Numbers and densities of states and partition functions from an efficient approach to phase space integration

2001 ◽  
Vol 3 (12) ◽  
pp. 2296-2305 ◽  
Author(s):  
Gerhard Taubmann ◽  
Stefan Schmatz
Keyword(s):  
Phase Space ◽  
Partition Functions ◽  
Efficient Approach ◽  
Densities Of States ◽  
Phase Space Integration

Hamilton-Jacobi Theory

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Phase Space ◽  
Bbgky Hierarchy ◽  
Distribution Functions ◽  
Symplectic Integrators ◽  
Partition Functions ◽  
Von Neumann ◽  
Equipartition Theorem ◽  
Recurrence Theorem ◽  
Phase Amplitude ◽  
Canonical Ensembles

This chapter focuses on Liouville’s theorem and classical statistical mechanics, deriving the classical propagator. The terms ‘phase space volume element’ and ‘Liouville operator’ are defined and an n-particle phase space probability density function is constructed to derive the Liouville equation. This is deconstructed into the BBGKY hierarchy, and radial distribution functions are used to develop n-body correlation functions. Koopman–von Neumann theory is investigated as a classical wavefunction approach. The chapter develops an operatorial mechanics based on classical Hilbert space, and discusses the de Broglie–Bohm formulation of quantum mechanics. Partition functions, ensemble averages and the virial theorem of Clausius are defined and Poincaré’s recurrence theorem, the Gibbs H-theorem and the Gibbs paradox are discussed. The chapter also discusses commuting observables, phase–amplitude decoupling, microcanonical ensembles, canonical ensembles, grand canonical ensembles, the Boltzmann factor, Mayer–Montroll cluster expansion and the equipartition theorem and investigates symplectic integrators, focusing on molecular dynamics.

scholarly journals Themis: A Software to Assess Association Free Energies via Direct Estimative of Partition Functions

2020 ◽  
Author(s):  
Felippe Mariano Colombari ◽  
Asdrubal Lozada-Blanco ◽  
Kalil Bernardino ◽  
Weverson Gomes ◽  
André Farias de Moura
Keyword(s):  
Phase Space ◽  
Statistical Mechanics ◽  
Excited States ◽  
Surface Adsorption ◽  
Chiral Discrimination ◽  
Partition Functions ◽  
Computer Implementation ◽  
Free Energies ◽  
Electronic Excited States ◽  
Discrete Grid

<div>We present the program <i>Themis</i> - a computer implementation of a standard statistical mechanics framework to compute free energies, average energies and entropic contributions for association processes of two atom-based structures. The partition functions are computed analytically using a discrete grid in the phase space, whose size and degree of coarseness can be controlled to allow efficient calculations and to achieve the desired level of accuracy. With this strategy, applications ranging from molecular recognition, chiral discrimination, surface adsorption and even the interactions involving molecules in electronic excited states can be handled.</div>

On a Direct Estimate of Densities of States and Partition Functions

1998 ◽  
Vol 12 (06) ◽  
pp. 697-707 ◽  
Author(s):  
R. Hashim ◽  
S. Romano
Keyword(s):  
General Type ◽  
Nearest Neighbor ◽  
One Dimension ◽  
Partition Functions ◽  
Spin Space ◽  
Classical Lattice ◽  
Direct Estimate ◽  
Densities Of States ◽  
Lattice Spin Models ◽  
Good Agreement

We report here an attempt of directly estimating densities of states, and hence partition functions, for classical lattice-spin models of a rather general type; the method has been applied to a few models in one dimension with nearest-neighbor interactions isotropic in spin space; their exact solutions, available in the literature, are used for unambiguous comparison. After obtaining the appropriate histograms, thermodynamic properties have been calculated over a range of temperatures. At sufficiently high temperatures, the resulting estimates are in very good agreement with available exact results; as expected, the agreement deteriorates at lower temperatures.

scholarly journals Generalised partition functions: inferences on phase space distributions

2016 ◽  
Vol 34 (6) ◽  
pp. 557-564 ◽  
Author(s):  
Rudolf A. Treumann ◽  
Wolfgang Baumjohann
Keyword(s):  
Phase Space ◽  
Partition Function ◽  
Bessel Functions ◽  
Chemical Potential ◽  
Temperature Limit ◽  
Large Temperature ◽  
Partition Functions ◽  
Functional Dependencies ◽  
Boltzmann Factor ◽  
Nonextensive Statistical Mechanics

Abstract. It is demonstrated that the statistical mechanical partition function can be used to construct various different forms of phase space distributions. This indicates that its structure is not restricted to the Gibbs–Boltzmann factor prescription which is based on counting statistics. With the widely used replacement of the Boltzmann factor by a generalised Lorentzian (also known as the q-deformed exponential function, where κ = 1∕|q − 1|, with κ, q ∈ R) both the kappa-Bose and kappa-Fermi partition functions are obtained in quite a straightforward way, from which the conventional Bose and Fermi distributions follow for κ → ∞. For κ ≠ ∞ these are subject to the restrictions that they can be used only at temperatures far from zero. They thus, as shown earlier, have little value for quantum physics. This is reasonable, because physical κ systems imply strong correlations which are absent at zero temperature where apart from stochastics all dynamical interactions are frozen. In the classical large temperature limit one obtains physically reasonable κ distributions which depend on energy respectively momentum as well as on chemical potential. Looking for other functional dependencies, we examine Bessel functions whether they can be used for obtaining valid distributions. Again and for the same reason, no Fermi and Bose distributions exist in the low temperature limit. However, a classical Bessel–Boltzmann distribution can be constructed which is a Bessel-modified Lorentzian distribution. Whether it makes any physical sense remains an open question. This is not investigated here. The choice of Bessel functions is motivated solely by their convergence properties and not by reference to any physical demands. This result suggests that the Gibbs–Boltzmann partition function is fundamental not only to Gibbs–Boltzmann but also to a large class of generalised Lorentzian distributions as well as to the corresponding nonextensive statistical mechanics.

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