Math 345 Intro to Math Biology Lecture 21: Diffusion

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1 Math 345 Intro to Math Biology Lecture 21: Diffusion Junping Shi College of William and Mary November 12, 2018

2 Functions of several variables z = f (x, y): two variables, one function value Domain: a subset of R 2, Range: a subset of R 1 Domain: a subset of R 1, Range: a subset of R 2 ) Examples: (1) z = x 2 + y 2 (distance of (x, y) to the origin) (2) The temperature T (x, y) of different locations of Virginia (3) The population density E(x, y, t) of deers in Virginia at time t and location (x, y) (4) Cobb-Douglas production function P(L, K) = al α K 1 α (L: labor, K: capital) Let u(x, y, z, t) be the density (concentration) of a substance. The density is defined population in O n by u(x, y, z, t) = lim, where O n is a sequence of domains n volume of O n containing (x, y, z), and the volume of O n tends to zero as n. (u : R 3 R R is a function of spatial location (x, y, z) and time t, and the dimension of u is ML 3 (M is mass or number, L is length (L 3 is volume)). Four ways of expressing a function: (a) verbally (a description in words); (b) algebraically (a formula) (c) numerically (a spread sheet); (d) visually (a graph)

3 Partial Derivative f f (x + h, y) f (x, y) (x, y) = Dx f (x, y) = fx (x, y) = lim x h 0 h partial derivative of f (x, y) with respect to x= rate of change of f (x, y) in the x-direction Meaning of partial derivative: when a function is determined by two or more variables, partial derivative shows the rate of change of the whole function when one of the variables changes. So it is a partial change. Second derivatives: f xx = 2 f x 2, fxy = 2 f y x,fyx = 2 f y x, fyy = 2 f y 2 But f xy = f yx if both are continuous Example Find f x, f t, f xx, f xt and f tt if f (x, t) = e x2 /t.

4 Diffusion Diffusion is the spontaneous spreading of matter (particles or molecules), heat, momentum, or light. Diffusion is one type of transport phenomenon. Diffusion is the movement of particles from higher chemical potential to lower chemical potential (chemical potential can in most cases be represented by a change in concentration). It is readily observed, for example, when dried food like spaghetti is cooked; water molecules diffuse into the spaghetti strings, making them thicker and more flexible. It is a physical process rather than a chemical reaction, which requires no net energy expenditure. In cell biology, diffusion is often described as a form of passive transport, by which substances cross membranes.

5 Brownian motion The concept of Brownian motion is closely related to diffusion. Brownian motion is the random movement of microscopic particles in a gas or liquid. This motion is caused by the collision of the microparticle with the moving atoms of the surrounding medium. 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)

6 Robert Brown and Albert Einstein Brownian motion is generally regarded as having been discovered by the botanist Robert Brown in It is believed that Brown was studying pollen particles floating in water under the microscope. He then observed minute particles within the vacuoles of the pollen grains executing a jittery motion. By repeating the experiment with particles of dust, he was able to rule out that the motion was due to pollen particles being alive, although the origin of the motion was yet to be explained. The first complete theory of Brownian motion was formulated by Albert Einstein in R. Brown (1828) A. Einstein (1905)

7 Random walk Considers a walker that takes steps of length x to the left or right along a line, and after each t time units, the walker will take one step. If the walker is at location x 0 at the time t 0, then at time t = t 0 + t, the walker will either be at x 0 x or x 0 + x. Normally the chances or going left or right should be equal, thus the probability of the walker going left or right is 1/2. Now we assume that many walkers are walking with the same time frame simultaneously, with the same step size and on the same lattice on the line. We define P(t, x) as the number of walkers at time t and location x. Then after one time step t, everyone who is at x 0 is gone now (go to x 0 x or x 0 + x), and half of those who are at x 0 x and half of those who are at x 0 + x move to x = x 0 now. Random walker

8 Brownian motion and diffusion ( x) 2 Diffusion equation: u t = du xx, where d = lim x 0, t 0 2 t Higher dimensional random walk produces diffusion equation u t = d x u xx + d y u yy, ( x) 2 where d x = lim x 0, t 0 2 t, dy = lim ( y) 2 y 0, t 0 2 t Brownian motion is the random moving of particles in a fluid (a liquid or a gas) resulting from their bombardment by the fast-moving atoms or molecules in the gas or liquid. It is a continuous-time stochastic processes, and the density of Brownian particles u(x, t) satisfies the diffusion equation. [Brown, 1827], [Bachelier, 1900], [Einstein, 1905] Louis Bachelier was a French mathematician at the turn of the 20th century. He is credited with being the first person to model the stochastic process now called Brownian motion, as part of his PhD thesis (published in 1900). Bacheliers Doctoral thesis, which introduced for the first time a mathematical model of Brownian motion and its use for valuing stock options, is historically the first paper to use advanced mathematics in the study of finance. Thus, Bachelier is considered as the forefather of mathematical finance and a pioneer in the study of stochastic processes. His work in finance is recognized as one of the foundations for the Black-Scholes model.

9 Diffusion equation P t (t, x) = D 2 P (t, x) x2 (The time-derivative equals to a constant times the second spatial-derivative) the rate of change w.r.t. time is caused by the the spatial movement, and it is the second derivative since the average of the first derivative is zero. It was first derived by Joseph Fourier in his treatise Théorie analytique de la chaleur, published in 1822, to describe the heat conduction; The particle diffusion equation was originally derived by Albert Einstein in Einstein used it in order to model Brownian motion. Diffusion constant D: dimension L 2 T 1 Example: oxygen (depends on temperature and media) Temperature ( C) Media D (cm 2 /sec) 0 air air water water

10 Discrete diffusion equation Suppose that t = 1, 2,, and x = 0, ±1, ±2,. Or equivalently t N and x Z. Let P(t, x) be the population at time t and location x. Then the random walk assumption gives equation: P(n + 1, x) = 1 2 P(n, x 1) + 1 P(n, x 1), n N, x Z. 2 Or we may assume that the probability of moving to the left (or right) neighbor is r, and the probability of staying in the original location is 1 2r, then the equation is or P(n + 1, x) = rp(n, x 1) + (1 2r)P(n, x) + rp(n, x 1), n N, x Z, P(n + 1, x) P(n, x) = r [P(n, x 1) 2P(n, x) + P(n, x 1)], n N, x Z. Here r is the dispersal rate and it is dimensionless (as a percentage).

11 Solution of discrete diffusion equation P(n + 1, x) P(n, x) = r [P(n, x 1) 2P(n, x) + P(n, x 1)], n N, x Z. Initial condition: P(0, 0) = 1 and P(0, x) = 0 for x 0 (population initially concentrates at x = 0) ( 2n Solution: P(n, x) = n x Here ( 2n n x ) = ) 1 (2n)! (n x)!(n + x)!. for n x n, and P(n, x) = 0 for x n n For fixed n, {P(n, x) : x Z} is a probability distribution so that n P(n, x) = 1. x= n

12 Binomial distribution If you flip a coin, the probability of getting a head is p and the probability of getting a tail is 1 p, then the probability of getting exactly k heads in m trials is given by ( m ) P(m, k) = p k (1 p) m k, 0 k m. k This is called binomial probability distribution function. So the solution of discrete diffusion equation at n is a binomial distribution P(2n, n x) with p = 1/2 (shifting from x [0, 2n] to x [ n, n]) If m is large, then the binomial distribution is approximately a normal distribution N(mp, mp(1 p)) (mean mp and variance mp(1 p)). So P(2n, n x) is approximately N(0, n/2) = 1 e x2 /(2n). 2nπ

13 Normal distribution P t (t, x) = D 2 P (t, x) x2 1 x 2 Solution: P(t, x) = e 4Dt (4πDt) 1/2 It is a normal distribution of mean 0 and variance 2Dt. Biological meaning: if a fixed amount of organism is released from a point (x = 0), then the dispersal of the organism follows the diffusion equation, and the distribution of the organism is a normal one with mean 0 and variance 2Dt.

14 Other solutions of diffusion equation u t = Du xx where u(x, t) is the solution trivial solution: u(x, t) = a + bx (steady state solution) separable solution: u(x, t) = U(t)V (x), u(x, t) = e dλt cos( λx) or u(x, t) = e dλt sin( λx) for any λ > 0. u(x, t) = e dλt e λx or u(x, t) = e dλt e λx for any λ > 0. (none of them are uniformly bounded for all x R, when t is fixed) self-similar solution: u(x, t) = t 1/2 U(x/t 1/2 ), u(x, t) = t 1/2 e x2 /(4dt) Φ(x, t) = (4πdt) 1/2 e x2 /(4dt) (so R udx = 1) fundamental solution (normal distribution function with mean 0 and variance 2dt)