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Register a new userTropical Limit in Statistical Physics
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Added by
Boris Konopelchenko
Publication date
2015
fields
Physics
and research's language is
English
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Tropical limit for macroscopic systems in equilibrium defined as the formal limit of Boltzmann constant k going to 0 is discussed. It is shown that such tropical limit is well-adapted to analyse properties of systems with highly degenerated energy levels, particularly of frustrated systems like spin ice and spin glasses. Tropical free energy is a piecewise linear function of temperature, tropical entropy is a piecewise constant function and the system has energy for which tropical Gibbs probability has maximum. Properties of systems in the points of jump of entropy are studied. Systems with finite and infinitely many energy levels and phenomena of limiting temperatures are discussed.
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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.
The probability that a point is to one side of a curve in Schramm-Loewner evolution (SLE) can be obtained alternatively using boundary conformal field theory (BCFT). We extend the BCFT approach to treat two curves, forming, for example, the left and right boundaries of a cluster. This proves to correspond to a generalisation to SLE(kappa,rho), with rho=2. We derive the probabilities that a given point lies between two curves or to one side of both. We find analytic solutions for the cases kappa=0,2,4,8/3,8. The result for kappa=6 leads to predictions for the current distribution at the plateau transition in the semiclassical approximation to the quantum Hall effect.
In this paper, we consider the problem of minimizing quantum free energies under the constraint that the density of particles is fixed at each point of Rd, for any d $ge$ 1. We are more particularly interested in the characterization of the minimizer, which is a self-adjoint nonnegative trace class operator, and will show that it is solution to a nonlinear self-consistent problem. This question of deriving quantum statistical equilibria is at the heart of the quantum hydrody-namical models introduced by Degond and Ringhofer. An original feature of the problem is the local nature of constraint, i.e. it depends on position, while more classical models consider the total number of particles in the system to be fixed. This raises difficulties in the derivation of the Euler-Lagrange equations and in the characterization of the minimizer, which are tackled in part by a careful parametrization of the feasible set.
We consider a kinetic model whose evolution is described by a Boltzmann-like equation for the one-particle phase space distribution $f(x,v,t)$. There are hard-sphere collisions between the particles as well as collisions with randomly fixed scatterers. As a result, this evolution does not conserve momentum but only mass and energy. We prove that the diffusively rescaled $f^varepsilon(x,v,t)=f(varepsilon^{-1}x,v,varepsilon^{-2}t)$, as $varepsilonto 0$ tends to a Maxwellian $M_{rho, 0, T}=frac{rho}{(2pi T)^{3/2}}exp[{-frac{|v|^2}{2T}}]$, where $rho$ and $T$ are solutions of coupled diffusion equations and estimate the error in $L^2_{x,v}$.
Statistical physics has proven to be a very fruitful framework to describe phenomena outside the realm of traditional physics. The last years have witnessed the attempt by physicists to study collective phenomena emerging from the interactions of individuals as elementary units in social structures. Here we review the state of the art by focusing on a wide list of topics ranging from opinion, cultural and language dynamics to crowd behavior, hierarchy formation, human dynamics, social spreading. We highlight the connections between these problems and other, more traditional, topics of statistical physics. We also emphasize the comparison of model results with empirical data from social systems.
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