Diffusion is the spreading and redistribution of particles through their random motion. In a homogeneous medium under ordinary conditions, it produces net transport from regions of higher concentration to regions of lower concentration, although individual particles move in both directions. It occurs in gases, liquids, and solids and connects microscopic motion with macroscopic changes in composition. Diffusion is important in physics, chemistry, and biology, particularly in mixing, material processing, and transport across biological membranes. (ocw.mit.edu)
Microscopic basis
Atoms, molecules, and other particles undergo irregular motion through collisions or thermally activated movements. Their trajectories can often be represented by a random walk: successive displacements accumulate without a preferred direction. The visible irregular motion of suspended particles, known as Brownian motion, provides a closely related example. A concentration difference does not make every particle move toward lower concentration; rather, more particles leave the more populated region than return from the less populated one. (ocw.mit.edu)
At equilibrium, particles continue moving, but opposing transfers balance and the net diffusive flux vanishes. Thus, the absence of net transport does not imply microscopic immobility. This distinction also allows self-diffusion—the movement of particles within a chemically uniform substance—to occur without a changing overall concentration profile. (openstax.org)
The familiar high-to-low-concentration description applies most directly to simple homogeneous systems. More general descriptions must consider electrical, chemical, thermal, and mechanical driving forces. In thermodynamics, chemical potential helps characterize the tendency of a component to redistribute; concentration alone need not determine transport in a nonideal or externally forced system. (publications.iupac.org)
Fick’s laws
The standard macroscopic description uses Fick’s laws of diffusion. The first law relates the diffusive flux to the concentration gradient:
[ \mathbf{J}=-D\nabla c. ]
Here, (\mathbf{J}) is the amount transported per unit area per unit time, (c) is concentration, and (D) is the diffusion coefficient, also called diffusivity. The minus sign indicates transport opposite to the direction of increasing concentration. Diffusivity has dimensions of length squared divided by time, conventionally expressed in square metres per second. (ocw.mit.edu)
Combining the first law with conservation of particles gives the diffusion equation, commonly called Fick’s second law:
[ \frac{\partial c}{\partial t}=D\nabla^2c. ]
This form assumes constant, isotropic diffusivity, no bulk flow, and no sources or sinks. If diffusivity varies in space, the corresponding expression is (\partial c/\partial t=\nabla\cdot(D\nabla c)). Initial concentration profiles and boundary conditions determine the solution. A maintained concentration difference can support a steady flux even when the concentration at each position no longer changes with time. (ocw.mit.edu)
As a differential equation, this model describes the smoothing of concentration differences. Its mathematical structure is analogous to the equation governing heat conduction, although the transported quantity and physical coefficient differ. (ocw.mit.edu)
Distance and time scales
For ordinary isotropic diffusion without drift, the mean squared displacement satisfies
[ \left\langle|\mathbf r(t)-\mathbf r(0)|^2\right\rangle=2dDt, ]
where (d) is the number of spatial dimensions and angle brackets denote an ensemble average. In one dimension, the displacement variance is (2Dt). The characteristic spreading distance therefore grows as the square root of elapsed time, rather than in direct proportion to time. (ocw.mit.edu)
Consequently, diffusion over a distance (L) requires a characteristic time of order (L^2/D). Increasing the distance tenfold increases this time scale roughly one hundredfold. This scaling makes diffusion effective over microscopic distances but comparatively slow over large ones. In an unbounded homogeneous medium, an initially localized pulse develops a Gaussian distribution whose width increases with time while its total amount remains conserved. (ocw.mit.edu)
Diffusivity and material structure
Diffusivity depends on the particle, medium, temperature, and interactions between them. For an isolated spherical particle in a dilute suspension within a continuum liquid, the Stokes–Einstein relation gives
[ D=\frac{k_{\mathrm B}T}{6\pi\eta a}, ]
where (k_{\mathrm B}) is the Boltzmann constant, (T) is absolute temperature, (\eta) is dynamic viscosity, and (a) is hydrodynamic radius. Within these assumptions, larger particles and more viscous liquids produce lower diffusivity. The relation is not a universal formula for all molecules and environments. (baldwinlab.chem.ox.ac.uk)
In crystalline solids, diffusion can proceed through atomic jumps into vacancies or between interstitial positions. These movements involve energy barriers, so diffusion commonly becomes faster as temperature rises. Surfaces and grain boundaries can provide pathways different from those through the crystal interior. Distinguishing these mechanisms is important in interpreting material-processing experiments. (ocw.mit.edu)
Biological transport
Across a cell membrane, simple diffusion allows suitable small molecules, including oxygen and carbon dioxide, to pass through the lipid bilayer. Facilitated diffusion instead uses membrane proteins, including channels and carriers, to permit transport of substances that cross the bilayer poorly. Both are passive processes: they do not require direct expenditure of cellular metabolic energy to drive transport down the relevant gradient. (openstax.org)
Osmosis concerns the movement of water across a selectively permeable membrane. Diffusion also redistributes substances within cells. Membrane permeability, available surface area, and diffusion distance affect how quickly exchange occurs, making spatial geometry important to biological transport. (openstax.org)
Related transport and computational uses
Diffusion differs from bulk transport by fluid motion: a flowing medium carries particles collectively, whereas molecular diffusion results from their irregular relative motion. Models can include both mechanisms, along with reactions or other sources and sinks. Identifying these contributions is necessary when interpreting observed spreading. (ocw.mit.edu)
In machine learning, a diffusion model uses a probabilistic corruption process, commonly adding noise progressively to data, and learns a reverse denoising process for generating samples. The terminology reflects connections to stochastic spreading and nonequilibrium thermodynamics; the generated data are not necessarily undergoing physical molecular diffusion. (arxiv.org)