(s) and its derivative, which were determined in the previous section by their Fourier transforms. 

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