A higher Boltzmann distribution

AbstractWe characterize the classical Boltzmann distribution as the unique solution to a combinatorial Hodge theory problem in homological degree zero on a finite graph. By substituting for the graph a CW complex X and a choice of degree $$d \le \dim X$$d≤dimX, we define by direct analogy a higher dimensional Boltzmann distribution $$\rho ^B$$ρB as a certain real-valued cellular $$(d-1)$$(d-1)-cycle. We then give an explicit formula for $$\rho ^B$$ρB. We explain how these ideas relate to the Higher Kirchhoff Network Theorem of Catanzaro et al. (Homol Homotopy Appl 17:165–189, 2015). We also deduce an improved version of the Higher Matrix-Tree Theorems of Catanzaro et al. (Homol Homotopy Appl 17:165–189, 2015).

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