How geometrically frustrated systems challenge our notion of thermodynamics

Although Boltzmann's definition of entropy and temperature are widely accepted, we will show scenarios which apparently are inconsistent with our normal notion of thermodynamics. We focus on generic geometrically frustrated systems (GFSs), which stay at constant negative Boltzmann temperatures, independent from their energetic state. Two weakly coupled GFSs at same temperature exhibit, in accordance with energy conservation, the same probability for all energetic combinations. Heat flow from a hot GFS to a cooler GFS or an ideal gas increases Boltzmann entropy of the combined system, however the maximum is non-local, which, in contrast to conventional thermodynamics, implies that both subsystems maintain different temperatures here. Re-parametrization can transform these non-local into local maxima with corresponding equivalence of re-defined temperatures. However, these temperatures cannot be assigned solely to a subsystem but describe combinations of both. The non-local maxima of entropy restrict the naive application of the zeroth law of thermodynamics. Reformulated this law is still valid with the consequence that a GFS at constant negative temperature can measure positive temperatures. Heat exchange between a GFS and a polarized paramagnetic spin gas, i.e. a system that may achieve besides positive also negative temperatures, drives the combined system to a local-, or non-local maximum of entropy, with equivalent or non-equivalent temperatures here. Energetic constraints determine which scenario results. In case of a local maximum, the spin gas can measure temperature of the GFS like a usual thermometer, however, this reveals no information about the energetic state of the GFS.

Automated summary

The article discusses the inconsistencies between Boltzmann's definition of entropy and temperature and the normal notion of thermodynamics in certain scenarios. It shows that under constant negative Boltzmann temperatures, two weakly coupled geometrically frustrated systems can maintain different temperatures, and the maximum entropy is non-local. Reparametrization can transform non-local maxima into local maxima, but these temperatures cannot be assigned solely to a subsystem.