Tropical limit in statistical physics

Angelelli, Mario; Konopelchenko, B. G.

doi:10.1016/j.physleta.2015.04.003

Cited by 11 publications

(26 citation statements)

References 33 publications

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“…Note that the tropical limit has been naturally adopted in statistical mechanics, e.g., free energy F = −kT ∑ E exp (−β E) with β → ∞ induces tropical limit, where F merely equals to minimum energy. 11 However, we emphasize here that the present tropical limit is completely different from such adoption. Because of the above character, we cannot simply apply tropical limit to the discrete dynamical system because (i) domain of Q includes negative value, and (ii) A (Q) can take positive, zero or negative sign, depending on Q.…”

Section: A Preparation For Derivationmentioning

confidence: 71%

Tropical Limit for Configurational Geometry in Discrete Thermodynamic Systems

Yuge,

Ohta

2019

Preprint

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For classical discrete systems with constant composition (typically referred to substitutional alloys) under thermodynamically equilibrium state, macroscopic structure should in principle depend on temperature and many-body interaction through Boltzmann factor, exp (−β E). Despite this fact, our recently find that (i) thermodynamic average for structure can be characterized by a set of special microscopic state whose structure is independent of energy and temperature, 1 and (ii) bidirectional-stability character for thermodynamic average between microscopic structure and potential energy surface is formulated without any information about temperature or many-body interaction. 2 These results strongly indicates the significant role of configurational geometry, where "anharmonicity in structural degree of freedom (ASDF)" that is a vector field on configuration space, plays central role, intuitively corresponding to nonlinearity in thermodynamic average depending only on configurational geometry. Although ASDF can be practically drawn by performing numerical simulation based on such as Monte Carlo simulation, it is still unclear how its entire character is dominated by geometry of underlying lattice. We here show that by applying the limit in tropical geometry (i.e., tropical limit) with special scale-transformation to discrete dynamical system for ASDF, we find tropical relationships between how nonlinearity in terms of configurational geometry expands or shrinks and geometric information about underlying lattice for binary system with a single structural degree of freedom (SDF). The proposed tropical limit will be powerful procedure to simplify analyzation of complicated nonlinear character for thermodynamic average with multiple SDF systems.

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Section: A Preparation For Derivationmentioning

confidence: 71%

Tropical Limit for Configurational Geometry in Discrete Thermodynamic Systems

Yuge,

Ohta

2019

Preprint

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“…The function β will be called tropical temperature (even if it plays the role of an inverse temperature) and the N microsystems are tropically thermalized. Generalizing the results in [9], the tropical free energy is defined as…”

Section: A Simple Example: Duality and Non-equilibriummentioning

confidence: 99%

“…This is a particular instance of a more general application of filters to probability. In fact, it is well known that a proper filter can be seen as a In this case, the limit process involves k B : idempotence for the probability W α = W ({α}) defined in [9] holds only at the lowest (zeroth) order in k B on the singular locus. This is evident in (8.2), where the statistical corrections depend on the cardinality λ 0 (T ) = #m 0 (T ) defined in (6.4).…”

Section: Global Tropical Symmetry and Statistical Amoebasmentioning

confidence: 99%

Tropical limit and a micro-macro correspondence in statistical physics

Angelelli¹

2017

J. Phys. A: Math. Theor.

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Tropical mathematics is used to establish a correspondence between certain microscopic and macroscopic objects in statistical models. Tropical algebra gives a common framework for macrosystems (subsets) and their elementary constituent (elements) that is well-behaved with respect to composition. This kind of connection is studied with maps that preserve a monoid structure. The approach highlights an underlying order relation that is explored through the concepts of filter and ideal. Main attention is paid to asymmetry and duality between max-and min-criteria. Physical implementations are presented through simple examples in thermodynamics and non-equilibrium physics. The phenomenon of ultrametricity, the notion of tropical equilibrium and the role of ground energy in non-equilibrium models are discussed. Tropical symmetry, i.e. idempotence, is investigated.

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“…The tropical limit of mean and scalar curvature is (n − 1)(n − 2) n(n + 1) (7.20) that coincide with maximum values of mean and scalar curvatures of super-ideal statistical hypersurface. Note that the tropical limit in statistical physics of macroscopic systems with highly degenerate energy levels studied in [27] corresponds to a very special, essentially onedimensional case of above consideration when all…”

Section: Tropical Limitmentioning

confidence: 99%

Geometry of basic statistical physics mapping

Angelelli¹,

Konopelchenko²

2016

J. Phys. A: Math. Theor.

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Geometry of hypersurfaces defined by the relation which generalizes classical formula for free energy in terms of microstates is studied. Induced metric, Riemann curvature tensor, Gauss-Kronecker curvature and associated entropy are calculated. Special class of ideal statistical hypersurfaces is analyzed in details. Non-ideal hypersurfaces and their singularities similar to those of the phase transitions are considered. Tropical limit of statistical hypersurfaces and double scaling tropical limit are discussed too.

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Tropical limit in statistical physics

Cited by 11 publications

References 33 publications

Tropical Limit for Configurational Geometry in Discrete Thermodynamic Systems

Tropical Limit for Configurational Geometry in Discrete Thermodynamic Systems

Tropical limit and a micro-macro correspondence in statistical physics

Geometry of basic statistical physics mapping

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