Examples where an ill-behaved function leads to surprising results?

Question

In mathematical derivations of physical identities, it is often more or less implicitly assumed that functions are well behaved. One example are the Maxwell identities in thermodynamics which assume that the order of partial derivatives of the thermodynamic potentials is irrelevant so one can write.

Also, it is often assumed that all interesting functions can be expanded in a Taylor series, which is important when one wants to define the function of an operator, for example $$e^{\hat A} = \sum_{n=0}^\infty \frac{(\hat A)^n}{n!}.$$

Are there some prominent examples where such assumptions of mathematically good behavior lead to wrong and surprising results? Such as... an operator $f(\hat A)$ where $f$ cannot be expanded in a power series?

Answer 1

I think, the most transparent example is phase transition: by definition it is when some thermodynamic value does not behave well.

AFAIK when Fourier showed that non-continuous function may be presented as an infinite sum of continuous, he had a hard time convincing people around that he is not crazy. That story might partially answer your question: as long as any not-so-well-behaved function may be presented as a sum of smooth ones, there is no much difference as long as good formulated laws are linear. Functions which are really bad behaved usually do not appear in real problems. If they do, there is some significant physics behind it (as with phase transition, shock wave, etc.) and one can not miss it.

For an operator it is better (for physicist) to think of function from operator as a function acting on its eigenvalues (if it is not diagonalizable, in physics it is bad behaviour). This is equivalent to power series definition, but works for any function.

Answer 2

I have had a surprizing result due to the wave function having different left and right derivatives at a point (see Chapter 2.1 and Appendix 3). Generally this article contains more surprizing results just due to implicit assumptions being wrong.

Answer 3

Well, I don't know if you want to count that, but QFT is full with functions that have poles which I'd call not well-behaved, and it does have lots of physical effects. If you're talking about observables only, you can approximate any discontinuous function to arbitrary precision with a continuous function, and you can push the difference below measurement precision. The reason one sometimes uses 'ill-behaved' functions (delta, heaviside, etc) is that they're easier to deal with.

Answer 4

This principle fails in the most startling way in second order phase transitions. This is a particularly clean example, because Landau predicted the critical exponents of second-order phase transitions using only the principle that the thermodynamic functions are analytic.

His argument is as follows: given a magnet going through the Curie point, where it loses its magnetization smoothly, the equilibrium magnetization should be the solution of some thermodynamic equation with the derivative some thermodynamic potential is set to zero.

$F(T,m)=0$

At temperatures lower than T_c, the magnetization is nonzero, and at temperatures higher than Tc, the magnetization is 0, and it goes to 0 in a continuous way. How does it go to zero?

Note that the magnetization m and -m are related by rotational symmetry. Shifting T_c to 0 by translating $f(t,m)= F(T_c - t,m)$, you get a new thermodynamic function, which has the property that f has only the trivial solution m=0 for negative t, and has two small nontrivial solutions in m for positive t.

Because m=0 is a solution at t=0, the function $f$ has no constant term in a Taylor expansion. By the symmetry of $m\rightarrow -m$, only even powers of m contribute to its Taylor series.

$f(t,m) = At + Bm^2 + Ct^2 + Dt^3 + E t m^2 ...$

Assuming that $f(t,m)$ is generic, A and B are not exactly zero. So for small enough t, for temperatures close enough to the critical point, you get that

$m \propto \sqrt{t}$

Further, this scaling only fails if one of the coefficients is zero. If A=0,

$m \propto |t|$

But m is then nonzero on both sides of the transition. If B=0, you get

$m \propto t^{1\over 4}$

and m is zero or if A,B,C are zero, in which case you get

$m \propto |t|^{3/4}$

And each of these cases requires fine tuning of parameters. So Landau predicted that the critical behavior of the magnetization will be as the square root of the temperature at the critical point, and that this behavior will be universal, it won't depend on the system, just on the existence of the phase transition. The Ising model should have the same critical exponent as the physical magnet, a square root dependence of the magnetization on the temperature, and the liquid gas transition will also have a bend in the curve of the density vs. temperature at the critical pressure which goes as the square root.

The exponent turned out to be universal, it was equal for the gas and liquid, and for the Ising model. But it wasn't 1/2, but more like .308 in three dimensions, and .125 in two dimensions, It only turned into Landau's 1/2 in 4 dimensions or higher. This means that Landau's argument fails, and that the thermodynamic function is conspiring to be non-analytic at exactly the place where Landau was expanding. Understanding why it is non-analytic exactly at the phase transition led to the modern renormalization theory.

In mathematics, Rene Thom proposed that a version of Landau's argument is a complete theory of the types of allowed phase transitions in nature. He called the phase transitions "catastrophes", because they showed a sudden change in behavior, and he predicted, based on catastrophe theory all sorts of scaling laws for natural transitions. This was the most ambitious attempt to exploit the observation that naturally occuring functions are nice. This fails for the same reason as Landau's argument: functions describing changes in the critical behavior of interesting systems at a transition point are rarely analytic at this point.

Answer 5

A nice example arises for the "rigorous coupled wave analysis" (RCWA) method (also called Fourier Modal Method), which is used as a Maxwell solver for diffraction gratings. The normal component of the electric field is discontinuous in the normal direction of a material interface. This leads to convergence problems of the RCWA method for TM polarization, because the discontinuous electric field component is expanded into a Fourier series and multiplied by another discontinuous function representing the grating geometry. Many modifications of the RCWA method to overcome this convergence problem where proposed, but the "correct" modification was only discovered in 1996 (by P. Lalanne and M. Morris?). Even so Lifeng Li didn't discover that "correct" modification, he wrote the famous paper "Use of Fourier series in the analysis of discontinuous periodic structures" (also in 1996) which analyzed mathematically what goes wrong (multiplication of "approximations of" discontinuous functions is dangerous) and why the latest proposed modification to the RCWA methods finally solved the convergence problem.

Today, the Fourier Modal Methods are the most efficient and accurate for many types of grating problems.

Answer 6

The partition function of a field theory is usually nonanalytic at zero coupling, which is why the Feynman expansion about zero coupling is an asymptotic rather than Taylor expansion and has zero radius of convergence. There are therefore nonperturbative effects in field theory that cannot be captured within a diagrammatic expansion, most notably instantons. (It took many decades for this to be fully appreciated.) See https://arxiv.org/abs/1210.2423 for more information.