The Memory Function Formulation

Our calculation of C(12) is based on a kinetic equation, which in turn is derived from the equation of motion for C(12). The derivation consists of a rather 

To find the equation of motion satisfied by C(12), we use the operator identity 

The operator L acts on the space of internal coordinates. Using the 5-function properties of / one finds that the action of L on the phase space densities can expressed in terms of operators that act upon the external field variables. 

In the formalism of fully renormalized kinetic theory (FRKT) one directs one s attention not to the phase space correlation function 0(12) but to the associated memory function (12), which is defined by  

The memory function is of interest because we expect it to be a less varied function of its arguments than the correlation function. The Fourier transform of the correlation function is resonant (has a pole structure) in the hydro-dynamical regime (small k and a ). The memory function, however, has a simple behavior in this regime and gives the placement and width of the hydrodynamical poles. Since z —Lq(1) is just the operator that describes the streaming of a single free particle, it is clear that the memory function describes the effects of the other N— 1 particles on this free-streaming particle. One may say that the memory function serves to renormalize the free-particle motion in the system. Since the memory function has to describe the effects of many-particle dynamics, it is not a simple quantity to calculate. 

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