scholarly journals A Review of the Classical Canonical Ensemble Treatment of Newton’s Gravitation

Entropy ◽  
2019 ◽  
Vol 21 (7) ◽  
pp. 677
Author(s):  
Flavia Pennini ◽  
Angel Plastino ◽  
Mario Rocca ◽  
Gustavo Ferri
Keyword(s):  
Partition Function ◽  
Statistical Mechanics ◽  
Exponential Distribution ◽  
Canonical Ensemble ◽  
Dimensional Regularization ◽  
Regularization Technique ◽  
Classical Statistical Mechanics ◽  
Analytical Extension

It is common lore that the canonical gravitational partition function Z associated with the classical Boltzmann-Gibbs (BG) exponential distribution cannot be built up because of mathematical pitfalls. The integral needed for writing up Z diverges. We review here how to avoid this pitfall and obtain a (classical) statistical mechanics of Newton’s gravitation. This is done using (1) the analytical extension treatment obtained of Gradshteyn and Rizhik and (2) the well known dimensional regularization technique.

A modernized version of Gibbs' use of the grand canonical ensemble

1938 ◽  
Vol 34 (3) ◽  
pp. 382-391 ◽  
Author(s):  
R. H. Fowler
Keyword(s):  
Phase Space ◽  
Partition Function ◽  
Statistical Mechanics ◽  
Canonical Ensemble ◽  
Grand Canonical Ensemble ◽  
Grand Canonical

In the last chapter of his Introduction to statistical mechanics Gibbs introduces the idea of the grand canonical ensemble. He had previously determined the properties of an assembly or a phase containing a given number of systems by averaging the properties of the assembly over an ensemble of examples canonically distributed in phase, keeping the number of systems in the assembly fixed. This means of course constructing what we now call the partition function for the assembly by summing or integrating e−E/kT over the whole of the accessible phase space.

Ensembles in Classical Statistical Mechanics

2019 ◽  
pp. 231-257
Author(s):  
Robert H. Swendsen
Keyword(s):  
Molecular Dynamics ◽  
Monte Carlo ◽  
Free Energy ◽  
Partition Function ◽  
Statistical Mechanics ◽  
Harmonic Oscillators ◽  
Canonical Partition Function ◽  
Classical Statistical Mechanics ◽  
Classical Models ◽  
The Relationship

This chapter explores more powerful methods of calculation than were seen previously. Among them are Molecular Dynamics (MD) and Monte Carlo (MC) computer simulations. Another is the canonical partition function, which is related to the Helmholtz free energy. The derivation of thermodynamic identities within statistical mechanics is illustrated by the relationship between the specific heat and the fluctuations of the energy. It is shown how the canonical ensemble allows us to integrate out the momentum variables for many classical models. The factorization of the partition function is presented as the best trick in statistical mechanics, because of its central role in solving problems. Finally, the problem of many simple harmonic oscillators is solved, both for its importance and as an illustration of the best trick.

scholarly journals Classical Partition Function for Non-Relativistic Gravity

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 121
Author(s):  
Mir Hameeda ◽  
Angelo Plastino ◽  
Mario Carlos Rocca ◽  
Javier Zamora
Keyword(s):  
Partition Function ◽  
Dimensional Regularization ◽  
Analytical Extension ◽  
Special Instance ◽  
Classical Partition Function

We considered the canonical gravitational partition function Z associated to the classical Boltzmann–Gibbs (BG) distribution e−βHZ. It is popularly thought that it cannot be built up because the integral involved in constructing Z diverges at the origin. Contrariwise, it was shown in (Physica A 497 (2018) 310), by appeal to sophisticated mathematics developed in the second half of the last century, that this is not so. Z can indeed be computed by recourse to (A) the analytical extension treatments of Gradshteyn and Rizhik and Guelfand and Shilov, that permit tackling some divergent integrals and (B) the dimensional regularization approach. Only one special instance was discussed in the above reference. In this work, we obtain the classical partition function for Newton’s gravity in the four cases that immediately come to mind.

A canonical ensemble derivation of the McMillan-Mayer solution theory

1983 ◽  
Vol 48 (10) ◽  
pp. 2888-2892 ◽  
Author(s):  
Vilém Kodýtek
Keyword(s):  
Free Energy ◽  
Partition Function ◽  
Energy Function ◽  
Special Function ◽  
Helmholtz Free Energy ◽  
Canonical Ensemble ◽  
Free Energy Function ◽  
Solution Theory ◽  
Pure Solvent ◽  
The Common

A special free energy function is defined for a solution in the osmotic equilibrium with pure solvent. The partition function of the solution is derived at the McMillan-Mayer level and it is related to this special function in the same manner as the common partition function of the system to its Helmholtz free energy.

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