104,119 research outputs found
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 (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
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
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