By N. H. March, M. P. Tosi (auth.)

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44) and use the identity - 1 -. 46) Liquid dynamics and time-dependent correlation functions 45 This is the precise statement of the fluctuation-dissipation theorem in the present case. Of course, the imaginary part of the response function describes energy dissipation in the system, while its real part describes reversible polarisation processes. (r', t') at times t' later than t, x(r, t) must vanish when t < 0. 43). In turn, this mathematical statement implies the existence of a dispersion relation of the form x(k, w)=~ Joo dw' 7r - 00 ~"(k, w').

The idea behind it is to use the fJk as collective coordinates. For them to afford an accurate approximation we must be able to express the Hamiltonian in terms of them and the corresponding momenta, and then to treat them as if they were 'almost' independent. There are 3N coordinates, R 1 ••• RN, in the original Hamiltonian and while, in a finite system with, say, periodic boundary conditions imposed over a large cube of side L, k has discrete though dense values, all such discrete k are allowed in enumerating the fJk.

Hydrodynamic forms of correlation functions and generalised hydrodynamics 57 We may now ask whether ¢(t) is related to the dissipative response of the fluid to some external perturbation, just as F,(k, t) is related to incoherent neutron inelastic scattering. 7) dz dz the first term being the diffusion current which is present even in the absence of external field provided that a density gradient dp(z)/dz is maintained in the fluid. 5) follows. 8) This relation shows that the mobility, which describes the response of the particles of the fluid to an applied field, is determined by the velocity correlations in time, which are a property of the fluid in the absence of the field.

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