2006/01/26 by H. J. de Vega, N. Sánchez, N. G. Sanchez
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Classical mechanics #Cosmology and Gravitation Theories #Limit (mathematics) #Material properties #Mathematical analysis #Mathematics #Phase (matter) #Phase diagram #Physics #Quantum mechanics #Statistical Mechanics and Entropy #Statistical mechanics #Statistical physics #Thermodynamic beta #Thermodynamic databases for pure substances #Thermodynamic limit #Thermodynamic process #Thermodynamics #astro-ph #cond-mat.stat-mech #gr-qc #hep-th
paper · pdf · doi:10.1016/j.crhy.2006.01.006
published as Comptes Rendus Physique 7 (2006) 391-397 · 12 pages, Invited lecture at `Statistical Mechanics of Non-Extensive Systems', Observatoire de Paris, October 2005, to be published in a Special issue of `Les Comptes rendus de l'Acade'mie des sciences', Elsevier
arxiv created 2006/01/26 · openalex publication_date 2006/04/01 · arxiv updated 2015/06/24 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We show that the self-gravitating gas at thermal equilibrium has an infinite volume limit in the three ensembles (GCE, CE, MCE) when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>,</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>→</mml:mo> <mml:mo>∞</mml:mo> </mml:math> , keeping <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>N</mml:mi> <mml:mo stretchy="false">/</mml:mo> <mml:msup> <mml:mi>V</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo stretchy="false">/</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> </mml:math> fixed, that is, with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>η</mml:mi> <mml:mo>≡</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>G</mml:mi> <mml:msup> <mml:mi>m</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>N</mml:mi> </mml:mrow> <mml:mrow> <mml:msup> <mml:mi>V</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo stretchy="false">/</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mi>T</mml:mi> </mml:mrow> </mml:mfrac> </mml:math> fixed. We develop Monte Carlo simulations, analytic mean field methods (MF) and low density expansions. We compute the equation of state and find it to be locally <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mover accent="true"> <mml:mi>r</mml:mi> <mml:mo>→</mml:mo> </mml:mover> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>T</mml:mi> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>V</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mover accent="true"> <mml:mi>r</mml:mi> <mml:mo>→</mml:mo> </mml:mover> <mml:mo stretchy="false">)</mml:mo> </mml:math> , that is a local ideal gas equation of state. The system is in a gaseous phase for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>η</mml:mi> <mml:mo><</mml:mo> <mml:msub> <mml:mi>η</mml:mi> <mml:mi>T</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1.51024</mml:mn> <mml:mo>…</mml:mo> </mml:math> and collapses into a very dense object for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>η</mml:mi> <mml:mo>></mml:mo> <mml:msub> <mml:mi>η</mml:mi> <mml:mi>T</mml:mi> </mml:msub> </mml:math> in the CE with the pressure becoming large and negative. The isothermal compressibility diverges at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>η</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>η</mml:mi> <mml:mi>T</mml:mi> </mml:msub> </mml:math> . We compute the fluctuations around mean field for the three ensembles. We show that the particle distribution can be described by a Haussdorf dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>1</mml:mn> <mml:mo><</mml:mo> <mml:mi>D</mml:mi> <mml:mo><</mml:mo> <mml:mn>3</mml:mn> </mml:math> .