Statistical ensembles (microcanonical and canonical ensembles, equivalence of ensembles)

Statistical mechanics

10.1093/oso/9780198803195.003.0002 ◽

2017 ◽

Author(s):

Michael P. Allen ◽

Dominic J. Tildesley

Keyword(s):

Statistical Mechanics ◽

Potential Model ◽

Quantum Corrections ◽

Inhomogeneous Systems ◽

Hard Constraints ◽

Complex Liquids ◽

Fluid Membranes ◽

Dynamical Properties ◽

Canonical Ensembles ◽

Grand Canonical

This chapter contains the essential statistical mechanics required to understand the inner workings of, and interpretation of results from, computer simulations. The microcanonical, canonical, isothermal–isobaric, semigrand and grand canonical ensembles are defined. Thermodynamic, structural, and dynamical properties of simple and complex liquids are related to appropriate functions of molecular positions and velocities. A number of important thermodynamic properties are defined in terms of fluctuations in these ensembles. The effect of the inclusion of hard constraints in the underlying potential model on the calculated properties is considered, and the addition of long-range and quantum corrections to classical simulations is presented. The extension of statistical mechanics to describe inhomogeneous systems such as the planar gas–liquid interface, fluid membranes, and liquid crystals, and its application in the simulation of these systems, are discussed.

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Hamilton-Jacobi Theory

10.1093/oso/9780198822370.003.0019 ◽

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.

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Microcanonical and Canonical Ensembles for fMRI Brain Networks in Alzheimer’s Disease

Entropy ◽

10.3390/e23020216 ◽

2021 ◽

Vol 23 (2) ◽

pp. 216 ◽

Cited By ~ 1

Author(s):

Jianjia Wang ◽

Xichen Wu ◽

Mingrui Li ◽

Hui Wu ◽

Edwin Hancock

Keyword(s):

Alzheimer’S Disease ◽

Alzheimer's Disease ◽

Ensemble Methods ◽

Brain Regions ◽

Fmri Data ◽

Thermal Systems ◽

Network Construction ◽

Network Characteristics ◽

Network Entropy ◽

Canonical Ensembles

This paper seeks to advance the state-of-the-art in analysing fMRI data to detect onset of Alzheimer’s disease and identify stages in the disease progression. We employ methods of network neuroscience to represent correlation across fMRI data arrays, and introduce novel techniques for network construction and analysis. In network construction, we vary thresholds in establishing BOLD time series correlation between nodes, yielding variations in topological and other network characteristics. For network analysis, we employ methods developed for modelling statistical ensembles of virtual particles in thermal systems. The microcanonical ensemble and the canonical ensemble are analogous to two different fMRI network representations. In the former case, there is zero variance in the number of edges in each network, while in the latter case the set of networks have a variance in the number of edges. Ensemble methods describe the macroscopic properties of a network by considering the underlying microscopic characterisations which are in turn closely related to the degree configuration and network entropy. When applied to fMRI data in populations of Alzheimer’s patients and controls, our methods demonstrated levels of sensitivity adequate for clinical purposes in both identifying brain regions undergoing pathological changes and in revealing the dynamics of such changes.

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NMR-Based Study of the Pore Types’ Contribution to the Elastic Response of the Reservoir Rock

Energies ◽

10.3390/en14051513 ◽

2021 ◽

Vol 14 (5) ◽

pp. 1513 ◽

Cited By ~ 1

Author(s):

Naser Golsanami ◽

Xuepeng Zhang ◽

Weichao Yan ◽

Linjun Yu ◽

Huaimin Dong ◽

...

Keyword(s):

Elastic Wave ◽

Wave Velocity ◽

Transverse Relaxation Time ◽

Seismic Attributes ◽

Elastic Wave Velocity ◽

Reservoir Rock ◽

Elastic Response ◽

Hydrocarbon Reservoir ◽

Statistical Ensembles ◽

Pore Types

Seismic data and nuclear magnetic resonance (NMR) data are two of the highly trustable kinds of information in hydrocarbon reservoir engineering. Reservoir fluids influence the elastic wave velocity and also determine the NMR response of the reservoir. The current study investigates different pore types, i.e., micro, meso, and macropores’ contribution to the elastic wave velocity using the laboratory NMR and elastic experiments on coal core samples under different fluid saturations. Once a meaningful relationship was observed in the lab, the idea was applied in the field scale and the NMR transverse relaxation time (T2) curves were synthesized artificially. This task was done by dividing the area under the T2 curve into eight porosity bins and estimating each bin’s value from the seismic attributes using neural networks (NN). Moreover, the functionality of two statistical ensembles, i.e., Bag and LSBoost, was investigated as an alternative tool to conventional estimation techniques of the petrophysical characteristics; and the results were compared with those from a deep learning network. Herein, NMR permeability was used as the estimation target and porosity was used as a benchmark to assess the reliability of the models. The final results indicated that by using the incremental porosity under the T2 curve, this curve could be synthesized using the seismic attributes. The results also proved the functionality of the selected statistical ensembles as reliable tools in the petrophysical characterization of the hydrocarbon reservoirs.

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