C. Example: The otherwise free particle In summary, we have determined an expression for the reduced density matrix of a system (consisting of a massive particle in a potential) interacting with a bath of oscillators. The coupling to the bath has been chosen in order to reproduce the results of classical Brownian motion in the high temperature limit. We have taken care to make the term in the expo- nential finite, by isolating the divergences through integration by parts. This is important for path integral Monte Carlo simulation to be possible for this functional integral. The reduced density matrix for a particle interacting with such a bath can be determined analytically for the otherwise free particle ( V$_{R}$ ( x ) = 0). First, write an arbitrary path as an expansion around the classical solution One can find an analytic solution by substituting this into Eq. 12 . We skip the details: after evaluating numerous integrals using contour integration, and then changing variables for the integration over the even Fourier coefficients, we find the reduced density matrix where Ψ( x ) is the digamma function and the overall normalization is determined by analytic continuation of β and requiring the propagator to conserve probability for purely imaginary β . One interesting physical result is easily obtained from the Fourier transform of the reduced density matrix Obtaining the analytic result for our reduced density matrix was reasonable for the oth- erwise free particle. For the simple harmonic oscillator, the analytic result exists but has a D. The path-integral Monte Carlo algorithm for the reduced density matrix 7 $^{〈}$p 2 〉 = M$_{eff}$ β , M$_{eff}$ = M + ηβ π [ log$^{(}$ ηβ 2 πM$^{)}$ − Ψ$^{(}$1 + ηβ 2 πM$^{)]}$ . (15) ρ$_{red}$ ( x$_{i}$, x$_{f}$ , β ) = √ M 2 πβ + η 2 π 2 [ log(2) + γ$_{E}$ + Ψ$^{(}$1 + ηβ 2 πM$^{)]}$ × exp$^{{}$ −$^{[}$ M 2 β + η 2 π [ log$^{(}$ ηβ 2 πM$^{)}$ − Ψ$^{(}$1 + ηβ 2 πM$^{)]]}$ ( x$_{i}$ − x$_{f}$ ) 2 } , (14) x ( τ ) = x$_{cl}$ ( τ ) + ξ ( τ ); x$_{cl}$ ( τ ) ≡ x$_{i}$ + ( x$_{f}$ − x$_{i}$ ) τ /β, ξ ( τ ) ≡ ∞ ∑ n =1 c$_{n}$ sin$^{(}$ nπτ β ) . (13)