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Topic: Equipartition theorem

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Kinetic energy

Statistical mechanics

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Ideal gas equation

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Thermodynamics

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	Equipartition of Energy
		The theorem of equipartition of energy states that molecules in thermal equilibrium have the same average energy associated with each independent degree of freedom of their motion and that the energy is
		Equipartition of energy also has implication for electromagnetic radiation when it is in equilibrium with matter, each mode of radiation having kT of energy in the Rayleigh-Jeans law.
		The average translational kinetic energy possessed by free particles given by equipartition of energy is sometimes called the thermal energy per particle.
	hyperphysics.phy-astr.gsu.edu /hbase/kinetic/eqpar.html (346 words)

	Equipartition theorem biography .ms (Site not responding. Last check: 2007-10-29)
		The Equipartition Theorem is a principle of classical (non-quantum) statistical mechanics which states that the internal energy of a system composed of a large number of particles will distribute itself evenly among each of the degrees of freedom allowed to the particles of the system.
		For example, in thermodynamics, the equipartition theorem says that the mean internal energy associated with each degree of freedom of a monatomic ideal gas is the same.
		The equipartition theorem is valid only in the classical limit of an energy continuum.
	equipartition-theorem.biography.ms (134 words)

	Thermodynamics of Electricity - Physics (Site not responding. Last check: 2007-10-29)
		The equipartition theorem would then tell us that the expected energy of the field in the region is equal to fkT/2, with f the number of degrees of freedom.
		The equipartition theorem would then tell us that > the expected energy of the field in the region is equal to fkT/2, with > f the number of degrees of freedom.
		Maxwell developed the equipartition theorem to describe the distribution of kinetic energy in a hard-sphere model of an ideal gas.
	www.okka.biz /Thermodynamics_of_Electricity-1614071-288-a.html (3517 words)

	[No title]
		This is an "essentially 4- dimensional" problem by the classification of V. Klee [Klee99], indicating that the answer is known and positive in all dimensions 3 and negative in di- mensions 5.
		Note that the proof of Theorem 5.1 is based on the "acci- dent" that the convex curve 4 and the associated solution manifold 0 are Z=2-spaces, where Z=2 is the group generated by the reflection RL.
		Equipartitions of continuous mass distributions on the sph* *ere and in the space (in Russian).
	hopf.math.purdue.edu /Zivaljevic/synergia.txt (7026 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		The classical equipartition theorem assigns a total > energy (note not just kinetic energy) of kT/2 per each term in the system's > Hamiltonian (momentum or position) that is quadratic.
		The equipartition theorem says that there will be kT/2 of (kinetic) energy for each degree of freedom whose momentum enters quadratically into the Hamiltonian, and kT/2 of (potential) energy for each degree of freedom whose coordinate enters quadratically into the Hamiltonian.
		As far as I know, the equipartition theorem is used in MD only for the momenta, not for the coordinates.
	www.ccl.net /cgi-bin/ccl/message.cgi?1996+11+06+010+raw (315 words)

	PHYSICS 323/324 EM1
		It also introduces the Maxwell-Boltzmann distribution, the equipartition theorem as applied to specific heats, and the exclusion principle and complex atoms, all of which have application to the theory of solids.
		, and the equipartition theorem thus predicts the average energy to be (3/2)k
		The equipartition theorem is reviewed on page 14 of Eisberg and Resnick and its application to solids is discussed on pages 421 and 422.
	www.physics.rutgers.edu /ugrad/323/SS.html (976 words)

	BLACK-BODY PROBLEMATIC: GENESIS AND STRUCTURE
		It must be emphasized that while his derivation of the irreversible increase of entropy made use of the statistical distribution of molecular velocities and energies, the result he asserted was supposed to be a certainty and not a probability.
		Loschmidt concluded that the H-Theorem could not be a deterministic theorem because there were some initial conditions from which H could for a time increase and entropy decrease.
		Applying a mathematical theorem published by Poincare five years ago, Zermelo argued that any mechanical system confined in a finite region of space would after a sufficiently long time ultimately return to its initial configuration.
	theoryandscience.icaap.org /content/vol003.001/roy.html (5927 words)

	Ideal gases (Site not responding. Last check: 2007-10-29)
		Quantitatively speaking, the equipartition theorem says that the translational kinetic energy of a typical particle is (3/2)kT.
		Yet again, the equipartition theorem says that the translational kinetic energy of a typical particle is (3/2)kT.
		According to the equipartition theorem, the total kinetic energy of a typical particle is (d/2)kT, where d is the total number of degrees of freedom.
	socrates.berkeley.edu /~phy7b/solutions/T1soln.html (1712 words)

	Equipartition theorem - TheBestLinks.com - Harmonic oscillator, Joule, Kelvin, Kinetic energy, ...
		Equipartition theorem - TheBestLinks.com - Harmonic oscillator, Joule, Kelvin, Kinetic energy,...
		Equipartition theorem, Harmonic oscillator, Joule, Kelvin, Kinetic energy...
		For example, in thermodynamics, the equipartition theorem says that the mean internal energy associated with each degree of freedom of a monoatomic ideal gas is the same.
	www.thebestlinks.com /Equipartition_theorem.html (207 words)

	Heat capacity - Wikipedia, the free encyclopedia
		According to the equipartition theorem from classical statistical mechanics, for a system made up of independent and quadratic degrees of freedom, any input of energy into a closed system composed of N molecules is evenly divided among the degrees of freedom available to each molecule.
		For matter in a crystalline solid phase, the Dulong-Petit law states that the dimensionless specific heat capacity assumes the value 3.
		The Dulong-Petit law is however based on the equipartition theorem, and as such is only valid in the classical limit of a microstate continuum, which is a high temperature limit.
	en.wikipedia.org /wiki/Heat_capacity (1513 words)

	Reading Quiz #24 (Site not responding. Last check: 2007-10-29)
		The equipartition theorem fails for quantum mechanical systems at low temperature because most of the energy states at low temperatures will be too small to appear on the graph.
		This equipartition theorem uses a Gaussian curve to approximate a bar graph.
		At lower energy levels the spacing between adjacent energies is much less than kT, and the equipartition theorem works when many distinct states contribute and in the high temperature, the spacing is unimportant.
	www.eg.bucknell.edu /physics/PHYS317/rQ/rq24.html (1367 words)

	The Role of the Asymptotic Equipartition Property in Noiseless Source Coding (ResearchIndex) (Site not responding. Last check: 2007-10-29)
		Abstract: The (noiseless) fixed-length source coding theorem states that, except for outcomes in a set of vanishing probability, a source can be encoded at its entropy but not more efficiently.
		It is well known that the Asymptotic Equipartition Property (AEP) is a sufficient condition for a source to be encodable at its entropy.
		This paper shows that the AEP is necessary for the source coding theorem to hold for nonzero-entropy finite-alphabet sources.
	citeseer.ist.psu.edu /verdu97role.html (326 words)

	[No title]
		For this purpose we proof a necessary and sufficient criterion for equipartition of energy and use this criterion to show equipartition for a system of partial differential equations with a coupled boundary condition.
		In the known results on equipartition the operator $A$ is assumed to be decomposable in operators with special properties, e.g.
		We have energy equipartition in the Cesaro sense for $A$ iff $\alpha=\beta$ or $\alpha=-1/\beta$.
	www.mat.ub.es /EMIS/journals/EJDE/Volumes/2000/70/boller-tex (2069 words)

	Ultraviolet catastrophe (Site not responding. Last check: 2007-10-29)
		The ultraviolet catastrophe results from the equipartition theorem of classical statistical mechanics which states that all modes (degrees of freedom) of a system at equilibrium have an average energy of
		Planck had postulated that electromagnetic energy did not follow the classical description, but could only oscillate or be emitted in discrete packets of energy proportional to the frequency (as given by Planck's law).
		In fact Planck never concerned himself with this aspect of the problem, because he did not believe that the equipartition theorem was fundamental - his motivation for introducing "quanta" was entirely different.
	www.worldhistory.com /wiki/U/Ultraviolet-catastrophe.htm (643 words)

	Harmonic Potential (Site not responding. Last check: 2007-10-29)
		As the temperature is increased the system becomes increasingly classical and hence the relationship between the total excitation energy and the temperature becomes linear, as predicted by the equipartition theorem (Eq.
		As the temperature is reduced to zero the quantum discretisation effects become increasingly important and the excitation energy becomes constant, due to the presence of zero point motion.
		Figure 6.4 shows a graph comparing the estimates of the expectation of the total energy as obtained by simulation with the theoretic values as predicted by classical and quantum statistical mechanics.
	www.ph.ed.ac.uk /~arjun/arjun_thesis/node96.html (466 words)

	lesson 1 (Site not responding. Last check: 2007-10-29)
		Aim: State the results of the equipartition theorem and when is it valid.
		Section 6.4 derives the equipartition theorem from the microcanonical ensemble.
		-> It turns out that the equipartition theorem is only valid for continuum energy spectrum (this will be clearer later when we derive it from the canonical ensemble).
	www2.hawaii.edu /~plam/ph730/lesson2.html (202 words)

	b_posets
		This is a kind of minimax theorem, in that it shows that the maximum size of an antichain in
		1.5 Corollary (P. Hall's theorem on distinct representatives) The condition (*) is a necessary and sufficient condition for the existence of an SDR.
		We say that a partition is an equipartition if all its blocks have the same cardinality.
	www.math.ucla.edu /~baker/222a.1.03w/handouts/b_posets/node3.html (181 words)

	Scientific Publications (Site not responding. Last check: 2007-10-29)
		We study the canonical transformations of generalised quantum dynamics, and show that C is a canonical invariant, as is the operator phase space volume element.
		The latter result is a generalisation of Liouville’s theorem, and permits the application of statistical mechanical methods to determine the canonical ensemble governing the equilibrium distribution of operator initial values.
		A structure theorem concerning projective quaternionic representations is stated and proved.
	www.crimron.net /papers.html (3301 words)

	Physics at Central Michigan University (Site not responding. Last check: 2007-10-29)
		4:00 PM A discussion and reexamination of the fundamental notions and concepts of thermodynamics and dynamics, such as temperature, entropy, specific heat, degrees of freedom, and equipartition theorem, as applied to finite systems will be presented.
		A new equipartition theorem will be formulated and the notion of dynamical degrees of freedom (distinct from the traditional, kinematical degrees of freedom) will be introduced.
		Illustrations of the use of the dynamical degrees of freedom as a powerful tool of dynamical analysis will be given.
	www.phy.cmich.edu /seminars/fall03/j-julius.htm (148 words)

	The equipartition theorem
		This is because the kinetic energy is usually a quadratic function of each momentum component, whereas the potential energy does not involve the momenta at all.
		This is the famous equipartition theorem of classical physics.
		If all terms in the energy are quadratic then the mean energy is spread equally over all degrees of freedom (hence the name ``equipartition'').
	farside.ph.utexas.edu /teaching/sm1/lectures/node67.html (282 words)

	Potential and kinetic energy (Site not responding. Last check: 2007-10-29)
		The most common is through the equipartition theorem, which states
		One way to realize this sample, particularly for molecular dynamics simulations, is to constrain the kinetic energy such that the equipartition relation above is satisfied for the set temperature.
		This module enforces this constraint in a very crude way---at every time step all velocities are scaled up or down by a common factor appropriate to enforce the constraint.
	www.ccr.buffalo.edu /etomica/app/modules/sites/Ljmd/Background4.html (400 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations
		The relationship between statistical ensembles (especially microcanonical ensemble) and dynamics, the equipartition theorem, and the notion of dynamical temperature are reexamined with an emphasis on finite size effects.
		A (dynamical) equipartition ansatz (postulate) is formulated and the notion of dynamical degrees of freedom is introduced.
		The utility of the dynamical degrees of freedom as an analysis tool is discussed and illustrated in applications to model aluminum clusters.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=20217183 (198 words)

	[No title]
		We used the Equipartition Theorem extensively in the early portions of this course to get a feeling for the concept of temperature and to help us understand thermal conductivity processes.
		b) Second, explain, on the basis of the Equipartition theorem, why energy flows from high temperature to low temperature when two blocks, at initially different temperatures, are placed in direct contact.
		Also explain, on the same basis, why the energy flow stops when the two blocks are at the same temperature.
	www-personal.umd.umich.edu /~devlin/phys406/w98examf.htm (1014 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		In so doing, he made clear that the second law is essentially statistical and that a system approaches a state of thermodynamic equilibrium (uniform energy distribution throughout) because equilibrium is overwhelmingly the most probable state of a material system.
		During these investigations Boltzmann worked out the general law for the distribution of energy among the various parts of a system at a specific temperature and derived the theorem of equipartition of energy (Maxwell-Boltzmann distribution law).
		This law states that the average amount of energy involved in each different direction of motion of an atom is the same.
	www.phy.bg.ac.yu /web_projects/giants/boltzmann.html (309 words)

	Physlet Physics by Christian and Belloni: Exploration 20.1
		This, divided by the number of particles, should be the same as the temperature of the system.
		This is the equipartition of energy theorem: The internal energy of a gas (the sum of the energy of all particles) is equal to (f/2)k
		NT, where f is the number of degrees of freedom for the atoms or molecules in a gas.
	webphysics.davidson.edu /physlet_resources/thermo_paper/thermo/examples/ex20_1.html (635 words)

	September 27 Assignment (Site not responding. Last check: 2007-10-29)
		Use the equipartition theorem to compute the average thermal energy of a methane molecule at 300 K. Do the same for He.
		Under conditions of constant volume, the heat capacity of a substance is defined as the derivative of energy with respect to temperature (yes, that's right, he's talking calculus): dU/dT.
		Use the equipartition theorem to compute the molar heat capacities (heat capacity per mole) of methane and helium.
	cas.bellarmine.edu /chem216/fall99/homework/sep27.htm (324 words)

	ACM Sigplan Notices 29, 4 (Apr 1994), 58-63.
		Under suitable conditions, the averaging effects and the central limit theorem allow nearly certain predictions of macroscopic properties and processes which are independent of the microscopic details.
		Liouville's Theorem is thus summarized by the assertion "phase space is incompressible", since the "density" of microstates in phase space cannot change over time.
		The energy "equipartition theorem" is then a tautology--every degree of freedom has the same average energy.
	www.pipeline.com /%7Ehbaker1/ThermoGC.html (7273 words)

	Equipartition Theorem (Site not responding. Last check: 2007-10-29)
		It didn't take much faith to believe the Equipartition Theorem of thermal energy between the identical (Cartesian) translations of the equilibrium Ideal Gas of the last lecture.
		It took a great deal more faith at the turn of the century to repair the theorem when it gave nonsensical answers to the equilibrium energy of a radiation field interacting with matter.
		But even untouched, the string is in thermal equilibrium; by the Equipartition Theorem there's kT (it's one harmonic oscillator not Avogadro's Number of them) each in the fundamental, the 1st overtone, the 2nd, the nth, etc. Mathematically, there's no end to the number of overtones; so there are infinitely many kT additions.
	www.utdallas.edu /~parr/chm5414/equipart.html (441 words)

	Southampton Physics and Astronomy PHYS1013 Course Diary
		to know the equipartition theorem and to be able to apply it to derive heat capacities
		Equipartition therem and specific heats of gases; failure of the equipartition theorem.
		Second law of thermodynamics: Kelvin-Planck and Clausius statements and their equivalence; heat engines, heat pumps and refrigerators; efficiencies and coefficients of performance; reversible engines, Carnot's theorem.
	www.hep.phys.soton.ac.uk /courses/phys1013/diary.html (1370 words)

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