R v v ( t 1 , t 2 ) ≡ ≡ ⟨ ⟨ v ( t 1 ) ⋅ ⋅ v ( t 2 ) ⟩ ⟩ = v 2 ( 0 ) e − − ( t 1 + t 2 ) / τ τ + ∫ ∫ 0 t 1 ∫ ∫ 0 t 2 R a a ( t 1 ′ , t 2 ′ ) e − − ( t 1 + t 2 − − t 1 ′ − − t 2 ′ ) / τ τ d t 1 ′ d t 2 ′ ≃ ≃ v 2 ( 0 ) e − − | t 2 − − t 1 | / τ τ + [ 3 k B T m − − v 2 ( 0 ) ] [ e − − | t 2 − − t 1 | / τ τ − − e − − ( t 1 + t 2 ) / τ τ ] , where we have used the property that the variables a ( t 1 ′ ) and a ( t 2 ′ ) become uncorrelated for time separations t 2 ′ − − t 1 ′ ≫ ≫ t c . Besides, the value of lim t → → ∞ ∞ ⟨ ⟨ v 2 ( t ) ⟩ ⟩ = lim t → → ∞ ∞ R v v ( t , t ) is set to be equal to 3 k B T / m such that it obeys the equipartition theorem. If the system is initially at thermal equilibrium already with v 2 ( 0 ) = 3 k B T / m , then ⟨ ⟨ v 2 ( t ) ⟩ ⟩ = 3 k B T / m for all t , meaning that the system remains at equilibrium at all times.