Application of Finite Orthogonal Polynomials to the Thermal Functions of Harmonic Oscillators. I. Reduced Partition Function of Isotopic Molecules

We consider noncommutative two-dimensional quantum harmonic oscillators and extend them to the case of twisted algebra. We obtained modified raising and lowering operators. Also we study statistical mechanics and thermodynamics and calculated partition function which yields the free energy of the system.

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Application of finite orthogonal polynomials to the thermal functions of harmonic oscillators. V. Isotope chemistry and molecular structure. Simplified theory of end atom isotope effects

Journal of the American Chemical Society ◽

10.1021/ja00800a001 ◽

1973 ◽

Vol 95 (19) ◽

pp. 6155-6157 ◽

Cited By ~ 11

Author(s):

Jacob. Bigeleisen ◽

Takanobu. Ishida

Keyword(s):

Molecular Structure ◽

Orthogonal Polynomials ◽

Isotope Effects ◽

Harmonic Oscillators

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Stability analysis of an ensemble of simple harmonic oscillators

International Journal of Modern Physics B ◽

10.1142/s021797922150034x ◽

2021 ◽

pp. 2150034

Author(s):

R. K. Thakur ◽

B. N. Tiwari ◽

R. Nigam ◽

Y. Xu ◽

P. K. Thiruvikraman

Keyword(s):

Partition Function ◽

Harmonic Oscillator ◽

Geometric Phase ◽

Hessian Matrix ◽

Harmonic Oscillators ◽

Real Numbers ◽

Oscillator Potential ◽

Harmonic Oscillator Potential ◽

The Stability ◽

Embedding Function

In this paper, we investigate the stability of the configurations of harmonic oscillator potential that are directly proportional to the square of the displacement. We derive expressions for fluctuations in partition function due to variations of the parameters, viz. the mass, temperature and the frequency of oscillators. Here, we introduce the Hessian matrix of the partition function as the model embedding function from the space of parameters to the set of real numbers. In this framework, we classify the regions in the parameter space of the harmonic oscillator fluctuations where they yield a stable statistical configuration. The mechanism of stability follows from the notion of the fluctuation theory. In Secs. ?? and ??, we provide the nature of local and global correlations and stability regions where the system yields a stable or unstable statistical basis, or it undergoes into geometric phase transitions. Finally, in Sec. ??, the comparison of results is provided with reference to other existing research.

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SOFT MATRIX MODELS AND CHERN–SIMONS PARTITION FUNCTIONS

Modern Physics Letters A ◽

10.1142/s0217732304014100 ◽

2004 ◽

Vol 19 (18) ◽

pp. 1365-1378 ◽

Cited By ~ 75

Author(s):

M. TIERZ

Keyword(s):

Partition Function ◽

Orthogonal Polynomials ◽

Matrix Models ◽

Simons Theory ◽

Partition Functions ◽

Soft Matrix ◽

Simple Mapping ◽

The Moment ◽

Chern Simons ◽

Chern Simons Theory

We study the properties of matrix models with soft confining potentials. Their precise mathematical characterization is that their weight function is not determined by its moments. We mainly rely on simple considerations based on orthogonal polynomials and the moment problem. In addition, some of these models are equivalent, by a simple mapping, to matrix models that appear in Chern–Simons theory. The models can be solved with q deformed orthogonal polynomials (Stieltjes–Wigert polynomials), and the deformation parameter turns out to be the usual q parameter in Chern–Simons theory. In this way, we give a matrix model computation of the Chern–Simons partition function on S3 and show that there are infinitely many matrix models with this partition function.

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Ensembles in Classical Statistical Mechanics

An Introduction to Statistical Mechanics and Thermodynamics ◽

10.1093/oso/9780198853237.003.0019 ◽

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.

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