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Topic: Canonical ensemble

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Statistical ensemble

Microcanonical ensemble

Derivation of the partition function

Statistical mechanics

Quantum statistical mechanics

Boltzmann factor

Physics

Derivative

World War II

	Statistical ensemble (mathematical physics) - Wikipedia, the free encyclopedia
		Willard Gibbs in 1878, an ensemble (also statistical ensemble or thermodynamic ensemble) is an idealization consisting of a large number of mental copies (possibly infinitely many) of a system, considered all at once, each of which represents a possible state that the real system might be in.
		Canonical ensemble -- an ensemble of systems, each of which can share its energy with a large heat reservoir or heat bath (in effect, fixing the temperature).
		Grand canonical ensemble -- an ensemble of systems, each of which can share both its energy and its particles with a reservoir (ie an open system, at a given temperature).
	en.wikipedia.org /wiki/Statistical_ensemble (1122 words)

	Canonical ensemble - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-29)
		A generalization of this is the grand canonical ensemble, in which the systems may share particles as well as energy.
		A full development of the concept of the canonical ensemble is given in the article on the partition function.
		An example of the mathematical formulation of the canonical ensemble as a probability measure expressed in the language of measure theory is given in the article on the Potts model.
	en.wikipedia.org /wiki/Canonical_ensemble (181 words)

	The canonical ensemble
		An ensemble is specified by fixing the list of allowed states, and the statistical weight for each state.
		This book relies primarily on the canonical ensemble, in which the number of particles N, system volume V, and temperature T are constraints.
		3.1 and the ensemble averaging concept cannot be derived (at least as of this writing) from the mechanical laws of motion, either Hamilton's or Schrodinger's.
	users.wpi.edu /~phillies/stat4/node2.html (519 words)

	P from a Canonical Ensemble Average
		While the atoms are moving, a correct representation of a single element of the ensemble shows atoms at a single instant in time, atoms at each instant having given, specified values of position and momentum.
		The ensemble average gives an average behavior, values of P in different elements of the ensemble being spread around the average value.
		Representative elements of the canonical ensemble for the N-atom gas.
	users.wpi.edu /~phillies/stat7/node1.html (812 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.
	www.ap.univie.ac.at /users/ves/sp_english/sp/node19.html (1234 words)

	lesson 6 (Site not responding. Last check: 2007-10-29)
		As long as the density fluctuation is small, the result derived with the grand canonical ensemble is the same as that of the canonical ensemble.
		The derivation is simlar to that for the canonical ensemble, see HW#3, problem 1.
		Grand partition function: As in the canonical ensemble case, various thermodynamic quanitities can be obtained from the derivatives of grand partition function.
	www2.hawaii.edu /~plam/ph730/lesson6.html (227 words)

	Partition function (statistical mechanics) (Site not responding. Last check: 2007-10-29)
		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.
	worldhistory.com /wiki/P/Partition-function-(statistical-mechanics).htm (1714 words)

	lesson 4 (Site not responding. Last check: 2007-10-29)
		Aim: Treat a system in contact with a heat bath (canonical ensemble).
		Note: Microcanonical Ensemble is a restricted counting problem (fixed energy) while the canonical ensemble is an un-restricted counting problem and hence is much easier.
		Classsical canonical ensemble method is even easier than than the quantum case because you need only to do an un-restricted integral over phase space.
	www2.hawaii.edu /~plam/ph730/lesson4.html (294 words)

	5.1 Definition of the System
		One of the most useful is the canonical ensemble.
		And to have all of our ensemble members at the same temperature we need walls that are diathermal, that is, walls that allow thermal energy to pass through them.
		Indeed, we expect that the energy of any particular ensemble member will change from time to time as energy enters or leaves the system.
	scholar.chem.nyu.edu /2600/classnotes/node66.html (237 words)

	PHS702 Statistical Thermodynamics, Lecture 4
		The term in the denominator is the grand canonical partition function.
		A grand canonical ensemble can be considered as a collection of canonical ensembles in thermal equilibrium each other and with all possible values of N.
		It is matter of convinience in the choice of ensemble one uses.
	phoenix.liu.edu /~nmatsuna/PHS702/statmech/lect4/stat.mech.4.html (506 words)

	Citebase - Bose-Einstein condensation in the canonical ensemble
		Authors: Bedingham, D. Large-volume, high-temperature Bose-Einstein condensation is illustrated for a relativistic O(2)-invariant scalar field with fixed charge using the canonical ensemble.
		The standard, grand canonical results are reproduced for the infinite-volume limit.
		Finite-volume corrections are calculated in the canonical ensemble and the results are found to differ from the finite-volume grand canonical approximation in a consistent qualitative way.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:hep-ph/0308067 (731 words)

	Advanced molecular dynamics parameters
		Users can choose between the micro-canonical (constant electronic energy) and the canonical (constant temperature) ensembles.
		The micro-canonical ensemble is integrated with the velocity Verlet algorithm.
		The canonical ensemble via Nose'-Hoover dynamics is integrated with a generalized leap frog method (J. Comp.
	www.nersc.gov /projects/paratec/DOC/node5.html (191 words)

	Citebase - Thermodynamics of charged anti-de Sitter black holes in canonical ensemble
		Authors: Mitra, P. As in the grand canonical treatment of Reissner - Nordstr\"{o}m fl holes in anti-de Sitter spacetime, the canonical ensemble formulation also shows that non-extremal fl holes tend to have lower action than extremal ones.
		The Euclidean action with boundary conditions appropriate to the canonical ensemble is shown to lead to zero entropy.
		The thermodynamical properties of the Reissner-Nordström-anti-de Sitter fl hole in the grand canonical ensemble are investigated using York's formalism.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:gr-qc/9903078 (984 words)

	Canonical ensemble treatment (Site not responding. Last check: 2007-10-29)
		The canonical partition function for the ideal gas is much easier to evaluate than the microcanonical partition function.
		Recall the expression for the canonical partition function Q(N,V,T):
		Thus, the canonical and microcanonical ensembles gives rise to exactly the same thermodynamics!!
	www.nyu.edu /classes/tuckerman/stat.mech/lectures/lecture_6/node4.html (110 words)

	Hybrid Monte Carlo with Adaptive Temperature in a Mixed-Canonical Ensemble: Efficient Conformational Analysis of RNA - ...
		Abstract: A hybrid Monte Carlo method with adaptive temperature choice is presented, which exactly generates the distribution of a mixed-canonical ensemble composed of two canonical ensembles at low and high temperature.
		The analysis of resulting Markov chains with the reweighting technique shows an efficient sampling of the canonical distribution at low temperature, whereas the high temperature component facilitates conformational transitions, which allows shorter simulation times.
		Hybrid Monte Carlo with adaptive temperature in a mixed{canonical ensemble: Ecient conformational analysis of RNA.
	citeseer.ist.psu.edu /fischer97hybrid.html (555 words)

	Monte Carlo Simulation of the canonical ensemble
		In this work, we will specialize on the simulation of Argon in the canonical ensemble.
		In this ensemble, the natural variables are the number of particles, N, the equilibrium temperature, T, and a fixed volume V.
		Further details on the canonical ensemble are found on textbooks [
	alan.aspuru.com /mcargon/node2.html (107 words)

	5.2 The Construction of the Ensemble
		We will want to compute the number of ensemble members in any particular energy state (remembering that there can be a spread of energies among the various systems in the ensemble.) As before the number of ensemble members in state j will be a
		First, we want to assume that all microstates are equally likely, subject to the constraint on the number of ensemble members (Equation 5.2.1 and to the constraint that the temperature is constant.
		where the first term is clearly the total energy of the ensemble members and the second the energy of the bath.
	scholar.chem.nyu.edu /2600/classnotes/node67.html (515 words)

	Canonical and Microcanonical Ensembles (Site not responding. Last check: 2007-10-29)
		(36) and (11) to show that in a canonical ensemble the probability
		(B) Show that for a microcanonical ensemble, where all of the states
		(23) and (45), the entropy of a microcanonical ensemble corresponds to the Boltzmann definition of entropy,
	xbeams.chem.yale.edu /~batista/vaa/node12.html (49 words)

	Energy fluctuations in the canonical ensemble (Site not responding. Last check: 2007-10-29)
		In the canonical ensemble, the total energy is not conserved.
		The energy fluctuations are given by the root mean square deviation of the Hamiltonian from its average
		Thus, in the thermodynamic limit, the canonical and microcanonical ensembles are equivalent, since the energy fluctuations become vanishingly small.
	www.nyu.edu /classes/tuckerman/stat.mech/lectures/lecture_5/node2.html (92 words)

	Complexity Digest - On the Derivation of Power-Law Distributions in the Canonical Ensemble
		On the Derivation of Power-Law Distributions in the Canonical Ensemble, SFI Working Papers
		Abstract: We note the possibility within classical thermodynamics to derive power-law distributions in the canonical ensemble.
		We do so by systematically finding a class of microcanonical distributions that allow for a separation of the total system energy from the reservoir.
	www.comdig.org /article.php?id_article=16519 (177 words)

	Grand canonical ensemble Monte Carlo simulation of the dCpG/proflavine crystal hydrate -- Resat and Mezei 71 (3): 1179 ...
		Grand canonical ensemble Monte Carlo simulation of the dCpG/proflavine crystal hydrate -- Resat and Mezei 71 (3): 1179 -- Biophysical Journal
		Grand canonical ensemble Monte Carlo simulation of the dCpG/proflavine crystal hydrate
		The grand canonical ensemble Monte Carlo molecular simulation method is
	www.biophysj.org /cgi/content/abstract/71/3/1179?maxtoshow= (225 words)

	Citebase - Dilaton black holes in grand canonical ensemble near the extreme state
		Dilaton fl holes in grand canonical ensemble near the extreme state
		Authors: Zaslavskii, O. Dilaton fl holes with a pure electric charge are considered in a framework of a grand canonical ensemble near the extreme state.
		It is shown that there exists such a subset of boundary data that the Hawking temperature smoothly goes to zero to an infinite value of a horizon radius but the horizon area and entropy are finite and differ from zero.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:gr-qc/9707019 (1116 words)

	Canonical Ensemble (ResearchIndex) (Site not responding. Last check: 2007-10-29)
		We have seen that we can use the microcanonical ensemble to calculate thermodynamic quantities from microscopic, mechanical variables such as the energy of individual components of a system.
		The microcanonical ensemble also requires the careful counting of all possible microstates of a system, and it is useful when the energy E, volume V, and numbers of "particles" N in a system are each fixed.
		0.1: Brownian Motion with Restoring Drift: Micro-canonical Ensemble and..
	citeseer.ist.psu.edu /273643.html (197 words)

	PHS702 Statistical Thermodynamics, Lecture 3
		The canonical ensemble - N, V, T fixed.
		The quantity in the denominator persists in the equations in the canonical ensemble.
		We have seen again, increasing the number of available system gives arise to spontaneous process in closed system.
	phoenix.liu.edu /~nmatsuna/PHS702/statmech/lect3/stat.mech.3.html (424 words)

	Canonical Ensemble - Eduseek (Site not responding. Last check: 2007-10-29)
		Subjects > Science > Science - 16+ > Physics - Physical Processes > Thermodynamics > Thermodynamic Ensembles > Canonical Ensemble
		The Canonical Ensemble - Detailed information about the Canonical Ensemble
		Canonical Ensemble - A thorough definition of canonical ensemble
	www.eduseek.com /static/navigate2364.html (72 words)

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