Integrated information in the thermodynamic limit

The capacity to integrate information is a prominent feature of biological, neural, and cognitive processes. Integrated Information Theory (IIT) provides mathematical tools for quantifying the level of integration in a system, but its computational cost generally precludes applications beyond relatively small models. In consequence, it is not yet well understood how integration scales up with the size of a system or with different temporal scales of activity, nor how a system maintains integration as it interacts with its environment. After revising some assumptions of the theory, we show for the first time how modified measures of information integration scale when a neural network becomes very large. Using kinetic Ising models and mean-field approximations, we show that information integration diverges in the thermodynamic limit at certain critical points. Moreover, by comparing different divergent tendencies of blocks that make up a system at these critical points, we can use information integration to delimit the boundary between an integrated unit and its environment. Finally, we present a model that adaptively maintains its integration despite changes in its environment by generating a critical surface where its integrity is preserved. We argue that the exploration of integrated information for these limit cases helps in addressing a variety of poorly understood questions about the organization of biological, neural, and cognitive systems

Statistical Mechanics of Dilute Batch Minority Games with Random External Information

We study the dynamics and statics of a dilute batch minority game with random external information. We focus on the case in which the number of connections per agent is infinite in the thermodynamic limit. The dynamical scenario of ergodicity breaking in this model is different from the phase transition in the standard minority game and is characterised by the onset of long-term memory at finite integrated response. We demonstrate that finite memory appears at the AT-line obtained from the corresponding replica calculation, and compare the behaviour of the dilute model with the minority game with market impact correction, which is known to exhibit similar features.Comment: 22 pages, 6 figures, text modified, references updated and added, figure added, typos correcte

The thermodynamic limit of the Whitham equations

The infinite-genus limit of the KdV-Whitham equations is derived. The limit involves special scaling for the associated spectral surface such that the integrated density of states remains finite as $N \to \infty$ (the thermodynamic type limit). The limiting integro-differential system describes slow evolution of the density of states and can be regarded as the kinetic equation for a soliton gas

Meaning of temperature in different thermostatistical ensembles

Depending on the exact experimental conditions, the thermodynamic properties of physical systems can be related to one or more thermostatistical ensembles. Here, we survey the notion of thermodynamic temperature in different statistical ensembles, focusing in particular on subtleties that arise when ensembles become non-equivalent. The 'mother' of all ensembles, the microcanonical ensemble, uses entropy and internal energy (the most fundamental, dynamically conserved quantity) to derive temperature as a secondary thermodynamic variable. Over the past century, some confusion has been caused by the fact that several competing microcanonical entropy definitions are used in the literature, most commonly the volume and surface entropies introduced by Gibbs. It can be proved, however, that only the volume entropy satisfies exactly the traditional form of the laws of thermodynamics for a broad class of physical systems, including all standard classical Hamiltonian systems, regardless of their size. This mathematically rigorous fact implies that negative 'absolute' temperatures and Carnot efficiencies $>1$ are not achievable within a standard thermodynamical framework. As an important offspring of microcanonical thermostatistics, we shall briefly consider the canonical ensemble and comment on the validity of the Boltzmann weight factor. We conclude by addressing open mathematical problems that arise for systems with discrete energy spectrum.Comment: 11 pages, 1 figur

Information theory in the study of anisotropic radiation

Information theory is used to perform a thermodynamic study of non equilibrium anisotropic radiation. We limit our analysis to a second-order truncation of the moments, obtaining a distribution function which leads to a natural closure of the hierarchy of radiative transfer equations in the so-called variable Eddington factor scheme. Some Eddington factors appearing in the literature can be recovered as particular cases of our two-parameter Eddington factor. We focus our attention in the study of the thermodynamic properties of such systems and relate it to recent nonequilibrium thermodynamic theories. Finally we comment the possibility of introducing a nonequilibrium chemical potential for photons.Comment: 1 eps figure upon request by e-mail, to appear in Journal of Physics

On quantum mean-field models and their quantum annealing

This paper deals with fully-connected mean-field models of quantum spins with p-body ferromagnetic interactions and a transverse field. For p=2 this corresponds to the quantum Curie-Weiss model (a special case of the Lipkin-Meshkov-Glick model) which exhibits a second-order phase transition, while for p>2 the transition is first order. We provide a refined analytical description both of the static and of the dynamic properties of these models. In particular we obtain analytically the exponential rate of decay of the gap at the first-order transition. We also study the slow annealing from the pure transverse field to the pure ferromagnet (and vice versa) and discuss the effect of the first-order transition and of the spinodal limit of metastability on the residual excitation energy, both for finite and exponentially divergent annealing times. In the quantum computation perspective this quantity would assess the efficiency of the quantum adiabatic procedure as an approximation algorithm.Comment: 44 pages, 23 figure