Linear Response and Equilibrium Dynamics

Abstract

Dynamical properties of a many-body system specified by a Hamiltonian H₀ in thermal equilibrium are most frequently expressed in terms of time-dependent correlation functions of the form <A₁(t)A₂(t′)> or related functions. Thermal equilibrium implies translational invariance with respect to time: <A₁(t)A₂(t′)> = <A₁(t−t′)A₂(0)>. For a canonical ensemble this correlation function is defined as follows:

$$ < A_1 (t)A_2 (0) > _0 = \frac{1} {{Z_0 }}Tr[e^{ - \beta H_0 } e^{iH_0 t/\hbar } A_1 e^{ - iH_0 t/\hbar } A_2 ] , $$

(2.1)

where Z₀ = Tr[exp(-βH₀)] is the partition function and β = 1/k _B T the inverse temperature. Here A₁, A₂ are two operators in whose time-delayed correlations we are interested.