Extending canonical Monte Carlo methods II

Previously, we have presented a methodology to extend canonical Monte Carlo methods inspired on a suitable extension of the canonical fluctuation relation $C=\beta^{2}$ compatible with negative heat capacities $C<0$. Now, we improve this methodology by introducing a better treatment of finite size effects affecting the precision of a direct determination of the microcanonical caloric curve $\beta (E) =\partial S(E) /\partial E$, as well as a better implementation of MC schemes. We shall show that despite the modifications considered, the extended canonical MC methods possibility an impressive overcome of the so-called \textit{super-critical slowing down} observed close to the region of a temperature driven first-order phase transition. In this case, the dependence of the decorrelation time $\tau$ with the system size $N$ is reduced from an exponential growth to a weak power-law behavior $\tau(N)\propto N^{\alpha}$, which is shown in the particular case of the 2D seven-state Potts model where the exponent $\alpha=0.14-0.18$.Comment: Version submitted to JSTA

Equilibrium fluctuation theorems compatible with anomalous response

Previously, we have derived a generalization of the canonical fluctuation relation between heat capacity and energy fluctuations $C=\beta^{2}<\delta U^{2}>$, which is able to describe the existence of macrostates with negative heat capacities $C<0$. In this work, we extend our previous results for an equilibrium situation with several control parameters to account for the existence of states with anomalous values in other response functions. Our analysis leads to the derivation of three different equilibrium fluctuation theorems: the \textit{fundamental and the complementary fluctuation theorems}, which represent the generalization of two fluctuation identities already obtained in previous works, and the \textit{associated fluctuation theorem}, a result that has no counterpart in the framework of Boltzmann-Gibbs distributions. These results are applied to study the anomalous susceptibility of a ferromagnetic system, in particular, the case of 2D Ising model.Comment: Extended version of the paper published in JSTA