A question came to us on twitter asking for “a natural phenomenon described by a differential equation with a general solution and a differential equation in interlace format”.

Ask Dr. Math is an awesome source of information regarding any kind of mathematics. It has an archive with 20 years-worth of mathematics questions. You can find out more about differential equations and general solutions with the links in this sentence. For any math related questions feel free to visit Ask Dr. Math! If you can’t find a solution there, our volunteer math doctors will be happy to help.

As far as a natural phenomenon is concerned, the first thing I can think of is an example from thermodynamics; the Clausius-Clapeyron relation.  The Clausius-Clapeyron relation is a model for a discontinuous transition in phase for any two states of matter of one material in terms of temperature and pressure. The relationship itself is a homogeneous  first order differential equation. This relation is the general solution of the state postulate or the Gibbs-Duhem equation. You can read the full derivations here. The general solution is:

;P= Pressure, T = Temperature, L = Latent heat, Δv = specific volume.

Latent heat, temperature and specific volume are all experimentally determined constants, therefore the only variable is Pressure. dT can be considered thermodynamically to be the rate of heat energy entering a system. Therefore, the variable P is expressed in terms of only one other variable with directly applicable constants and is a general solution

Science note: Latent heat is the heat energy absorbed by a volume of matter during a change in phase. For example, imagine a bowl of ice. In every thermodynamic situation, there’s a system (the ice) and the surroundings (the bowl, atmosphere). We only need to concern ourselves with the actions of the ice as it reacts to external heat energy.  The ice, to remain solid for a determinable amount of time is some temperature below 273 K (Oº C, 32°F). It’s safe to say that unless we’re looking at a bowl of ice in outer space or outside during the winter, the external temperature is above the freezing point of water. Since the temperature of the surroundings is greater than the temperature of the ice, the ice will absorb energy and eventually melt. As the ice changes in phase to liquid water (or as any matter changes phase), it has been experimentally determined that the temperature remains constant. Even if it were hot enough to boil water, the temperature of the melting ice remains constant as it is changing in phase, even if for a brief time.

Specific volume is the rate at a set quantity of matter expands or contracts given an input of energy in the form of heat or pressure.

Why is the Clausius-Clapeyron Relation important? It allows us to ice skate! The relation suggests that a solid can melt when it is below its freezing point with sufficient pressure. Most ice skating rinks keep the air well below freezing point lest it become as swimming pool. Ice on it’s own is slippery, but as a solid it does not have the minimal friction required to achieve the speeds at which we can skate. This is where the Clausius-Clapeyron relation comes in. Solving for ΔP, we get the equation:

. Mathematically, we can substitute ΔP with ΔT at will. Scientifically, this says that in terms of changing the phase of a material, pressure and temperature contribute work in equal ways. Considering how ice skates work, the sharp blades are essentially for putting all the weight of a skater over a very small area which creates a huge amount of pressure. As a skater applies this pressure to the ice, it melts creating lubricant such that he or she can fly across the ice.

As far as Interlace format is concerned, I’m not familiar. I looked for it in the Ask Dr. Math archives but found no mention. Perhaps one of our volunteer math doctors knows. I looked around the internet and found a few papers about using interlace format to solve stochastic differential equations. Essentially, Interlacing is applying multiple systems of differential equations into multiple matrices via eigen values and vectors, and then combining them to find a point of convergence.

In Materials Science, we use complicated mathematics to create computer simulations of how crystalline structures react to forces, stresses and strains down to the individual atoms in a structure. Here is an example of such a simulation. In the video the all the atoms that stresses are not applied to are invisible, they will appear when a force is applied to them and when they are moved away from their position in the crystal lattice. Essentially, what you’re watching is how atoms move erratically through their crystal structures when forces are applied. In complex systems of dislocated crystals, there is a lot of mathematics at play. Materials scientists may use interlaced differential systems to streamline the calculations used in these simulations.