Finite-temperature asymptotic expansions of Gibbs-preserving semigroups and preorders

Derive asymptotic expansions of the semigroup of Gibbs-preserving stochastic matrices and of the preorder induced by Gibbs-preserving transformations in powers of e^{-\beta}, in order to characterize how BT-majorisation deforms into thermomajorisation away from the zero-temperature limit.

Background

The paper characterizes the zero-temperature limit of semigroups of Gibbs-preserving stochastic matrices. For a Gibbs family whose energy levels have degeneracy composition $\mu=(m_1,\ldots,m_k)$, the limiting semigroup consists of block-upper-triangular stochastic matrices with column-substochastic diagonal blocks, called BT-stochastic matrices. The associated preorder, BT-majorisation, interpolates between ordinary majorisation and upper-triangular majorisation.

The authors note that the exact finite-temperature structure between the zero- and infinite-temperature regimes is more difficult to characterize. They identify asymptotic expansions in powers of $e^{-\beta}$ as a concrete unresolved direction that could explain how the zero-temperature BT-majorisation relation continuously deforms into finite-temperature thermomajorisation.