Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1017developed into the more useful probability distribution for the order parametersM G at long–time macroscopic time event t = (u + 1)θ + t 0 , interms of a Stratonovich–Riemannian path–integral over mesoscopic Gaussianconditional probabilities. Here, macroscopic variables are defined asthe long–time limit of the evolving mesoscopic system.The corresponding Schrödinger–type equation is∂ t P = 1 2 (gGG′ P ), GG ′ −(g G P ), G +V, (6.20)g GG′ = k T δ jk ĝ G j ĝ G′k ,g G = f G + 1 2 δjk ĝ G′j ĝ G k,G ′,[.], G = ∂[.]/∂M G .This is properly referred to as a Fokker–Planck equation when V = 0.Note that although the partial differential equation (6.20) contains equivalentinformation regarding the order parameters M G as in the stochasticdifferential equation (6.19), all references to j have been properly averagedover, i.e., ĝjG in (6.19) is an entity with parameters in both microscopic andmesoscopic spaces, but M is a purely mesoscopic variable, and this is moreclearly reflected in (6.20).Now, the path integral representation is given in terms of the LagrangianL = L(Ṁ G , M G , t), as∫P [M t |M t0 ] dM(t) = D[M] exp(−A[M])δ[M(t 0 ) = M 0 ]δ[M(t) = M t ],where the action A[M] is given by(6.21)∫ tA[M] = k −1Tmin dt ′ L(Ṁ G , M G , t),t 0and the path measure D[M] is equal toD[M] = limu→∞The Lagrangian L is given byu+1∏ρ=1g 1/2 ∏ G(2πθ) −1/2 dM G ρ .L(Ṁ G , M G , t) = 1 2 (Ṁ G − h G )g GG ′(Ṁ G′ − h G′ ) + 1 2 hG ;G + R/6 − V,