Einstein's Theory of Brownian Motion | Derivation & Notes

Fick's First Law of Diffusion describes how particles move from areas of high concentration to low concentration, stating that the flux (J) of particles across a surface is proportional to the negative of the concentration gradient (ΔC/ΔZ), with the proportionality constant being the diffusion coefficient (D_AB): J = -D_AB × (ΔC/ΔZ). The negative sign ensures that flux is positive when particles move down their concentration gradient, and the diffusion coefficient is unique to each pair of diffusing substances and determines the rate of diffusion.

Diffusion is the movement of particles down a concentration gradient, described by Fick's First Law (J = -D(dC/dx)) for steady-state diffusion and Fick's Second Law (∂C/∂t = D(∂²C/∂x²)) for non-steady-state diffusion; in solids, diffusion depends on crystal structure, activation energy barriers, and temperature, with interstitial diffusion being faster than vacancy diffusion due to smaller atomic sizes fitting into crystal voids.

Fick's law of diffusion states that the rate of particle movement across a barrier is directly proportional to the pressure difference (ΔP), surface area (A), and diffusion constant (D), while being inversely proportional to the thickness (T) of the barrier; this can be expressed as V̇ = (ΔP × A × D) / T, where reducing thickness, increasing pressure difference, using smaller molecules (lower molecular weight), and expanding surface area all increase the rate of diffusion.

Fick's law states that the particle flux (number of particles crossing a unit area per unit time) is proportional to the concentration gradient: J = -D∂n/∂x, where D is the diffusion coefficient. For a gas with molecular diameter d, concentration n, and average speed v, the diffusion coefficient is D = (1/3)vλ = (1/3)v/(nσ). This law describes how particles spread from regions of high concentration to regions of low concentration due to random molecular motion.

Fick's law states that flux j is proportional to the negative concentration gradient: j = -D(∂n/∂x), where D is the diffusion coefficient. The negative sign ensures flux moves from high to low concentration. Substituting Fick's law into the particle balance equation yields the standard diffusion equation: ∂n/∂t = D(∂²n/∂x²). The second derivative arises because the first derivative of flux (containing a first derivative of concentration) produces a second derivative when substituted. This equation governs how concentration profiles evolve over time in diffusive systems.

Stokes' Law states that the viscous drag force on a spherical particle falling through a viscous fluid is given by F = 6πηrv, where η is the coefficient of viscosity, r is the radius of the sphere, and v is the terminal velocity. Additionally, the terminal velocity of a spherical particle in a viscous fluid is given by v = (2/9)(ρ - σ)gr²/η, where ρ is the density of the particle, σ is the density of the fluid, g is the acceleration due to gravity, and η is the coefficient of viscosity.

Stokes' Law describes the drag force experienced by a spherical object moving through a viscous fluid, given by the formula F = 6πrηv, where F is the drag force, r is the radius of the sphere, η is the viscosity of the fluid, and v is the velocity of the object. The drag force depends on three factors: the radius of the sphere (larger radius means more drag), the velocity of the object (higher velocity means more drag), and the viscosity of the medium (higher viscosity means more drag). This law applies specifically to spherical objects and is valid for laminar flow conditions.

Stokes' Law gives the viscous drag force on a sphere moving through a viscous fluid: F = 6πηrv, where η is the viscosity, r is the sphere radius, and v is the velocity. This law applies for small spheres moving slowly through viscous fluids (low Reynolds number). The force is proportional to the radius, velocity, and viscosity.

Stokes' Law describes the viscous drag force experienced by a spherical object moving through a viscous fluid. The drag force is given by F = 6πrηv, where r is the radius of the sphere, η is the coefficient of viscosity (also called coefficient of friction), and v is the velocity of the sphere. The coefficient of viscosity measures the fluid's resistance to flow - higher viscosity means more resistance (like honey or mud). This law applies to laminar flow conditions where the Reynolds number is low.

Stokes' Law states that the drag force on a spherical object is given by: FD = 6π η R V, where η is the coefficient of viscosity, R is the radius of the sphere, and V is the velocity of the object. This formula shows that drag force is directly proportional to viscosity, radius, and velocity.

Random walks model particles taking steps of fixed length in random directions. In one dimension, a walker moves +1 or -1 with probabilities p and q (p+q=1). The binomial distribution P(k) = C(n,k)p^kq^(n-k) gives the probability of k right steps in n trials. For symmetric walks (p=q=½), average displacement is zero but mean squared displacement equals n, growing as √n. This applies to atomic motion in gases, stock markets, and polymer chains. Mathematically, random walks use independent identically distributed random variables X₁,X₂,...,Xₙ where each Xᵢ = ±1 with probability ½. The position after n steps is Sₙ = ΣXᵢ. For symmetric walks, the probability of returning to origin after 2m steps is P(2m) = C(2m,m)(½)^(2m). First return requires never touching the origin before step 2m, excluding paths that cross the origin multiple times.

This section establishes the theoretical foundation of random walk in particle systems. It begins by defining random walk as the motion of particles in liquids and gases where particles move in straight lines until colliding, then change direction randomly. The net displacement after n jumps is the vector sum of individual jump vectors. In systems of many particles, each moves randomly in all directions with independent jump directions and magnitudes, causing isotropic spreading. However, average net displacement is zero due to cancellation between positive and negative displacements. Einstein established that mean square displacement (MSD), denoted ⟨r²⟩, is the correct parameter to measure particle distribution spreading. Mathematically, MSD expands to include diagonal terms (squares of individual jumps) and off-diagonal terms (dot products between different jumps), requiring careful summation over all pairs.

This extensive section develops random walk theory from probability fundamentals to practical applications. The instructor explains independence (one outcome doesn't affect another), covariance and correlation (measuring how variables vary together), and identically distributed variables (sharing the same probability distribution). Combined with independence, this gives the i.i.d. assumption fundamental to random walk models. For an unbiased random walk starting at the origin, the mean displacement <x(t)> = 0 because at each step, the expected change is zero. Two-dimensional random walks can be simulated by choosing random step lengths in both x and y directions. Mean Square Displacement (MSD) captures the actual spread of trajectories by averaging squared displacement from the origin: MSD(t) = <(x(t) - x(0))²>. Unlike mean displacement, MSD cannot be zero and provides a meaningful measure of how far organisms have moved from their starting point.

The probability distribution f_N(n) describes likelihood of being at displacement n after N steps. For small step counts, all outcomes can be enumerated: 0 steps yields 100% at origin; 1 step yields 50% at ±1; 2 steps yield 25% at ±2 and 50% at 0; 3 steps yield 12.5% at ±3, 37.5% at ±1, and 12.5% at 0. These probabilities follow binomial coefficients and correspond to Pascal's triangle rows, where each entry equals the sum of two diagonal entries above it. The coefficient for k heads in N steps is C(N, k) = N!/(k!(N-k)!), representing the number of paths leading to that outcome. In unbiased random walks, the mean displacement from origin is always zero because left and right movements cancel symmetrically. However, the mean squared displacement equals the number of steps taken, because squaring eliminates sign differences. This relationship—mean squared displacement equals time—is a fundamental property of diffusion processes.

A random walk is a model where an object moves in discrete steps with random directions. In a symmetric random walk (equal left/right probability), the mean position after n steps is zero because for every path to +d, there's a symmetric path to -d. The mean squared displacement after n steps is exactly n, derived by considering how squared displacement changes with each step. The root mean squared displacement (RMSD) is √n, representing the typical distance from the origin. This growth as √n, not linearly with n, is a key property of random walks.