  Ordered chains (such as chains of amino acids) are ubiquitous in biological
cells, and these chains perform specific functions contingent on the sequence
of their components. Using the existence and general properties of such
sequences as a theoretical motivation, we study the statistical physics of
systems whose state space is defined by the possible permutations of an ordered
list, i.e., the symmetric group, and whose energy is a function of how certain
permutations deviate from some chosen correct ordering. Such a non-factorizable
state space is quite different from the state spaces typically considered in
statistical physics systems and consequently has novel behavior in systems with
interacting and even non-interacting Hamiltonians. Various parameter choices of
a mean-field model reveal the system to contain five different physical regimes
defined by two transition temperatures, a triple point, and a quadruple point.
Finally, we conclude by discussing how the general analysis can be extended to
state spaces with more complex combinatorial properties and to other standard
questions of statistical mechanics models.
