Results 1  10
of
66
The fluctuation theorem as a Gibbs property
 J. Stat. Phys
, 1999
"... Dedicated to the memory of Edwin T. Jaynes. Common ground to recent studies exploiting relations between dynamical systems and nonequilibrium statistical mechanics is, so we argue, the standard Gibbs formalism applied on the level of spacetime histories. The assumptions (chaoticity principle) unde ..."
Abstract

Cited by 91 (27 self)
 Add to MetaCart
(Show Context)
Dedicated to the memory of Edwin T. Jaynes. Common ground to recent studies exploiting relations between dynamical systems and nonequilibrium statistical mechanics is, so we argue, the standard Gibbs formalism applied on the level of spacetime histories. The assumptions (chaoticity principle) underlying the GallavottiCohen fluctuation theorem make it possible, using symbolic dynamics, to employ the theory of onedimensional lattice spin systems. The Kurchan and LebowitzSpohn analysis of this fluctuation theorem for stochastic dynamics can be restated on the level of the spacetime measure which is a Gibbs measure for an interaction determined by the transition probabilities. In this note we understand the fluctuation theorem as a Gibbs property as it follows from the very definition of Gibbs state. We give a local version of the fluctuation theorem in the Gibbsian context and we derive from this a version also for some class of spatially extended stochastic dynamics.
Robustness of the nonGibbsian property: some examples
 J. Phys. A
, 1997
"... We discuss some examples of measures on lattice systems, which lack the property of being a Gibbs measure in a rather strong sense. 1 Introduction In recent years extensive research has been done on the occurrence of states (probability measures) on lattice systems which are not of Gibbsian type. ..."
Abstract

Cited by 24 (11 self)
 Add to MetaCart
We discuss some examples of measures on lattice systems, which lack the property of being a Gibbs measure in a rather strong sense. 1 Introduction In recent years extensive research has been done on the occurrence of states (probability measures) on lattice systems which are not of Gibbsian type. Such measures occur for example in renormalizationgroup studies [17, 18, 21, 8, 9, 10, 11, 12, 13, 40], nonequilibrium statistical mechanical models [42, 26, 33, 38], image analysis [5, 15, 34], probabilistic cellular automata [25, 33] and random cluster models [19, 39]. The possibility of their occurrence and their properties have been considered by various authors [1, 7, 14, 20, 22, 24, 28, 29, 30, 31, 32, 36, 37, 41, 44]. This nonGibbsian behaviour has often been considered `pathological'  undesirable , and there have been various attempts to control the nonGibbsianness. One approach, advocated by Martinelli and Olivieri [36, 37], is to study how the nonGibbsian measures behav...
Reversible quantum cellular automata
"... Abstract. We define a class of dynamical maps on the quasilocal algebra of a quantum spin system, which are quantum analogues of probabilistic cellular automata. We develop criteria for such a system to be ergodic, i.e., to possess a unique invariant state. Intuitively, ergodicity obtains if the lo ..."
Abstract

Cited by 23 (3 self)
 Add to MetaCart
(Show Context)
Abstract. We define a class of dynamical maps on the quasilocal algebra of a quantum spin system, which are quantum analogues of probabilistic cellular automata. We develop criteria for such a system to be ergodic, i.e., to possess a unique invariant state. Intuitively, ergodicity obtains if the local transition operators exhibit sufficiently large disorder. The ergodicity criteria also imply bounds for the exponential decay of correlations in the unique invariant state. The main technical tool is a quantum version of oscillation norms, defined in the classical case as the sum over all sites of the variations of an observable with respect to local spinflips.
Networks and the Epidemiology of Infectious Disease
, 2011
"... The science of networks has revolutionised research into the dynamics of interacting elements. It could be argued that epidemiology in particular has embraced the potential of network theory more than any other discipline. Here we review the growing body of research concerning the spread of infecti ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
The science of networks has revolutionised research into the dynamics of interacting elements. It could be argued that epidemiology in particular has embraced the potential of network theory more than any other discipline. Here we review the growing body of research concerning the spread of infectious diseases on networks, focusing on the interplay between network theory and epidemiology. The review is split into four main sections, which examine: the types of network relevant to epidemiology; the multitude of ways these networks can be characterised; the statistical methods that can be applied to infer the epidemiological parameters on a realised network; and finally simulation and analytical methods to determine epidemic dynamics on a given network. Given the breadth of areas covered and the everexpanding number of publications, a comprehensive review of all work is impossible. Instead, we provide a personalised overview into the areas of network epidemiology that have seen the greatest progress in recent years or have the greatest potential to provide novel insights. As such, considerable importance is placed on analytical approaches and statistical methods which are both rapidly expanding fields. Throughout this review we restrict our
Quasilocality of Projected Gibbs Measures through Analyticity Techniques
 Helv. Phys. Acta
, 1995
"... . We present two examples of projections of Gibbs measures. In the first example we prove hightemperature complete analyticity for the qstate Potts model. As a consequence we obtain complete analyticity also for the decimated Potts model. In the second example we prove quasilocality of the project ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
(Show Context)
. We present two examples of projections of Gibbs measures. In the first example we prove hightemperature complete analyticity for the qstate Potts model. As a consequence we obtain complete analyticity also for the decimated Potts model. In the second example we prove quasilocality of the projections to the line of the pure phases of the two dimensional standard Ising system in the whole uniqueness region, and indicate why nonGibbsianness can be expected to occur for higher dimensions. 1 Introduction Over the last few years there has been a revival of interest in the mathematical welldefinedness of real space renormalization group transformations applied to lattice spin systems. This problem originally dates back to the late `70s when Griffiths and Pearce, and Israel noticed that in certain renormalization examples one cannot take for granted the existence of an effective potential [22, 23, 31]. Later a comprehensive investigation of these so called renormalizationpathologies has...
From Global to Local Fluctuation Theorems
 Moscow Mathematical Journal
, 2001
"... The GallavottiCohen uctuation theorem suggests a general symmetry in the fluctuations of the entropy production, a basic concept in the theory of irreversible processes, based on results in the theory of strongly chaotic maps. We study this symmetry for some standard models of nonequilibrium steady ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
(Show Context)
The GallavottiCohen uctuation theorem suggests a general symmetry in the fluctuations of the entropy production, a basic concept in the theory of irreversible processes, based on results in the theory of strongly chaotic maps. We study this symmetry for some standard models of nonequilibrium steady states. We give a general strategy to derive a local uctuation theorem exploiting the Gibbsian features of the stationary spacetime distribution. This is applied to spin ip processes and to the asymmetric exclusion process.
Asymptotics of Locally Interacting Markov Chains with Global Signals
 Advances in Applied Probability
, 2002
"... We study the long run behaviour of interactive Markov chains on in nite product spaces. The behaviour at a single site is in
uenced by the local situation in some neighborhood and by a random signal about the average situation throughout the whole system. The asymptotic behaviour of such Markov cha ..."
Abstract

Cited by 9 (6 self)
 Add to MetaCart
We study the long run behaviour of interactive Markov chains on in nite product spaces. The behaviour at a single site is in
uenced by the local situation in some neighborhood and by a random signal about the average situation throughout the whole system. The asymptotic behaviour of such Markov chains is analyzed on the microscopic level and on the macroscopic level of empirical elds. We give suÆcient conditions for convergence on the macroscopic level. Combining a convergence result from the theory of random systems with complete connections with a perturbation of the DobrushinVasserstein contraction technique we show that macroscopic convergence implies that the underlying Microscopic process has local asymptotic loss of memory. Key Words: Markov chains on innite product spaces, local asymptotic loss of memory, contraction techniques, Gibbs measures
Steady state thermodynamics
, 2004
"... We propose a thermodynamic formalism that is expected to apply to a large class of nonequilibrium steady states including a heat conducting fluid, a sheared fluid, and an electrically conducting fluid. We call our theory steady state thermodynamics (SST) after Oono and Paniconi’s original proposal. ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
(Show Context)
We propose a thermodynamic formalism that is expected to apply to a large class of nonequilibrium steady states including a heat conducting fluid, a sheared fluid, and an electrically conducting fluid. We call our theory steady state thermodynamics (SST) after Oono and Paniconi’s original proposal. The construction of SST is based on a careful examination of how the basic notions in thermodynamics should be modified in nonequilibrium steady states. We define all thermodynamic quantities through operational procedures, which can be (in principle) realized experimentally. Based on SST thus constructed, we make some nontrivial predictions, including an extension of Einstein’s formula on density fluctuation, an extension of the minimum work principle, the existence of a new osmotic pressure of a purely nonequilibrium origin, and a shift of coexistence temperature. All these predictions may be checked experimentally to test SST for its quantitative validity. Contents
Phase transitions in a piecewise expanding coupled map lattice with linear nearest neighbour coupling, Nonlinearity
, 2006
"... Abstract. We construct a mixing continuous piecewise linear map on [−1, 1] with the property that a twodimensional lattice made of these maps with a linear north and east nearest neighbour coupling admits a phase transition. We also provide a modification of this construction where the local map is ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We construct a mixing continuous piecewise linear map on [−1, 1] with the property that a twodimensional lattice made of these maps with a linear north and east nearest neighbour coupling admits a phase transition. We also provide a modification of this construction where the local map is an expanding analytic circle map. The basic strategy is borroughed from [10], namely we compare the dynamics of the CML to those of a probabilistic cellular automaton of Toom’s type, see [24] for a detailed discussion. 1.