Possible loss and recovery of Gibbsianness during the stochastic evolution of Gibbs measures

We consider Ising-spin systems starting from an initial Gibbs measure $\nu$ and evolving under a spin-flip dynamics towards a reversible Gibbs measure $\mu\not=\nu$. Both $\nu$ and $\mu$ are assumed to have a finite-range interaction. We study the Gibbsian character of the measure $\nu S(t)$ at time $t$ and show the following: (1) For all $\nu$ and $\mu$, $\nu S(t)$ is Gibbs for small $t$. (2) If both $\nu$ and $\mu$ have a high or infinite temperature, then $\nu S(t)$ is Gibbs for all $t>0$. (3) If $\nu$ has a low non-zero temperature and a zero magnetic field and $\mu$ has a high or infinite temperature, then $\nu S(t)$ is Gibbs for small $t$ and non-Gibbs for large $t$. (4) If $\nu$ has a low non-zero temperature and a non-zero magnetic field and $\mu$ has a high or infinite temperature, then $\nu S(t)$ is Gibbs for small $t$, non-Gibbs for intermediate $t$, and Gibbs for large $t$. The regime where $\mu$ has a low or zero temperature and $t$ is not small remains open. This regime presumably allows for many different scenarios

On the Variational Principle for Generalized Gibbs Measures

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.Comment: To appear in Markov Processes and Related Fields, Proceedings workshop Gibbs-nonGibb

Sensitive dependence of geometric Gibbs states

For quadratic-like maps, we show a phenomenon of sensitive dependence of geometric Gibbs states: There are analytic families of quadratic-like maps for which an arbitrarily small perturbation of the parameter can have a definite effect on the low-temperature geometric Gibbs states. Furthermore, this phenomenon is robust: There is an open set of analytic 2-parameter families of quadratic-like maps that exhibit sensitive dependence of geometric Gibbs states. We introduce a geometric version of the Peierls condition for contour models ensuring that the low-temperature Gibbs states are concentrated near the critical orbit.Comment: Minor change

Symmetric Gibbs measures

We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic transformation (the same as equality after a permutation of finitely many coordinates). The relations we consider are defined by cocycles taking values in groups, including some nonabelian ones. This generalizes (half of) the identification of the invariant ergodic probability measures for the Pascal adic transformation as exactly the Bernoulli measures---a version of de Finetti's Theorem. Generalizing the other half, we characterize the measures on subshifts of finite type that are invariant under both the adic and the shift as the Gibbs measures whose potential functions depend on only a single coordinate. There are connections with and implications for exchangeability, ratio limit theorems for transient Markov chains, interval splitting procedures, `canonical' Gibbs states, and the triviality of remote sigma-fields finer than the usual tail field

Thermodynamics and time-average

For a dynamical system far from equilibrium, one has to deal with empirical probabilities defined through time-averages, and the main problem is then how to formulate an appropriate statistical thermodynamics. The common answer is that the standard functional expression of Boltzmann-Gibbs for the entropy should be used, the empirical probabilities being substituted for the Gibbs measure. Other functional expressions have been suggested, but apparently with no clear mechanical foundation. Here it is shown how a natural extension of the original procedure employed by Gibbs and Khinchin in defining entropy, with the only proviso of using the empirical probabilities, leads for the entropy to a functional expression which is in general different from that of Boltzmann--Gibbs. In particular, the Gibbs entropy is recovered for empirical probabilities of Poisson type, while the Tsallis entropies are recovered for a deformation of the Poisson distribution.Comment: 8 pages, LaTex source. Corrected some misprint

Adaptive Gibbs samplers

We consider various versions of adaptive Gibbs and Metropolis- within-Gibbs samplers, which update their selection probabilities (and perhaps also their proposal distributions) on the fly during a run, by learning as they go in an attempt to optimise the algorithm. We present a cautionary example of how even a simple-seeming adaptive Gibbs sampler may fail to converge. We then present various positive results guaranteeing convergence of adaptive Gibbs samplers under certain conditions