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Divergence and curl of $-\nabla p+\mu {\nabla }^{2}u=0$ with $\nabla \cdot u=0$ give ${\nabla }^{2}p=0$ and ${\nabla }^2\omega =0$.

Taking the curl of the planar Stokes equation makes vorticity harmonic. Since vorticity is minus the Laplacian of the stream function,

For an incompressible Newtonian fluid, the nonnegative rate at which viscosity converts mechanical energy into heat is

For two Stokes velocity-stress fields in the same domain,It follows by integrating the divergence of the corresponding cross-work flux and using symmetry of the Newtonian stress.

At fixed orientation, linearity of Stokes flow relates a rigid body's translational velocity $\mathbf{U}$ to the hydrodynamic force $\mathbf{F}$ byThe Lorentz reciprocal theorem makes $\mathbf{R}$ symmetric, and positive viscous dissipation makes it positive definite.

Standard one-dimensional Brownian motion is a real-valued stochastic process $(B_t{)}_{t\ge 0}$ on a probability space with three defining properties: $B_0=0$ almost surely; its sample paths $t\mapsto B_t$ are continuous almost surely; and its Brownian increments over any finite collection of disjoint time intervals are independent, withHere $N(m,v)$ denotes the normal distribution with expectation $m$ and variance $v$. The alternative symbol $W_t$ denotes the same kind of stochastic process. Continuity is a simultaneous statement about an entire sample path, rather than a separate assertion at each fixed time. Relative to its natural Brownian filtration, a future Brownian increment is independent of all observations up to its starting time.

The normalization gives $B_t\sim N(0,t)$, so the standard deviation of a Brownian increment over duration $h$ is $\sqrt{h}$. For $h=1$, $0.01$, and $0.0001$, the corresponding standard deviations are $1$, $0.1$, and $0.01$. Time becoming one hundred times shorter makes typical displacement ten times smaller. Brownian scaling expresses this rule for the whole stochastic process, while the Brownian covariance kernel describes how positions at different times remain related despite independent increments.

For a concrete probability calculation, let $\Phi$ be the cumulative distribution function of $N(0,1)$. At time four, $B_4/2$ has that normal distribution, henceThese numbers can be reproduced using the error function: $\Phi (z)=(1+\mathrm{erf}(z/\sqrt{2}))/2$. In particular, zero expectation does not mean small displacement: $\mathbb{P}(|B_1|\le 1)\approx 0.6826894921$, but the same fixed interval contains only about $38.3\%$ of the probability at time four.

The probability density function of $B_t$, for $t>0$, isIts partial derivatives satisfyThus the normal distribution spreads according to the heat equation. Starting at a fixed position $x_0$ replaces $x$ by $x-x_0$. This is a useful bridge between individual random trajectories and the deterministic evolution of their probability density function: the same Brownian motion provides both descriptions.

Physical diffusion uses its own scale. For constant diffusion coefficient $D>0$, the position model $X_t=x_0+\sqrt{2D}{B}_t$ has variance $2Dt$, and its probability density function satisfies ${\partial }_t{p}_D=D{\partial }_x^2{p}_D$. For example, $D=0.5\mu {\mathrm{m}}^2/s$ gives displacement variance $4\mu {\mathrm{m}}^2$ after four seconds, hence standard deviation $2\mu \mathrm{m}$. Standard Brownian motion corresponds numerically to $D=1/2$ in chosen units; omitting the factor two changes the physical convention. Einstein's 1905 paper relates diffusion to the mean squared displacement in precisely this convention. ^[ref]

Continuity does not supply a smooth physical velocity. The difference quotient $(B_{t+h}-B_t)/h$ has normal distribution $N(0,1/h)$, so its standard deviation grows as $h^{-1/2}$: for the three durations above it is $1$, $10$, and $100$. This calculation motivates the nowhere differentiability of Brownian motion, but it alone is not a proof of the simultaneous assertion at every time. The theorem says that, with probability one, no finite derivative exists anywhere along the sample path. A drawn polygon only displays a finite grid; its straight segments cannot establish this theorem. See Theorem 7.1.6 in Probability: Theory and Examples (fifth edition) by Rick Durrett.

For physical particles, Brownian motion is a model of position on suitable observation scales. It does not claim that microscopic velocity is a derivative of this idealized path. Einstein explicitly restricts his assumption of independent increments to intervals that are not too short. ^[ref]

Another way to measure the roughness is quadratic variation. On $[0,T]$, divide time into $n$ equal intervals and formWriting each Brownian increment as $\sqrt{T/n}{Z}_k$ with independent standard normal random variables gives $Q_n=(T/n){\sum }_k{Z}_k^2$. Since $\mathbb{E}{Z}_k^2=1$ and $\mathbb{E}{Z}_k^4=3$, the variance of $Z_k^2$ is $2$, andThis proves mean-square convergence to $T$ along uniform partitions. For $T=2$ and $n=100$, the expectation is $2$, the variance is $0.08$, and the root mean square of the error is $\sqrt{0.08}\approx 0.2828427125$. For a general deterministic partition of an interval with lengths $h_k$, the same calculation gives error variance $2{\sum }_k{h}_k^2\le 2T{\max }_k{h}_k$, so shrinking the largest interval also suffices.

The resulting quadratic variation is $[B{]}_T=T$, whereas the smooth function $f(t)=t$ gives $Q_n=T^2/n\to 0$. This difference explains why the ordinary rules of calculus need the second-order correction in the Itô formula. In the figure, coarser sums reuse the same finest-grid Brownian increments; their individual values can oscillate with refinement even though the error tends to zero in mean-square convergence. Refining a fixed interpolating polygon forever would instead give zero quadratic variation, so the finite-grid illustration must not be confused with the limiting Brownian motion.

Historically, Brown's 1828 report describes microscopic observations made in 1827, starting with particles released from pollen and extending to inorganic materials; this challenged an explanation based solely on living matter. His observations and the later mathematical model are distinct contributions. ^[ref] Einstein's 1905 paper derived a quantitative connection between kinetic theory, diffusion, and displacement. ^[ref] Wiener's 1923 Differential-Space developed integration on spaces of paths, a foundation for Wiener measure; it also acknowledges earlier work and his own previous papers. These dates mark documented contributions rather than an assertion that each subject had a single discoverer. ^[ref]

A Gaussian process is a stochastic process whose every finite vector of values has a multivariate normal distribution. Its law is determined by its mean and covariance functions.

A Lévy process starts at zero, has independent stationary increments, is stochastically continuous, and is conventionally taken to have càdlàg paths.

A stochastic process $X$ is stochastically continuous when $X_s\to X_t$ in probability as $s\to t$.

Stochastic calculus extends integration and differential calculus to stochastic processes such as Brownian motion and semimartingales.

Processes $X_n$ converge uniformly on compacts in probability to $X$ when, for every $T<\infty$ and $\epsilon >0$,

The natural filtration of a stochastic process $X$ records its history:possibly completed and made right-continuous when the usual conditions are required.

The graphical Stirling number ${\left\{\begin{array}{c}n\\ k\end{array}\right\}}_G$ counts partitions of the vertices of a graph $G$ into $k$ nonempty independent sets. Equivalently, it counts proper colourings with $k$ unlabeled nonempty colour classes.

The mass-luminosity relation connects a star's mass to its luminosity. Its slope depends on opacity, nuclear burning, pressure support, composition, and evolutionary state.

In a radiative stellar region, photons carry luminosity down the temperature gradient. Dimensional scaling of the radiative-diffusion equation gives $L\sim R{T}_c^4/({\kappa }_c{\rho }_c)$.

Homologous stellar models have the same dimensionless radial profiles after mass, radius, density, pressure, and temperature are scaled by characteristic values.

A stellar polytrope obeys $P=K{\rho }^{1+1/n}$ for constant $K$ and polytropic index $n$. Its dimensionless density profile satisfies the Lane-Emden equation.

The CNO cycle catalyses hydrogen fusion through carbon, nitrogen, and oxygen nuclei. Near $2\times 10^7\mathrm{K}$, its specific energy generation is approximately proportional to $\rho T^{17}$ and is therefore strongly concentrated near a star's centre.

The proton–proton chain converts hydrogen to helium and dominates hydrogen burning in stars near and below the Sun's mass. Near $2\times 10^7\mathrm{K}$, its specific energy generation is roughly proportional to $\rho T^4$.

If Steiner symmetrization preserves area and cannot lower the perimeter of a minimizer, equality forces an axis of symmetry in the chosen direction. Applying every direction gives symmetry axes in all directions.