We obtain time dependent $q$-Gaussian wave-packet solutions to a non linear Schrodinger 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 soluti
ons ($q$-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 Schrodinger 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 Schrodinger equation. In the present work we also show that there are other families of nonlinear Schrodinger-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.
Recently (arXiv:0910.2870), we have derived a fluctuation theorem for systems in thermodynamic equilibrium compatible with anomalous response functions, e.g. the existence of states with textit{negative heat capacities} $C<0$. In this work, we show t
hat the present approach of the fluctuation theory introduces new insights in the understanding of textit{critical phenomena}. Specifically, the new theorem predicts that the environmental influence can radically affect critical behavior of systems, e.g. to provoke a suppression of the divergence of correlation length $xi$ and some of its associated phenomena as spontaneous symmetry breaking. Our analysis reveals that while response functions and state equations are emph{intrinsic properties} for a given system, critical behaviors are always emph{relative phenomena}, that is, their existence crucially depend on the underlying environmental influence.
Previously, we have presented a methodology to extend canonical Monte Carlo methods inspired on a suitable extension of the canonical fluctuation relation $C=beta^{2}<delta E^{2}>$ 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 $alpha=0.14-0.18$.
In this work, we discuss the implications of a recently obtained equilibrium fluctuation-dissipation relation on the extension of the available Monte Carlo methods based on the consideration of the Gibbs canonical ensemble to account for the existenc
e of an anomalous regime with negative heat capacities $C<0$. The resulting framework appears as a suitable generalization of the methodology associated with the so-called textit{dynamical ensemble}, which is applied to the extension of two well-known Monte Carlo methods: the Metropolis importance sample and the Swendsen-Wang clusters algorithm. These Monte Carlo algorithms are employed to study the anomalous thermodynamic behavior of the Potts models with many spin states $q$ defined on a $d$-dimensional hypercubic lattice with periodic boundary conditions, which successfully reduce the exponential divergence of decorrelation time $tau$ with the increase of the system size $N$ to a weak power-law divergence $taupropto N^{alpha}$ with $alphaapprox0.2$ for the particular case of the 2D 10-state Potts model.
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.
Recently, we have presented some simple arguments supporting the existence of certain complementarity between thermodynamic quantities of temperature and energy, an idea suggested by Bohr and Heinsenberg in the early days of Quantum Mechanics. Such a
complementarity is expressed as the impossibility of perform an exact simultaneous determination of the system energy and temperature by using an experimental procedure based on the thermal equilibrium with other system regarded as a measure apparatus (thermometer). In this work, we provide a simple generalization of this latter approach with the consideration of a thermodynamic situation with several control parameters.
We propose a statistical mechanics for a general class of stationary and metastable equilibrium states. For this purpose, the Gibbs extremal conditions are slightly modified in order to be applied to a wide class of non-equilibrium states. As usual,
it is assumed that the system maximizes the entropy functional $S$, subjected to the standard conditions; i.e., constant energy and normalization of the probability distribution. However, an extra conserved constraint function $F$ is also assumed to exist, which forces the system to remain in the metastable configuration. Further, after assuming additivity for two quasi-independent subsystems, and that the new constraint commutes with density matrix $rho$, it is argued that F should be an homogeneous function of the density matrix, at least for systems in which the spectrum is sufficiently dense to be considered as continuous. The explicit form of $F$ turns to be $F(p_{i})=p_{i}^{q}$, where $p_i$ are the eigenvalues of the density matrix and $q$ is a real number to be determined. This $q$ number appears as a kind of Tsallis parameter having the interpretation of the order of homogeneity of the constraint $F$. The procedure is applied to describe the results of the plasma experiment of Huang and Driscoll. The experimentally measured density is predicted with a similar precision as it is done with the use of the extremum of the enstrophy and Tsallis procedures. However, the present results define the density at all the radial positions. In particular, the smooth tail shown by the experimental distribution turns to be predicted by the procedure. In this way, the scheme avoids the non-analyticity of the density profile at large distances arising in both of the mentioned alternative procedures.
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 non
extensive 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 $alphageq0$. Here, we review old nonextensive scaling. In addition, we propose a new Hamiltonian scaled by $2frac{(N/2)^{1-alpha}-1}{1-alpha}$ 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.
