The System in Thermal Contact with a Reservoir: The Canonical and Grand Canonical Ensembles

2010 ◽  
pp. 22-43
Keyword(s):  
Thermal Contact ◽  
Canonical Ensembles ◽  
Grand Canonical

Statistical mechanics

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.

Coherent alloy phase separation: Differences in canonical and grand canonical ensembles

1997 ◽  
Vol 55 (21) ◽  
pp. 14222-14229 ◽  
Author(s):  
E. M. Vandeworp ◽  
Kathie E. Newman
Keyword(s):  
Phase Separation ◽  
Canonical Ensembles ◽  
Grand Canonical

Canonical and Grand–Canonical Ensembles for Trapped Bose Gases

2010 ◽  
pp. 3-14
Author(s):  
Abel Camacho ◽  
Alfredo Macías ◽  
Abel Camacho–Galván
Keyword(s):  
Bose Gases ◽  
Canonical Ensembles ◽  
Grand Canonical

scholarly journals GROSS–PITAEVSKII MODEL FOR NONZERO TEMPERATURE BOSE–EINSTEIN CONDENSATES

2006 ◽  
Vol 16 (09) ◽  
pp. 2713-2719 ◽  
Author(s):  
KESTUTIS STALIUNAS
Keyword(s):  
Power Spectra ◽  
Pitaevskii Equation ◽  
Nonzero Temperature ◽  
Ginzburg Landau ◽  
Long Wavelength ◽  
Occupation Numbers ◽  
Ginzburg Landau Equations ◽  
Bose Einstein ◽  
Canonical Ensembles ◽  
Grand Canonical

Momentum distributions and temporal power spectra of nonzero temperature Bose–Einstein condensates are calculated using a Gross–Pitaevskii model. The distributions are obtained for micro-canonical ensembles (conservative Gross–Pitaevskii equation) and for grand-canonical ensembles (Gross–Pitaevskii equation with fluctuations and dissipation terms). Use is made of equivalence between statistics of the solutions of conservative Gross–Pitaevskii and dissipative complex Ginzburg–Landau equations. In all cases the occupation numbers of modes follow a 〈Nk〉 ∝ k-2 dependence, which corresponds in the long wavelength limit (k → 0) to Bose–Einstein distributions. The temporal power spectra are of 1/fα form, where: α = 2 - D/2 with D the dimension of space.

scholarly journals Thermodynamic properties of rod-like chains: Entropic sampling simulations

2016 ◽  
Vol 30 (31) ◽  
pp. 1650378 ◽  
Author(s):  
L. S. Ferreira ◽  
L. N. Jorge ◽  
A. A. Caparica ◽  
D. A. Nascimento ◽  
Minos A. Neto ◽  
...  
Keyword(s):  
Thermodynamic Properties ◽  
Exact Solutions ◽  
Single Molecule ◽  
Energy Formula ◽  
Three State Model ◽  
Entropic Sampling ◽  
Null State ◽  
Canonical Ensembles ◽  
State 1 ◽  
Grand Canonical

In this work, we apply entropic sampling simulations to a three-state model which has exact solutions in the microcanonical and grand-canonical ensembles. We consider N chains placed on an unidimensional lattice, such that each site may assume one of the three states: empty (state 1), with a single molecule energetically null (state 2), and with a single molecule with energy [Formula: see text] (state 3). Each molecule, which we will treat here as dimers, consists of two monomers connected one to each other by a rod. The thermodynamic properties such as internal energy, densities of dimers and specific heat were obtained as functions of temperature, where the analytic results in the microcanonical and grand-canonical ensembles were successfully confirmed by the entropic sampling simulations.

The perfect Bose-Einstein gas in the theory of the quantum-mechanical grand canonical ensembles

1952 ◽  
Vol 212 (1111) ◽  
pp. 552-558 ◽  
Keyword(s):  
Equation Of State ◽  
Energy Levels ◽  
Einstein Condensation ◽  
Quantum Mechanical ◽  
Condensation Phenomenon ◽  
Bose Einstein ◽  
Canonical Ensembles ◽  
Grand Canonical ◽  
Different Parts ◽  
Number Of Particles

The theory of quantum-mechanical grand canonical ensembles is used to derive for the case of a perfect Bose-Einstein gas the average number of particles in the different energy levels, the fluctuations in these numbers and the equation of state. The Einstein condensation phenomenon is then discussed, and it is shown that in a p-v diagram (v being the specific volume) the isotherm consists of two analytically different parts in the limit where the number of particles in the system, JV, goes to infinity. It is also shown that for finite N at the critical volume ∂ n p/∂v n is of the order N1/3 (n-2) in accordance with a result obtained by Wergeland & Hove-Storhoug.

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