Diffusive processes and Brownian motionA liquid or gas consists of particles----atoms ormolecules----that are free to move. We shall con-sider a subset of particles, such as a dissolved soluteor a suspension, characterized by a number density∆N∆V = n(x, y, z, t) (1)that in general depends on position and time.The flux of particles across a plane perpendicularto the x-axis1 is the number density times the meanvelocity in that direction, jx =df n 〈vx〉 . (2)If the particles are moving randomly, then it is clearthat the number that cross the plane (moving inthe negative x-direction) must be proportional tothe density immediately to the right of the plane(say, at position x + δx ⁄ 2); conversely, those mov-ing to the right must be proportional to the densityjust to the left of the plane (say, at x − δx ⁄ 2) so theflux will bejx δx = D n(x − δx ⁄ 2) − n(x + δx ⁄ 2)≈ −D ∂n∂x δx .That is, the equationjx = −D ∂n∂x(3)defines the diffusion constant, D.Since there is nothing special about the x-direc-tion2 we can express the flux of particles resultingfrom diffusion as3j→ = −D ∇n . (3′)However, if the particles are neither created nordestroyed they must obey a conservation law4∂n∂t + ∇ ⋅ j→ = 0 ; (4)hence by substituting the flux resulting from diffu-sion into the conservation equation we obtain thediffusion equation∂n∂t = D ∇2n . (5)Many physical phenomena are described by equa-tions of this form, including heat conduction in asolid, transport of radiation through a dense at-mosphere, and movement of neutrons throughfissionable material (or shielding, for that matter).Since the underlying physical behavior (that thediffusion equation models) is the same in theseexamples, it is not surprising they can be describedby the same equation.1. Random walk model of diffusionTo get a feeling for the physical meaning of diffu-sion we shall now examine several different waysto describe this phenomenon. A diffusing particleis subjected to a variety of collisions that we canconsider random, in the sense that each such eventPhysics of the Human Body 65Chapter 7 Diffusive processes and Brownian motion1. That is, the number of particles per unit area per unit time that cross the surface.2. Here is another example of the use of symmetry to generalize a result.3.Eq. 3′ is known as Fick’s Law.4. The conservation law can be verified using Gauss’s Theorem.is virtually unrelated to its predecessor. It makes nodifference whether the particle is a molecule ofperfume diffusing in air, a solute molecule in asolution, a colloid in a suspension, a neutron in anuclear reactor, or a photon beating its way out-ward from the center of a star.One mathematical model used to describe such aprocess is the random walk5. In one dimension itlooks something like this: a clock ticks at intervalsδt; at each tick, the particle moves one step to theleft (with probability α), one step to the right (withprobability β), or remains where it is (with prob-ability6 1 − α − β). Thus, using the rules for com-bining probabilities we find that the probability forthe particle to be at x (literally, in the interval[x, x + δx]) at time t+ δt isp(x, t + δt) = (6)α p(x + δx, t) + β p(x − δx, t) + p(x, t) 1 − α − β .For now we shall consider the case where leftwardand rightward movements are equally likely, i.e.α = β : in that case, expanding both sides in Tay-lor’s series we find∂p∂t = α (δ x)2δ t ∂2p∂ x2 + O (δt) + O (δx4)thus, if we define a diffusion constant D = α (δ x)2δ tand extend the same idea to 3 dimensions, wederive the diffusion equation∂p∂t = D ∇2p . (Einstein) (7)Of course this is the equation for the probability ofa single particle being in the volume elementdx dy dz around the point x→ . We may well ask howthe probability p(x→, t) is related to the numberdensity n(x→, t) . In some sense it is obvious that ifthe behavior of a single particle is described byp(x→, t), then a system of N independently movingparticles is carrying out a measurement of p(x→, t),and that we can estimate this probability in termsof the number density viap(x→, t) ≈ n(x→, t)N . (8)The mean and variance of the positionIn order for p(x→, t) to represent a probability den-sity, it must satisfy several restrictions. First, it mustbe positive, since it is no more possible to define anegative probability than a negative concentra-tion. Second, the probability of the particle beingsomewhere is unity, i.e.∫∫∫ d3x p(x→, t) = 1 . (9)Equation 9 is sometimes called a normalization con-dition.The mean, or expected, value of----say----the x-co-ordinate of a particle is defined to be66 Physics of the Human BodyRandom walk model of diffusion5. Sometimes called a ‘‘drunkard’s walk’’.6. Since it can only perform three mutually exclusive actions, the probabilities must sum to unity.〈x〉 =df ∫∫∫ d3x p(x→, t) x ; (10)from the diffusion equation we can deduce thatddt 〈x〉 = D ∫ −∞∞dx x ∂2q(x, t)∂x2 = 0where we have definedq(x, t) = ∫∫ dy dz p(x, y, z, , t) .Sinceddt 〈x〉 = 0we see that〈x〉 = constant.However, even though the average position of adiffusing particle does not change, the variance ofits position is a function of time. The variance isdefined byVar(x) =df 〈(x − 〈x〉)2 〉 (11)≡ 〈x2〉 − 〈x〉2 .Once again working directly with Eq. 7 (the diffu-sion equation) we see thatddt Var(x) = D ∫ −∞∞dx ∂2q(x, t)∂x2 (x − 〈x〉)2≡ 2D , (12)orVar(x) = 2D t . (13)That is, the expected mean-square deviation of aparticle from its initial position increases linearlywith time. Put another way, the distance a diffus-ing particle travels in time t is proportional to √t.2. Stochastic differential equationWe can take another approach to the random walkby analyzing the motion of an object subject to atime-dependent random force f(t). Newton’s Sec-ond Law yields the differential equationmx.. + γ x. = f(t) (Langevin) (14)where we have included viscous drag7. Changingvariables to v = x. we may write the solution to theequationv. + γm v =df v. + 1τ v = f(t)masv(t) = v0 e−t ⁄ τ + 1m e−t ⁄ τ ∫du eu ⁄ τ f(u)0 t.Now what do we mean by a ‘‘random force’’? If wecould observe a system described by the aboveequation many times, each time the force wouldbe a different function of time. Suppose we wereto