Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1023Let ρ(y i , τ|x i , t) with ρ(y i , t|x i , t) = δ(y i − x i ) describe a diffusion processfor τ > t defined by the Fokker–Planck equation∂ τ ρ = H † ρ = − V λ ρ − ∂ x i(b iρ) + 1 2 ν ij∂ x i x j J(xi , t)ρ (6.29)with H † the Hermitian–conjugateof H. Then A(τ) = ∫ dy i ρ(y i , τ|x i , t)ψ(y i , τ) is independent of τ and inparticular A(t) = A(t f ). It immediately follows that∫ψ(x i , t) = dy i ρ(y i , t f |x i , t) exp(−φ(y i )/λ) (6.30)We arrive at the important conclusion that ψ(x i , t) can be computed eitherby backward integration using (6.22) or by forward integration of a diffusionprocess given by (6.29).We can write the integral in (6.30) as a path integral. Following [Kappen(2006)] we can divide the time interval t → t f in n 1 intervals and writeρ(y i , t f |x i , t) = ∏ n 1i=1 ρ(xi i , t i|x i i−1 , t i−1) and let n 1 → ∞. The result is∫(ψ(x i , t) = [dx i ] x i exp − 1 )λ S(xi (t → t f )) (6.31)with ∫ [dx i ] x i an integral over all paths x i (t → t f ) that start at x i and with∫ tfS(x i (t → t f )) = φ(x i (t f )+ dτ( 1t 2 (ẋi −b i (x i , τ))R(ẋ i −b i (x i , τ))+V (x i , τ))(6.32)the Action associated with a path. From (6.27) and (6.31), the cost–to–goJ(x, t) becomes a log partition sum (i.e., a free energy) with temperatureλ.6.3.5.2 Monte Carlo SamplingThe path integral (6.31) can be estimated by stochastic integration from tto t f of the diffusion process (6.29) in which particles get annihilated at arate V (x i , t)/λ [Kappen (2006)]:x i = x i + b i (x i , t)dt + dξ i , with probability 1 − V dt/λx i = †, with probability V dt/λ (6.33)where † denotes that the particle is taken out of the simulation. Denotethe trajectories by x i α(t → t f ), (α = 1, . . . , N). Then, ψ(x i , t) and u i are