# Probabilistic (and other mathematical) methods of physics without the physics?

Many of the methods of physics are vastly more general than their use in that discipline. For example, information theory overlaps with a lot of statistical mechanics, and the latter actually developed first. ET Jaynes wrote a famous paper illustrating the connections. However, each is comprehensible without the language and intuition of the other (though I do not deny that a richer understanding comes from knowing both).

What other methods of physics (particularly those with a statistical or computational bent) have interpretations (Please mention useful introductory texts!) that are completely physics free? I understand that various field theories meet this criterion; any good non-physics introductions?

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I seem to recall that quaternions were originally intended for use in physics. Today they're used for computer graphics. Does that count as physics-free? – Michael Hardy Jul 25 '11 at 6:10
You might enjoy Michel Talagrand's introductory article on spin glasses: people.math.jussieu.fr/~talagran/spinglasses/roma.ps – Simon Lyons Aug 22 '11 at 18:52

Percolation (several kinds), Ising model and other probabilistic models on arbitrary (transitive) graphs came from physics and are purely mathematical theories now.

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Distilled to its linear algebraic core, Quantum Computing can be presented and understood in a completely physics-free way easily accessible to Computer Scientists. Two papers taking this point of view are Fortnow and Fenner.

Driving this linear algebraic point of view even further, one can see multilinear algebra, which deals with the contraction of tensor networks as a core concept, as fundamental. It suffices to model quantum computing, simulation of quantum systems (Projected Entangled Pair States, or PEPS), statistical mechanical models (partition functions), etc. A good text to get started is probably this. Tensor networks have also an intuitive yet precise graphical calculus as explored by Bob Coecke, et al. which abstracts manipulations of tensor networks to operations in compact closed monoidal categories.

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Hamiltonian Monte Carlo/Hybrid Monte Carlo (HMC) uses Hamiltonian dynamics to construct MCMC algorithms for sampling from complicated probability distributions. It is quite useful in Bayesian statistics.

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Euclidean geometry was originally supposed to model physical space.

Now suppose $\overline{X} = (1/n)\sum_{i=1}^n X_i$ and $S^2 = (1/(n-1))\sum_{i=1}^n(X_i - \overline{X})^2$ are the sample mean and sample variance of an i.i.d. sample from a normally distributed (or "Gaussian") population. How do we know that $\overline{X}$ is probabilistically independent of $S^2$? Maybe the quickest way is via two facts from Euclidean geometry: (1) the two mappings $(X_1,\ldots,X_n) \mapsto (\overline{X},\ldots,\overline{X})$ and $(X_1,\ldots,X_n) \mapsto (X_1-\overline{X},\ldots,X_n-\overline{X})$ are complementary orthogonal projections, and (2) the probability distribution of $(X_1,\ldots,X_n)$ is spherically symmetric, i.e the density depends on the coordinates only through the sum of their squares.

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Statisticians use Tracy-Widom laws in the new developments in random matrix theory. The asymptotic behavior of spectra of random matrices was understood (by statisticians) back in the 1960s when the rows of the matrix are independent and identically distributed row-vectors from a fixed distribution (most importantly, with a fixed dimension), and produced asymptotically normal/Gaussian laws typical for the Central Limit Theorem and its generalizations. Tracy-Widom laws apply to matrices in which both row and column dimensions are allowed to grow to infinity (proportionally to one another). See e.g. doi:10.1214/aos/1009210544.

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I found Complexity and Criticality by Christensen and Moloney to by quite excellent. It gives a much more computational approach to the percolation phase transition, the ising phase change and issues of self organized criticality (via the sand pile and rice pile model).

As a computer science student, I found this book to be invaluable for my work on phase transitions of NP-Complete problems.

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This was a "comment", but I may as well make it an "answer": Quaternions were intended for use in physics. Today they are used in computer graphics.

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This is interesting because I originally learned about Quaternions solely in their context as an interesting mathematical construct, a non-commutative algebra of historical interest. – DoubleJay Jul 29 '11 at 16:47