Perturbation and variational-perturbation method for the free energy of anharmonic oscillators

2007 ◽  
Vol 85 (1) ◽  
pp. 13-30 ◽  
Author(s):  
K Vlachos ◽  
V Papatheou ◽  
A Okopińska
Keyword(s):  
Free Energy ◽  
Partition Function ◽  
Perturbation Method ◽  
Perturbation Methods ◽  
Anharmonic Oscillator ◽  
Numerical Calculations ◽  
Anharmonic Oscillators ◽  
One Dimensional ◽  
Variational Perturbation ◽  
Fifth Order

The perturbation and the variational-perturbation methods are applied for calculating the partition function of one-dimensional oscillators with anharmonicity x2n. New formally simple expressions for the free energy and for the Rayleigh–Schrodinger energy corrections are derived. It is shown that the variational-perturbation method overcomes all the deficiencies of the conventional perturbation method. The results of fifth-order numerical calculations for the free energy of the quartic, quartic–sextic, and octic anharmonic oscillator are highly accurate in the whole range of temperatures. PACS Nos.: 03.65.–w, 05.30.–d

scholarly journals Linearly Coupled Anharmonic Oscillators and Integrability

1987 ◽  
Vol 40 (5) ◽  
pp. 587
Author(s):  
W-H Steeb ◽  
JA Louw ◽  
CM Villet
Keyword(s):  
Lyapunov Exponent ◽  
Anharmonic Oscillator ◽  
First Integrals ◽  
Chaotic Behaviour ◽  
Painlevé Test ◽  
Anharmonic Oscillators ◽  
One Dimensional

The Painleve test for a linearly coupled anharmonic oscillator is performed. We show that this system does not pass the Painleve test. This suggests that this system is not integrable. Moreover, we apply Ziglin's (1983) theorem which provides a criterion for non-existence of first integrals besides the Hamiltonian. Calculating numerically the maximal one-dimensional Lyapunov exponent, we find regions with positive exponents. Thus, the system can show chaotic behaviour. Finally we compare our results with the quartic coupled anharmonic oscillator.

scholarly journals Exact Solution of a One-dimensional Spin System

1970 ◽  
Vol 23 (5) ◽  
pp. 927 ◽  
Author(s):  
RW Gibberd
Keyword(s):  
Phase Transition ◽  
Exact Solution ◽  
Free Energy ◽  
Partition Function ◽  
Thermodynamic Limit ◽  
Spin System ◽  
Spin Systems ◽  
One Dimensional ◽  
Gibb’S Free Energy ◽  
Theory Of Superconductivity

The partition function and the Gibb's free energy are calculated exactly in the thermodynamic limit, using techniques which are well known in the theory of superconductivity. This calculation illustrates explicitly the similarity between the phase transition in superconductivity and the molecular field transitions in spin systems.

scholarly journals Confinement-deconfinement transition and Z2 symmetry in Z2+Higgs theory

2021 ◽  
Vol 36 (30) ◽  
Author(s):  
Minati Biswal ◽  
Sanatan Digal ◽  
Vinod Mamale ◽  
Sabiar Shaikh
Keyword(s):  
Monte Carlo ◽  
Free Energy ◽  
Partition Function ◽  
Density Of States ◽  
Dimensional Space ◽  
Polyakov Loop ◽  
Dimensional Model ◽  
One Dimensional ◽  
Original Action ◽  
Symmetric Phase

In this paper, we study the Polyakov loop and the [Formula: see text] symmetry in the lattice [Formula: see text] theory in four-dimensional space using Monte Carlo simulations. The results show that this symmetry is realized in the Higgs symmetric phase for large number of “temporal” lattice sites. To understand this dependence on the number of “temporal” sites, we consider a one-dimensional model by keeping terms of the original action corresponding to a single spatial site. In this approximation, the partition function can be calculated exactly as a function of the Polyakov loop. The resulting free energy is found to have the [Formula: see text] symmetry in the limit of large temporal sites. We argue that this is due to [Formula: see text] invariance as well as dominance of the distribution or density of states corresponding to the action.

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.

scholarly journals Approximate solution for fractional attractor one-dimensional Keller-Segel equations using homotopy perturbation sumudu transform method

2020 ◽  
Vol 9 (1) ◽  
pp. 370-381
Author(s):  
Dinkar Sharma ◽  
Gurpinder Singh Samra ◽  
Prince Singh
Keyword(s):  
Approximate Solution ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Simple Technique ◽  
Nonlinear Partial Differential Equations ◽  
Transform Method ◽  
Present Scheme ◽  
One Dimensional ◽  
Homotopy Perturbation ◽  
Sumudu Transform

AbstractIn this paper, homotopy perturbation sumudu transform method (HPSTM) is proposed to solve fractional attractor one-dimensional Keller-Segel equations. The HPSTM is a combined form of homotopy perturbation method (HPM) and sumudu transform using He’s polynomials. The result shows that the HPSTM is very efficient and simple technique for solving nonlinear partial differential equations. Test examples are considered to illustrate the present scheme.

scholarly journals The bulk, surface and corner free energies of the anisotropic triangular Ising model

2020 ◽  
Vol 476 (2234) ◽  
pp. 20190713
Author(s):  
Rodney J. Baxter
Keyword(s):  
Free Energy ◽  
Ising Model ◽  
Partition Function ◽  
Temperature Series ◽  
Isotropic Case ◽  
Series Expansions ◽  
Exact Results ◽  
Free Energies ◽  
Bulk Surface ◽  
Bulk Free Energy

We consider the anisotropic Ising model on the triangular lattice with finite boundaries, and use Kaufman’s spinor method to calculate low-temperature series expansions for the partition function to high order. From these, we can obtain 108-term series expansions for the bulk, surface and corner free energies. We extrapolate these to all terms and thereby conjecture the exact results for each. Our results agree with the exactly known bulk-free energy and with Cardy and Peschel’s conformal invariance predictions for the dominant behaviour at criticality. For the isotropic case, they also agree with Vernier and Jacobsen’s conjecture for the 60 ° corners.

Phase equilibria in solutions of rod-like particles

1956 ◽  
Vol 234 (1196) ◽  
pp. 73-89 ◽  
Keyword(s):  
Free Energy ◽  
Phase Separation ◽  
Partition Function ◽  
Colloidal Particles ◽  
Double Layers ◽  
Positive Energy ◽  
Isotropic Phase ◽  
Electric Double Layers ◽  
Rigid Rod ◽  
Anisotropic Phase

A partition function for a system of rigid rod-like particles with partial orientation about an axis is derived through the use of a modified lattice model. In the limit of perfect orientation the partition function reduces to the ideal mixing law ; for complete disorientation it corresponds to the polymer mixing law for rigid chains. A general expression is given for the free energy of mixing as a function of the mole numbers, the axis ratio of the solute particles, and a disorientation parameter. This function passes through a minimum followed by a maximum with increase in the disorientation parameter, provided the latter exceeds a critical value which is 2e for the pure solute and which increases with dilution. Assigning this parameter the value which minimizes the free energy, the chemical potentials display discontinuities a t the concentration a t which the minimum first appears. Separation into an isotropic phase and a some what more concentrated anisotropic phase arises because of the discontinuity, in confirmation of the theories of Onsager and Isihara, which treat only the second virial coefficient. Phase separation thus arises as a consequence of particle asymmetry, unassisted by an energy term . Whereas for a large-particle asymmetry both phases in equilibrium are predicted to be fairly dilute when mixing is athermal, a comparatively small positive energy of interaction causes the concentration in the anisotropic phase to increase sharply, while the concentration in the isotropic phase becomes vanishingly small. The theory offers a statistical mechanical basis for interpreting precipitation of rod-like colloidal particles with the formation of fibrillar structures such as are prominent in the fibrous proteins. The asymmetry of tobacco mosaic virus particles (with or without inclusion of their electric double layers) is insufficient alone to explain the well-known phase separation which occurs from their dilute solutions at very low ionic strengths. Higher-order interaction between electric double layers appears to be a major factor in bringing about dilute phase separation for these and other asymmetric colloidal particles bearing large charges, as was pointed out previously by Oster.

scholarly journals THE CRITICAL POINT OF UNORIENTED RANDOM SURFACES WITH A NONEVEN POTENTIAL

1993 ◽  
Vol 08 (06) ◽  
pp. 1139-1152
Author(s):  
M.A. MARTÍN-DELGADO
Keyword(s):  
Free Energy ◽  
Partition Function ◽  
Critical Point ◽  
Discrete Model ◽  
Recursion Relation ◽  
Scaling Limit ◽  
Random Surfaces ◽  
Matrix Ensemble ◽  
Quartic Interaction ◽  
Double Scaling Limit

The discrete model of the real symmetric one-matrix ensemble is analyzed with a cubic interaction. The partition function is found to satisfy a recursion relation that solves the model. The double scaling-limit of the recursion relation leads to a Miura transformation relating the contributions to the free energy coming from oriented and unoriented random surfaces. This transformation is the same kind as found with a quartic interaction.

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