Brownian motion is the irregular movement of small particles suspended in a liquid or gas, caused by interactions with the fluid’s thermally moving molecules. It also denotes a mathematical stochastic process that idealizes this movement through continuous paths and independent Gaussian increments. The phenomenon connects microscopic thermal agitation with observable diffusion, while its mathematical model provides a foundation for describing randomness evolving continuously in time. The physical phenomenon and its idealized model are closely related but are not identical at every timescale. (nobelprize.org)
Historical development
The name commemorates Robert Brown, who in 1827 used a microscope to examine minute particles contained in pollen from Clarkia pulchella, immersed in water. He observed persistent, irregular movements and subsequently found similar behavior in particles obtained from inorganic materials. His account, published in 1828, established that the movement was not restricted to living material. The objects he observed were microscopic particles, not individual molecules in the modern chemical sense. (jxshix.people.wm.edu)
In 1905, Albert Einstein developed a quantitative theory relating the displacement of suspended particles to diffusion and molecular thermal motion. Marian Smoluchowski independently developed a kinetic explanation. Their work made experimentally testable predictions rather than attempting to calculate each individual collision. Jean Perrin subsequently tested these predictions, providing evidence for the physical existence of atoms and molecules and determining the Avogadro constant. Perrin received the 1926 Nobel Prize in Physics for work on the discontinuous structure of matter, particularly sedimentation equilibrium. (nobelprize.org)
Physical mechanism
A suspended particle exchanges momentum continually with surrounding fluid molecules. Although the average molecular bombardment is balanced in a homogeneous fluid at equilibrium, instantaneous imbalances produce fluctuating forces. Viscous resistance damps the resulting movement. Brownian motion therefore persists even when the fluid has no bulk flow: macroscopic rest does not imply microscopic stillness. This distinction is central to statistical mechanics. (nobelprize.org)
For an isolated spherical particle of radius (a), the Stokes–Einstein relation is
[ D=\frac{k_{\mathrm B}T}{6\pi\eta a}, ]
where (D) is the translational diffusion coefficient, (k_{\mathrm B}) is the Boltzmann constant, (T) is absolute temperature, and (\eta) is the fluid’s dynamic viscosity. The relation assumes conditions under which Stokes drag applies, including a continuum fluid, low Reynolds number, and a no-slip particle surface. At fixed viscosity, increasing temperature increases (D); at fixed temperature and viscosity, larger particles diffuse more slowly. Nearby walls can modify the drag and reduce diffusion. (nobelprize.org)
Displacement and diffusion
For free, unbiased Brownian motion in one spatial dimension, the expected displacement is zero, but its mean-square displacement grows linearly:
[ \mathbb E[X(t)-X(0)]=0,\qquad \mathbb E[(X(t)-X(0))^2]=2Dt. ]
In (d) dimensions, isotropic diffusion gives
[ \mathbb E[|\mathbf X(t)-\mathbf X(0)|^2]=2dDt. ]
Thus the root-mean-square displacement grows as (\sqrt{t}), not as (t). These are statements about ensembles or displacement statistics, not predictions that each particle travels a prescribed distance. (nobelprize.org)
For particles initially concentrated at the origin, the one-dimensional probability density is
[ p(x,t)=\frac{1}{\sqrt{4\pi Dt}} \exp!\left(-\frac{x^2}{4Dt}\right). ]
It satisfies the diffusion equation,
[ \frac{\partial p}{\partial t} =D\frac{\partial^2p}{\partial x^2}, ]
which has the same mathematical form as the heat equation. Brownian motion describes individual random trajectories; diffusion describes how their distribution spreads. The density is a normal distribution with variance (2Dt). (echo-old.mpiwg-berlin.mpg.de)
Mathematical model
Standard mathematical Brownian motion, also called the Wiener process, is a real-valued process (W_t), (t\geq0), with these defining properties:
- (W_0=0).
- Its sample paths are continuous with probability one.
- Increments over disjoint time intervals have statistical independence.
- For (0\leq s<t), (W_t-W_s) is normally distributed with mean zero and variance (t-s).
A free physical diffusion process with constant coefficient (D) is represented by (X_t=X_0+\sqrt{2D},W_t). The standard process is a mathematical normalization, not a universal numerical diffusion coefficient for physical particles. (math.uchicago.edu)
Brownian motion is a Gaussian process, with covariance (\mathbb E[W_sW_t]=\min(s,t)). It has the Markov property: given its present position, its future evolution does not require its earlier trajectory. Although its paths are continuous, they are almost surely nowhere differentiable. Consequently, the mathematical path has no ordinary instantaneous velocity. Its increments are stationary, but the process itself is not stationary, because its variance grows with time. (math.uchicago.edu)
Dynamical description and limits
The Langevin equation provides a complementary physical description,
[ m\frac{dv}{dt}=-\gamma v+\xi(t), ]
where (m) is particle mass, (\gamma) is a drag coefficient, and (\xi(t)) represents a fluctuating force. Unlike the overdamped position model, it retains inertia. At times much shorter than the velocity-relaxation time (m/\gamma), displacement is approximately ballistic; at longer times, ordinary diffusive behavior emerges. The Wiener model is therefore a coarse-grained approximation, not a literal description at arbitrarily short intervals. (galileo.phys.virginia.edu)
Brownian motion is also related to the random walk, whose appropriately rescaled paths can converge to the continuous process. Its irregularity motivates stochastic integration rather than ordinary differentiation, and the idealized time derivative of a Wiener process is white noise. Confinement, external forces, particle interactions, and fluid memory require modifications to the simplest free-diffusion model; boundaries already played an important role in interpreting Perrin’s measurements. (sia.mit.edu)