Statistical Mechanics, Canonical Ensemble, Probability and the Boltzmann Factor, and Canonical Partition Function

2020 ◽  
pp. 411-417
Author(s):  
Daniel Blankschtein
Keyword(s):  
Partition Function ◽  
Statistical Mechanics ◽  
Canonical Ensemble ◽  
Boltzmann Factor ◽  
Canonical Partition Function

Classical Ensembles: Grand and Otherwise

2019 ◽  
pp. 258-265
Author(s):  
Robert H. Swendsen
Keyword(s):  
Partition Function ◽  
Particle Number ◽  
Canonical Ensemble ◽  
Ideal Gas ◽  
Grand Canonical Ensemble ◽  
Physical Situation ◽  
Canonical Partition Function ◽  
Grand Canonical Partition Function ◽  
Grand Canonical ◽  
The Ideal

The chapter introduces the grand canonical ensemble as a means of describing systems that exchange particles with a reservoir. The grand canonical partition function is defined in general and calculated for the ideal gas in particular. Other ensembles are described and their relationship to the grand canonical ensemble is shown. The physical situation described by the grand canonical ensemble is that of a system that can exchange both energy and particles with a reservoir. As usual, we assume that the reservoir is much larger than the system of interest, so that its properties are not signifficantly affected by relatively small changes in its energy or particle number.

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 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 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.

Supradegeneracy and the Second Law of Thermodynamics

2020 ◽  
Vol 45 (2) ◽  
pp. 121-132
Author(s):  
Daniel P. Sheehan
Keyword(s):  
Steady State ◽  
Statistical Mechanics ◽  
Population Inversion ◽  
Laboratory Experiments ◽  
Second Law Of Thermodynamics ◽  
Second Law ◽  
Boltzmann Factor ◽  
Energy Currents ◽  
Non Equilibrium

AbstractCanonical statistical mechanics hinges on two quantities, i. e., state degeneracy and the Boltzmann factor, the latter of which usually dominates thermodynamic behaviors. A recently identified phenomenon (supradegeneracy) reverses this order of dominance and predicts effects for equilibrium that are normally associated with non-equilibrium, including population inversion and steady-state particle and energy currents. This study examines two thermodynamic paradoxes that arise from supradegeneracy and proposes laboratory experiments by which they might be resolved.

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