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.

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 REMARK ON PRIME DIVISORS OF PARTITION FUNCTIONS

2014 ◽  
Vol 10 (01) ◽  
pp. 125-131
Author(s):  
PAUL POLLACK
Keyword(s):  
Partition Function ◽  
Wide Class ◽  
Analogous Result ◽  
Partition Functions ◽  
Prime Divisors ◽  
Classical Partition Function

Schinzel showed that the set of primes that divide some value of the classical partition function is infinite. For a wide class of sets 𝒜, we prove an analogous result for the function p𝒜(n) that counts partitions of n into terms belonging to 𝒜.

Thermodynamic Functions of Pendular Molecules

1993 ◽  
Vol 58 (10) ◽  
pp. 2458-2473 ◽  
Author(s):  
Břetislav Friedrich ◽  
Dudley R. Herschbach
Keyword(s):  
Partition Function ◽  
Thermodynamic Functions ◽  
Energy Levels ◽  
Quantum Statistical Mechanics ◽  
Rotational Constant ◽  
Chemical Equilibria ◽  
Rotational States ◽  
Dipolar Molecules ◽  
Linear Molecules ◽  
Classical Partition Function

External electric or magnetic fields can hybridize rotational states of individual dipolar molecules and thus create pendular states whose field-dependent eigenproperties differ qualitatively from those of a rotor or an oscilator. The pendular eigenfunctions are directional, so the molecular axis id oriented. Here we use quantum statistical mechanics to evaluate ensamble properties of the pendular states. For linear molecules, the partition function and the averages that determine the thermodynamic functions can be specified by two reduced variables involving the dipole moment, field strength, rotational constant, and temperature. We examine a simple approximation due to Pitzer that employs the classical partition function with quantum corrections. This provides explicit analytic formulas which permit thermodynamic properties to be evaluated to good accuracy without computing energy levels. As applications we evaluate the high-field average orientation of the molecular dipoles and field-induced shifts of chemical equilibria.

scholarly journals On the Complementarity of the Harmonic Oscillator Model and the Classical Wigner–Kirkwood Corrected Partition Functions of Diatomic Molecules

Entropy ◽  
2020 ◽  
Vol 22 (8) ◽  
pp. 853
Author(s):  
Marcin Buchowiecki
Keyword(s):  
Phase Space ◽  
Partition Function ◽  
Harmonic Oscillator ◽  
High Temperatures ◽  
Low Temperatures ◽  
Diatomic Molecules ◽  
Partition Functions ◽  
Harmonic Oscillator Model ◽  
Harmonic Oscillator Approximation ◽  
Classical Partition Function

The vibrational and rovibrational partition functions of diatomic molecules are considered in the regime of intermediate temperatures. The low temperatures are those at which the harmonic oscillator approximation is appropriate, and the high temperatures are those at which classical partition function (with Wigner–Kirkwood correction) is applicable. The complementarity of the harmonic oscillator and classical integration over the phase space approaches is investigated for the CO and H2+ molecules showing that those two approaches are complementary in the sense that they smoothly overlap.

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