The fluctuation-dissipation theorem and the autocorrelation function of thermal radiation

Abstract

A new relatively simple derivation of the fluctuation-dissipation theorem (FDT) is presented. The generalized coordinate of the system is changed by an external force and is expressed by means of causal susceptibility, its Fourier transform – the transfer function, generalized impedance and active resistance. These characteristics describe heat dissipation on the resistor and the result is generalized to the dissipative system which is under the action of macroscopic force. The fluctuation voltage on the resistor is obtained by decomposing the thermal chaotic motion of free charges along the conductor into a Fourier series. The number of standing waves and the average energy of the quantum oscillation state at a fixed temperature give the thermal power of charge transfer. By comparing with the Joule-Lenz law and by generalizing the result to an arbitrary isothermal system, the mean square of the fluctuating force and dispersion of the generalized coordinate caused by the thermal motion are obtained. The autocorrelation functions of the generalized coordinate and the random force, and their spectral densities are expressed through the considered characteristics. The content of FDT is that the power of heat release, the spectral densities of the fluctuating force and the autocorrelation are proportional to the imaginary part of the transfer function of the system. The result is used for thermal radiation in a cavity the walls of which contain electric dipoles excited by thermal motion. The transfer function, the fluctuating force acting on the charge, the dispersion of the electric field strength, time autocorrelation of the electric field strength and its spectral density are obtained. Real and imaginary components, the modulus and phase are found for complex relative autocorrelation of the electric field strength and the coherence time is determined.