scholarly journals Twisted Conformal Algebra and Quantum Statistics of Harmonic Oscillators

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
J. Naji ◽  
S. Heydari ◽  
R. Darabi
Keyword(s):  
Free Energy ◽  
Partition Function ◽  
Statistical Mechanics ◽  
Quantum Statistics ◽  
Conformal Algebra ◽  
Harmonic Oscillators ◽  
Two Dimensional ◽  
Raising And Lowering Operators ◽  
Lowering Operators

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.

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 Parity and the modular bootstrap

2018 ◽  
Vol 5 (3) ◽  
Author(s):  
Tarek Anous ◽  
Raghu Mahajan ◽  
Edgar Shaghoulian
Keyword(s):  
Free Energy ◽  
Partition Function ◽  
Fixed Points ◽  
Continuous Family ◽  
Two Dimensional ◽  
Modular Invariant ◽  
Parity Symmetry ◽  
Operator Spectrum ◽  
Left And Right ◽  
Modular Inversion

We consider unitary, modular invariant, two-dimensional CFTs which are invariant under the parity transformation PP. Combining PP with modular inversion SS leads to a continuous family of fixed points of the SPSP transformation. A particular subset of this locus of fixed points exists along the line of positive left- and right-moving temperatures satisfying \beta_L \beta_R = 4\pi^2βLβR=4π2. We use this fixed locus to prove a conjecture of Hartman, Keller, and Stoica that the free energy of a large-cc CFT_22 with a suitably sparse low-lying spectrum matches that of AdS_33 gravity at all temperatures and all angular potentials. We also use the fixed locus to generalize the modular bootstrap equations, obtaining novel constraints on the operator spectrum and providing a new proof of the statement that the twist gap is smaller than (c-1)/12(c−1)/12 when c>1c>1. At large cc we show that the operator dimension of the first excited primary lies in a region in the (h,\overline{h})(h,h¯)-plane that is significantly smaller than h+\overline{h}<c/6h+h¯>c/6. Our results for the free energy and constraints on the operator spectrum extend to theories without parity symmetry through the construction of an auxiliary parity-invariant partition function.

Statistical Mechanics of Rubber

1953 ◽  
Vol 26 (2) ◽  
pp. 302-310
Author(s):  
F. W. Boggs
Keyword(s):  
Free Energy ◽  
Partition Function ◽  
Thermodynamic Properties ◽  
Statistical Mechanics ◽  
Partial Pressure ◽  
Solvent Mixtures ◽  
Perfect Gas ◽  
Second Order Transition ◽  
Statistical Studies ◽  
Solvent Systems

Abstract It is shown in this paper that, above the second-order transition, the partition function of a rubber can be represented to a reasonably good approximation by the product of the partition function of a liquid formed of molecules similar to the chain elements and the partition function of the noninteracting chains divided by the partition function of a perfect gas. In terms of the free energy this means that the free energy of rubber is equal to the sum of the free energy of the chain network without interactions and that of a liquid formed of the chain elements alone minus the free energy of a perfect gas having the same number of molecules as the rubber has chain elements. This same procedure has made possible the calculation of the free energy and partial pressure of a rubber solution. This is, however, by no means the only possible application. All the thermodynamic properties of elastomer solvent systems can be obtained. This is a generalization of the methods previously used in the statistical studies of rubber and rubber solvent mixtures.

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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