The diffusion equation plays an integral role modelling reality. Whenever there is a system containing a physical quantity that wants to smooth out and become homogeneously distributed then the diffusion term will model that happening. A temperature fluctuation in the atmosphere, a drop of dye in a pool of water, a magnetic field in a resistive fluid. All of these phenomena evolve according to the diffusion equation. Before going into the specifics of this equation, let’s take a quick look at it. If Q is a physical quantity (scalar, vector, or tensor) then the diffusion equation reads:
Here D is a scalar known as the diffusivity constant. We can gain a quick understanding of this equation by considering the geometrical interpretation of the Laplacian operator on the rhs. Second spatial derivatives correspond to the concavity of a function, with minima (like the bottom of a bowl) having positive concavity, and maxima (like an overturned bowl) having negative concavity. Since the diffusivity constant is positive, the diffusion equation causes the temporal evolution of the quantity in question to be negative for maxima and positive for minima. This means that regions where there is an over abundance of Q, evolve to decrease the amount of Q, and conversely for regions sparse in the amount of Q. When Q is distributed homogeneously, the concavity vanishes, and the system stops evolving.
For conserved quantities, the diffusion equation is a consequence of the continuity equation coupled with an ansatz for the current density in the form of Fick’s Law.
Fick’s law simply states that a current will establish itself in the direction opposite the gradient of the quantity. This makes sense, just think of temperature. Heat energy flows from hot to cold, against the temperature gradient. Here the diffusivity can be a function of the field quantity, but it must be positive definite (otherwise we would have a clumping equation). Plugging the latter into the former we find a modified diffusion equation:
If the diffusivity is a constant function, then the first term vanishes and we have the diffusion equation. Conversely, for any quantity that satisfies the diffusion equations, a current density can be defined in such a way so as to satisfy the continuity equation.
Now even though following Q is what we set out to do with the diffusion equation, in many physical contexts we want to follow the energy associated with Q as well. It is common for the energy density to depend quadratically on Q. This is the case with the kinetic energy associated with velocity, as well as the energy contained in electric and magnetic fields. If the underlying quantity satisfies the diffusion equation, then the continuity equation for the energy will develop a sink term, which corresponds to the dissipative effects of the diffusion. Let’s see how this works for a scalar. Define the energy density as:
We then find the temporal derivative of this quantity, and use the diffusion equation:
If we define the term whose divergence is being taken on the rhs as the negative of the energy current, and treat the diffusivity as a constant, then we get a continuity equation with a negative definite (sink) term on the lhs:
This is a fascinating result, because it tells us that diffusion acts to dissipate the energy contained in a scalar field. As the field homogenizes, some of the energy contained in it is lost. Note that once the gradient of the field is zero, and homogeneity has been established, the energy contained in the field becomes conserved.
For a vector field, things get a little more complicated. In the following we make use of an index notation along with the Einstein summation convention. Let A be a vector quantity, then the energy density is defined as:
We once again find the time rate of change of the energy density, and use a vector identity:
This result motivates our definition of the energy density current as:
The first term in this current can be attributed to the gradient in the energy density, whereas the second comes from a tension inherent in the field lines of A. The resulting continuity equation now reads:
Once again we find that the right hand side is negative definite, and hence diffusion of a vector field is a sink for the energy contained int he field. Note that once the field settles down into a configuration that has no curl the sink vanishes and energy is conserved.
This energy sink is incredibly important in magnetohydrodynamics. Within the theory, diffusion terms appear for the velocity and magnetic fields, as well as the thermal energy field. In the latter case energy is conserved and causes the diffusion of energy through heat. In the first two, however, energy is drained away from the fields and is dissipated into heat. For the case of velocity, the diffusion constant is called the viscosity, and for the magnetic field it is called the resistivity. These sinks need to be taken into account in order to properly model the evolution of plasmas, which occur in many astrophysical phenomena, including stars, accretion disks, and the ISM and IGM. Hopefully the above analysis has shed light on this fascinating topic.