Thermodynamics from a scaling Hamiltonian

There are problems with defining the thermodynamic limit of systems with long-range interactions; as a result, the thermodynamic behavior of these types of systems is anomalous. In the present work, we review some concepts from both extensive and nonextensive thermodynamic perspectives. We use a model, whose Hamiltonian takes into account spins ferromagnetically coupled in a chain via a power law that decays at large interparticle distance $r$ as $1/r^{\alpha}$ for $\alpha\geq0$. Here, we review old nonextensive scaling. In addition, we propose a new Hamiltonian scaled by $2\frac{(N/2)^{1-\alpha}-1}{1-\alpha}$ that explicitly includes symmetry of the lattice and dependence on the size, $N$, of the system. The new approach enabled us to improve upon previous results. A numerical test is conducted through Monte Carlo simulations. In the model, periodic boundary conditions are adopted to eliminate surface effects.Comment: 12 pages, 2 figures, submitted for publication to Phys. Rev.

Fisher information, Wehrl entropy, and Landau Diamagnetism

Using information theoretic quantities like the Wehrl entropy and Fisher's information measure we study the thermodynamics of the problem leading to Landau's diamagnetism, namely, a free spinless electron in a uniform magnetic field. It is shown that such a problem can be "translated" into that of the thermal harmonic oscillator. We discover a new Fisher-uncertainty relation, derived via the Cramer-Rao inequality, that involves phase space localization and energy fluctuations.Comment: no figures. Physical Review B (2005) in pres

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

A Family of Exact, Analytic Time Dependent Wave Packet Solutions to a Nonlinear Schroedinger Equation

We obtain time dependent $q$-Gaussian wave-packet solutions to a non linear Schr\"odinger equation recently advanced by Nobre, Rego-Montero and Tsallis (NRT) [Phys. Rev. Lett. 106 (2011) 10601]. The NRT non-linear equation admits plane wave-like solutions ($q$-plane waves) compatible with the celebrated de Broglie relations connecting wave number and frequency, respectively, with energy and momentum. The NRT equation, inspired in the $q$-generalized thermostatistical formalism, is characterized by a parameter $q$, and in the limit $q \to 1$ reduces to the standard, linear Schr\"odinger equation. The $q$-Gaussian solutions to the NRT equation investigated here admit as a particular instance the previously known $q$-plane wave solutions. The present work thus extends the range of possible processes yielded by the NRT dynamics that admit an analytical, exact treatment. In the $q \to 1$ limit the $q$-Gaussian solutions correspond to the Gaussian wave packet solutions to the free particle linear Schr\"odinger equation. In the present work we also show that there are other families of nonlinear Schr\"odinger-like equations, besides the NRT one, exhibiting a dynamics compatible with the de Broglie relations. Remarkably, however, the existence of time dependent Gaussian-like wave packet solutions is a unique feature of the NRT equation not shared by the aforementioned, more general, families of nonlinear evolution equations

Geometrical aspects and connections of the energy-temperature fluctuation relation

Recently, we have derived a generalization of the known canonical fluctuation relation $k_{B}C=\beta^{2}$ between heat capacity $C$ and energy fluctuations, which can account for the existence of macrostates with negative heat capacities $C<0$. In this work, we presented a panoramic overview of direct implications and connections of this fluctuation theorem with other developments of statistical mechanics, such as the extension of canonical Monte Carlo methods, the geometric formulations of fluctuation theory and the relevance of a geometric extension of the Gibbs canonical ensemble that has been recently proposed in the literature.Comment: Version accepted for publication in J. Phys. A: Math and The

Thermodynamic fluctuation relation for temperature and energy

The present work extends the well-known thermodynamic relation $C=\beta ^{2}< \delta {E^{2}}>$ for the canonical ensemble. We start from the general situation of the thermodynamic equilibrium between a large but finite system of interest and a generalized thermostat, which we define in the course of the paper. The resulting identity $=1+< \delta {E^{2}}> \partial ^{2}S(E) /\partial {E^{2}}$ can account for thermodynamic states with a negative heat capacity $C<0$; at the same time, it represents a thermodynamic fluctuation relation that imposes some restrictions on the determination of the microcanonical caloric curve $\beta (E) =\partial S(E) /\partial E$. Finally, we comment briefly on the implications of the present result for the development of new Monte Carlo methods and an apparent analogy with quantum mechanics.Comment: Version accepted for publication in J. Phys. A: Math and The

Delocalization and the semiclassical description of molecular rotation

We discuss phase-space delocalization for the rigid rotator within a semiclassical context by recourse to the Husimi distributions of both the linear and the $3D-$anisotropic instances. Our treatment is based upon the concomitant Fisher information measures. The pertinent Wehrl entropy is also investigated in the linear case.Comment: 6 pages, 3 figure