Negative temperature

Abstract: The circumstances under which negative absolute temperatures can occur are discussed, and principles of thermodynamics and statistical mechanics at negative temperatures are developed. If the entropy of a thermodynamic system is not a monotonically increasing function of its internal energy, it possesses a negative temperature whenever ${(\frac{\ensuremath{\partial}S}{\ensuremath{\partial}U})}_{X}$ is negative. Negative temperatures are hotter than positive temperatures. When account is taken of the possibility of negative temperatures, various modifications of conventional thermodynamics statements are required. For example, heat can be extracted from a negative-temperature reservoir with no other effect than the performance of an equivalent amount of work. One of the standard formulations of the second law of thermodynamics must be altered to the following: It is impossible to construct an engine that will operate in a closed cycle and provide no effect other than (1) the extraction of heat from a positive-temperature reservoir with the performance of an equivalent amount of work or (2) the rejection of heat into a negative-temperature reservoir with the corresponding work being done on the engine. A thermodynamic system that is in internal thermodynamic equilibrium, that is otherwise essentially isolated, and that has an energetic upper limit to its allowed states can possess a negative temperature. The statistical mechanics of such a system are discussed and the results are applied to nuclear spin systems.

Abstract: Theoretical development and numerical simulation of the two-dimensional electrostatic guiding-centre plasma with positive total interaction energy are presented. Equilibrium statistical mechanics predicts that no spatially homogeneous thermal equilibrium state exists for this system. This non-existence is associated with the phenomenon of ‘negative temperatures’. Quasi-stable, spatially inhomogeneous states are shown to form, and are characterized by macroscopic spatially-separated vortex structures.

Abstract: The dynamics of two-dimensional interacting line vortices is identical to that of the two-dimensional electrostatic guiding center plasma. Both are Hamiltonian systems and are therefore susceptible to statistical mechanical treatments. The predictions of the microcanonical ensemble are explored for this system. Interest focuses primarily on the regime of total positive interaction energy, which should be above the Onsager negative temperature threshold. Calculations of the probability distribution for a component by means of the central limit theorem are carried out in the manner of Khinchin. The probability distribution of a component reduced to the usual Gibbs distribution in the regime of positive temperatures, and is still explicitly calculable for negative temperatures. The negative temperature states are neither quiescent nor spatially uniform. Expressions for the temperature are explicitly provided in terms of the total particle energy and particle number. A BBGKY hierarchy can be derived for both temperature regimes. Numerical simulations involving solutions of the equations of motion of 4008 particles are presented.