Which kinds of systems are described by a heat equation?

Question

Every extensive thermodynamic variable has a continuity equation associated to it:

$$\frac{\partial \rho}{\partial t}+\vec \nabla \cdot \vec J=0$$

where $\rho$ is the density of said variable and $\vec J$ is its density flux. Nonetheless, I realised the sufficient condition for a system to be described by a heat equation is that the flux density is equal to the gradient of the density:

$$\vec J = \vec \nabla \rho$$

Substituting this in the continuity equation yields a heat equation for $\rho$:

$$\frac{\partial \rho}{\partial t}+\nabla^2\rho=0$$

Therefore, this leads to the following question: what kinds of systems follow a heat equation understood as a specific case of a continuity equation? I know a material with some temperature gradient does follow it (i.e: your usual heat equation, where $\rho=T$ and $\vec J=\vec \nabla T$), but I wonder whether there are other systems fulfill this as well.

Answer 1

So you’ve asked for other examples where the “heat equation” comes up. This is really a diffusion equation in general.

The first thing worth mentioning is, your derivation is making a particular assumption. In general the Fick’s law equation is going to look like $$J = -\nabla(D~ \rho)$$ for some diffusivity $D$, and we only can simplify this to $-D~ \nabla\rho$ by assuming that this diffusivity $D$ is constant over space. So for example when you do this for $\rho$-as-temperature, the actual conserved quantity is energy, $D$ takes the form of a specific heat or heat capacity times a coupling constant or so--you are making a bit of an assumption here that the specific heat is not changing throughout the system!

The second thing worth mentioning is, often $J$ also contains a term for the stuff flowing downstream in some external velocity field, and perhaps another term for stuff entering the system from outside at the end, so that a more general $J$ is, $$J = \rho~\mathbf v - \nabla(D~\rho),\\ \frac{\partial \rho}{\partial t} + (\mathbf v \cdot \nabla) \rho = D ~\nabla^2 \rho + \rho\big(\nabla D -\nabla\cdot\mathbf v\big) + \Omega $$ This is the transport equation, also called the convection-diffusion equation. The left hand side can be looked as a change in the count of the amount of stuff in a box that flows downstream along a streamline, as partial derivatives would say that $$f(x + v_x\delta t, y + v_y\delta t, z + v_z \delta t, t + \delta t) \approx f(x, y, z, t) + \delta t \left( \frac{\partial f}{\partial t} + (\mathbf v \cdot \nabla) f\right).$$ So the transport equation says “a box flows downstream, the time rate of change of stuff in the box is equal to the diffusion from higher concentrations from nearby boxes, plus any gradients in the diffusivity, plus any effects where the fluid itself is compressing into the point, plus any effects $\Omega$ where, say, a chemical reaction or external heat source is inserting this substance into the fluid when it wasn’t there before.”

Now, where does the transport equation pop up? Just about everywhere. And if the transport equation pops up then there is a “heat equation” (I’d just say ‘diffusion’) case where $\mathbf v$ is zero and $\Omega$ is zero and $D$ is constant. It pops up so frequently that the units of $D$, $\text{m}^2/\text{s},$ become an “alert signal” when you see them that you should see if there’s a diffusion equation hiding here.

• Momentum diffuses across a fluid, and this is called “viscosity.” (Kinematic viscosity is measured in cm²/s.) The Navier-Stokes equations are in fact three transport equations for $\rho = p_x, p_y, p_z$, the momentum of the fluid itself in each direction. (This is also why the Navier-Stokes equations are nonlinear and why there's a million dollar prize for understanding them better.) The standard Newtonian definition of viscosity (put a fluid between two plates, create relative motion, measure the force) is then the transport equation where the convective transport $\mathbf v\cdot\nabla$ is in the $x$-direction, say, but because we're looking at how the momentum diffuses in the $y$-direction, this convective transport doesn't matter to that study and all we have left is the diffusivity created by viscosity.

• You've already identified temperature and the heat equation, as long as we don't get into the Stefan-Boltzmann law (You can do a little bit of Stefan-Boltzmann this way too, but $D\propto T^3$ can be a minor challenge and if the radiation isn't immediately absorbed by nearby neighboring packets of fluid then it in principle needs to be described by a whole separate vector field, which is a more major challenge.)

• Chemical concentrations work this way. Although, often they drive fluid flows in turn. (I once had a professor who put it this way: “If it weren’t for convective diffusion I wouldn’t care if someone smoked cigarettes in my class. I mean, it would also be lethal to them as they’d burn up all the oxygen that they needed to breathe, so I guess I would have to call emergency services, but at least I wouldn’t have to breathe in the smoke while making the call.”) But if you try to describe a tube of length $L$ where the thing starts out with clean water and then you inject some saline on the left hand side and monitor the salinity on the right hand side, you can probably describe this with a diffusion equation. And if the salt concentrations are driving too much in the way of convective transport, you can always study in a supramolecular context -- these are bigger structures like hydrogels and dendrimers that "slow down" a lot of those convective processes simply by occupying the space and reducing water's freedom to move.

• Related to the former, if you're trying to dope a semiconductor then you need to introduce dopants, they need to travel into the bulk of the semiconductor given that they're kept at a higher concentration in the surface: diffusion would describe how the dopants penetrate into the semiconductor.

• Diffusion currents are important when we're talking about semiconductor junctions, especially diodes. Solar cells in particular depend pretty crucially on a diffusion length before electrons and holes recombine.

• Voltage in a resistor works this way; voltage drives a current which spreads the voltage to nearby places. If you try to do problems with infinite networks of resistors, I'd expect that the voltage spreads out to infinity with an approximate 2D diffusion equation?

So yeah, there's a lot of places where the same idea comes out. Some universities have dedicated physics courses on “transport phenomena” and others roll it into a fluid mechanics class.

Answer 2

The following is intended to expand on hyportnex's comment in one particular direction; you may or may not find it useful.

Your path from the conservation equation of an extensive quantity $X$ (I'll use $\rho$ for something else later)

$$\frac{\partial X}{\partial t}+ \vec\nabla\cdot\vec J_X=0\tag{1}$$

straight to a heat-equation-like relation where $X$ is replaced with the temperature $T$ arguably requires clarification, as the temperature is not extensive or conserved.

A better link might start with conservation of energy density $u$ in a control volume. Then, $du=\rho c\,dT$ (with density $\rho$ and heat capacity $c$), and if only conduction is considered, Fourier's law gives heat flux $\vec J_Q=-k\nabla T$, yielding

$$\frac{\partial T}{\partial t}- \alpha\nabla^2 T=0,\tag{2}$$

where the thermal diffusivity $\alpha=\frac{k}{c\rho}$, where material properties are assumed to be constant, and where $T=T(x,y,z,t)$. This is the conduction heat equation.

One could instead work with the even simpler

$$\frac{\partial T}{\partial t}- \nabla^2 T=0,\tag{3}$$

given the implicit acceptance that now $T=T\left(x,y,z,\alpha t\right)$ or $T=T\left(\frac{x}{\sqrt{\alpha}},\frac{y}{\sqrt{\alpha}},\frac{z}{\sqrt{\alpha}},t\right)$.

Returning to the conservation equation, we might ask: What is the extensive conjugate pair to temperature? It is entropy $S$ (or entropy density $s$ in a control volume). However, entropy isn't conserved but rather is paraconserved, so to speak; put another way, $dS\ge 0$ in an isolated system—the Second Law. The following is adapted from Balluffi, Allen, and Carter's Kinetics of Materials. We could write a paraconservation equation

$$\frac{\partial s}{\partial t}+ \vec\nabla\cdot\vec J_S=\dot\sigma,\tag{4}$$

where $\dot\sigma\ge 0$ is the entropy density generation rate. From the time derivative of the First Law (considering heat transfer only, so $du=T\,ds$, and with minimal temperature changes) and then applying the energy conservation equation,

$$\frac{du}{dt}=T\frac{ds}{dt};$$ $$\frac{\partial s}{\partial t}=\frac{1}{T}\left(\frac{\partial u}{\partial t}\right)=-\frac{1}{T}\left(\vec\nabla\cdot\vec J_Q\right),$$ and because

$$-\frac{1}{T}\left(\vec\nabla\cdot \vec J_Q\right)=-\vec\nabla\cdot\left(\frac{\vec J_Q}{T}\right)+\nabla\left(\frac{1}{T}\right)\cdot \vec J_Q=-\vec\nabla\cdot\left(\frac{-k\nabla T}{T}\right)+\frac{k}{T^2}\left(\nabla T\right)^2,$$

we have

$$\frac{\partial s}{\partial t}+\vec\nabla\cdot\left(\frac{-k\nabla T}{T}\right)=\frac{k}{T^2}\left(\nabla T\right)^2.\tag{5}$$

Matching the terms to (4), we associate the entropy flux $\vec J_S$ with $\frac{-k\nabla T}{T}$ and estimate the entropy density generation rate as

$$\dot\sigma=\frac{k}{T^2}\left(\nabla T\right)^2;$$

thus, this rate scales up with the square of the temperature gradient down which heat is flowing. In this way, additionally, the First and Second Laws prohibit a negative thermal conductivity $k$. (5) provides the paraconservation equation for heat transfer.

Thus, although I don't agree that moving from (1) to (3) is as simple as replacing conserved quantity $X$ with temperature $T$, one can certainly apply the (para)conservation equations (1) and (4) to a variety of systems. In the case of diffusion, for example, matter is conserved, and the entropy density generation rate scales up with the square of the chemical potential gradient (in ideal systems, the square of the concentration gradient), with the key material property—in this case, the mobility—again always positive. Or one could apply the same treatment to (conserved) charge, for instance, and confirm an entropy generation rate of $\frac{V^2}{RT}$ at temperature $T$, consistent with the well-known power dissipation $I^2R$ of resistive heating.