Statistical mechanics of systems with long-range interactions and negative absolute temperature

Abstract

A Hamiltonian model living in a bounded phase space and with long-range interactions is studied. It is shown, by analytical computations, that there exists an energy interval in which the microcanonical entropy is a decreasing convex function of the total energy, meaning that ensemble equivalence is violated in a negative-temperature regime. The equilibrium properties of the model are then investigated by molecular dynamics simulations: first, the caloric curve is reconstructed for the microcanonical ensemble and compared to the analytical prediction, and a generalized Maxwell-Boltzmann distribution for the momenta is observed; then, the nonequivalence between the microcanonical and canonical descriptions is explicitly shown. Moreover, the validity of Fluctuation-Dissipation Theorem is verified through a numerical study, also at negative temperature and in the region where the two ensembles are nonequivalent.

A Topological Approach to Shrinking Higher Dimensions of Space to Observable Space-Time: Can the Dimensional Anisotropy of Space Satisfy Mach’s Principle?

“…The existence of negative temperatures [60,61,62,63,64,65] suggests a possible relationship between vacuum energy and temperature [66]. Recent studies show that in highpressure regions, energy input, such as an increase of temperature, must increase entropy but decrease entropy on negative temperature vacuum (Figure 2) [67,68,69].…”

Statistical Mechanics of Systems with Negative Temperature

“…Two choices of the parameters are considered: J = 0.5, K = 1, 4 (panel (a)) and J = −0.5, K = −1, 4 (panel (b)). For both cases, numerical simulations in the microcanonical ensemble (light blue diamonds) are compared with the analytical equilibrium prediction (blue solid line), obtained with large-deviations techniques [149]. The values of β are obtained from the analysis of averages of suitable observables, depending on the singleparticle momentum distribution.…”

“…The GHMF model has been widely investigated as a paradigmatic example of a long-range interacting system which exhibits inequivalence between statistical ensembles in correspondence of a magnetic transition [148]. As discussed in [149], the modified kinetic term in (129) allows one to obtain negative-temperature equilibrium states, because of the boundedness of the phase-space yielding an energy range with decreasing entropy. It is worth recalling again here that, as discussed in other sections of this review (see I C, III C), this peculiar feature of the phase-space is a basic ingredient for observing negative-temperature equilibrium states in similar models.…”

“…The behaviour of the caloric curve of model ( 129), i.e. β vs. = H/N , has been obtained by large-deviation techniques and further checked by numerical simulations of the Hamiltonian dynamics (see [149]: as discussed in this reference the non-quadratic form of the kinetic term demands the use of a suitable relation for obtaining the numerical estimate of β). For positive values of the parameters J and K the magnetic transition is found to occur at positive temperature (see Fig.…”

“…15(b) a case with long-range interactions is considered, discussed in Ref. [149]. The system is the bounded space-phase Hamiltonian with mean field interactions already introduced in Section IV C 1.…”