Statistical Mechanics of the Parallel Hard Squares in Canonical Ensemble

1972 ◽  
Vol 56 (11) ◽  
pp. 5434-5444 ◽  
Author(s):  
Francis H. Ree ◽  
Taikyue Ree
Keyword(s):  
Statistical Mechanics ◽  
Canonical Ensemble ◽  
Hard Squares

scholarly journals Hard Squares with Negative Activity and Rhombus Tilings of the Plane

10.37236/1093 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Jakob Jonsson
Keyword(s):  
Partition Function ◽  
Statistical Mechanics ◽  
Simplicial Complex ◽  
Transfer Matrix ◽  
Euler Characteristic ◽  
Independent Sets ◽  
Roots Of Unity ◽  
Hard Squares ◽  
Vertex Set

Let $S_{m,n}$ be the graph on the vertex set ${\Bbb Z}_m \times {\Bbb Z}_n$ in which there is an edge between $(a,b)$ and $(c,d)$ if and only if either $(a,b) = (c,d\pm 1)$ or $(a,b) = (c \pm 1,d)$ modulo $(m,n)$. We present a formula for the Euler characteristic of the simplicial complex $\Sigma_{m,n}$ of independent sets in $S_{m,n}$. In particular, we show that the unreduced Euler characteristic of $\Sigma_{m,n}$ vanishes whenever $m$ and $n$ are coprime, thereby settling a conjecture in statistical mechanics due to Fendley, Schoutens and van Eerten. For general $m$ and $n$, we relate the Euler characteristic of $\Sigma_{m,n}$ to certain periodic rhombus tilings of the plane. Using this correspondence, we settle another conjecture due to Fendley et al., which states that all roots of $\det (xI-T_m)$ are roots of unity, where $T_m$ is a certain transfer matrix associated to $\{\Sigma_{m,n} : n \ge 1\}$. In the language of statistical mechanics, the reduced Euler characteristic of $\Sigma_{m,n}$ coincides with minus the partition function of the corresponding hard square model with activity $-1$.

Statistical mechanics of Fermi-Pasta-Ulam chains with the canonical ensemble

1997 ◽  
Vol 55 (3) ◽  
pp. 3727-3730
Author(s):  
Melik C. Demirel ◽  
Mehmet Sayar ◽  
Ali R. Atılgan
Keyword(s):  
Statistical Mechanics ◽  
Canonical Ensemble

scholarly journals Derivation of the entropic formula for the statistical mechanics of space plasmas

2018 ◽  
Vol 25 (1) ◽  
pp. 77-88 ◽  
Author(s):  
George Livadiotis
Keyword(s):  
Statistical Mechanics ◽  
Canonical Ensemble ◽  
Space Plasmas ◽  
Kappa Distribution ◽  
Physical Origin ◽  
Nonextensive Statistical Mechanics

Abstract. Kappa distributions describe velocities and energies of plasma populations in space plasmas. The statistical origin of these distributions is associated with the framework of nonextensive statistical mechanics. Indeed, the kappa distribution is derived by maximizing the q entropy of Tsallis, under the constraints of the canonical ensemble. However, the question remains as to what the physical origin of this entropic formulation is. This paper shows that the q entropy can be derived by adapting the additivity of energy and entropy.

scholarly journals Derivation of the entropic formula for the statistical mechanics of space plasmas

2017 ◽  
Author(s):  
George Livadiotis
Keyword(s):  
Statistical Mechanics ◽  
Canonical Ensemble ◽  
Space Plasmas ◽  
Kappa Distribution ◽  
Physical Origin

Abstract. Kappa distributions describe velocities and energies of plasma populations in space plasmas. The statistical origin of these distributions is the non-extensive statistical mechanics. Indeed, the kappa distribution is derived by maximizing the q-entropy of Tsallis under the constraints of canonical ensemble. However, there remains the question what is the physical origin of this entropic formulation. This paper shows that the q-entropy can be derived by adapting the additivity of energy and entropy.

Quantum Canonical Ensemble

2019 ◽  
pp. 297-321
Author(s):  
Robert H. Swendsen
Keyword(s):  
Statistical Mechanics ◽  
Canonical Ensemble ◽  
Quantum Statistical Mechanics ◽  
Harmonic Oscillators ◽  
Quantum Mechanical ◽  
The Third ◽  
Two Level Systems ◽  
Third Law ◽  
Quantum Statistical ◽  
The Relationship

This chapter introduces the quantum mechanical canonical ensemble, which is used for the majority of problems in quantum statistical mechanics. The ensemble is derived and analogies with the classical ensemble are presented. A useful expression for the quantum entropy is derived. The origin of the Third Law is explained. The relationship between fluctuations and derivatives found in classical statistical mechanics is shown to have counterparts in quantum statistical mechanics. The factorization of the partition function is re-introduced as the best trick in quantum statistical mechanics. Due to their importance in later chapters, basic calculations of the properties of two-level systems and simple harmonic oscillators are derived.

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