We prove the Gibbs variational formula in terms of quantum relative entropy density that characterizes translation invariant thermal equilibrium states in quantum lattice systems. It is a natural quantum extension of the similar statement established by Föllmer for classical systems. In this work we advocate the viewpoint: Avoid the unique-phase assumption if it is not essential.
We present a novel approach to establishing the variational principle for Gibbs and generalized (weak and almost) Gibbs states. Limitations of a thermodynamical formalism for generalized Gibbs states will be discussed. A new class of intuitively weak Gibbs measures is introduced, and a typical example is studied. Finally, we present a new example of a non-Gibbsian measure arising from an industrial application.
We study the thermodynamic formalism for generalized Gibbs measures, such as renormalization group transformations of Gibbs measures or joint measures of disordered spin systems. We first show existence of the relative entropy density and obtain a familiar expression in terms of entropy and relative energy for "almost Gibbsian measures" (almost sure continuity of conditional probabilities). We also describe these measures as equilibrium states and establish an extension of the usual variational principle. As a corollary, we obtain a full variational principle for quasilocal measures. For the j…
We continue our study of Gibbs-non-Gibbs dynamical transitions. In the present paper we consider a system of Ising spins on a large discrete torus with a Kac-type interaction subject to an independent spin-flip dynamics (infinite-temperature Glauber dynamics). We show that, in accordance with the program outlined in \cite{vEFedHoRe10}, in the thermodynamic limit Gibbs-non-Gibbs dynamical transitions are \emph{equivalent} to bifurcations in the set of global minima of the large-deviation rate function for the trajectories of the empirical density \emph{conditional} on their endpoint. More preci…
The maximum entropy formalism developed by Jaynes determines the relevant ensemble in nonequilibrium statistical mechanics by maximising the entropy functional subject to the constraints imposed by the available information. We present an alternative derivation of the relevant ensemble based on the Kullback-Leibler divergence from equilibrium. If the equilibrium ensemble is already known, then calculation of the relevant ensemble is considerably simplified. The constraints must be chosen with care in order to avoid contradictions between the two alternative derivations. The relative entropy fu…
From the study of a functional equation relating the Gibbs measures at two different tempratures we prove that the specific entropy of the Gibbs measure of the Sherrington-Kirkpatrick Spin Glass Model vanishes at the inverse temperature $β=4 \log2$.
We present conditions on families of diffeomorphisms that guarantee statistical stability and SRB entropy continuity. They rely on the existence of horseshoe-like sets with infinitely many branches and variable return times. As an application we consider the family of Henon maps within the set of Benedicks-Carleson parameters.
We prove a metastability result for finitary microstates which are good models for a Gibbs measure for a nearest-neighbor interaction on a finitely-generated group. This is used to show that any maximal-entropy joining of two such Gibbs states is a relative product over the tail $σ$-algebra, except in degenerate cases. We also use results on extremal cuts of random graphs to further investigate optimal self-joinings of the Ising model on a free group.
We perform a detailed study of Gibbs-non-Gibbs transitions for the Curie-Weiss model subject to independent spin-flip dynamics ("infinite-temperature" dynamics). We show that, in this setup, the program outlined in van Enter, Fernández, den Hollander and Redig can be fully completed, namely that Gibbs-non-Gibbs transitions are equivalent to bifurcations in the set of global minima of the large-deviation rate function for the trajectories of the magnetization conditioned on their endpoint. As a consequence, we show that the time-evolved model is non-Gibbs if and only if this set is not a single…
This paper considers the problem of data compression for dependent quantum systems. It is the second in a series under the same title. As in the previous paper, we are interested in Lempel--Ziv encoding for quantum Gibbs ensembles. Here, we consider the canonical ideal lattice Bose- and Fermi-ensembles. We prove that as in the case of the grand canonical ensemble, the (limiting) von Neumann entropy rate $h$ can be assessed, via the classical Lempel--Ziv universal coding algorithm, from a single eigenvector of the density matrix.
A model of a quantum information source is proposed, based on the Gibbs ensemble of ideal (free) particles (bosons or fermions). We identify the (thermodynamic) von Neumann entropy as the information rate and establish the classical Lempel--Ziv universal coding algorithm in Grassberger's form for such a source. This generalises the Schumacher theorem to the case of non-IID qubits.
We examine the entropy of non-equilibrium stationary states of boundary driven totally asymmetric simple exclusion processes. As a consequence, we obtain that the Gibbs-Shannon entropy of the non equilibrium stationary state converges to the Gibbs-Shannon entropy of the local equilibrium state. Moreover, we prove that its fluctuations are Gaussian, except when the mean displacement of particles produced by the bulk dynamics agrees with the particle flux induced by the density reservoirs in the maximal phase regime.
This work is a follow-up on [GOVW]. In that previous work a two-scale approach was used to prove the logarithmic Sobolev inequality for a system of spins with fixed mean whose potential is a bounded perturbation of a Gaussian, and to derive an abstract theorem for the convergence to the hydrodynamic limit. This strategy was then successfully applied to Kawasaki dynamics. Here we shall use again this two-scale approach to show that the microscopic variable in such a model behaves according to a local Gibbs state. As a consequence, we shall prove the convergence of the microscopic entropy to the…
Let $X$ be a finite set and $Ω=\{1,...,d\}^{\mathbb{N}}$ be the Bernoulli space. Denote by $σ$ the shift map acting on $Ω$. For a fixed probability $μ$ on $X$ with supp($μ$)$=X$, define $Π(μ,σ)$ as the set of all Borel probabilities $π\in P(X\times Ω)$ such that the $x$-marginal of $π$ is $μ$ and the $y$-marginal of $π$ is $σ$-invariant. We consider a fixed Lipschitz cost function $c: X \times Ω\to \mathbb{R}$ and an associated Ruelle operator. We introduce the concept of Gibbs plan, which is a probability on $X \times Ω$. Moreover, we define entropy, pressure and equilibrium plans. The study …
We consider the Sherrington-Kirkpatrick model of spin glasses with ferromagnetically biased couplings. For a specific choice of the couplings mean, the resulting Gibbs measure is equivalent to the Bayesian posterior for a high-dimensional estimation problem known as `$Z_2$ synchronization'. Statistical physics suggests to compute the expectation with respect to this Gibbs measure (the posterior mean in the synchronization problem), by minimizing the so-called Thouless-Anderson-Palmer (TAP) free energy, instead of the mean field (MF) free energy. We prove that this identification is correct, pr…
