arXiv subjects

Artur O. Lopes

Publications and source records attributed to Artur O. Lopes.

At least 55 records · Page 3Linked to original sources

arXiv2018

Quantum Spin probabilities at positive temperature are Hölder Gibbs probabilities

We consider the KMS state associated to the Hamiltonian $H= σ^x \otimes σ^x$ over the quantum spin lattice $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes ...$. For a fixed observable of the form $L \otimes L \otimes L \otimes ...$, where $L:\mathbb{C}^2 \to \mathbb{C}^2 $ is self adjoint, and for positive temperature $T$ one can get a naturally defined stationary probability $μ_T$ on the Bernoulli space $\{1,2\}^\mathbb{N}$. The Jacobian of $μ_T$ can be expressed via a certain continued fraction expansion. We will show that this probability is a Gibbs probability for a Hölder potential. Therefore, this probability is mixing for the shift map. For such probability $μ_T$ we will show the explicit deviation function for a certain class of functions. When decreasing temperature we will be able to exhibit the explicit transition value $T_c$ where the set of values of the Jacobian of the Gibbs probability $μ_T$ changes from being a Cantor set to being an interval. We also present some properties for quantum spin probabilities at zero temperature (for instance, the explicit value of the entropy).

math.DS↗

arXiv2018

Selection of calibrated subaction when temperature goes to zero in the discounted problem

Consider $T(x)= d \, x$ (mod 1) acting on $S^1$, a Lipschitz potential $A:S^1 \to \mathbb{R}$, $0<λ<1$ and the unique function $b_λ:S^1 \to \mathbb{R}$ satisfying $ b_λ(x) = \max_{T(y)=x} \{ λ\, b_λ(y) + A(y)\}.$ We will show that, when $λ\to 1$, the function $b_λ- \frac{m(A)}{1-λ}$ converges uniformly to the calibrated subaction $V(x) = \max_{μ\in \mathcal{ M}} \int S(y,x) \, d μ(y)$, where $S$ is the Mañe potential, $\mathcal{ M}$ is the set of invariant probabilities with support on the Aubry set and $m(A)= \sup_{μ\in \mathcal{M}} \int A\,dμ$. For $β>0$ and $λ\in (0,1)$, there exists a unique fixed point $u_{λ,β} :S^1\to \mathbb{R}$ for the equation $e^{u_{λ,β}(x)} = \sum_{T(y)=x}e^{βA(y) +λu_{λ,β}(y)}$. It is known that as $λ\to 1$ the family $e^{[u_{λ,β}- \sup u_{λ,β}]}$ converges uniformly to the main eigenfuntion $ϕ_β$ for the Ruelle operator associated to $βA$. We consider $λ=λ(β)$, $β(1-λ(β))\to+\infty$ and $λ(β) \to 1$, as $β\to\infty$. Under these hypothesis we will show that $\frac{1}β(u_{λ,β}-\frac{P(βA)}{1-λ})$ converges uniformly to the above $V$, as $β\to \infty$. The parameter $β$ represents the inverse of temperature in Statistical Mechanics and $β\to \infty$ means that we are considering that the temperature goes to zero. Under these conditions we get selection of subaction when $β\to \infty$.

math.DS↗

arXiv2018

A formula for the Entropy of the Convolution of Gibbs probabilities on the circle

Consider the transformation $T:S^1 \to S^1$, such that $T(x)=2\, x$ (mod 1), and where $S^1$ is the unitary circle. Suppose $J:S^1 \to \mathbb{R}$ is Holder continuous and positive, and moreover that, for any $y\in S^1$, we have that $\sum_{x\,\,\text{such that}\,\,\, T(x)= y} \, J(x)=1.$ We say that $ρ$ is a Gibbs probability for the Holder continuous potential $\log J$, if $\mathcal{L}_{\log J}^* \,(ρ)=ρ,$ where $\mathcal{L}_{\log J}$ is the Ruelle operator for $\log J$. We call $J$ the Jacobian of $ρ$. Suppose $ν=μ_1*μ_2$ is the convolution of two Gibbs probabilities $μ_1$ and $μ_2$ associated, respectively, to $\log J_1$ and $\log J_2$. We show that $ν$ is also Gibbs and its Jacobian $\tilde{J}$ is given by $\tilde{J}(u) = \int J_1(u-x) d μ_2(x)$ In this case, the entropy $h(ν)$ is given by the expression $$ h(ν) = - \int\,[\,\,\int\, \log \,(\,\int J_1(r+s-x) d μ_2(x)\,) \, d μ_2(r)\,\, ]\,\,d μ_1 (s).$$ For a fixed $μ_2$ we consider differentiable variations $μ_1^t$, $t \in (-ε,ε)$, of $μ_1$ on the Banach manifold of Gibbs probabilities, where $μ_1^0=μ_1$, and we estimate the derivative of the entropy $h(μ_1^t * μ_2)$ at $t=0$. We also present an expression for the Jacobian of the convolution of a Gibbs probability $ρ$ with the invariant probability with support on a periodic orbit of period two. This expression is based on the Jacobian of $ρ$ and two Radon-Nidodym derivatives.

math.DS↗

arXiv2017

A general renormalization procedure on the one-dimensional lattice and decay of correlations

We present a general form of Renormalization operator $\mathcal{R}$ acting on potentials $V:\{0,1\}^\mathbb{N} \to \mathbb{R}$. We exhibit the analytical expression of the fixed point potential $V$ for such operator $\mathcal{R}$. This potential can be expressed in a naturally way in terms of a certain integral over the Hausdorff probability on a Cantor type set on the interval $[0,1]$. This result generalizes a previous one by A. Baraviera, R. Leplaideur and A. Lopes where the fixed point potential $V$ was of Hofbauer type. For the potentials of Hofbauer type (a well known case of phase transition) the decay is like $n^{-γ}$, $γ>0$. Among other things we present the estimation of the decay of correlation of the equilibrium probability associated to the fixed potential $V$ of our general renormalization procedure. In some cases we get polynomial decay like $n^{-γ}$, $γ>0$, and in others a decay faster than $n \,e^{ -\, \sqrt{n}}$, when $n \to \infty$. The potentials $g$ we consider here are elements of the so called family of Walters potentials on $\{0,1\}^\mathbb{N} $ which generalizes the potentials considered initially by F. Hofbauer. For these potentials some explicit expressions for the eigenfunctions are known. In a final section we also show that given any choice $d_n \to 0$ of real numbers varying with $n \in \mathbb{N}$ there exist a potential $g$ on the class defined by Walters which has a invariant probability with such numbers as the coefficients of correlation (for a certain explicit observable function).

math.DS↗

arXiv2017

Large Deviations for Quantum Spin probabilities at temperature zero

We consider certain self-adjoint observables for the KMS state associated to the Hamiltonian $H= σ^x \otimes σ^x$ over the quantum spin lattice $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes ...$. For a fixed observable of the form $L \otimes L \otimes L \otimes ...$, where $L:\mathbb{C}^2 \to \mathbb{C}^2 $, and for the zero temperature limit one can get a naturally defined stationary probability $μ$ on the Bernoulli space $\{1,2\}^\mathbb{N}$. This probability is ergodic but it is not mixing for the shift map. It is not a Gibbs state for a continuous normalized potential but its Jacobian assume only two values almost everywhere. Anyway, for such probability $μ$ we can show that a Large Deviation Principle is true for a certain class of functions. The result is derived by showing the explicit form of the free energy which is differentiable.

math.DS↗

arXiv2017

Correlation Inequalities and Monotonicity Properties of the Ruelle Operator

Let $X = \{1,-1\}^\mathbb{N}$ be the symbolic space endowed with the product order. A Borel probability measure $μ$ over $X$ is said to satisfy the FKG inequality if for any pair of continuous increasing functions $f$ and $g$ we have $μ(fg)-μ(f)μ(g)\geq 0$. In the first part of the paper we prove the validity of the FKG inequality on Thermodynamic Formalism setting for a class of eigenmeasures of the dual of the Ruelle operator, including several examples of interest in Statistical Mechanics. In addition to deducing this inequality in cases not covered by classical results about attractive specifications our proof has advantage of to be easily adapted for suitable subshifts. We review (and provide proofs in our setting) some classical results about the long-range Ising model on the lattice $\mathbb{N}$ and use them to deduce some monotonicity properties of the associated Ruelle operator and their relations with phase transitions. As is widely known, for some continuous potentials does not exists a positive continuous eigenfunction associated to the spectral radius of the Ruelle operator acting on $C(X)$. Here we employed some ideas related to the involution kernel in order to solve the main eigenvalue problem in a suitable sense - for a class of potentials having low regularity. From this we obtain an explicit tight upper bound for the main eigenvalue (consequently for the pressure) of the Ruelle operator associated to Ising models with $1/r^{2+\varepsilon}$ interaction energy. Extensions of the Ruelle operator to suitable Hilbert Spaces are considered and a theorem solving to the main eigenvalue problem (in a weak sense) is obtained by using the Lions-Lax-Milgram theorem.

math.DS↗

arXiv2017

Interactions, Specifications, DLR probabilities and the Ruelle Operator in the One-Dimensional Lattice

In this paper, we describe several different meanings for the concept of Gibbs measure on the lattice $\mathbb{N}$ in the context of finite alphabets (or state space). We compare and analyze these "in principle" distinct notions: DLR-Gibbs measures, Thermodynamic Limit and eigenprobabilities for the dual of the Ruelle operator (also called conformal measures). Among other things we extended the classical notion of a Gibbsian specification on $\mathbb{N}$ in such way that the similarity of many results in Statistical Mechanics and Dynamical System becomes apparent. One of our main result claims that the construction of the conformal Measures in Dynamical Systems for Walters potentials, using the Ruelle operator, can be formulated in terms of Specification. We also describe the Ising model, with $1/r^{2+\varepsilon}$ interaction energy, in the Thermodynamic Formalism setting and prove that its associated potential is in Walters space - we present an explicit expression. We also provide an alternative way for obtaining the uniqueness of the DLR-Gibbs measures.

math.DS↗

arXiv2016

On the thin boundary of the fat attractor

For, $0<λ<1$, consider the transformation $T(x) = d x $ (mod 1) on the circle $S^1$, a $C^1$ function $A:S^1 \to \mathbb{R}$, and, the map $F(x,s) = ( T(x) , λ\, s + A(x))$, $(x,s)\in S^1 \times \mathbb{R}$. We denote $\mathcal{B}= \mathcal{B}_λ$ the upper boundary of the attractor (known as fat attractor). We are interested in the regularity of $\mathcal{B}_λ$, and, also in what happens in the limit when $λ\to 1$. We also address the analysis of the following conjecture which were proposed by R. Bamón, J. Kiwi, J. Rivera-Letelier and R. Urzúa: for any fixed $λ$, $C^1$ generically on the potential $A$, the upper boundary $\mathcal{B}_λ$ is formed by a finite number of pieces of smooth unstable manifolds of periodic orbits for $F$. We show the proof of the conjecture for the class of $C^2$ potentials $A(x)$ satisfying the twist condition (plus a combinatorial condition). We do not need the generic hypothesis for this result. We present explicit examples. On the other hand, when $λ$ is close to $1$ and the potential $A$ is generic a more precise description can be done. In this case the finite number of pieces of $C^1$ curves on the boundary have some special properties. Having a finite number of pieces on this boundary is an important issue in a problem related to semi-classical limits and micro-support. This was consider in a recent published work by A. Lopes and J. Mohr. Finally, we present the general analysis of the case where $A$ is Lipschitz and its relation with Ergodic Transport

math.DS↗

arXiv2016

Semiclassical limits, Lagrangian states and coboundary equations

Assume that $f$ is a continuous transformation $f:S^1 \to S^1$. We consider here the cases where $f$ is either the transformation $f(z)=z^2$ or $f$ is a smooth diffeomorphism of the circle $S^1$. Consider a fixed continuous potential $τ:S^1=[0,1) \to \mathbb{R}$, $ν\in \mathbb{R}$ and $φ:S^1 \to \mathbb{C}$ (a quantum state). The transformation $\hat F_ν$ acting on $φ:S^1 \to \mathbb{C}$, $\hat F_ν(φ) = ϕ$, defined by $\displaystyle ϕ(z) = \hat F_ν (φ(z)) = φ(f(z))e^{iντ(z)}$ describes a discrete time dynamical evolution of the quantum state $φ$. Given $S: \mathbb{R}\to \mathbb{R}$ we define the Lagrangian state $$φ_{x}^S(z) = \sum_{k\in\mathbb{Z}} e^{\frac{iS (z-k)}{\hbar}} e^{-\frac{(z-k-x)^2}{4\hbar}}.$$ In this case $\hat F_ν(φ_{x}^S(z)) = \sum_{k\in\mathbb{Z}}e^{\frac{iS (f(z)-k)}{\hbar}}e^{-\frac{(f(z)-k-x)^2}{4\hbar}}e^{iντ(z)}$. Under suitable conditions on $S$ the micro-support of $φ^S_x (z)$, when $\hbar \to 0$, is $(x,S'(x))$. One of meanings of the semiclassical limit in Quantum Mechanics is to take $ν=\frac{1}{\hbar}$ and $\hbar \to 0$. We address the question of finding $S$ such that $φ^S_x$ satisfies the property: $ \forall x$, we have that $\hat{F}_ν(φ^S_x)$ has micro-support on the graph of $y\to S'(y)$ (which is the micro-support of $φ^S_x$). In other words: which $S$ is such that $\hat{F}_ν$ leaves the micro-support of $φ^S_x$ invariant? This is related to a coboundary equation for $τ$, twist conditions and the boundary of the fat attractor.

math.DS↗

arXiv2016

Transport and large deviations for Schrodinger operators and Mather measures

In this mainly survey paper we consider the Lagrangian $ L(x,v) = \frac{1}{2} \, |v|^2 - V(x) $, and a closed form $w$ on the torus $ \mathbb{T}^n $. For the associated Hamiltonian we consider the the Schrodinger operator ${\bf H}_β=\, -\,\frac{1}{2 β^2} \, Δ+V$ where $β$ is large real parameter. Moreover, for the given form $β\, w$ we consider the associated twist operator ${\bf H}_β^w$. We denote by $({\bf H}_β^w)^*$ the corresponding backward operator. We are interested in the positive eigenfunction $ ψ_β$ associated to the the eigenvalue $ E_β$ for the operator ${\bf H}_β^{w} $. We denote $ ψ_β^*$ the positive eigenfunction associated to the the eigenvalue $ E_β$ for the operator $({\bf H}_β^{w})^* $. Finally, we analyze the asymptotic limit of the probability $ν_β= ψ_β\, ψ_β^*$ on the torus when $β\to \infty$. The limit probability is a Mather measure. We consider Large deviations properties and we derive a result on Transport Theory. We denote $L^{-}(x,v) = \frac{1}{2} \, |v|^2 - V(x) - w_x(v) $ and $L^{+}(x,v) = \frac{1}{2} \, |v|^2 - V(x) + w_x(v) $. We are interest in the transport problem from $μ_{-}$ (the Mather measure for $L^{-}$) to $μ_{+}$ (the Mather measure for $L^{+}$) for some natural cost function. In the case the maximizing probability is unique we use a Large Deviation Principle due to N. Anantharaman in order to show that the conjugated sub-solutions $u$ and $u^*$ define an admissible pair which is optimal for the dual Kantorovich problem.

math.DS↗

arXiv2016

Phase Transitions in One-dimensional Translation Invariant Systems: a Ruelle Operator Approach

We consider a family of potentials f, derived from the Hofbauer potentials, on the symbolic space Omega=\{0,1\}^\mathbb{N} and the shift mapping $σ$ acting on it. A Ruelle operator framework is employed to show there is a phase transition when the temperature varies in the following senses: the pressure is not analytic, there are multiple eigenprobabilities for the dual of the Ruelle operator, the DLR-Gibbs measure is not unique and finally the Thermodynamic Limit is not unique. Additionally, we explicitly calculate the critical points for these phase transitions. Some examples which are not of Hofbauer type are also considered. The non-uniqueness of the Thermodynamic Limit is proved by considering a version of a Renewal Equation. We also show that the correlations decay polynomially and that each one of these Hofbauer potentials is a fixed point for a certain renormalization transformation.

math.DS↗

arXiv2015

A detailed proof of the von Neumann's Quantum Ergodic Theorem

We present a simplified proof of the von Neumann's Quantum Ergodic Theorem. This important result was initially published in german by J. von Neumann in 1929. We are interested here in the time evolution $ψ_t$, $t\geq 0$, (for large times) under the Schrodinger equation associated to a given fixed Hamiltonian $H : \mathcal{H} \to \mathcal{H}$ and a general initial condition $ψ_0$. The dimension of the Hilbert space $\mathcal{H}$ is finite.

quant-ph↗

arXiv2015

Duality between Eigenfunctions and Eigendistributions of Ruelle and Koopman operators via an integral kernel

We consider the classical dynamics given by a one sided shift on the Bernoulli space of $d$ symbols. We study, on the space of Hölder functions, the eigendistributions of the Ruelle operator with a given potential. Our main theorem shows that for any isolated eigenvalue, the eigendistributions of such Ruelle operator are dual to eigenvectors of a Ruelle operator with a conjugate potential. We also show that the eigenfunctions and eigendistributions of the Koopman operator satisfy a similar relationship. To show such results we employ an integral kernel technique, where the kernel used is the involution kernel.

math.DS↗

arXiv2015

Generic properties for random repeated quantum iterations

We denote by $M^n$ the set of $n$ by $n$ complex matrices. Given a fixed density matrix $β:\mathbb{C}^n \to \mathbb{C}^n$ and a fixed unitary operator $U : \mathbb{C}^n \otimes \mathbb{C}^n \to \mathbb{C}^n \otimes \mathbb{C}^n$, the transformation $Φ: M^n \to M^n$ $$ Q \to Φ(Q) =\, \text{Tr}_2 (\,U \, ( Q \otimes β)\, U^*\,)$$ describes the interaction of $Q$ with the external source $β$. The result of this is $Φ(Q)$. If $Q$ is a density operator then $Φ(Q)$ is also a density operator. The main interest is to know what happen when we repeat several times the action of $Φ$ in an initial fixed density operator $Q_0$. This procedure is known as random repeated quantum iterations and is of course related to the existence of one or more fixed points for $Φ$. In \cite{NP}, among other things, the authors show that for a fixed $β$ there exists a set of full probability for the Haar measure such that the unitary operator $U$ satisfies the property that for the associated $Φ$ there is a unique fixed point $ Q_Φ$. Moreover, there exists convergence of the iterates $Φ^n (Q_0) \to Q_Φ$, when $n \to \infty$, for any given $Q_0$ We show here that there is an open and dense set of unitary operators $U: \mathbb{C}^n \otimes \mathbb{C}^n \to \mathbb{C}^n \otimes \mathbb{C}^n $ such that the associated $Φ$ has a unique fixed point. We will also consider a detailed analysis of the case when $n=2$. We will be able to show explicit results. We consider the $C^0$ topology on the coefficients of $U$. In this case we will exhibit the explicit expression on the coefficients of $U$ which assures the existence of a unique fixed point for $Φ$. Moreover, we present the explicit expression of the fixed point $Q_Φ$

math-ph↗

arXiv2015

On Bertelson-Gromov Dynamical Morse Entropy

In this mainly expository paper we present a detailed proof of several results contained in a paper by M. Bertelson and M. Gromov on Dynamical Morse Entropy. This is an introduction to the ideas presented in that work. Suppose $M$ is compact oriented connected $C^\infty$ manifold of finite dimension. Assume that $f_0 :M \to [0,1]$ is a surjective Morse function. For a given natural number $n$, consider the set $M^n$ and for $x=(x_0,x_1,...,x_{n-1}) \in M^n$, denote $ f_n (x) = \frac{1}{n} \, \sum_{j=0}^{n-1} f_0 (x_j).$ The Dynamical Morse Entropy describes for a fixed interval $I\subset [0,1]$ the asymptotic growth of the number of critical points of $f_n$ in $I$, when $n \to \infty$. The part related to the Betti number entropy does not requires the differentiable structure. One can describe generic properties of potentials defined in the $XY$ model of Statistical Mechanics with this machinery.

math.DS↗

arXiv2014

The Dual Potential, the involution kernel and Transport in Ergodic Optimization

Consider the shift $σ$ acting on the Bernoulli space $Σ={1,2,...,n}^\mathbb{N}$. We denote $\hatΣ= {1,2,...,n}^\mathbb{Z}$. We analyze several properties of the maximizing probability $μ_{\infty,A}$ of a Holder potential $A: Σ\to \mathbb{R}$. Associated to $A(x)$, via the involution kernel, $W: \hatΣ \to \mathbb{R}$, it is known that can we get the dual potential $A^*(y)$, where $(x,y)\in \hatΣ$. Consider $μ_{\infty, A^*}$ a maximizing probability for $A^*$. We would like to consider the transport problem from $μ_{\infty,A}$ to $μ_{\infty,A^*}$. In this case, it is natural to consider the cost function $c(x,y) = I(x) - W(x,y) +γ$, where $I$ is the deviation function. The pair of functions for the Kantorovich Transport dual Problem are $(-V,-V^*$), where we denote the two calibrated sub-actions by $V$ and $V^*$, respectively, for $A$ and $A^*$ for $μ_{\infty,A}$. We analyze the graph property for the optimal plan $\hatμ$.

math.DS↗

arXiv2014

Large Deviations for stationary probabilities of a family of continuous time Markov chains via Aubry-Mather theory

We consider a family of continuous time symmetric random walks indexed by $k\in \mathbb{N}$, $\{X_k(t),\,t\geq 0\}$. For each $k\in \mathbb{N}$ the matching random walk take values in the finite set of states $Γ_k=\frac{1}{k}(\mathbb{Z}/k\mathbb{Z})$ which is a subset of the unitary circle. The stationary probability for such process converges to the uniform distribution on the circle, when $k\to \infty$. We disturb the system considering a fixed $C^2$ potential $V: \mathbb{S}^1 \to \mathbb{R}$ and we will denote by $V_k$ the restriction of $V$ to $Γ_k$. Then, we define a non-stochastic semigroup generated by the matrix $k\,\, L_k + k\,\, V_k$, where $k\,\, L_k $ is the infinifesimal generator of $\{X_k(t),\,t\geq 0\}$. From the continuous time Perron's Theorem one can normalized such semigroup, and, then we get another stochastic semigroup which generates a continuous time Markov Chain taking values on $Γ_k$. The stationary probability vector for such Markov Chain is denoted by $π_{k,V}$. We assume that the maximum of $V$ is attained in a unique point $x_0$ of $\mathbb{S}^1$, and from this will follow that $π_{k,V}\to δ_{x_0}$. Our main goal is to analyze the large deviation principle for the family $π_{k,V}$, when $k \to\infty$. The deviation function $I^V$, which is defined on $ \mathbb{S}^1$, will be obtained from a procedure based on fixed points of the Lax-Oleinik operator and Aubry-Mather theory.

math.DS↗

arXiv2013

Entropy and Variational Principle for one-dimensional Lattice Systems with a general a-priori probability: positive and zero temperature

We generalize several results of the classical theory of Thermodynamic Formalism by considering a compact metric space $M$ as the state space. We analyze the shift acting on $M^\mathbb{N}$ and consider a general a-priori probability for defining the Transfer (Ruelle) operator. We study potentials $A$ which can depend on the infinite set of coordinates in $M^\mathbb{N}.$ We define entropy and by its very nature it is always a nonpositive number. The concepts of entropy and transfer operator are linked. If M is not a finite set there exist Gibbs states with arbitrary negative value of entropy. Invariant probabilities with support in a fixed point will have entropy equal to minus infinity. In the case $M=S^1$, and the a-priori measure is Lebesgue $dx$, the infinite product of $dx$ on $(S^1)^\mathbb{N}$ will have zero entropy. We analyze the Pressure problem for a Hölder potential $A$ and its relation with eigenfunctions and eigenprobabilities of the Ruelle operator. Among other things we analyze the case where temperature goes to zero and we show some selection results. Our general setting can be adapted in order to analyze the Thermodynamic Formalism for the Bernoulli space with countable infinite symbols. Moreover, the so called XY model also fits under our setting. In this last case M is the unitary circle $S^1$. We explore the differentiable structure of $(S^1)^\mathbb{N}$ by considering potentials which are of class $C^1$ and we show some properties of the corresponding main eigenfunctions.

math.DS↗