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Topic: Grand canonical ensemble

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	The grand canonical ensemble (Site not responding. Last check: 2007-10-20)
		In the grand canonical ensemble, the control variables are the chemical potential
		It is therefore related to the canonical ensemble by a Legendre transformation with respect to the particle number
		Its utility lies in the fact that it closely represents the conditions under which experiments are often performed and, as we shall see, it gives direct access to the equation of state.
	www.nyu.edu /classes/tuckerman/stat.mechII/lectures/lecture_5/node1.html (62 words)

	Other Ensembles
		If the canonical ensemble is envisioned as a volume, a microcanonical ensemble may be envisioned as a sheet (a surface for the surface ensemble, or an exceedingly thin volume for the quantum microcanonical ensemble) cut out from the canonical ensemble.
		Equivalently, the canonical ensemble may be envisioned as being constructed from microcanonical ensembles the way an onion is constructed from concentric shells of onionskin, each layer of onion corresponding to those states of the canonical ensemble which have a particular value of the energy.
		The grand canonical, isobaric-isothermal, and isodynamic-polythermal ensembles differ from the canonical ensemble in the variables that are held fixed, and in the statistical weights assigned to different states.
	users.wpi.edu /~phillies/stat4/node3.html (1549 words)

	exchap07
		Show that the average quadratic deviation of the number of particles in the grand canonical ensemble may be written as
		is the average quadratic deviation of energy in the canonical ensemble.
		The grand partition function for a simplified statistical model is given by the expression
	fge.if.usp.br /~ssalinas/exchap07/index.html (405 words)

	Ensemble, nash ensemble, ensemble studio theater (Site not responding. Last check: 2007-10-20)
		Ensemble Dastan Ensemble, nash ensemble, ensemble studio theater..
		Ensemble is the next generation of the Horus group communication toolkit.
		Ensemble Ensemble 21 is one of the finest young ensembles dedicated to contemporary classical music.
	www.risk-control.org /ensemble.html (816 words)

	Statistical mechanics Summary
		This ensemble uses in its derivation the fact that temperatures are constant at equilibrium.
		To clarify, this is not a grand canonical ensemble.
		In the micro-canonical ensemble, the system exchanges no energy with the outside world, and is therefore not subject to energy fluctuations, while in the canonical ensemble, the system is free to exchange energy with the outside in the form of heat.
	www.bookrags.com /Statistical_mechanics (3157 words)

	Canonical ensemble - Wikipedia, the free encyclopedia
		A generalization of this is the grand canonical ensemble, in which the systems may share particles as well as energy.
		Mathematical treatments are given in the articles on the Potts model, where the canonical ensemble as a probability measure expressed in the language of measure theory, and quantum statistical mechanics.
		The canonical ensemble is used in statistical thermodynamics to calculate various thermodynamic quantities.
	en.wikipedia.org /wiki/Canonical_ensemble (916 words)

	Grand canonical ensemble - Wikipedia, the free encyclopedia
		In statistical mechanics, the grand canonical ensemble is a statistical ensemble, that means a set of identically prepared systems, each of which is in equilibrium with an external bath with respect to particle and energy exchange.
		The partition function of the grand canonical ensemble with a matrix/operator formalism is:
		The characteristic state function for the grand canonical ensemble is the quantity PV.
	en.wikipedia.org /wiki/Grand_canonical_ensemble (233 words)

	Grand Canonical Ensemble -- from Eric Weisstein's World of Physics
		Grand Canonical Ensemble -- from Eric Weisstein's World of Physics
		An ensemble of systems in contact with a heat reservoir for which
		In open systems such as a mixture of reacting gases, the number of particles of one type is not fixed and
	scienceworld.wolfram.com /physics/GrandCanonicalEnsemble.html (70 words)

	relativistic (grand) canonical ensemble
		Prechtl Josef wrote: > > > The canonical distribution is given by > > > > R=1/Z exp[-beta(p n)], > > > > where n is a 4-vector with n^2=1 which is the velocity of the heat bath > > relative to an inertial frame (I restrict the discussion to special > > relativity).
		In local equilibrium, the relativistic canonical ensemble has rho = 1/Z exp(-W), where W = integral dx n(x)^* J(x) w(x), integrated over a spacelike surface with timelike normal n(x) (defining the observer), J(x) is the energy-momentum tensor, w(x) is a timelike vector field, and * denotes the adjoint in the -+++ metric.
		The grand canonical ensemble has an additional term integral dx n(x)^* j_l(x) alpha_l(x) for each current j_l(x) (a vector field) deemed relevant, with associated scalar chemical potential field mu_l(x) = alpha_l(x) T(x).
	www.lns.cornell.edu /spr/2003-12/msg0057409.html (351 words)

	Lecture 1, 2: Basic thermodynamics and Kinetic theory of the ideal gas
		Canonical ensemble can describe most of the physical systems in contact with heat baths.
		in the canonical ensemble is analogous to the
		There are several ways to calculate partition function in the canonical ensemble framework.
	www.usu.edu /nanolab/phyx6410/files/web-abs-hw.htm (1775 words)

	More on Partition Function
		There are actually several different types of partition function, each corresponding to different types of statistical ensemble (or, equivalently, different types of free energy.) The canonical partition function applies to a canonical ensemble, in which the system is allowed to exchange heat with the environment at fixed temperature, volume, and number of particles.
		The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential.
		In a manner similar to the definition of the canonical partition function for the canonical ensemble, we can define a grand canonical partition function for a grand canonical ensemble, a system that can exchange both heat and particles with the environment, which has a constant temperature T, volume V, and chemical potential μ.
	www.artilifes.com /partition-function.htm (1965 words)

	4.2 Canonical ensemble
		But this means that we may calculate averages of physical quantities either in the microcanonical or in the canonical ensemble, according to mathematical convenience.
		We have derived the properties of the canonical ensemble using a Taylor expansion of the entropy.
		First of all, it is a normalizing factor in the calculation of averages over the canonical ensemble.
	homepage.univie.ac.at /Franz.Vesely/sp_english/sp/node19.html (1234 words)

	Department of Chemical Engineering ::: CALTECH
		Distribution function (derivation), evaluation of Lagrange multipliers using second law, expressions for thermodynamic functions in terms of canonical partition function, third law and entropy, remark on microcanonical ensemble, physical meaning of partition function.
		Relation of canonical partition function to molecular partition function and thermodynamic properties, monatomic, diatomic and polyatomic gases, symmetry of wave functions, ortho-para hydrogen, chemical equilibria, phase equilibria.
		Fluctuations in energy (in canonical ensemble), in magnetization (canonical ensemble), in number of particles (grand canonical ensemble and, more generally, in other properties ("thermodynamically conjugate variables"), smallness of fluctuations, and equivalence of different ensembles.
	www.che.caltech.edu /courses/ch164/syllabus.html (512 words)

	Free Energy
		An ensemble is the assembly of all possible configurations that are consistent with the constraints that we impose on the system.
		The microcanonical ensemble (NVE) is the assembly of all states for a system with fixed total energy, E, number of molecules, N, and volume, V. In the canonical ensemble (NVT) the energy can fluctuate.
		The isothermal-isobaric ensemble (NPT), with a constant number of particles, pressure, and temperature, is used to obtain the Gibbs free energy.
	www.biochem.vt.edu /modeling/deltaG.html (1445 words)

	Detailed Employment
		By ``grand canonical,'' we mean molecules diffuse microcanonically in a large periodic box but perform analysis on subsystems, so that a particle is ``created'' when it drifts into the subsystem and ``destroyed'' when it drifts out.
		The grand-canonical ensemble does not lead to important simplifications as had been hoped; however, it does lead to some long-time power-law tails in certain dynamic correlation functions, including the dynamic structure function.
		The grand canonical ensemble is appropriate in describing scattering experiments, since molecules can drift into and out of the incident beam, so these long-time tails might be experimentally observable.
	faculty.tamu-commerce.edu /hrichards/employment.html (2058 words)

	Isothermal compressibility
		(1.38), where the statistical averages are evaluated in an ensemble where the number of particles is not fixed (e.g.
		the grand canonical ensemble), which is the relevant ensemble for, e.g., light scattering which probes volumes large compared with the tipical intermolecular distances.
		If the volume integral is evaluated in the gran-canonical ensemble, the first integral is the mean value of the number of pairs in the volume
	puccini.che.pitt.edu /~gio/PhD_thesis/node15.html (162 words)

	Method and computer program product for drug discovery using weighted grand canonical metropolis Monte Carlo sampling - ...
		Making use of the properties of the grand canonical ensemble, the affinity of fragments for different regions on the protein surface can be efficiently computed, using a so-called "simulated annealing of the chemical potential" process.
		Note that in the framework of the grand canonical ensemble, the number of fragments in the system fluctuates from one system state to another.
		The fact that the simulation system is considered in the framework of the grand canonical ensemble, instead of the canonical ensemble, enables through simulated annealing of the chemical potential an efficient estimation of the free energy of binding of a given fragment type for various binding modes on the protein surface.
	www.freepatentsonline.com /20040267456.html (7223 words)

	Thermodynamics
		In the canonical ensemble, the Helmholtz free energy
		is the proper free energy of the grand canonical ensemble from thermodynamic considerations, we need to introduce a mathematical theorem, known as Euler's theorem:
		is the natural free energy in the grand canonical ensemble, and, unlike the other ensembles, it is not given a special name or symbol!
	www.nyu.edu /classes/tuckerman/stat.mechII/lectures/lecture_5/node2.html (203 words)

	4.3 Grand canonical ensemble
		From the grand partition function we can easily derive expressions for the various thermodynamic observables.
		Thus the grand ensemble is again equivalent to others ensembles of statistical mechanics.
		The states within the grand ensemble may again be sampled in a random manner.
	homepage.univie.ac.at /Franz.Vesely/sp_english/sp/node20.html (290 words)

	Grand Canonical MD (Site not responding. Last check: 2007-10-20)
		We give the ansatz Lagrangians and equiations of motion for both the isothermal and adiabatic forms of grand molecular dynamics.
		The averages calculated over the trajectories generated by these equations of motion represent the classical grand canonical ensemble (muVT) and the constant chemical potential adiabatic ensemble (muVL) averages, respectively.
		The microcanonical phase space densities of the adiabatic and isothermal forms of the molecular dynamics method are shown to be equivalent to adiabatic constant chemical potential adiabatic ensemble, and grand canonical ensemble partition functions.
	www.wag.caltech.edu /home-pages/tahir/paper_ms06.html (237 words)

	Lecture Notes
		Lecture 8 - The entropy of mixing and Gibbs' paradox; indistinguishable particles; the canonical ensemble and partition function
		Lecture 9 - The canoncial partition function and the Helmholtz free energy; the equivalence of the canonical and microcanonical ensembles; the ideal gas in the canonical ensemble
		Lecture 13 - Grand canonical ensemble for non-interacting degrees of freedom; Quantum ensembles and the density operator
	spider.pas.rochester.edu:8080 /phy418S04/lectures (566 words)

	Semigrand ensemble
		A better choice is a grand canonical simulation in which particles of different species are inserted and removed according to the configurational energy and the imposed chemical potential of that component.
		The semigrand ensemble provides an alternative representation that combines the best features of the canonical and grand canonical ensembles for the study of continuous mixtures [85,86].
		The isobaric semigrand canonical free energy Y is defined by a Legendre transform of the Gibbs free energy G.
	www.science.uva.nl /~bolhuis/ths/pages/node37.html (553 words)

	Partition function and energy fluctuation of a canonical ensemble
		But there is no such requirement on the total energy when the system is in equilibrium with a heat bath, as is the case in canonical ensembles, rather than being isolated as in the microcanonical ensembles.
		If we take air as our grand canonical ensemble we will reach a conclusion that due to the density fluctuations we have a blue sky [8].
		If there were no fluctuations in density, sunlight would not be appreciably scattered by the atmosphere; it would go in straight lines -as it does, for example, in the vacuum over the surface of the moon- and the sky would be nearly fl.
	physnet.physik.uni-oldenburg.de /~hilf/vortraege/twente99/stat12/node4.html (790 words)

	Designing Non-Hamiltonian Equations of Motion
		It will now be shown how the basic principles of non-Hamiltonian statistical mechanics can be employed to design of MD equations of motion for generating the canonical and isothermal-isobaric ensembles.
		The Nosé-Hoover chain (NHC) dynamics method [29] is a non-Hamiltonian MD scheme for generating the canonical ensemble.
		Thus, in designing a MD algorithm for the NPT ensemble, one should both incorporate the volume as a dynamical variable [24], and employ a thermostat to ensure that both instantaneous temperature and pressure fluctuations are generated properly.
	pages.nyu.edu /~mt33/jpc_feat/node5.html (735 words)

	Stacey Winter (Site not responding. Last check: 2007-10-20)
		For a single-component, Lennard-Jones thin-film, canonical ensemble Monte Carlo is shown to be insufficient for predicting disjoining pressures.
		Next, a modified Gibbs ensemble Monte Carlo simulation is applied whereby the disjoining pressure is fixed and the resulting film thickness and composition are measured.
		We isolate the thin film from disturbances using a canonical ensemble simulation of the flat portion of the film.
	www.cchem.berkeley.edu /~cjrgrp/res/stacey.html (372 words)

	Monte Carlo simulations in lipid bilayers
		The determination of canonical and microcanonical ensemble averages in homogeneous disordered systems via computer simulation is by now a well established technique, using either the probabilistic Monte Carlo method or the deterministic molecular dynamics method.
		Currently work is in progress on this system in the canonical, grand-canonical and isothermal-isobaric ensemble, with independently varying prism length and edge.
		This is in agreement with the change in simulation cell's shape that was produced by the istothermal-isobaric ensemble run: there is still "too much" space in the bilayer.
	fulcrum.physbio.mssm.edu /~mezei/proj/scri (2456 words)

	CiteULike: A Canonical Ensemble Approach to the Fermion/Boson Random Point Processes and its Applications (Site not responding. Last check: 2007-10-20)
		That is, we consider quantum statistical mechanics of canonical ensemble for a fixed number of particles which obey Bose-Einstein, Fermi-Dirac statistics, respectively, in a finite volume.
		This argument is a realization of the equivalence of ensembles, since resulting processes are considered to describe a grand canonical ensemble of points.
		Focusing on the distribution of positions of the particles, we have point processes of the fixed number of points in a bounded domain.
	www.citeulike.org /article/86792 (349 words)

	Dr. Otto Anatole von Lilienfeld-Toal
		My postdoctoral research is mostly concerned with rational compound design, chemical space, and grand-canonical ensemble theory.
		More recent work attempts to establish a molecular grand-canonical ensemble density functional theory as a physically rigorous link between compound design and chemical space [7, 8].
		I've also studied discotic materials [5], physisorption [6], and am currently expanding my expertise to the field of path-integral studies on nuclear quantum effects and organic reaction mechanisms.
	homepages.nyu.edu /~ovt203 (433 words)

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