# Relationship between the diffusion equation and the heat equation

In physics we have the heat equation which describes the propagation of heat

$$\dfrac{\partial u}{\partial t} = \kappa \dfrac{\partial^2 u}{\partial x^2},$$

while in biomathematics we have the diffusion equation which describes the random diffusion of a particle across space with time

$$\dfrac{\partial P}{\partial t} = D \dfrac{\partial ^2 P}{\partial x^2}.$$

Clearly these two equations are of similar forms, but I'm having some trouble trying to see why heat and diffusion would be related. I've always thought of heat as quite "orderly" - i.e. it propagates out in a more or less regular manner from some point of heat. However, I've always thought of diffusion as more "random" with a particle performing Brownian motion and the like to eventually reach a point ($P$ is normally related to a probability and it has a discrete recursive form, $P_{x, t+1}$).

If anyone could provide some insight, that would be great. Is this purely a coincidence or is there some basic condition which means we get analogous equations?

• These literally are the same thing. Heat is the diffusion of thermal energy, which is random in nature (e.g. the motion of every particle in a medium in which heat is propagating is random). Nov 7 '14 at 1:23
• Isn't $\kappa$ a constant but $D$ a function?
– lhf
Nov 7 '14 at 1:24
• $D$ is supposed to depend on the medium of diffusion, so it can be regarded as a constant for each particular medium, but we also have "fractional diffusion" and the like where $D=D(t) = t^{-1/2}$ for example. But $\kappa$ is similarly defined and is characteristic of the medium propagating heat as well... So I guess they could potentially both be regarded as functions of the properties of the media? Nov 7 '14 at 1:30
• @ArturoDonJuan But then why is the probability for a particle's position of the same form as the equation that governs its motion? I'm just surprised that they're exactly the same and not even slightly different. Is the temperature supposed to be analogous to the probability of a bit of heat-energy being there? Nov 7 '14 at 1:33
• The diffusion equation tells you how the probability density function of a Wiener process evolves over time. It's probably obvious why this also describes evolution of temperature if you know the definition of temperature. But mathematically they are the exact same differential equation. Nov 7 '14 at 1:42

In the heat equation, what is the base assumption? It is that the heat flow $$\frac{\partial u}{\partial t}$$ is proportional to the second spacial-derivative (laplacian) of the temperature at the point.

$$\frac{\partial u}{\partial t}\propto\nabla^2 u$$

In the diffusion equation with the diffusivity $$D$$ being constant, what is the base assumption? It is that the diffusion (rate of change of the density of particles with respect to time) of the particles at a point is proportional to the second spacial-derivative (laplacian) of the density of those particles at that point.

$$\frac{\partial P}{\partial t}\propto\nabla^2 P$$

So how are they related? Well, they literally revolve around the same base assumption - that the amount of whatever they're dealing with (e.g. temperature or density) is conserved. Thus, both of these are applications of the Continuity Equation.

If we have a density of "something" $$H$$ in space, which has a local current density $$\mathbf{J}$$ (i.e. amount of $$H$$ passing through a sufficiently small volume per unit time, divided by that volume), then the assumption that "no $$H$$ can be created or destroyed" is equivalent to the continuity equation,

$$\frac{\partial H}{\partial t} + \nabla\cdot \mathbf{J} = 0$$

If we would like to make contact with the heat equation, we define $$H$$ to be local measure of thermal energy (i.e. temperature - we assume thermal energy can neither be created nor destroyed, but only transferred from one place to another), and make use of Fick's law of diffusion, which states that the current density of a diffusive material/stuff is proportional to the negative of its gradient.

$$\mathbf{J}\propto -\nabla H$$

Letting the proportionality constant be equal to $$D$$, we get the heat equation.

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

This is really just the steady-state diffusion equation, which can be generalized to any situation which revolves around the same basic assumptions. An example of this is Schrodinger's equation for a free-particle.

$$\frac{\partial \psi}{\partial t}=i\nabla^2 \psi$$