Callen–Welton formula for slow temporal and spatial processes

Using the Langevin approach and the multiscale technique, a kinetic theory of the time and space non-local fluctuations in the collisional plasma is constructed. In local equilibrium a generalized version of the Callen–Welton theorem is derived. It is shown that not only the dissipation but also the time and space derivatives of the dispersion determine the amplitude and the width of the spectrum lines of the electrostatic field fluctuations, as well as the form factor. There appear significant differences with respect to the non-uniform plasma. In the kinetic regime the form factor is more sensible to space gradient than the spectral function of the electrostatic field fluctuations. As a result of the inhomogeneity, these properties became asymmetric with respect to the inversion of the frequency sign. The differences in amplitude of peaks could become a.o. a new tool to diagnose slow time and space variations in the plasma.