the force autocorrelator for the heavy particle is proportional to δ ( t − t $^{′}$) at high temperatures. In this “white noise” limit, the density matrix describes an ensemble of particles interacting according to the Langevin equation, which has been used to describe Brownian motion for a long time. It is a stochastic differential equation, is irreversible, and evolves any ensemble towards the thermal phase space distribution. We now take this density of states and substitute it into Eq. 8 The divergences of this action can be isolated by integrating by parts twice where lnsin( x ) ≡ ln [sin ( x )] and γ$_{E}$ is the Euler-Mascheroni constant. The first two terms on the right-hand side correspond to a renormalization of the potential for the heavy par- ticle, always necessary when considering the interaction of a particle with infinitely many additional degrees of freedom. They are temperature-independent in the limit of large Ω, and may be introduced as temperature-independent modifications of the Lagrangian for the particle. The final three terms are finite and well-behaved, and have been evaluated in the limit Ω → ∞ . The final form for the reduced density matrix becomes 6 ρ$_{red}$ ( x$_{i}$, x$_{f}$ , β ) = ∫ x ( β )= x$_{f}$ x (0)= x$_{i}$ D x exp$^{{}$ − S E S [ x ] − η 2 π ( x$_{i}$ − x$_{f}$ ) 2 [ γ$_{E}$ + ln$^{(}$ ηβ πM )] + η π ( x$_{i}$ − x$_{f}$ )$^{∫}$ β 0 dτ ˙ x ( τ ) lnsin$^{(}$ πτ β ) + η π ∫ β 0 dτ ∫ τ 0 ds ˙ x ( τ ) ˙ x ( s ) lnsin$^{(}$ π ( τ − s ) β ) } . (12) ∫ Ω 0 dω ∫ β 0 dτ ∫ τ 0 ds x ( τ ) x ( s ) ω cosh [ ω ( τ − s − β/ 2)] sinh( ωβ 2 ) = Ω$^{∫}$ β 0 dτ ( x ( τ )) 2 − 1 2 ( x$_{i}$ − x$_{f}$ ) $^{2}$ln( M Ω /η ) cosh(Ω β/ 2) − 1 sinh(Ω β/ 2) − 1 2 ( x$_{i}$ − x$_{f}$ ) 2 [ γ$_{E}$ + ln$^{(}$ ηβ πM )] + ( x$_{i}$ − x$_{f}$ )$^{∫}$ β 0 dτ ˙ x ( τ ) lnsin$^{(}$ πτ β ) +$^{∫}$ β 0 dτ ∫ τ 0 ds ˙ x ( τ ) ˙ x ( s ) lnsin$^{(}$ π ( τ − s ) β ) , (11) ρ$_{red}$ ( x$_{i}$, x$_{f}$ , β ) = ∫ x ( β )= x$_{f}$ x (0)= x$_{i}$ D x × exp$^{(}$ − S E S [ x ] + η π ∫ Ω 0 dω ∫ β 0 dτ ∫ τ 0 ds x ( τ ) x ( s ) ω cosh [ ω ( τ − s − β/ 2)] sinh( ωβ 2 ) ) . (10)