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SCIENTIA MA NU NTE E T ME Fractional Diffusion – Theory and Applications – Part I 22nd Canberra International Physics Summer School 2008 Bruce Henry (Trevor Langlands, Peter Straka, Susan Wearne, Claire Delides) School of Mathematics and Statistics The University of New South Wales Sydney NSW 2052 Australia Fractional Diffusion – Theory and Applications – Part I – p. 1/3 SCIENTIA The mathematical description of diffusion has many formulations: MA NU NTE E T ME conservation of mass and constitutive laws probabilistic models based on random walks and central limit theorems microscopic stochastic models based on Brownian motion and Langevin equations mesoscopic stochastic models based on master equations and Fokker-Planck equations Fractional Diffusion – Theory and Applications – Part I – p. 2/3 SCIENTIA The mathematical description of diffusion has many formulations: MA NU NTE E T ME conservation of mass and constitutive laws probabilistic models based on random walks and central limit theorems microscopic stochastic models based on Brownian motion and Langevin equations mesoscopic stochastic models based on master equations and Fokker-Planck equations The mean square displacement h(X − hXi)2 i of a diffusing particle scales linearly with time Fractional Diffusion – Theory and Applications – Part I – p. 2/3 SCIENTIA Numerous experimental measurements in spatially complex systems have revealed anomalous diffusion in which The mean square displacement scales as a fractional order power law in time MA NU NTE E T ME Fractional Diffusion – Theory and Applications – Part I – p. 3/3 SCIENTIA Numerous experimental measurements in spatially complex systems have revealed anomalous diffusion in which The mean square displacement scales as a fractional order power law in time MA NU NTE E T ME Over the past two decades a new mathematical description has been formulated, linked together by tools of fractional calculus: fractional constitutive laws probabilistic models based on continuous time random walks and generalized central limit theorems fractional Langevin equations, fractional Brownian motions fractional diffusion, fractional Fokker-Planck equations Fractional Diffusion – Theory and Applications – Part I – p. 3/3 A Triptych Overview SCIENTIA MA NU NTE E T ME Fractional Diffusion – Theory and Applications – Part I – p. 4/3 A Triptych Overview SCIENTIA MA NU NTE E T ME Brownian motion standard random walks standard diffusion Fractional Diffusion – Theory and Applications – Part I – p. 4/3 A Triptych Overview SCIENTIA MA NU NTE E T ME Brownian motion standard random walks standard diffusion fractional calculus Fractional Diffusion – Theory and Applications – Part I – p. 4/3 A Triptych Overview SCIENTIA MA NU NTE E T ME Brownian motion standard random walks standard diffusion fractional calculus fractional Brownian motion continuous time random walks fractional diffusion Fractional Diffusion – Theory and Applications – Part I – p. 4/3 Mathematical Models for Diffusion SCIENTIA MA NU NTE E T ME Having found motion in the particles of the pollen of all the living plants which I had examined, I was led next to inquire whether this property continued after the death of the plant, and for what length of time it was retained. Robert Brown (1828) Fractional Diffusion – Theory and Applications – Part I – p. 5/3 SCIENTIA Einstein’s Relations (1905) MA NU NTE E T ME It is possible that the movements to be discussed here are identical with the so-called “Brownian molecular motion” ; however, the information available to me regarding the latter is so lacking in precision, that I can form no judgment in the matter Fractional Diffusion – Theory and Applications – Part I – p. 6/3 SCIENTIA Einstein’s Relations (1905) MA NU NTE E T ME It is possible that the movements to be discussed here are identical with the so-called “Brownian molecular motion” ; however, the information available to me regarding the latter is so lacking in precision, that I can form no judgment in the matter The motion of the suspended particle arises as a consequence of random buffeting from the thermal motions of the enormous numbers of molecules that comprise the fluid Fractional Diffusion – Theory and Applications – Part I – p. 6/3 SCIENTIA Einstein’s Relations (1905) MA NU NTE E T ME It is possible that the movements to be discussed here are identical with the so-called “Brownian molecular motion” ; however, the information available to me regarding the latter is so lacking in precision, that I can form no judgment in the matter The motion of the suspended particle arises as a consequence of random buffeting from the thermal motions of the enormous numbers of molecules that comprise the fluid kB T = hr2 (t)i = 2Dt D = 6NRT Einstein Relations πaη γ Fractional Diffusion – Theory and Applications – Part I – p. 6/3 SCIENTIA Einstein’s Relations (1905) MA NU NTE E T ME It is possible that the movements to be discussed here are identical with the so-called “Brownian molecular motion” ; however, the information available to me regarding the latter is so lacking in precision, that I can form no judgment in the matter The motion of the suspended particle arises as a consequence of random buffeting from the thermal motions of the enormous numbers of molecules that comprise the fluid kB T = hr2 (t)i = 2Dt D = 6NRT Einstein Relations πaη γ γ = 6πηa (Stokes), T temperature of fluid, R = N kB gas constant, a particle radius, η fluid viscosity, N Avogadro’s number Fractional Diffusion – Theory and Applications – Part I – p. 6/3 SCIENTIA Langevin’s Equation (1908) d2 x m 2 = dt MA NU F (t) |{z} random force − NTE E T ME dx γ dt |{z} viscous drag Fractional Diffusion – Theory and Applications – Part I – p. 7/3 SCIENTIA Langevin’s Equation (1908) d2 x m 2 = dt MA NU F (t) |{z} random force hF (t)i = 0, hF (0)F (t)i = Dδ(t) − NTE E T ME dx γ dt |{z} viscous drag hx(t)F (t)i = hx(t)ihF (t)i = 0 Fractional Diffusion – Theory and Applications – Part I – p. 7/3 SCIENTIA Langevin’s Equation (1908) d2 x m 2 = dt MA NU F (t) |{z} random force − NTE E T ME dx γ dt |{z} viscous drag hF (t)i = 0, hF (0)F (t)i = Dδ(t) hx(t)F (t)i = hx(t)ihF (t)i = 0 * + 1 dx 2 1 m = kB T Equipartition of energy (1D) – Boltzmann (1896) 2 dt 2 Fractional Diffusion – Theory and Applications – Part I – p. 7/3 SCIENTIA Langevin’s Equation (1908) d2 x m 2 = dt MA NU F (t) |{z} dx γ dt |{z} − random force NTE E T ME viscous drag hF (t)i = 0, hF (0)F (t)i = Dδ(t) hx(t)F (t)i = hx(t)ihF (t)i = 0 * + 1 dx 2 1 m = kB T Equipartition of energy (1D) – Boltzmann (1896) 2 dt 2 d 2 2kB T γ hx i = 1 − exp(− t) dt γ m ⇒ |{z} “ ” t≫ m γ d 2 2kB T hx i ≈ dt γ ≈10−8 Fractional Diffusion – Theory and Applications – Part I – p. 7/3 SCIENTIA Langevin’s Equation (1908) d2 x m 2 = dt MA NU F (t) |{z} dx γ dt |{z} − random force NTE E T ME viscous drag hF (t)i = 0, hF (0)F (t)i = Dδ(t) hx(t)F (t)i = hx(t)ihF (t)i = 0 * + 1 dx 2 1 m = kB T Equipartition of energy (1D) – Boltzmann (1896) 2 dt 2 d 2 2kB T γ hx i = 1 − exp(− t) dt γ m hx2 i ∼ 2Dt ⇒ |{z} “ ” t≫ m γ d 2 2kB T hx i ≈ dt γ ≈10−8 Fractional Diffusion – Theory and Applications – Part I – p. 7/3 SCIENTIA Random Walks MA NU NTE E T ME A man starts from a point O and walks ℓ yards in a straight line; he then turns through any angle whatever and walks another ℓ yards in a second straight line. He repeats this process n times. I require the probability that after these n stretches he is at a distance between r and r + δr from his starting point, O. Karl Pearson (1905) Fractional Diffusion – Theory and Applications – Part I – p. 8/3 SCIENTIA Random Walks MA NU NTE E T ME A man starts from a point O and walks ℓ yards in a straight line; he then turns through any angle whatever and walks another ℓ yards in a second straight line. He repeats this process n times. I require the probability that after these n stretches he is at a distance between r and r + δr from his starting point, O. Karl Pearson (1905) If n be very great, the probability sought is 2 −r2 /n e r dr n Lord Rayleigh (1905) Fractional Diffusion – Theory and Applications – Part I – p. 8/3 SCIENTIA The lesson of Lord Rayleigh’s solution is that in open country the most probable place to find a drunken man who is at all capable of keeping on his feet is somewhere near his starting point! Karl Pearson (1905) MA NU NTE E T ME Fractional Diffusion – Theory and Applications – Part I – p. 9/3 SCIENTIA The lesson of Lord Rayleigh’s solution is that in open country the most probable place to find a drunken man who is at all capable of keeping on his feet is somewhere near his starting point! Karl Pearson (1905) MA NU NTE E T ME ... the consideration of true prices permits the statement of the fundamental principle – The mathematical expectation of the speculator is zero. Louis Bachelier (1900) Fractional Diffusion – Theory and Applications – Part I – p. 9/3 SCIENTIA Random Walks and the Binomial Distribution MA NU NTE E T ME A particle starts from an origin and at each time step ∆t the particle has an equal probability of jumping an equal distance ∆x to the left or the right. What is the probability Pm,n that the particle will be at position x = m∆x at time t = n∆t? Fractional Diffusion – Theory and Applications – Part I – p. 10/3 SCIENTIA Random Walks and the Binomial Distribution MA NU NTE E T ME A particle starts from an origin and at each time step ∆t the particle has an equal probability of jumping an equal distance ∆x to the left or the right. What is the probability Pm,n that the particle will be at position x = m∆x at time t = n∆t? 1/2 m−1 m m+1 1/2 Fractional Diffusion – Theory and Applications – Part I – p. 10/3 SCIENTIA Random Walks and the Binomial Distribution MA NU NTE E T ME A particle starts from an origin and at each time step ∆t the particle has an equal probability of jumping an equal distance ∆x to the left or the right. What is the probability Pm,n that the particle will be at position x = m∆x at time t = n∆t? 1/2 m−1 m m+1 1/2 1 1 Pm,n = Pm−1,n−1 + Pm+1,n−1 , 2 2 k X P0,0 = 1, Pj,k = 1 k = 0, 1, 2, . . . n j=−k Fractional Diffusion – Theory and Applications – Part I – p. 10/3 SCIENTIA Enumeration MA NU NTE E T ME Suppose an n step walk from 0 to m has k steps to the right and n − k = k − m steps to the left. Fractional Diffusion – Theory and Applications – Part I – p. 11/3 SCIENTIA Enumeration MA NU NTE E T ME Suppose an n step walk from 0 to m has k steps to the right and n − k = k − m steps to the left. n n! There are C(n, k) = = ways of distributing k!(n − k)! k these k steps among the n steps. Fractional Diffusion – Theory and Applications – Part I – p. 11/3 SCIENTIA Enumeration MA NU NTE E T ME Suppose an n step walk from 0 to m has k steps to the right and n − k = k − m steps to the left. n n! There are C(n, k) = = ways of distributing k!(n − k)! k these k steps among the n steps. There are 2n possible paths in an n step walk. Fractional Diffusion – Theory and Applications – Part I – p. 11/3 SCIENTIA Enumeration MA NU NTE E T ME Suppose an n step walk from 0 to m has k steps to the right and n − k = k − m steps to the left. n n! There are C(n, k) = = ways of distributing k!(n − k)! k these k steps among the n steps. There are 2n possible paths in an n step walk. C(n, k) p(m(k), n) = 2n Fractional Diffusion – Theory and Applications – Part I – p. 11/3 SCIENTIA Enumeration MA NU NTE E T ME Suppose an n step walk from 0 to m has k steps to the right and n − k = k − m steps to the left. n n! There are C(n, k) = = ways of distributing k!(n − k)! k these k steps among the n steps. There are 2n possible paths in an n step walk. C(n, k) p(m(k), n) = 2n But n+m k= 2 ⇒ p(m, n) = 2n n! n+m ! 2 n−m 2 ! Fractional Diffusion – Theory and Applications – Part I – p. 11/3 SCIENTIA Random Walks and the Normal Distribution MA NU NTE E T ME In n step walks from 0 to m the av. # steps to the right is n hki = . 2 n+m n m − = How are fluctuations X = k − hki = 2 2 2 distributed? Fractional Diffusion – Theory and Applications – Part I – p. 12/3 SCIENTIA Random Walks and the Normal Distribution MA NU NTE E T ME In n step walks from 0 to m the av. # steps to the right is n hki = . 2 n+m n m − = How are fluctuations X = k − hki = 2 2 2 distributed? n! n! p(m(X), n) = n n+m n−m = n n n ( − X)!( + X)!2 2 ! ! 2 2 2 2 Fractional Diffusion – Theory and Applications – Part I – p. 12/3 SCIENTIA Random Walks and the Normal Distribution MA NU NTE E T ME In n step walks from 0 to m the av. # steps to the right is n hki = . 2 n+m n m − = How are fluctuations X = k − hki = 2 2 2 distributed? n! n! p(m(X), n) = n n+m n−m = n n n ( − X)!( + X)!2 2 ! ! 2 2 2 2 √ n −n n! ≈ 2πnn e }⇒ | {z q Stirling’s approx. 2 nπ P (X, n) = 1− 1 n 2X ( 2 −X+ 2 ) n 1+ 1 n 2X ( 2 +X+ 2 ) n Fractional Diffusion – Theory and Applications – Part I – p. 12/3 SCIENTIA Re-write P (X, n) = MA NU exp ( n2 −X + 1 2 ) ln(1 − q 2 nπ 2X n ) + ( n2 +X + 1 2 ) ln(1 Carry out a series expansion of the log terms in powers of P (X, n) ∼ r 2 − 2X 2 e n = nπ NTE E T ME r + 2X n ) 2X n 2 − m2 e 2n nπ Recall m is the final position of the walker. The probability density function for unbiased standard random walks in the long time limit is the Gaussian or normal distribution. Fractional Diffusion – Theory and Applications – Part I – p. 13/3 SCIENTIA Random Walks in the Continuum Approximation MA NU Write P (m, n) = P (x, t), NTE E T ME x = m∆x, t = n∆t 1 1 P (x, t) = P (x − ∆x, t − ∆t) + P (x + ∆x, t − ∆t) 2 2 Fractional Diffusion – Theory and Applications – Part I – p. 14/3 SCIENTIA Random Walks in the Continuum Approximation MA NU Write P (m, n) = P (x, t), NTE E T ME x = m∆x, t = n∆t 1 1 P (x, t) = P (x − ∆x, t − ∆t) + P (x + ∆x, t − ∆t) 2 2 Taylor expansions P (x ± ∆x, t − ∆t) ≈ ∂P (∆x)2 ∂ 2 P (∆t)2 ∂ 2 P ∂P − ∆t + + P (x, t) ± ∆x ∂x ∂t 2 ∂x2 2 ∂t2 ∂2P ∓∆t∆x + O((∆t)3 ) + O((∆x)3 ), ∂x∂t Fractional Diffusion – Theory and Applications – Part I – p. 14/3 SCIENTIA Random Walks in the Continuum Approximation MA NU Write P (m, n) = P (x, t), NTE E T ME x = m∆x, t = n∆t 1 1 P (x, t) = P (x − ∆x, t − ∆t) + P (x + ∆x, t − ∆t) 2 2 Taylor expansions P (x ± ∆x, t − ∆t) ≈ ∂P (∆x)2 ∂ 2 P (∆t)2 ∂ 2 P ∂P − ∆t + + P (x, t) ± ∆x ∂x ∂t 2 ∂x2 2 ∂t2 ∂2P ∓∆t∆x + O((∆t)3 ) + O((∆x)3 ), ∂x∂t Retain leading order terms in ∆t and ∆x then ∂P ∂2P =D 2, ∂t ∂x (∆x)2 = constant D= lim ∆t→0,∆x→0 2∆t Fractional Diffusion – Theory and Applications – Part I – p. 14/3 SCIENTIA Fundamental Solution and Mean Square Displacement Green’s solution G(x, t), G(x, 0) = δ(x). MA NU NTE E T ME Fractional Diffusion – Theory and Applications – Part I – p. 15/3 SCIENTIA Fundamental Solution and Mean Square Displacement Green’s solution G(x, t), G(x, 0) = δ(x). MA NU NTE E T ME dĜ(q, t) = −Dq 2 Ĝ(q, t), Ĝ(q, 0) = δ̂(q) = 1 Fourier transform dt Fractional Diffusion – Theory and Applications – Part I – p. 15/3 SCIENTIA Fundamental Solution and Mean Square Displacement Green’s solution G(x, t), G(x, 0) = δ(x). MA NU NTE E T ME dĜ(q, t) = −Dq 2 Ĝ(q, t), Ĝ(q, 0) = δ̂(q) = 1 Fourier transform dt Fourier solution Ĝ(q, t) = e −Dq 2 t Fractional Diffusion – Theory and Applications – Part I – p. 15/3 SCIENTIA Fundamental Solution and Mean Square Displacement Green’s solution G(x, t), G(x, 0) = δ(x). MA NU NTE E T ME dĜ(q, t) = −Dq 2 Ĝ(q, t), Ĝ(q, 0) = δ̂(q) = 1 Fourier transform dt Fourier solution Ĝ(q, t) = e −Dq 2 t Inverse Fourier transform 1 G(x, t) = 2π Z +∞ −Dq 2 t + iqx | {z } e complete the square −∞ dq x2 1 − 4Dt e =√ 4πDt Fractional Diffusion – Theory and Applications – Part I – p. 15/3 SCIENTIA Fundamental Solution and Mean Square Displacement Green’s solution G(x, t), G(x, 0) = δ(x). MA NU NTE E T ME dĜ(q, t) = −Dq 2 Ĝ(q, t), Ĝ(q, 0) = δ̂(q) = 1 Fourier transform dt Fourier solution Ĝ(q, t) = e −Dq 2 t Inverse Fourier transform 1 G(x, t) = 2π Z +∞ −Dq 2 t + iqx | {z } e complete the square −∞ dq x2 1 − 4Dt e =√ 4πDt Mean square displacement Z +∞ d2 2 2 2 hx i = x G(x, t) dx or hx i = lim − 2 Ĝ(q, t) = 2Dt. q→0 dq −∞ Fractional Diffusion – Theory and Applications – Part I – p. 15/3 SCIENTIA Central Limit Theorem Each step of a random walk is a random variable ∆xi . The mean of ∆xi after N steps is the position x divided by N . MA NU NTE E T ME Fractional Diffusion – Theory and Applications – Part I – p. 16/3 SCIENTIA Central Limit Theorem Each step of a random walk is a random variable ∆xi . The mean of ∆xi after N steps is the position x divided by N . MA NU NTE E T ME If we sample (infinitely many) random walks of length N (sufficiently large) then the sampling distribution of random variables X = Nx should approach a normal distribution µ)2 1 (x − √ P (X ∈ dx) = exp − 2 2σ 2 2πσ with µ = hXi equal to the mean of the sampled set and variance σ 2 = hX 2 i − hXi2 equal to the variance of the sampled set divided by N . Fractional Diffusion – Theory and Applications – Part I – p. 16/3 SCIENTIA Central Limit Theorem Each step of a random walk is a random variable ∆xi . The mean of ∆xi after N steps is the position x divided by N . MA NU NTE E T ME If we sample (infinitely many) random walks of length N (sufficiently large) then the sampling distribution of random variables X = Nx should approach a normal distribution µ)2 1 (x − √ P (X ∈ dx) = exp − 2 2σ 2 2πσ with µ = hXi equal to the mean of the sampled set and variance σ 2 = hX 2 i − hXi2 equal to the variance of the sampled set divided by N . Consider µ = 0 and σ 2 = 2Dt. Fractional Diffusion – Theory and Applications – Part I – p. 16/3 SCIENTIA Diffusion Equation MA NU NTE E T ME If there are N non-interacting walkers then they all have the same probability P (x, t) of being at x at time t and hence the number per unit volume (concentration) at x at time t is c(x, t) = N P (x, t) ∂2c ∂c =D 2 ∂t ∂x Fractional Diffusion – Theory and Applications – Part I – p. 17/3 SCIENTIA Diffusion Equation MA NU NTE E T ME If there are N non-interacting walkers then they all have the same probability P (x, t) of being at x at time t and hence the number per unit volume (concentration) at x at time t is c(x, t) = N P (x, t) ∂2c ∂c =D 2 ∂t ∂x The diffusion equation can be derived from conservation of mass and Fick’s Law. ∂c ∂q =− |∂t {z ∂x} conservation of mass ∂c q=− | {z ∂x} Fick’s law Fractional Diffusion – Theory and Applications – Part I – p. 17/3 SCIENTIA Fick’s Law (1855) MA NU NTE E T ME Define q(x, t) as the number of particles per unit time that pass through a test area perpendicular to the flow in the positive x direction. Fractional Diffusion – Theory and Applications – Part I – p. 18/3 SCIENTIA Fick’s Law (1855) MA NU NTE E T ME Define q(x, t) as the number of particles per unit time that pass through a test area perpendicular to the flow in the positive x direction. The net flow of diffusing particles is from regions of high concentration to regions of low concentration and the magnitude of this flow is proportional to the concentration gradient. ∂c q(x, t) = −D | {z } ∂x flux Fractional Diffusion – Theory and Applications – Part I – p. 18/3 SCIENTIA Conservation of Mass 1111 0000 00 11 0000 1111 00 11 0000 1111 00 11 0000 1111 00 11 0000 1111 00 11 0000 1111 00 11 0000 1111 00 11 0000 1111 00 11 A x 0 1 0 MA NU NTE E T ME x+ δx 1 0 1 net number of particles number of particles number of particles C C B C B B B in volume V at time C = B in volume V at time C+B entering volume V between C A A @ A @ @ times t, t + δt t t + δt (c(x, t + δt)Aδx) = (c(x, t)Aδx) + (q(x, t)Aδt − q(x + δx, t)Aδt) c(x, t + δt) − c(x, t) q(x + δx) − q(x, t) ⇒ =− δt δx ∂c ∂q ⇒ =− ∂t ∂x Fractional Diffusion – Theory and Applications – Part I – p. 19/3 SCIENTIA Generalizations If the media is spatially heterogeneous then replace the diffusion constant in Fick’s law with a space dependent function. ∂ ∂c = ∂t ∂x ∂c D(x) ∂x MA NU NTE E T ME diffusion 1D Fractional Diffusion – Theory and Applications – Part I – p. 20/3 SCIENTIA Generalizations If the media is spatially heterogeneous then replace the diffusion constant in Fick’s law with a space dependent function. ∂ ∂c = ∂t ∂x ∂ ∂c = ∂t ∂x ∂c D(x) ∂x ∂c D(x) ∂x MA NU NTE E T ME diffusion 1D ∂ − (v(x) c) ∂x advection-diffusion 1 dim Fractional Diffusion – Theory and Applications – Part I – p. 20/3 SCIENTIA Generalizations If the media is spatially heterogeneous then replace the diffusion constant in Fick’s law with a space dependent function. ∂ ∂c = ∂t ∂x ∂ ∂c = ∂t ∂x ∂c D(x) ∂x ∂c D(x) ∂x MA NU NTE E T ME diffusion 1D ∂ − (v(x) c) ∂x ∂ ∂c ∂ ∂u ∂c = D(x) −χ c ∂t ∂x ∂x ∂x ∂x advection-diffusion 1 dim cemotactic-diffusion 1 dim Fractional Diffusion – Theory and Applications – Part I – p. 20/3 SCIENTIA Generalizations If the media is spatially heterogeneous then replace the diffusion constant in Fick’s law with a space dependent function. ∂ ∂c = ∂t ∂x ∂ ∂c = ∂t ∂x ∂c D(x) ∂x ∂c D(x) ∂x ∂ − (v(x) c) ∂x 1 ∂ ∂c = d−1 ∂t r ∂r r d−1 ∂c D(r) ∂r NU NTE E T ME diffusion 1D ∂ ∂c ∂ ∂u ∂c = D(x) −χ c ∂t ∂x ∂x ∂x ∂x MA advection-diffusion 1 dim cemotactic-diffusion 1 dim radially symmetric d dim Fractional Diffusion – Theory and Applications – Part I – p. 20/3 SCIENTIA Master Equations and the Fokker-Planck Equation P (x, t) = Z MA NU NTE E T ME +∞ −∞ λ(∆x)P (x − ∆x, t − ∆t) d∆x P (x, t) probability density for walker to be at x at time t λ(∆x) probability density function for a jump of length ∆x ∆t discrete time between jumps For spatially heterogeneous media generalize λ = λ(∆x, x) the step length density varies with position Z +∞ P (x, t) = λ(∆x, x − ∆x)P (x − ∆x, t − ∆t) d∆x −∞ Fractional Diffusion – Theory and Applications – Part I – p. 21/3 SCIENTIA Continuum limit ∆t → 0 and ∆x → 0 P |(x,t−∆t) ˛ ∂P ˛˛ + ∆t ∂t ˛(x,t−∆t) Z Z Z ≈ × +∞ MA NU +∞ ˛ ∆x2 ∂λ ˛˛ + λ|(∆x,x) − ∆x ˛ ∂x 2 −∞ (∆x,x) ˛ ∂P ˛˛ (∆x)2 P |(x,t−∆t) − ∆x + ∂x ˛(x,t−∆t) 2 Z λ(∆x, x) d∆x = 1, ∆x λ(∆x, x) d∆x = h∆x(x)i, ∆x2 λ(∆x, x) d∆x = h∆x2 (x)i. ∂2λ ˛ ˛ ˛ 2 ∂x ˛ (∆x,x) NTE E T ME ! ! ˛ ˛ ˛ 2 ∂x ˛(x,t−∆t) ∂2P −∞ +∞ −∞ +∞ −∞ Retain leading order terms to derive the general Fokker-Planck equation Fractional Diffusion – Theory and Applications – Part I – p. 22/3 SCIENTIA General Fokker-Planck Equation MA NU NTE E T ME ∂2 ∂ ∂P = (D(x)P (x, t)) − (v(x)P (x, t)) ∂t ∂x2 ∂x h∆x(x)i v(x) = drift ∆t h∆x2 (x)i D(x) = diffusivity 2∆t Fractional Diffusion – Theory and Applications – Part I – p. 23/3 SCIENTIA General Fokker-Planck Equation MA NU NTE E T ME ∂2 ∂ ∂P = (D(x)P (x, t)) − (v(x)P (x, t)) ∂t ∂x2 ∂x h∆x(x)i v(x) = drift ∆t h∆x2 (x)i D(x) = diffusivity 2∆t Compare the generalized Fickian diffusion equation ∂ ∂P (x, t) ∂ ∂P = D(x) − (v(x)P (x, t)) ∂t ∂x ∂x ∂x Fractional Diffusion – Theory and Applications – Part I – p. 23/3 SCIENTIA The Chapman-Kolmogorov Equation, Markov Processes MA NU NTE E T ME In the limit where the time increment approaches zero the sequence of jumps {Xt } in a random walk defines a stochastic process. A realization defines a trajectory x(t). Fractional Diffusion – Theory and Applications – Part I – p. 24/3 SCIENTIA The Chapman-Kolmogorov Equation, Markov Processes MA NU NTE E T ME In the limit where the time increment approaches zero the sequence of jumps {Xt } in a random walk defines a stochastic process. A realization defines a trajectory x(t). Markov property if at any t the distribution of all Xu , u > t only depends on Xt and not on any Xs , s < t. Fractional Diffusion – Theory and Applications – Part I – p. 24/3 SCIENTIA The Chapman-Kolmogorov Equation, Markov Processes MA NU NTE E T ME In the limit where the time increment approaches zero the sequence of jumps {Xt } in a random walk defines a stochastic process. A realization defines a trajectory x(t). Markov property if at any t the distribution of all Xu , u > t only depends on Xt and not on any Xs , s < t. If Markov then the conditional probability q(x, t|x′ , t′ ) that Xt lies in x, x + dx given that Xt′ lies in x′ , x′ + dx′ satisfies Z q(x, t|x′′ , t′′ ) = q(x, t|x′ , t′ )q(x′ , t′ |x′′ , t′′ ) dx′ Bachelier (1990) Note the times t > t′ > t′′ are discrete. Fractional Diffusion – Theory and Applications – Part I – p. 24/3 SCIENTIA Wiener Process, Brownian Motion MA NU One solution of Bachelier’s equation is (x−x′ )2 1 − e 2(t−t′ ) , t > t′ q(x, t|x′ , t′ ) = p 2π(t − t′ ) NTE E T ME Wiener process Fractional Diffusion – Theory and Applications – Part I – p. 25/3 SCIENTIA Wiener Process, Brownian Motion MA NU One solution of Bachelier’s equation is (x−x′ )2 1 − e 2(t−t′ ) , t > t′ q(x, t|x′ , t′ ) = p 2π(t − t′ ) NTE E T ME Wiener process Limiting behaviour of a random walk in the limit where the time increment approaches zero (Wiener, 1923) – also referred to as Brownian motion Bt . Fractional Diffusion – Theory and Applications – Part I – p. 25/3 SCIENTIA Wiener Process, Brownian Motion MA NU One solution of Bachelier’s equation is (x−x′ )2 1 − e 2(t−t′ ) , t > t′ q(x, t|x′ , t′ ) = p 2π(t − t′ ) NTE E T ME Wiener process Limiting behaviour of a random walk in the limit where the time increment approaches zero (Wiener, 1923) – also referred to as Brownian motion Bt . (i) Realizations xB (t) of Bt are continuous but nowhere differentiable – xB (t) versus t is a fractal graph with dimension d = 3/2. (ii) The increments Bt − Bt′ are normally distributed with mean 0 and variance t − t′ for t > t′ . (iii) The increments Bt − Bt′ and Bs − Bs′ are independent for t > t′ ≥ s ≥ s′ ≥ 0. Fractional Diffusion – Theory and Applications – Part I – p. 25/3 SCIENTIA Standard diffusion MA NU NTE E T ME The mean square displacement scales linearly with time and the probability density function is the Gaussian normal distribution Fractional Diffusion – Theory and Applications – Part I – p. 26/3 SCIENTIA Standard diffusion MA NU NTE E T ME The mean square displacement scales linearly with time and the probability density function is the Gaussian normal distribution Theoretical support from random walks, central limit theorem, Langevin equation, diffusion equation, master equations, Weiner processes. Fractional Diffusion – Theory and Applications – Part I – p. 26/3 SCIENTIA Standard diffusion MA NU NTE E T ME The mean square displacement scales linearly with time and the probability density function is the Gaussian normal distribution Theoretical support from random walks, central limit theorem, Langevin equation, diffusion equation, master equations, Weiner processes. Experimental support widespread, e.g., Perrin (1909) measured mean square displacements and used Einstein’s relations to determine Avogadro’s number, the constant number of molecules in any mole of substance, thus consolidating the atomistic description of nature. Fractional Diffusion – Theory and Applications – Part I – p. 26/3 SCIENTIA In the theory of Brownian motion the first concern has always been the calculation of the mean square displacement of the particle, because this could be immediately observed. George Uhlenbeck and Leonard Ornstein (1930) MA NU NTE E T ME Fractional Diffusion – Theory and Applications – Part I – p. 27/3 SCIENTIA In the theory of Brownian motion the first concern has always been the calculation of the mean square displacement of the particle, because this could be immediately observed. George Uhlenbeck and Leonard Ornstein (1930) MA NU NTE E T ME In the atmosphere a spreading dot will not serve as an element from which general distributions can be built up. Lewis Richardson (1926) Fractional Diffusion – Theory and Applications – Part I – p. 27/3 SCIENTIA Anomalous Diffusion MA NU NTE E T ME There have been numerous experimental measurements of anomalous diffusion – the mean square displacement does not scale linearly with time. D E h∆X 2 (t)i = (X(t) − hX(t)i)2 ≁ t Fractional Diffusion – Theory and Applications – Part I – p. 28/3 SCIENTIA Anomalous Diffusion MA NU NTE E T ME There have been numerous experimental measurements of anomalous diffusion – the mean square displacement does not scale linearly with time. D E h∆X 2 (t)i = (X(t) − hX(t)i)2 ≁ t Anomalous diffusion is ‘normal’ in spatially disordered systems, porous media, fractal media, turbulent fluids and plasmas, biological media with traps, binding sites or macro-molecular crowding, stock price movements. Fractional Diffusion – Theory and Applications – Part I – p. 28/3 SCIENTIA Anomalous Scaling h∆X 2 i ∼ t (ln t)κ MA NU ultraslow diffusion 1<κ<4 NTE E T ME Sinai diffusion deterministic diffusion h∆X 2 i ∼ tα subdiffusion disordered solids 0<α<1 biological media fractal media porous media 8 < tα 2 h∆X i ∼ : t t<τ t>τ h∆X 2 i ∼ t h∆X 2 i ∼ tβ 1<β<2 transient subdiffusion biological media standard diffusion homogeneous media superdiffusion turbulent plasmas Levy flights transport in polymers h∆ℓ2 i ∼ t3 Richardson diffusion atmospheric turbulence Fractional Diffusion – Theory and Applications – Part I – p. 29/3 SCIENTIA Fractional Diffusion MA NU NTE E T ME Over the past decade a new theoretical framework has been developed to model anomalous diffusion. The new framework is based around the physics of continuous time random walks and the mathematics of fractional calculus. One can ask what would be a differential having as its exponent a fraction. Although this seems removed from Geometry . . . it appears that one day these paradoxes will yield useful consequences. Gottfried Leibniz (1695) Fractional Diffusion – Theory and Applications – Part I – p. 30/3