Boltzmann and the Heuristics of Representation in Statistical Mechanics

Boltzmann’s work in physics has been studied almost always opposing a strictly mechanical approach of the 2nd law of thermodynamics – attributed to his first works in kinetic – molecular gas theory (1866-1871) – to a probabilistic approach, built and developed in his later works (1872-1884). The analysis of the use of these different approaches covers a spectrum of positions ranging from the recognition of an intrinsic incoherence to Boltzmann’s thinking, go through a radical change in the development of his work, until the adoption of pluralistic strategies as justifications for their methodological options. The purpose of this paper is to explore Boltzmann’s research program from the view of what we characterize as heuristics of representation, highlighting the tools used he used for the solution of problems related to thermal phenomena. We will argue that what in the standard historiographical analysis is understood as a radical turn in Boltzmann’s work – probabilistic “turn point”, that is, the use of an overtly statistical terminology (combinatorial formalism, 1877) instead of a kinetic language (kinetic formalism, 1872) in the analysis of evolution toward the thermal equilibrium (Maxwell’s distribution) – could be better understood as a change of representation within the same conceptual framework.

A Quantum Algebra Approach to Multivariate Askey–Wilson Polynomials

We study matrix elements of a change of basis between two different bases of representations of the quantum algebra ${\mathcal{U}}_q(\mathfrak{s}\mathfrak{u}(1,1))$. The two bases, which are multivariate versions of Al-Salam–Chihara polynomials, are eigenfunctions of iterated coproducts of twisted primitive elements. The matrix elements are identified with Gasper and Rahman’s multivariate Askey–Wilson polynomials, and from this interpretation we derive their orthogonality relations. Furthermore, the matrix elements are shown to be eigenfunctions of the twisted primitive elements after a change of representation, which gives a quantum algebraic derivation of the fact that the multivariate Askey–Wilson polynomials are solutions of a multivariate bispectral $q$-difference problem.

A representation theorem for second-order functionals

Representation theorems relate seemingly complex objects to concrete, more tractable ones. In this paper, we take advantage of the abstraction power of category theory and provide a datatype-generic representation theorem. More precisely, we prove a representation theorem for a wide class of second-order functionals which are polymorphic over a class of functors. Types polymorphic over a class of functors are easily representable in languages such as Haskell, but are difficult to analyse and reason about. The concrete representation provided by the theorem is easier to analyse, but it might not be as convenient to implement. Therefore, depending on the task at hand, the change of representation may prove valuable in one direction or the other. We showcase the usefulness of the representation theorem with a range of examples. Concretely, we show how the representation theorem can be used to prove that traversable functors are finitary containers, how coalgebras of a parameterised store comonad relate to very well-behaved lenses, and how algebraic effects might be implemented in a functional language.