Diffusion
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
 Anisotropic diffusion, also known as the PeronaMalik equation, enhances high gradients
 Anomalous diffusion in porous medium
 Atomic diffusion, in solids
 Eddy diffusion, in coarsegrained description of turbulent flow
 Effusion of a gas through small holes
 ElectronicsElectronic 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 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, and we have the system of m diffusion equations. For one component we get Fick's law and linear equations because . 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 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 = L^{2} / 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:
MEAN SQUARED DISPLACEMENT
 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 ^{units²}⁄_{step} ≈ 0.1 ^{µm²}⁄_{s}. It was found that an XY random stepsize of µ=0.4 (σ=.2) units produced an MSE ≈ 0.1 ^{units²}⁄_{step}. Then, the arbitrary 0.5 units were given meaning (converted to 0.5 µm) by scaling the model according to realworld 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 3subunits (.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 stepsize 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 lowerbound would be extremely rare).