We analyse the properties of a very simple "balls-in-boxes" model which can exhibit a phase transition between a fluid and a condensed phase, similar to behaviour encountered in models of random geometries in one, two and four dimensions. This model can be viewed as a generalisation of the backgammon model introduced by Ritort as an example of glassy behaviour without disorder.
We discuss a Pareto macro-economy (a) in a closed system with fixed total wealth and (b) in an open system with average mean wealth and compare our results to a similar analysis in a super-open system (c) with unbounded wealth [1]. Wealth condensation takes place in the social phase for closed and open economies, while it occurs in the liberal phase for superopen economies. In the first two cases, the condensation is related to a mechanism known from the balls-in-boxes model, while in the last case to the non-integrable tails of the Pareto distribution. For a closed macro-economy in the social phase, we point to the emergence of a "corruption" phenomenon: a sizeable fraction of the total wealth is always amassed by a single individual.
Multiplicative logarithmic corrections to scaling are frequently encountered in the critical behavior of certain statistical-mechanical systems. Here, a Lee-Yang zero approach is used to systematically analyse the exponents of such logarithms and to propose scaling relations between them. These proposed relations are then confronted with a variety of results from the literature. 05.50.+q, 05.70.Jk, 75.10.Hk Conventional leading scaling behavior at a secondorder phase transition is described by power laws in the reduced temperature t and field h. With h = 0, the correlation length, specific heat, and susceptibility behave as ξ ∞ (t) ∼ |t| −ν , C ∞ (t) ∼ |t| −α , and χ ∞ (t) ∼ |t| −γ , while the magnetization in the broken phase has m ∞ (t) ∼ |t| β . Here the subscript indicates the extent of the system. At t = 0 the magnetization scales as m ∞ (h) ∼ h 1/δ while the anomalous dimension η characterizes the correlation function at criticality. In the 1960's, it was shown that these six critical exponents are related via four scaling relations (see e.g. Ref.[1] and references therein), which are now firmly established and fundamentally important in the theory of critical phenomena. With d representing the dimensionality of the system, the scaling relations areIn the conventional scaling scenario, (2) and (3) can, in fact, be deduced from the Widom scaling hypothesis that the Helmholtz free energy is a homogeneous function [2]. Widom scaling and the remaining two laws can, in turn, be derived from the Kadanoff block-spin construction [3] and ultimately from Wilson's renormalization group (RG) [4]. The relation (1) can also be derived from the hyperscaling hypothesis, namely, that the free energy behaves near criticality as the inverse correlation volume:. Twice differentiating this relation recovers (1).The scaling relations, (2) and (3), were both rederived using an alternative route by Abe [5] and Suzuki [6] exploiting the fact that the even and odd scaling fields can be linked by Lee-Yang zeros [7]. The locus of these zeros in the magnetic-field plane is controlled by the temperature. In the t > 0 (disordered) phase this locus terminates at the Yang-Lee edge [7], the distance of which from the critical point is denoted by r YL (t). At a conventional second-order phase transition r YL (t) ∼ t ∆ for t > 0, and the gap exponent ∆ is related to the other exponents through [5,6] Logarithmic corrections are characteristic of a number of marginal scenarios (see, e.g., Ref.[8] and references therein). Hyperscaling fails at and above the upper critical dimension d c and, while (1) holds there, it too fails above d c , where mean-field behavior (which is independent of d) prevails. At d c itself, multiplicative logarithmic corrections to scaling are manifest. Such corrections are found in marginal d < d c situations too [8,9]. The q-state Potts model in d = 2 dimensions possesses a firstorder transition for q > 4 and a second-order one when q < 4. The q = 4 case is also characterized by a transition of second order, alb...
The introduction of a metric onto the space of parameters in models in Statistical Mechanics and beyond gives an alternative perspective on their phase structure. In such a geometrization, the scalar curvature, R, plays a central role. A noninteracting model has a flat geometry (R = 0), while R diverges at the critical point of an interacting one. Here, the information geometry is studied for a number of solvable statistical-mechanical models.
We investigate a generalized Ising action containing nearest neighbour, next to nearest neighbour and plaquette terms that has been suggested as a potential string worldsheet discretization on cubic lattices by Savvidy and Wegner. We use both mean field techniques and Monte-Carlo simulations to sketch out the phase diagram.The Gonihedric (Savvidy-Wegner) model has a symmetry that allows any plane of spins to be flipped with zero energy cost, which gives a highly degenerate vacuum state. We choose boundary conditions in the simulations that eliminate this degeneracy and allow the definition of a simple ferromagnetic order parameter. This in turn allows us to extract the magnetic critical exponents of the system.(a) Permanent Address:
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