Classical FluctuationRelaxation Theorem
Abstract
A general expression is derived for the average infinitesimalimpulseresponse matrix of a conservative classical system in a canonical ensemble. The equations of motion are taken as ẋ_{n}=X_{n}, where the X's, as well as the energy E, are functions of the x's but not of their time derivatives. The Liouville equation Σ∂ẋ_{n}∂x_{n}=0 is assumed, but it is not required that the equations of motion be derivable from a Hamiltonian. If they are, the x's are the canonical coordinates and momenta. The result found is g_{mn}(τ)=Φ_{mn}(τ)kT, (τ>=0), where ∊g_{mn}(τ) is the average increment in x_{m} at time t resulting from an infinitesimal increment ∊ externally induced in x_{n} at time tτ, Φ_{mn}(τ) is the covariance <x_{m}(t)(∂E∂x_{n})'>, where the prime denotes argument tτ, k and T are Boltzmann's constant and absolute temperature. This relation is derived as a direct consequence of the fact that two initially isolated systems in equilibrium at identical temperatures remain in equilibrium when weakly coupled to each other.
 Publication:

Physical Review
 Pub Date:
 March 1959
 DOI:
 10.1103/PhysRev.113.1181
 Bibcode:
 1959PhRv..113.1181K