Heat Equation — Physica

Equivalent forms

u_t = D u_{xx}

u(x,t) = \frac{1}{\sqrt{4\pi D t}}\,e^{-x^2/4Dt}\quad(\text{point-source / heat kernel})

First order in time, so it has an arrow: it smooths forward but is ill-posed backward — the mathematics of irreversibility in one symbol.

Symbol	Name	SI unit	Dimension	Range
$u$	Temperatureoutput Field value (temperature or concentration)	1	1	0 – 2
$D$	Diffusivity Thermal/mass diffusion coefficient	m^2/s	L^2 T^-1	0.05 – 3
$x$	Position Coordinate along the rod	m	L	-4 – 4
$t$	Time Elapsed diffusion time	s	T	0.05 – 6

Unit systems

Symbol	SI	CGS	Imperial
$D$	m^2/s	cm^2/s	ft^2/s
$x$	m	cm	ft
$t$	s	s	s
$u$	1	1	1

Where it holds

Linear diffusion with constant D; continuum (non-relativistic) heat/mass transport.

Dimensional analysis

u_t \to [u]/[T]

D u_xx \to [L^{2}/T]\cdot [u]/[L^{2}] = [u]/[T]

The diffusivity's length

^{2}

cancels the two spatial derivatives, leaving one inverse time on each side.

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Discovery

Joseph Fourier · 1822

Fourier derived the equation to model heat conduction in solids and invented Fourier series specifically to solve it. Later, Einstein (1905) and Bachelier showed the same equation governs Brownian motion and diffusion of particles — and of stock prices.

Try this

Touch one end of a cold metal rod to a flame. Why does the far end warm so slowly, and why can you never un-blur the heat once it spreads? One diffusion law answers both.

Drop a hot spike onto a rod and watch it relax into a spreading Gaussian — sharp gradients punished, peaks flattened, time's arrow visible.

Research status: stable

Real-world applications

Thermal management and heat-sink design
Diffusion of dopants in semiconductor fabrication
Brownian motion and the Black–Scholes equation in finance
Image denoising and scale-space (Gaussian) smoothing

Common misconceptions

Heat travels at finite speed in this model — the classical equation has infinite propagation speed (a known flaw fixed by hyperbolic corrections).
Diffusion is reversible — forward smoothing is well-posed, the reverse is not.
It only describes heat — the identical equation governs particle diffusion, Brownian motion, and option pricing (Black–Scholes).

Experimental verification

Measured temperature relaxation in rods, ink diffusing in water, and Einstein's prediction ⟨

x^{2}

⟩

= 2Dt

for Brownian motion (confirmed by Perrin, Nobel 1926) all match the equation quantitatively.

Derivation

Combine conservation of energy

(u_t = -\partial q/\partial x

per unit volume) with Fourier's law

q = -D u_x

.

Substituting gives

u_t = D u_xx

.

In Fourier space each mode obeys û_

t = -Dk^{2}

û, so û(k,

t) =

û(k,

0)e^{-Dk^{2}t}

— high-k detail decays fastest, which is why diffusion smooths.

Limiting cases

t \to 0

⟶ Approaches the initial spike

\delta (x)

The heat kernel narrows to a delta function as time rewinds.

t \to \infty

⟶

u \to

uniform averageAll gradients vanish; the system reaches thermal equilibrium.

Run time backward⟶ Ill-posed (blows up)High-frequency noise amplifies — you cannot reconstruct the past sharp profile.

What if…

What if D is negative?

The equation sharpens instead of smooths and becomes catastrophically unstable — backward heat flow is ill-posed.

What if I add a drift term?

$u_t = D u_xx - v u_x$ is the advection–diffusion (Fokker–Planck) equation: spreading plus transport.

1

Spreading width

Given ·

D:: 1
t:: 4

Find · Gaussian width

\sigma

Steps

Heat kernel variance $= 2Dt$
$\sigma = \sqrt{2\cdot 1\cdot 4} = \sqrt{8}$
$\approx 2.83$ — width grows like $\sqrt{t}$ , not t

Result ·

\sigma = \sqrt{2Dt} = \sqrt{8} \approx 2.83

2

Mode decay

Given ·

k:: 2
D:: 1
t:: 1

Find · amplitude factor

Steps

Each Fourier mode decays as $e^{-Dk^{2}t}$
$k = 2 \to e^{-1\cdot 4\cdot 1} = e^{-4}$
$\approx 0.018$ — fine detail (large k) vanishes fastest

Result ·

e^{-Dk^{2}t} = e^{-4} \approx 0.018

Laplacian→Wave Equation→Fourier Series→Gaussian (Normal) Distribution→