In this work an approximate analytic expression for the quantum partition function of the quartic oscillator described by the potential V(x)=12 omega 2x2+gx4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepacka...
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In this work an approximate analytic expression for the quantum partition function of the quartic oscillator described by the potential V(x)=12 omega 2x2+gx4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(x) = \fracpartition functionpartition function \omega <^>2 x<^>2 + g x<^>4$$\end{document} is presented. Using a path integral formalism, the exact partition function is approximated by the partition function of a harmonic oscillator with an effective frequency depending both on the temperature and coupling constant g. By invoking a Principle of Minimal Sensitivity (PMS) of the path integral to the effective frequency, we derive a mathematically well-defined analytic formula for the partition function. Quite remarkably, the formula reproduces qualitatively and quantitatively the key features of the exact partition function. The free energy is accurate to a few percent over the entire range of temperatures and coupling strengths g. Both the harmonic (g -> 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g\rightarrow 0$$\end{document}) and classical (high-temperature) limits are exactly recovered. The divergence of the power series of the ground-state energy at weak coupling, characterized by a factorial growth of the perturbational energies, is reproduced as well as the functional form of the strong-coupling expansion along with accurate coefficients. Explicit accurate expressions for the ground- and first-excited state energies, E0(g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{
The saddle point approximation to formal quantum gravitational partition functions has yielded plausible computations of horizon entropy in various settings, but it stands on shaky ground. In this paper we visit some ...
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The saddle point approximation to formal quantum gravitational partition functions has yielded plausible computations of horizon entropy in various settings, but it stands on shaky ground. In this paper we visit some of that shaky ground, address some foundational questions, and describe efforts toward a more solid footing. We focus on the case of de Sitter horizon entropy which, it has been argued, corresponds to the dimension of the Hilbert space of a ball of space surrounded by the cosmological horizon.
Almost nothing is known about the parity of the partition function p(n), which is conjectured to be random. Despite this expectation, Ono [10] surprisingly proved the existence of infinitely many linear dependence con...
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Almost nothing is known about the parity of the partition function p(n), which is conjectured to be random. Despite this expectation, Ono [10] surprisingly proved the existence of infinitely many linear dependence congruence relations modulo 4 for p(n), indicating that the parity of the partition function cannot be truly random. Answering a question of Ono, we explicitly exhibit the first examples of these relations which he proved theoretically exist. The first two relations invoke 131 (resp. 198) different discriminants D <= 24k-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ D \le 24 k - 1 $$\end{document} for k=309\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ k = 309 $$\end{document} (resp. k=312\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ k = 312 $$\end{document});new relations occur for k=316,317,319,321,322,326,& mldr;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ k = 316,\, 317,\, 319,\, 321,\, 322,\, 326, \, \ldots $$\end{document}.
The partition function for two-dimensional nearest neighbour Ising model in a non-zero magnetic field have been derived for a finite square lattice of 16, 25, 36 and 64 sites with the help of graph theoretical procedu...
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The partition function for two-dimensional nearest neighbour Ising model in a non-zero magnetic field have been derived for a finite square lattice of 16, 25, 36 and 64 sites with the help of graph theoretical procedures, show-bit algorithm, enumeration of configurations and transfer matrix methods.
The partition function is a fundamental concept of equilibrium thermodynamics. It encodes the particle distribution at different energy levels within a thermal system. By modifying the partition function of a particle...
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The partition function is a fundamental concept of equilibrium thermodynamics. It encodes the particle distribution at different energy levels within a thermal system. By modifying the partition function of a particle, the partition function of a protein conformational state has been obtained. According to the protein thermodynamic structure theory, a protein conformational state represents one thermal system within a protein. The partition function of it encodes the thermodynamic relation among the different conformational states of a protein. Many experimental observations and models in enzymology, biochemistry, and molecular biology have been summarized and reexamined by applying this concept. This paper discusses the following: (1) the partition function of a protein conformational state;(2) the conformation distribution between the two states;(3) the protein conformation distribution along the temperature gradient and the protein conformation stability curve;(4) the agonist efficiency and conformational distribution of a receptor along voltage;(5) the enzyme active conformation distribution along the temperature gradient;(6) the enzyme active conformation distribution with respect to protein flexibility;(7) a revision of the Arrhenius equation for enzymatic reactions;and (8) the biophysical nature of conformational equilibrium and deformation.
In this work, the atomic parameters of Ag XLIV (Be-like Ag) are examined and evaluated by implementing the GRASP2K package with the multi- configuration Dirac-Hartree-Fock (MCDHF) method for the calculation of wave-fu...
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In this work, the atomic parameters of Ag XLIV (Be-like Ag) are examined and evaluated by implementing the GRASP2K package with the multi- configuration Dirac-Hartree-Fock (MCDHF) method for the calculation of wave-functions. We have listed fine structure energy levels of the lowest 170 levels with radiative data for multipole moments, such as electric dipole (E1), electric quadrupole (E2), magnetic dipole (M1), and magnetic quadrupole (M2) transitions, that lie in the region of extreme ultraviolet (EUV) and soft X-ray (SXR) for Ag XLIV from the ground state within the lowest 170 levels. We have compared our GRASP2K and FAC results with theoretical results available in the literature for some levels. Additionally, we have also calculated partition function and thermodynamic quantities for temperature ranges from 104 to 107 K. We believe that our presented details and data may be beneficial not only in plasma modeling but also in imaging of nanostructure as well as in medicine, semiconductors, and EUV and SXR laser applications.
The distribution of the zeros of the partition function in the complex temperature plane (Fisher zeros) of the two-dimensional Q-state Ports model is studied for non-integer Q. On L x L self-dual lattices studied (L l...
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The distribution of the zeros of the partition function in the complex temperature plane (Fisher zeros) of the two-dimensional Q-state Ports model is studied for non-integer Q. On L x L self-dual lattices studied (L less than or equal to 8). no Fisher zero lies on the unit circle p(0) = e(10) in the complex p = (e(beta J) - 1) root Q plane for Q < 1, while some of the Fisher zeros lie on the unit circle fnr Q > 1 and the number of such zeros increases with increasing e. The ferromagnetic and antiferromagnetic properties of the Ports model are investigated using the distribution of the Fisher zeros. For the Potts ferromagnet we verify the den Nijs formula for the thermal exponent gamma(1). For the Potts antiferromagnet we also verify the Baxter conjecture for the critical temperature and present new results for the thermal exponents in the range 0 < Q < 3. (C) 2000 Elsevier Science B.V. All rights reserved.
Writing the hindered rotor (hr) partition function as the trace of (rho) over cap = e(-beta(H) over cap hr), we approximate it by the sum of contributions from a set of points in position space. The contribution of th...
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Writing the hindered rotor (hr) partition function as the trace of (rho) over cap = e(-beta(H) over cap hr), we approximate it by the sum of contributions from a set of points in position space. The contribution of the density matrix from each point is approximated by performing a local harmonic expansion around it. The highlight of this method is that it can be easily extended to multidimensional systems. Local harmonic expansion leads to a breakdown of the method a low temperatures. In order to calculate the partition function at low temperatures, we suggest a matrix multiplication procedure. The results obtained using these methods closely agree with the exact partition function at all temperature ranges. Our method bypasses the evaluation of eigenvalues and eigenfunctions and evaluates the density matrix for internal rotation directly. We also suggest a procedure to account for the antisymmetry of the total wavefunction in the same. (C) 2012 Elsevier B.V. All rights reserved.
The partition function for one-dimensional nearest neighbour Ising models is estimated by summing all the energy terms in the Hamiltonian for N sites. The algebraic expression for the partition function is then employ...
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The partition function for one-dimensional nearest neighbour Ising models is estimated by summing all the energy terms in the Hamiltonian for N sites. The algebraic expression for the partition function is then employed to deduce the eigenvalues of the basic 2 x 2 matrix and the corresponding Hermitian Toeplitz matrix is derived using the Discrete Fourier Transform. A new recurrence relation pertaining to the partition function for two-dimensional Ising models in zero magnetic field is also proposed.
We present a closed-form expression of the vibrational partition function for the improved Tietz potential energy model. The vibrational partition functions for the improved Manning-Rosen potential and improved Rosen-...
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We present a closed-form expression of the vibrational partition function for the improved Tietz potential energy model. The vibrational partition functions for the improved Manning-Rosen potential and improved Rosen-Morse potential are easily deduced as two special cases, The obtained results are applicable to many issues in chemical physics and engineering. (C) 2017 Elsevier B.V. All rights reserved.
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