scholarly journals Algebras in higher-dimensional statistical mechanics - the exceptional partition (mean field) algebras

1994 ◽  
Vol 30 (3) ◽  
pp. 179-185 ◽  
Author(s):  
Paul Martin ◽  
Hubert Saleur
Keyword(s):  
Statistical Mechanics ◽  
Mean Field ◽  
Higher Dimensional

Nonextensive Effects in Hamiltonian Systems

2004 ◽  
Author(s):  
Andrea Rapisarda ◽  
Vito Latora
Keyword(s):  
Statistical Mechanics ◽  
Thermodynamic Formalism ◽  
Mean Field ◽  
Fractal Structures ◽  
Range Interaction ◽  
Equilibrium Statistical Mechanics ◽  
Long Range Correlations ◽  
Equilibrium Dynamics ◽  
Nonextensive Thermodynamics ◽  
Non Gaussian

The Boltzmann-Gibbs formulation of equilibrium statistical mechanics depends crucially on the nature of the Hamiltonian of the JV-body system under study, but this fact is clearly stated only in the introductions of textbooks and, in general, it is very soon neglected. In particular, the very same basic postulate of equilibrium statistical mechanics, the famous Boltzmann principle S = k log W of the microcanonical ensemble, assumes that dynamics can be automatically an easily taken into account, although this is not always justified, as Einstein himself realized [20]. On the other hand, the Boltzmann-Gibbs canonical ensemble is valid only for sufficiently short-range interactions and does not necessarily apply, for example, to gravitational or unscreened Colombian fields for which the usually assumed entropy extensivity postulate is not valid [5]. In 1988, Constantino Tsallis proposed a generalized thermostatistics formalism based on a nonextensive entropic form [24]. Since then, this new theory has been encountering an increasing number of successful applications in different fields (for some recent examples see Abe and Suzuki [1], Baldovin and Robledo [4], Beck et al. [8], Kaniadakis et al. [12], Latora et al. [16], and Tsallis et al. [25]) and seems to be the best candidate for a generalized thermodynamic formalism which should be valid when nonextensivity, long-range correlations, and fractal structures in phase space cannot be neglected: in other words, when the dynamics play a nontrivial role [11] and fluctuations are quite large and non-Gaussian [6, 7, 8, 24, 26]. In this contribution we consider a nonextensive JV-body classical Hamiltonian system, with infinite range interaction, the so-called Hamiltonian mean field (HMF) model, which has been intensively studied in the last several years [3, 13, 14, 15, 17, 18, 19]. The out-of-equilibrium dynamics of the model exhibits a series of anomalies like negative specific heat, metastable states, vanishing Lyapunov exponents, and non-Gaussian velocity distributions. After a brief overview of these anomalies, we show how they can be interpreted in terms of nonextensive thermodynamics according to the present understanding.

scholarly journals Chaos and statistical mechanics in the Hamiltonian mean field model

1999 ◽  
Vol 131 (1-4) ◽  
pp. 38-54 ◽  
Author(s):  
Vito Latora ◽  
Andrea Rapisarda ◽  
Stefano Ruffo
Keyword(s):  
Statistical Mechanics ◽  
Field Model ◽  
Mean Field ◽  
Mean Field Model

scholarly journals PDE/Statistical Mechanics Duality: Relation Between Guerra’s Interpolated p-Spin Ferromagnets and the Burgers Hierarchy

2021 ◽  
Vol 183 (1) ◽  
Author(s):  
Alberto Fachechi
Keyword(s):  
Statistical Mechanics ◽  
Mean Field ◽  
Finite Size ◽  
Interpolation Scheme ◽  
Equilibrium Dynamics ◽  
Mechanics Model ◽  
The Mean ◽  
Burgers Hierarchy ◽  
Generalized Partition ◽  
Effective Description

AbstractWe examine the duality relating the equilibrium dynamics of the mean-field p-spin ferromagnets at finite size in the Guerra’s interpolation scheme and the Burgers hierarchy. In particular, we prove that—for fixed p—the expectation value of the order parameter on the first side w.r.t. the generalized partition function satisfies the $$p-1$$ p - 1 -th element in the aforementioned class of nonlinear equations. In the light of this duality, we interpret the phase transitions in the thermodynamic limit of the statistical mechanics model with the development of shock waves in the PDE side. We also obtain the solutions for the p-spin ferromagnets at fixed N, allowing us to easily generate specific solutions of the corresponding equation in the Burgers hierarchy. Finally, we obtain an effective description of the finite N equilibrium dynamics of the $$p=2$$ p = 2 model with some standard tools in PDE side.

scholarly journals On the Functional Form of the Transition Probabilities in the Monte Carlo Method as Used in Statistical Mechanics

1979 ◽  
Vol 32 (5) ◽  
pp. 455
Author(s):  
CHJ Johnson
Keyword(s):  
Monte Carlo ◽  
Statistical Mechanics ◽  
Linear Functional ◽  
Functional Form ◽  
Transition Probabilities ◽  
Mean Field ◽  
Linear Functional Equation ◽  
Analytical Comparison ◽  
Simulation Process ◽  
The Monte Carlo Method

The transition probabilities W in the traditional Monte Carlo simulation process used in statistical mechanics are shown to satisfy a linear functional equation. General classes of solution to this equation are presented. A simple one-particle mean-field Ising model of a ferromagnet is used in an analytical comparison of the various possible forms of W.

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