[tMcGrath]

There's also an interesting set of applications in statistical physics: many systems can be modelled by Markov chains whose rate parameters can change through time. Changing these parameters non-quasistatically (i.e. so that the system does not relax to the appropriate steady state) gives rise to distances on parameter space that are related to the amount of energy dissipated along the path through parameter space [0,1].

What's nice about this is that the derivation of a suitable metric allows us to compute trajectories that minimise quantities we care about (e.g. minimise energy dissipated), so this has clear potential to be useful. Some cool examples are in spin systems [2] and a harmonic trap [3].

For any differential geometers reading this: it seems to me that a good geometric way to think about this is as a fibre bundle, with the parameter space being the base space and the simplex being a vector bundle over it (see [4] on the simplex being a vector space).

[0] https://arxiv.org/pdf/0706.0559v2.pdf [1] https://arxiv.org/pdf/1201.4166.pdf [2] https://arxiv.org/pdf/1607.07425v1.pdf [3] http://journals.aps.org/pre/abstract/10.1103/PhysRevE.86.041... [4] https://golem.ph.utexas.edu/category/2016/06/how_the_simplex...