Statistical mechanics of exploding phase spaces: Ontic open systems

Pazuki, Roozbeh H.; Pruessner, Gunnar; Tempesta, Piergiulio

doi:10.48550/arxiv.1609.02065

Cited by 2 publications

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“…The exponential growth is essentially responsible for the logarithm in Boltzmann's famous formula for the entropy, S = k log W, and has many further implications on statistical mechanics. A few theoretical examples where classical configuration space grows faster than exponentially with N have been discussed in the mathematical literature [3,4], but the link to physical models is not evident. To devise a many-body system with a subexponentially growing configuration space or Hilbert space, an obvious strategy is to impose a sufficient number of constraints such that the usual exponential growth is impeded.…”

Section: Introductionmentioning

confidence: 99%

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Subexponentially Growing Hilbert Space and Nonconcentrating Distributions in a Constrained Spin Model

Webster

Kastner

2018

J Stat Phys

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Motivated by recent experiments with two-component Bose-Einstein condensates, we study fully-connected spin models subject to an additional constraint. The constraint is responsible for the Hilbert space dimension to scale only linearly with the system size. We discuss the unconventional statistical physical and thermodynamic properties of such a system, in particular the absence of concentration of the underlying probability distributions. As a consequence, expectation values are less suitable to characterize such systems, and full distribution functions are required instead. Sharp signatures of phase transitions do not occur in such a setting, but transitions from singly peaked to doubly peaked distribution functions of an "order parameter" may be present.

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Section: Introductionmentioning

confidence: 99%

“…Fig 4. Canonical probability distributions P(m z ) of the magnetization density m z for the fullyconnected XX model (blue) and the fully-connected TFIM (orange).…”

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confidence: 99%

Subexponentially Growing Hilbert Space and Nonconcentrating Distributions in a Constrained Spin Model

Webster

Kastner

2018

J Stat Phys

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“…A recent preprint [12] proposes the study of "exploding" phase spaces: statistical systems such that the cardinality of the space of configurations grows faster than k n , the combination of n components that can occupy k states. The total grassmannians Gr(n) = Gr(n, F q ) are an example, since their cardinality grows like q n 2 4 +o(n 2 ) .…”

Section: Further Remarksmentioning

confidence: 99%

“…4 There is extensive empirical evidence about the pertinence of the predictions made by nonextensive statistical mechanics [23]. However, very few papers address the microscopical foundations of the theory (for instance, [8], [12], [21]). We present here a novel approach in this direction, based on the combinatorics of flags, but only for the case α = 2.…”

Section: Introduction a Two Faces Of Entropymentioning

confidence: 99%

Information Theory With Finite Vector Spaces

Vigneaux¹

2019

IEEE Trans. Inform. Theory

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While Shannon entropy is related to the growth rate of multinomial coefficients, we show that the quadratic entropy (Tsallis 2-entropy) is connected to their q-version; when q is a prime power, these coefficients count flags of finite vector spaces with prescribed length and dimensions. In particular, the q-binomial coefficients count vector subspaces of given dimension. We obtain this way a combinatorial explanation for the non-additivity of the quadratic entropy. We show that statistical systems whose configurations are described by flags provide a frequentist justification for the maximum entropy principle with Tsallis statistics. We introduce then a discrete-time stochastic process associated to the q-binomial distribution, that generates at time n a vector subspace of F n q (here F q is the finite field of order q). The concentration of measure on certain "typical subspaces" allows us to extend the asymptotic equipartition property to this setting. We discuss the applications to Shannon theory, particularly to source coding, when messages correspond to vector spaces.

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Statistical mechanics of exploding phase spaces: Ontic open systems

Cited by 2 publications

References 0 publications

Subexponentially Growing Hilbert Space and Nonconcentrating Distributions in a Constrained Spin Model

Subexponentially Growing Hilbert Space and Nonconcentrating Distributions in a Constrained Spin Model

Information Theory With Finite Vector Spaces

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