01 December 2007

Boiling eggs and dimensional analysis

A formula for boiling an egg. We learn that the time it takes to cook an egg is
t_{cooked} = {M^{2/3} c \rho^{1/3} \over K \pi^2 (4\pi/3)^{2/3}} \log \left[ 0.76 {T_{egg}-T_{water} \over T_{yolk}-T_{water} }\right]

where ρ is density, c specific heat capacity, and K thermal conductivity of egg. Tegg is the initial temperature of the egg, Twater is the temperature of the cooking water, and Tyolk is the temperature of the yolk-white boundary when the egg is done. (The egg is modeled as a homogeneous, spherical object; the yolk-white boundary is just a proxy for a certain distance from the center of the egg.)

What I keep noticing about formulas that purport to model real-world situations is that no matter how complicated they are, when numerical exponents appear they are always rational numbers. This is also true for purely mathematical phenomena. If someone told me that some statistic of some combinatorial object, for large n, had mean an and standard deviation bn7/22, for some bizarre, probably-transcendental constants a and b, I would think "ah, that must be a complicated proof, to generate such a large denominator." But if they said that this statistic had mean n and standard deviation n1/π, I would know they were pulling my leg. Of course, these would be fairly difficult to distinguish "experimentally". I suspect the reason that such exponents are rational in physics problems is for reasons having to do with dimensional analysis. For example, in the case of this problem, someone with sufficient physical intuition would probably guess that, fixing the temperatures, the important variables are the mass, density, thermal conductivity, and specific heat capacity of the egg. How can we multiply these together in some way to get time? Well:

  • mass (m) has units of kg

  • density (ρ) has units of kg m-3

  • thermal conductivity (K) has units W/(m × K), or kg m K-1 s-3

  • specific heat capacity (c) has units J/(kg × K), or m^2 s-2 K-1

Thus mα ρβ Kγ cδ has units kgα+β+γ m-3β+γ+2δ K-γ-δ s-3γ-2δ. But we know this is a time, so it must have units of seconds; thus we have the system of equations

α + β + γ = 0, -3β + γ + 2δ = 0, -γ - δ = 0, -3γ-2δ=1

which has the solution α = 2/3, β = 1/3, γ = -1, δ = 1, exactly the exponents we see. Similar analyses are possible for many physics problems, and in the end one gets a system in which integer combinations of the exponents are integers; thus the exponents themselves must be rational. And that's the really important part of the problem. In fact, all that really matters is that α=2/3. In practice, if you're cooking eggs, the thermal properties will be the same from egg to egg, but some will be larger than others; the exponent of mass tells you how to adjust for that, and everything else can be calibrated experimentally.

Mathematicians don't worry about this so much... but lots of these sorts of problems have some sort of physical model, which probably explains why rational exponents are common in combinatorial problems but I can't ever remember seeing an irrational exponent. (That's not to say they don't exist -- just that the solution to any such problem would be quite unconventional.)

(I learned about this from this post at Backreaction, about the phase diagram of water.)


Theo said...

We do, occasionally, see irrational exponents. Most notably, natural logs of numbers occasionally appear, because we are writing the formulas in perhaps the wrong way. For example, the natural log of -1 is a multiple of π.

Having only rational exponents means that all our relationships are in fact polynomial equations (a famous result in the beginnings of any undergraduate Galois theory class says that any solution to a polynomial with algebraic coefficients is itself an algebraic number). Perhaps, then, the Greeks were right, and the physical world has only algebraic numbers? (Well, the Greeks would want only rationals, but this is the next best thing, and much more plausible.) Alas, derivatives are important, and thus there are transcendental numbers, but certainly it's not unreasonable to expect that most equations "in the real world" are polynomial?

(Or at worst that differential equations are polynomial in jet space? Even that hope fails: the physicists want to use operators like e^{-(\hbar/2m)(d^q/dq^2) + V(q)}. But such terms are rare, and I've never met a physical formula worse than the exponential of a polynomial.)

Kevin said...

This reminds me of an interesting result of Shelah and Spencer on Random Graphs (mentioned in Chapter 10 of Alon and Spencer's Probabilistic Method book).

If you look at the Erdos Renyi random graph G(n,p), then only certain p can serve as threshold functions(*). Among the p which can never be a threshold function is n^(-\alpha), where \alpha is any irrational number.

In other words, while interesting things happen to the graph at such times as p=1/n (the giant component shows up), p=ln n/n (the graph becomes connected), and p=n^(-2/7) (complete graphs on 8 vertices appear for the first time), nothing interesting ever happens at p=n^(-1/pi).

(*) This may not exactly be the correct interpretation. More formally, I think they show that if p=n^(-\alpha), then G(n,p) satisfies a 0-1 law, while normally at a threshold you expect to be able to see a probability strictly between 0 and 1).

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