Diffusion Equation

The Equation

$\frac{\partial \phi}{\partial t} = D\frac{\partial^2 \phi}{\partial x^2}$

Or: $\phi_t = D\phi_{xx}$

$D$ is the diffusion coefficient (or thermal diffusivity for heat).

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Physical Derivation

Fick’s law: Diffusive flux $J = -D\frac{\partial C}{\partial x}$ (down concentration gradient).

Conservation: Rate of change = -net flux out:

$\frac{\partial C}{\partial t} = D\frac{\partial^2 C}{\partial x^2}$

Heat equation: From Fourier’s law ($J = -k\nabla T$):

$\frac{\partial T}{\partial t} = D\nabla^2 T$

where $D = k/(\rho c_p)$.

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Three Fundamental Solutions

1. Diffusion from Step Function

Initially: $\phi(x, 0) = \begin{cases} 0 & x < 0 \\ 1 & x > 0 \end{cases}$

Solution:

$\phi(x, t) = \frac{1}{2}\left[1 - \text{erf}\left(\frac{x}{2\sqrt{Dt}}\right)\right]$

where $\text{erf}(z) = \frac{2}{\sqrt{\pi}}\int_0^z e^{-u^2}du$.

Interpretation: Smooth spreading of the initial discontinuity.

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2. Point Source (Green’s Function)

Initially: $\phi(x, 0) = \delta(x)$ (concentrated at origin)

Solution:

$\phi(x, t) = \frac{1}{\sqrt{4\pi Dt}}\exp\left(-\frac{x^2}{4Dt}\right)$

Properties:

• Gaussian shape at all times
• Spreads: width $\sim \sqrt{Dt}$
• Mass conserved: $\int_{-\infty}^{\infty}\phi\,dx = 1$

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3. Semi-Infinite System with Absorbing Wall

Domain: $0 < x < L$ with $\phi(0, t) = 0$ (absorbing boundary).

Solution:

$\phi(x, t) = \sum_{n\text{ odd}} A_n \sin\left(\frac{n\pi x}{2L}\right) \exp\left(-\frac{n^2\pi^2 D t}{4L^2}\right)$

Long-time behaviour: Dominated by $n = 1$ mode, decays as $e^{-\pi^2 Dt/4L^2}$.

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Separation of Variables

Try $\phi(x, t) = X(x)T(t)$:

$XT' = DX''T$

$\frac{T'}{DT} = \frac{X''}{X} = -k^2$

Time: $T(t) = e^{-Dk^2 t}$

Space: $X(x) = A\sin(kx) + B\cos(kx)$

Apply boundary conditions to determine $k$.

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Example: Finite Rod with Fixed Ends

$\phi(0, t) = \phi(L, t) = 0$

$X(0) = 0 \Rightarrow B = 0$

$X(L) = 0 \Rightarrow \sin(kL) = 0$, so $k = n\pi/L$

Solution:

$\phi(x, t) = \sum_{n=1}^{\infty} A_n \sin\left(\frac{n\pi x}{L}\right) \exp\left(-\frac{n^2\pi^2 Dt}{L^2}\right)$

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Fourier Coefficients

For initial condition $\phi(x, 0) = f(x)$:

$A_n = \frac{2}{L}\int_0^L f(x)\sin\left(\frac{n\pi x}{L}\right) dx$

Each mode decays exponentially; higher modes decay faster.

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Long-Time Behaviour

At large $t$, the $n = 1$ mode dominates:

$\phi \propto \sin(\pi x/L) \exp(-\pi^2 Dt/L^2)$

Characteristic decay time: $\tau = L^2/(\pi^2 D)$

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Comparison with Wave Equation

Property	Wave	Diffusion
Equation	$y_{tt} = c^2 y_{xx}$	$\phi_t = D\phi_{xx}$
Order in $t$	Second	First
Energy	Conserved	Dissipated
Information	Propagates at speed $c$	Diffuses, spreads

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Why the Diffusion Equation Matters

• Heat conduction
• Particle diffusion
• Chemical transport
• Financial mathematics (Black-Scholes)
• Biological processes (morphogen gradients)

The diffusion equation describes how things spread out and dissipate — a universal phenomenon in nature.