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The “unreasonable effectiveness of mathematics in the physical sciences,” a phrase coined by Eugene Wigner, is often mentioned as an exotic property of mathematics. It is an expression of the wonder that models constructed from purely theoretical considerations and reasoned out can predict real systems with great precision. A modern instance of this wonder is found in the writings of Subrahmanyan Chandrasekhar: “In my entire scientific life, the most shattering experience has been the realization that an exact solution of Einstein’s equations of general relativity, discovered by Roy Kerr, provides the absolutely exact representation of untold numbers of massive black holes that populate the Universe.”

In science fiction writer Stanley G Weinbaum’s 1935 short story “The Lotus Eaters” the ultimate law of physics is declared to be the law of chance. Given the overwhelming power of statistical mechanics and of quantum mechanics this assessment is hard to dispute. The study of probability combines beautiful reasoning beginning from abstract first principles and describes the observable world with remarkable clarity. So before moving into Monte Carlo methods it is worthwhile reviewing the fundamentals of probability.

One of the goals in this book is finding of an optimum state, an arrangement of site values subject to certain constraints. At first one may suppose these questions can be perfectly answered by analytic methods. Our energy function has only a scalar value, which is continuous and smooth (provided no two points have the same position). Its variables may be the strengths of each site, or they may be the positions of each site, but they are just an n-tuple (an ordered set, as in a vector) of real-valued coordinates. This is almost as well-behaved as one could hope for a function.

What will we find if we measure the energy of a system? The focus of this chapter is finding the expectation value of any random variable - energy or another property — by Monte Carlo methods.

In this chapter we introduce spectral methods. The principle is familiar to anyone who has used Fourier series to solve a differential equation: we represent the unknown but presumed-to-exist solution as the summation of a (possibly infinite) number of constants times base functions, and convert the problem of solving a differential equation into solving a set of equations for the coefficients.

With some basis now in statistical mechanics, we turn to fluid mechanics. We want a representation of a fluid’s flow compatible with our Monte Carlo tools. We will develop these along two lines of thought. Our first is to represent the vorticity field of a fluid flow as a collection of point particles. This is the vortex gas model, a dynamical system we can treat just as we do any ordinary problem of mechanics. In the next chapter we will create the lattice gas model, a system based on approximating the vorticity field with a function continuous almost everywhere.

We built a model of fluid flow by representing the vorticity field of the fluid as a set of movable point vortices. But we know we cannot use this to represent every interesting phenomenon.We cannot consider vortices of mixed sign. Our inability to consider negative temperatures is another drawback.

We have now introduced most of the components we need, and with an understanding of how to conduct each experiment we can set up a Monte Carlo simulator. Pick a mesh size , pick an inverse temperature β, and conduct a large number of sweeps. Each sweep consists of experiments. Each experiment consists of picking three distinct sites and attempting to vary their strength as described in the above section. The change in enthalpy from an experiment is calculated, and a random number is drawn from the interval [0, 1]; the change in site values is accepted if is less than exp() and is rejected otherwise.

The general problem of how to place point vortices on the sphere to minimize energy is unsolved. There are some known results, though, for example that we can minimize the energy of four point vortices by placing all four at approximately 109.4 degrees apart from the other points. That number is suggestive: we know those are the internal angles of the vertices of a tetrahedron. Were we to draw lines between the equilibrium points we would get the tetrahedron.

In this chapter we continue exploring the vortex gas problem, partly for its dynamic interest, and partly to apply Monte Carlo methods to it. The vortex gas system has been analytically explored and quite a few relative equilibria are known by Lim, Montaldi, and Roberts to exist [270] and have been classified in shape and in dynamic stability. We will be interested here in the shape, and use the problem to easily generate meshes for other numerical computations.