Fluctuation geometry was recently proposed as a counterpart approach of Riemannian geometry of inference theory. This theory describes the geometric features of the statistical manifold $mathcal{M}$ of random events that are described by a family of
continuous distributions $dp(x|theta)$. A main goal of this work is to clarify the statistical relevance of Levi-Civita curvature tensor $R_{ijkl}(x|theta)$ of the statistical manifold $mathcal{M}$. For this purpose, the notion of emph{irreducible statistical correlations} is introduced. Specifically, a distribution $dp(x|theta)$ exhibits irreducible statistical correlations if every distribution $dp(check{x}|theta)$ obtained from $dp(x|theta)$ by considering a coordinate change $check{x}=phi(x)$ cannot be factorized into independent distributions as $dp(check{x}|theta)=prod_{i}dp^{(i)}(check{x}^{i}|theta)$. It is shown that the curvature tensor $R_{ijkl}(x|theta)$ arises as a direct indicator about the existence of irreducible statistical correlations. Moreover, the curvature scalar $R(x|theta)$ allows to introduce a criterium for the applicability of the emph{gaussian approximation} of a given distribution function. This type of asymptotic result is obtained in the framework of the second-order geometric expansion of the distributions family $dp(x|theta)$, which appears as a counterpart development of the high-order asymptotic theory of statistical estimation. In physics, fluctuation geometry represents the mathematical apparatus of a Riemannian extension for Einsteins fluctuation theory of statistical mechanics. Some exact results of fluctuation geometry are now employed to derive the emph{invariant fluctuation theorems}.
Recently, Velazquez and Curilef have proposed a methodology to extend Monte Carlo algorithms based on canonical ensemble, which is aimed to overcome slow sampling problems associated with temperature-driven discontinuous phase transitions. We show in
this work that Monte Carlo algorithms extended with this methodology also exhibit a remarkable efficiency near a critical point. Our study is performed for the particular case of 2D four-state Potts model on the square lattice with periodic boundary conditions. This analysis reveals that the extended version of Metropolis importance sample is more efficient than the usual Swendsen-Wang and Wolff cluster algorithms. These results demonstrate the effectiveness of this methodology to improve the efficiency of MC simulations of systems that undergo any type of temperature-driven phase transition.
Quantum mechanics and classical statistical mechanics are two physical theories that share several analogies in their mathematical apparatus and physical foundations. In particular, classical statistical mechanics is hallmarked by the complementarity
between two descriptions that are unified in thermodynamics: (i) the parametrization of the system macrostate in terms of mechanical macroscopic observables $I={I^{i}}$; and (ii) the dynamical description that explains the evolution of a system towards the thermodynamic equilibrium. As expected, such a complementarity is related to the uncertainty relations of classical statistical mechanics $Delta I^{i}Delta eta_{i}geq k$. Here, $k$ is the Boltzmanns constant, $eta_{i}=partial mathcal{S}(I|theta)/partial I^{i}$ are the restituting generalized forces derived from the entropy $mathcal{S}(I|theta)$ of a closed system, which is found in an equilibrium situation driven by certain control parameters $theta={theta^{alpha}}$. These arguments constitute the central ingredients of a reformulation of classical statistical mechanics from the notion of complementarity. In this new framework, Einstein postulate of classical fluctuation theory $dp(I|theta)simexp[mathcal{S}(I|theta)/k]dI$ appears as the correspondence principle between classical statistical mechanics and thermodynamics in the limit $krightarrow0$, while the existence of uncertainty relations can be associated with the non-commuting character of certain operators.
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.
Starting from an axiomatic perspective, emph{fluctuation geometry} is developed as a counterpart approach of inference geometry. This approach is inspired on the existence of a notable analogy between the general theorems of emph{inference theory} an
d the the emph{general fluctuation theorems} associated with a parametric family of distribution functions $dp(I|theta)=rho(I|theta)dI$, which describes the behavior of a set of emph{continuous stochastic variables} driven by a set of control parameters $theta$. In this approach, statistical properties are rephrased as purely geometric notions derived from the emph{Riemannian structure} on the manifold $mathcal{M}_{theta}$ of stochastic variables $I$. Consequently, this theory arises as an alternative framework for applying the powerful methods of differential geometry for the statistical analysis. Fluctuation geometry has direct implications on statistics and physics. This geometric approach inspires a Riemannian reformulation of Einstein fluctuation theory as well as a geometric redefinition of the information entropy for a continuous distribution.
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.
