content_image/332655.jpg content_image/332654.jpg content_image/332653.jpg
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 

<img src='content_image/332655.jpg'>

The divergences of this action can be isolated by integrating by parts twice 

<img src='content_image/332654.jpg'>

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 

<img src='content_image/332653.jpg'>

