The heat equation: where it comes from

Let \( u(x,t) \) be the temperature in a thin rod with density \( \rho \), specific heat \( c \) and conductivity \( K \). The heat in a segment \( [a, b] \) is \( \int_a^b c\rho\,u\,dx \), and it changes only through the flux \( \phi \) at the ends: \( \frac{d}{dt}\int_a^b c\rho\,u\,dx = \phi(a,t) - \phi(b,t) = -\int_a^b \phi_x\,dx \). Since the segment is arbitrary, \( c\rho\,u_t = -\phi_x \). Fourier's law says heat flows down the gradient, \( \phi = -K u_x \), and together they give \[ u_t = k\,u_{xx}, \qquad k = \frac{K}{c\rho}. \] In a region of space the same argument with the divergence theorem gives \( u_t = k\,\Delta u \).

The equation has a second derivation that explains its universality. A particle hops a distance \( h \) left or right every \( \tau \) seconds. The probability \( u \) of finding it at \( x \) obeys \( u(x, t + \tau) = \tfrac12 u(x - h, t) + \tfrac12 u(x + h, t) \); expanding both sides in Taylor series gives \( \tau u_t \approx \tfrac12 h^2 u_{xx} \), the heat equation with \( k = h^2/(2\tau) \). Diffusion of ink, of heat and of probability are one equation.

The equation contains its own length scale. It is unchanged by \( x \mapsto \lambda x,\ t \mapsto \lambda^2 t \), so the only length that can be built from \( k \) and a time \( t \) is \( \sqrt{kt} \): in four times as long, heat spreads twice as far. The example checks that \( u = e^{-t}\sin x \) solves \( u_t = u_{xx} \): the sine shape stays put while its amplitude decays.

Picture it: \( u_{xx} \) measures bending. Where the temperature graph is concave down (a local hot spot), \( u_{xx} < 0 \) and the temperature falls; where it is concave up, the point warms. Every bump flattens.

Think it: for a rod with fixed ends, \( \frac{d}{dt}\int \tfrac12 u_x^2\,dx = \int u_x u_{xt}\,dx = -\int u_{xx}u_t\,dx = -\int u_t^2\,dx \le 0 \). The heat equation runs downhill on the Dirichlet energy as fast as possible, a gradient flow. The viscous term \( \nu\Delta u \) in Navier-Stokes comes from the same structure: a conservation law (momentum) closed by a constitutive law (stress proportional to velocity gradient).