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GREEN'S RETARDED AND ADVANCED FUNCTIONS IN STATISTICAL PHYSICS

N. N. Bogolyubov, Sergei Vladimirovich Tyablikov

Published May 1, 1959• 1 citation
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Abstract

An approximation method is outlined for resolving problems in statistical physics, based on the analysis of Green's two-dimensional retarded and advanced functions, with the application of the Lehmann-Callen spectral representation. (R.V.J.) with alternating conductance, when the magnetic field rises in transition through the shock wave. (R.V.J.)

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An approximation method is outlined for resolving problems in statistical physics, based on the analysis of Green's two-dimensional retarded and advanced functions, with the application of the Lehmann-Callen spectral representation. (R.V.J.) with alternating conductance, when the magnetic field rises in transition through the shock wave. (R.V.J.)
In this work we will establish the definitions of the Green’s functions in statistical mechanics and their basic properties. These functions depend on double-time and temperature. It can be observed by mean of theier definitions, where we can find the medium values of the operators product, this average is done over the grand canonical ensemble, for instance, (FAB(t − t0) ≡ hAˆ(t)Bˆ(t0)i ou FBA(t − t0) ≡ hBˆ(t0)Aˆ(t)i), with β = 1/kBT, and T is the absolute temperature. The operators involved in theses functions satisfy the Heisenberg equation of motion, which permit us describe the evolution equations for Green’s functions. By mean of spectral representation of the time correlation functions(FAB(t−t0) ≡ hAˆ(t)Bˆ(t0 )i or FBA(t−t0) ≡ hBˆ(t0)Aˆ(t)i), that is done by the introduction of a Fourier transform to change the system from time-space to frequency-space, we can obtain the spectral representations for the retarded Gr, advanced Gr e causal Gc green’s functions. Lastly, we will make the use of retarded green’s function to describe the electrical conductivity of a system of electrons under a external time-dependent electric field, in others words, we will write the electrical conductivity tensor in terms of Gr and, at last, we will calculate the electrical conductivity of a system of electrons and phonons.