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As a particle travels, the molecule is jostled by collisions with other molecules which prevent it from following a straight line. If the path is examined in close detail, it will be the approximation of a random walk. Mathematically, a random walk is a series of steps, one after another, where each step is taken in a completely random direction from the one before. This kind of path was famously analyzed by Albert Einstein in a study of Brownian motion and he showed that the mean square of the distance traveled by particle following a random walk is proportional to the time elapsed. In two dimensions this relationship can be written as:

For a list of common mathematical concepts used to model diffusion see: Diffusion Mathematics

Types of Diffusion

SURFACE DIFFUSION -- Model of a single adatom diffusing across a square surface lattice. Note the frequency of vibration of the adatom is greater than the jump rate to nearby sites. Also, the model displays examples of both nearest-neighbor jumps (straight) and next-nearest-neighbor jumps (diagonal). Not to scale on a spatial or temporal basis.
  • Anisotropic diffusion, also known as the Perona-Malik equation, enhances high gradients
  • Anomalous diffusion in porous medium
  • Atomic diffusion, in solids
  • Eddy diffusion, in coarse-grained description of turbulent flow
  • Effusion of a gas through small holes
  • Electronics|Electronic diffusion, resulting in an current (electricity)|electric current called the diffusion current
  • Facilitated diffusion, present in some organisms
  • Gaseous diffusion, used for isotope separation
  • Heat equation, diffusion of thermal energy
  • Itō diffusion, mathematisation of Brownian motion, continuous stochastic process.
  • Knudsen diffusion of gas in long pores with frequent wall collisions
  • Momentum diffusion ex. the diffusion of the hydrodynamic velocity field
  • Photon diffusion
  • Plasma diffusion
  • Random walk
  • Reverse diffusion, against the concentration gradient, in phase separation
  • Rotational diffusion, random reorientations of molecules
  • Surface diffusion, diffusion of adparticles on a surface
  • Turbulent diffusion, transport of mass, heat, or momentum within a turbulent fluid

Slot Diffusion

Diffusion in the monolayer: oscillations near temporary equilibrium positions and jumps to the nearest free places.

Diffusion of reagents on the surface of a catalyst may play an important role in heterogeneous catalysis. The model of diffusion in the ideal monolayer is based on the jumps of the reagents on the nearest free places. This model was used for CO on Pt oxidation under low gas pressure.

The system includes several reagents (A₁ , A₂ , ... A⩋) on the surface. Their surface concentrations are (c₁ , c₂ , ... c⩋). The surface is a lattice of the adsorption places. Each reagent molecule fills a place on the surface. Some of the places are free. The concentration of the free paces is z=c₀. The sum of all cᵢ (including free places) is constant, the density of adsorption places b.

The jump model gives for the diffusion flux of Aᵢ (i=1,...,n):


The corresponding diffusion equation is:


Due to the conservation law, Diffeq2.png and we have the system of m diffusion equations. For one component we get Fick's law and linear equations because Diffeq3.png. For two and more components the equations are nonlinear.

If all particles can exchange their positions with their closest neighbours then a simple generalization gives


where Diffeq6.png is a symmetric matrix of coefficients which characterize the intensities of jumps. The free places (vacancies) should be considered as special "particles" with concentration c₀.

Various versions of these jump models are also suitable for simple diffusion mechanisms in solids.

Brownian Motion

Brownian Motion describes the stochastic movement of particles as they travel through space. This type of random movement is often referred to as a random walk, which is typical of particles that diffuse about a 1D 2D or 3D space filled with other particles or barriers. To understand Brownian motion, lets start by characterizing this phenomenon in 2D space.
The mathematical description of this process often includes these terms:

D  (Δ): diffusion rate coefficient

  • D = L2 / 2d⋅t
L : step length
d : dimensions
t : time
D (in units: µm²/s) is the mean diffusion rate per unit time, often in µm²/s for biological motion on a molecular level. This refers to how fast, on average, the particle moves along its trajectories. This value is often of ultimate interest, particularly for simulating Brownian motion; however under most circumstances, the diffusion rate cannot be observed directly in empirical experiments - this would require the ability to visualize all the microscopic particles and collisions that dictate the particle's movement, and would have to be done on a nanosecond timescale. In the animation on the right, the diffusion rate can actually be quantified directly; but what is often seen when observing particle diffusion through a microscope would more closely resemble this:


Brownian Motion Mean Squared Displacement
  • The goal of this calculation is to relate the simulated particle diffusion to real world values, namely velocity.
  • Particle velocity will be a function of MSD x units ²⁄s which scales on space (units) and time (s) parameters.
  • Space and time in the model are defined arbitrarily as Step_Size and Step where each Step a particle moves a distance randomly chosen from a normal distribution (µ=1,σ=.2)
  • a step size of 1 unit/step will produce a brownian motion MSD of ~0.52 ±0.2 units ²/s
  • empirical observations show that reasonable values for MSD are:
    • PSD 0.01 µm ²/s
    • synaptic 0.05 µm ²/s
    • extrasynaptic 0.1 µm ²/s
  • given an MSD of 0.52 ±0.2 units ²/s at the current parameters: 1 step = 1 unit (at µ=1,σ=.2), the model will need to be scaled such that particles move at an extrasynaptic rate of 0.1 µm ²/s.
  • spines are on average 1 to 10 µm apart, if the model is comparing two spines 1 µm apart, they should be separated by 5 units of model space. This is because the current particle diffusion rate of the model is .5 µm ²/s and the empirical MSD is .1 µm ²/s

to make 0.1 units²step0.1 µm²s. It was found that an XY random step-size of µ=0.4 (σ=.2) units produced an MSE ≈ 0.1 units²step. Then, the arbitrary 0.5 units were given meaning (converted to 0.5 µm) by scaling the model according to real-world values (see below) by making 1 unit = 1 µm; as a convention, a subunit will be 1/10th of a unit, thus 1 subunit = 0.1 µm). The PSD areas were set to 3-subunits (.3 µm) square, 20 subunits (2 µm) apart, within a rectangular field 20 subunits (2 µm) wide and 60 subunits (6 µm) long. Given these scaled dimensions where 10 subunits ≈ 1 µm, a particle with an XY step-size of 0.5 units moving in a straight line, could theoretically go from PSD1 to PSD2 in 4 steps (obviously given the simulated particles are moving with Brownian motion, this lower-bound would be extremely rare).