Fractional Feynman-Kac Equation for non-Brownian Functionals IntroductionResults Applications See also: L. Turgeman, S. Carmi, and E. Barkai, Phys. Rev. Lett. 103, (2009). Lior Turgeman, Shai Carmi, Eli Barkai Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel Random walk functionals A functional of a random walk: x(t) is the path, U(x) is some function. Occupation time- How long is the particle at x>0 ? Example: U(x)=Θ(x). Functionals in nature: Chemical reactions NMR Turbulent flow Surface growth Stock prices Climate Complexity of algorithms Brownian functionals G(x,A,t): the joint PDF of the particle to be at x and the functional to equal A. G(x,p,t): the Laplace transform of G(x,A,t) (A → p). For Brownian motion (normal diffusion: ~t), Feynman-Kac equation: Anomalous diffusion ~t α In many physical, biological, and other systems diffusion is anomalous. What is the equation for the distribution of non-Brownian functionals? Model: Continuous-time random-walk (CTRW) Lattice spacing a, jumps to nearest neighbors with equal probability. Waiting times between jumps distributed according to ψ(t)~t. -(1+α) For 0 ~t α. Fractional Feynman-Kac equation D t 1-α is the fractional substantial derivative operator In Laplace space (t → s), D t 1-α equals [s+pU(x)] 1- α. This is a non-Markovian operator- The evolution of G(x,p,t) depends on the entire history. Variants Backward equation: In the presence of a force field F(x), replace Laplacian with Fokker- Planck operator: Distribution of occupation times Consider the occupation time in half space, usually denoted with. Boundary conditions: For x → ∞, G(A,t)=δ(A-t)  G(p,s)=1/(s+p) (particle is always at x>0). For x → -∞, G(A,t)=δ(A)  G(p,s)=1/s (particle is never at x>0). The distribution of f ≡ T + /t, the fraction of time spent at x>0: The particle trajectory is almost never symmetric: It usually sticks to one side. Weak ergodicity breaking Consider the time average, where. Assume harmonic potential. For normal diffusion, the system is ergodic, that is for t → ∞:. For sub-diffusion, the time average is a random variable even in the long time limit - weak ergodicity breaking. Fluctuations in time average for t → ∞,. What are the fluctuations of the time average for all t? Use the Fractional Feynman-Kac equation: No fluctuations for α=1. α<1 : Fluctuations exist- the system does not uniformly sample all available states. Mittag-Leffler function Q n (x,A,t)dxdA: the probability to arrive into [(x,x+dx),(A,A+dA)] after n jumps. The time the particle performed the last jump in the sequence is (t- τ ). The particle is at (x,A) at time t if it was on [x,A- τ U(x)] at (t- τ ) and did not move since. The probability the particle did not move during (t- τ,t) is Thus, G and Q are related via: To arrive into (x,A) at t the particle must have arrived into either [x+a,A- τ U(x+a)] or [x-a,A- τ U(x-a)] at (t- τ ), and then jumped after waiting time τ. Thus, a recursion relation exists for Q n : Solving in Laplace-Fourier space and taking the continuum limit, a → 0, we get the Analysis