Fluctuation-dissipation theorems classical limit

The fluctuation dissipation theorem relates the dissipative part of the response fiinction (x") to the correlation of fluctuations (A, for any system in themial equilibrium. The left-hand side describes the dissipative behaviour of a many-body system all or part of the work done by the external forces is irreversibly distributed mto the infinitely many degrees of freedom of the themial system. The correlation fiinction on the right-hand side describes the maimer m which a fluctuation arising spontaneously in a system in themial equilibrium, even in the absence of external forces, may dissipate in time. In the classical limit, the fluctuation dissipation theorem becomes / /., w) = w). 

Equation (20) (or Eq. (21) in the classical limit) constitute the formulation of the fluctuation-dissipation theorem of the second kind or second FDT (using the Kubo terminology [30,31]). This theorem applies to the random force F(t), which is a bath dynamical variable. It expresses the fact that the bath is in equilibrium. 

Let us first consider spectroscopy. Linear-response theory, in particular the fluctuation dissipation theorem - which relates the absorption of an incident monochromatic field to the correlation function of (e.g. dipole) fluctuations in equilibrium - has changed our perspective on spectroscopy of dense media. It has moved away from a static Schrodinger picture -phrased in terms of transitions between immutable (but usually incomputable) quantum levels - to a dynamic Heisenberg picture, in which the spectral line shape is related by Fourier transform to a correlation function that describes the decay of fluctuations. Of course, any property that cannot be computed in the Schrodinger picture, cannot be computed in the Heisenberg picture either however, correlation functions, unlike wave-functions, have a clear meaning in the classical limit. This makes it much easier to come up with simple (semi) classical interpretations and approximations. 

The friction coefficient is the inverse particle s relaxation time, jS = 9py/(2pp ), where py is the fluid s dynamic viscosity. Since the Langevin equations are linear, particle velocity and position may be formally solved as functionals of the random force, and in the diffusive limit f >> i. e., for times much larger than the particle relaxation time, they allow for the analytical evaluation of ensemble averaged products of particle position and velocity and two-point correlation functions, in terms of the random-force strength q. The authors carefully justify why they use the classical (equilibrium) form of the fluctuation-dissipation theorem (FDT) in a Langevin description the time scale of the white noise is considered to be much shorter than the time scale of the imjxjsed flow. Thus, the non-equilibrium corrections would be of the order of the ratio of the fluid molecular relaxation time to the time scale of the imposed shear and may be neglected. In this case both the time scales are clearly separated and q may be determined solely from the classical form of the FDT,... 

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