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Topic: Partition function

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Partition function (number theory)

Number theory

Quantum field theory

Fluctuation theorem

Srinivasa Ramanujan

Thermodynamics

Third law of thermodynamics

Pressure

	Partition function (statistical mechanics) - Wikipedia, the free encyclopedia
		In statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium.
		There are actually several different types of partition functions, 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.
		An specific example of the partition function, expressed in terms of the mathematical formalism of measure theory, is presented in the article on the Potts model.
	en.wikipedia.org /wiki/Partition_function_(statistical_mechanics) (1596 words)

	Partition function (number theory) - Open Encyclopedia (Site not responding. Last check: 2007-11-06)
		The partition function p(n) represents the number of possible partitions of a natural number n, which is to say the number of distinct (and order independent) ways of representing n as a sum of natural numbers.
		More generally, the generating function for the partitions of n into numbers from a set A can be found by taking only those terms in the product where k is an element of A This result is due to Euler
		The formulation of the generating function is similar to the product formulation of many modular forms, giving some idea of the connection between the two.
	www.open-encyclopedia.com /Partition_function_%28number_theory%29 (433 words)

	Statistical mechanics - Wikipedia, the free encyclopedia
		The partition function is a measure of the number of states accessible to the system at a given temperature.
		See derivation of the partition function for a proof of Boltzmann's factor and the form of the partition function from first principles.
		The partition function can be used to find the expected (average) value of any microscopic property of the system, which can then be related to macroscopic variables.
	www.wikipedia.org /wiki/Statistical_mechanics (1174 words)

	Encyclopedia: Statistical mechanics
		The entropy of a macroscopic state is proportional to the logarithm of the number of microscopic states corresponding to it.
		The probabilities of the various microstates must add to one, and the normalization factor is the partition function: A canonical ensemble in statistical mechanics is an ensemble of dynamically similar systems, each of which can share its energy with a large heat reservoir, or heat bath.
		In thermodynamics the Gibbs free energy is a state function of any system defined as G = H − T·S where G is the Gibbs free energy, measured in joules H is the enthalpy, measured in joules T is the temperature, measured in kelvins S is the entropy, measured in joules...
	www.nationmaster.com /encyclopedia/Statistical-mechanics (2411 words)

	Partition function - Wikipedia, the free encyclopedia
		In number theory, see partition function (number theory).
		In statistical mechanics, see partition function (statistical mechanics).
		In quantum field theory, see partition function (quantum field theory).
	www.wikipedia.org /wiki/Partition_function (91 words)

	Partition function (number theory) (Site not responding. Last check: 2007-11-06)
		The partition function p(n) is a non-multiplicative function andrepresents the number of possible partitions of a natural number n,which is to say the number of distinct (and order independent) ways of representing n as a sum of natural numbers.
		The number of partitions meeting the second condition is p(k+1,n) since a partition into parts of at leastk which contains no parts of at exactly k must have all parts at least k+1.
		More generally, the generating function for the partitions of n intonumbers from a set A can be found by taking only those terms in the product where k is an element of A Thisresult is due to Euler
	www.therfcc.org /partition-function-number-theory--151116.html (430 words)

	TC-Tutorial: partition function
		In the definition of the partition function, however, energies are part of the exponent; therefore, the individual partition functions have to be multiplied:
		The translational partition function does not only depend on quantities of the partical under scrutiny, it also depends on the volume.
		The above partition function therefore has to be divided by a factor of 2 for homonuclear molecules, since only each second energy level has to be considered.
	www.ptc.tugraz.at /quanten/tczustandssummeE.html (463 words)

	Partition function (statistical mechanics) (Site not responding. Last check: 2007-11-06)
		It depends on thephysical system under consideration and is a function of temperature as wellas other parameters (such as volume enclosing a gas etc.).
		Qualitatively, Z grows when the temperature rises, because then the exponential weights increase for states of larger energy.Roughly, Z is a measure of how many different energy states are populated appreciably in thermal equilibrium (at least when wesuppose the ground state energy to be zero).
		If one is interested in the average of an operator that does not appear in the Hamiltonian, one often adds it artificially tothe Hamiltonian, calculates Z as a function of the extra new parameter and sets the parameter equal to zero afterdifferentiation.
	www.therfcc.org /partition-function-statistical-mechanics--143041.html (447 words)

	Ivars Peterson's MathLand
		One can prove that for a given whole number, the number of partitions in which all the parts are odd always equals the number of partitions in which all the parts are distinct.
		In general, the partition function p(n) is the number of partitions of n.
		Here are the values of the partition function for the integers from 1 to 21: 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, and 792.
	www.maa.org /mathland/mathland_3_24.html (816 words)

	Partition function (number theory) (Site not responding. Last check: 2007-11-06)
		The number of partitions meeting the second is p(k +1 n) since a partition into parts of least k which contains no parts of at k must have all parts at least k+1.
		More generally the generating function for partitions of n into numbers from a set A can be found by taking only terms in the product where k is an element of A This result is due to Euler
		The formulation of the generating function is to the product formulation of many modular forms giving some idea of the connection the two.
	www.freeglossary.com /Partition_function_(number_theory) (559 words)

	Density of partition function zeroes and phase transition strength (Site not responding. Last check: 2007-11-06)
		Density of partition function zeroes and phase transition strength
		A new method to extract the density of partition function zeroes (a continuous function) from their distribution for finite lattices (a discrete data set) is presented.
		The efficiacy of the technique is demonstrated by its application to a number of models in the case of Fisher zeroes and to the $XY$ model in the case of Lee-Yang zeroes.
	www.maths.tcd.ie /report_series/abstracts/tcdm0117.html (111 words)

	Books on Partition Function (Site not responding. Last check: 2007-11-06)
		A partition function with the prime modulus P>3
		Partition functions and thermodynamic properties of argon plasma
		Zeros of the grand partition function of a hard core lattice gas
	books.bankhacker.com /Partition+Function (84 words)

	partition function (statistical mechanics) (Site not responding. Last check: 2007-11-06)
		However, the total partition function for the system containing N sub-systems is of the form
		where $\zeta_j$ is the partition function for the j:th sub-system.
		is the factorial and ζ is the "common" partition function for a sub-system.
	www.yourencyclopedia.net /Partition_function_(statistical_mechanics).html (615 words)

	APS - 2005 APS March Meeting PostDeadline - Event - Form of the exact partition function for the generalized Ising model (Site not responding. Last check: 2007-11-06)
		A closed form expression is obtained for the exact partition function per spin in terms of the energy levels of this cluster and the degeneracies are functions of temperature.
		This raises a new possibility that the partition function may be determined as a sum of finite number of terms, which may not sum to a single term expression.
		Seven functions need to be determined to describe the exact partition function of the 3D Ising model.
	meetings.aps.org /Meeting/MAR05/Event/26170 (238 words)

	Statistical mechanics: the Riemann zeta function interpreted as a partition function (Site not responding. Last check: 2007-11-06)
		In the theory of the distribution of primes, the fundamental object is the Riemann zeta function.
		The Green's function is defined on a cylinder of radius R and we show that the condition R = a yields the Riemann zeta function as a quantum transition amplitude for the fermion.
		A recursion relation for the partition function allows to calculate the mean density of states from the asymptotic expansion for the single particle density.
	www.maths.ex.ac.uk /~mwatkins/zeta/physics2.htm (6942 words)

	Learn more about Partition function (statistical mechanics) in the online encyclopedia. (Site not responding. Last check: 2007-11-06)
		Learn more about Partition function (statistical mechanics) in the online encyclopedia.
		where is the partition function for the j:th sub-system.
		=Quantum mechanical partition function= More formally, the partition function Z of a quantum-mechanical system may be written as a trace over all states (which may be carried out in any basis, as the trace is basis-independent):
	www.onlineencyclopedia.org /p/pa/partition_function__statistical_mechanics_.html (577 words)

	Papers (Site not responding. Last check: 2007-11-06)
		Abstract: The Ising partition function for a graph counts the number of bipartitions of the vertices with given sizes, with a given size of the induced edge cut.
		Expressed as a 2-variable generating function it is easily translatable into the corresponding partition function studied in statistical physics.
		From the physical partition function we extract quantities such as magnetisation and susceptibility and study their asymptotic behaviour at the critical temperature.
	www.math.umu.se /~phl/Text/index.html (554 words)

	Derivation of the partition function (Site not responding. Last check: 2007-11-06)
		In order to understand the partition function, how it can be derived, and why it works, it is important to recognize that these bulk thermodynamic properties reflect the average behavior of the atoms and molecules.
		The partition function provides a way to determine the most likely average behavior of atoms and molecules given information about the microscopic properties of the material.
		It uses material from the wikipedia article Derivation of the partition function.
	www.eurofreehost.com /de/Derivation_of_the_partition_function.html (503 words)

	Learn more about Thermodynamics in the online encyclopedia. (Site not responding. Last check: 2007-11-06)
		Given U and V (or the density ρ) as a function of T and P, we can define the Helmholtz energy as before and the Gibbs energy as G=U-TS+PV and the enthalpy as H=U+PV.
		The values of these properties are a function of the state of the system and are independent of the path by which the system arrived at that state.
		The number of properties that must be specified to describe the state of a given system is given by Gibbs phase rule.
	www.onlineencyclopedia.org /t/th/thermodynamics.html (1185 words)

	Partition Function - Numericana
		A partition of an integer n is an unordered collection of positive integers (not necessarily distinct) which add up to n.
		The number of such partitions is usually denoted p(n).
		The function p is called the partition function and its first values are tabulated below.
	home.att.net /~numericana/data/partition.htm (80 words)

	Partition function (Site not responding. Last check: 2007-11-06)
		The partition function described here is part of number theory.
		See the next section for the partition function of statistical mechanics.
		The number of partitions meeting the second condition is p(k+1,n).
	www.eurofreehost.com /pa/Partition_function.html (272 words)

	More Partition Function (Site not responding. Last check: 2007-11-06)
		The partition function is the number of add only sums that can represent n, where each part is greater than 0.
		This happens to be the convolution of the natural numbers with the partition function, as we shall see.
		Therefore if we know the values of the partition function, we can calculate the sigma values, and hence determine the prime distribution (sigma(n)=n+1 iff n is prime).
	www.users.globalnet.co.uk /~perry/maths/morepartitionfunction/morepartitionfunction.htm (779 words)

	Kimberly L. Tripp: Improving my SQL skills through your questions! - Clarifying LEFT and RIGHT in the defintion of a ...
		In SQL Server 2005, you can create truly Partitioned Objects (objects are inclusive to Tables and Indexes) and to create a partitioned table you must base that table on a Partition Scheme (PS) and the PS must be based on a Partition Function (PF).
		When a partition function goes through a merge of a boundary point, that boundary point is essentially removed.
		So, if this partitioned table is NEVER going to be modified and you never need to plan for a merge or split, then you can choose whatever definition is easier for you to use.
	www.sqlskills.com /blogs/kimberly/PermaLink.aspx?guid=cf223632-c93b-4242-b0f2-af493e051266 (1886 words)

	ALCOMFT-TR-03-46 (Site not responding. Last check: 2007-11-06)
		We study the complexity of (approximately) computing the partition function in terms of these parameters.
		Exact computation of the partition function Z is NP-hard except in the trivial case beta gamma=1, so we concentrate on the issue of whether Z can be computed within small relative error in polynomial time.
		In one direction, we provide an FPRAS for the partition function within a region which extends well away from the hyperbola.
	www.brics.dk /ALCOM-FT/TR/ALCOMFT-TR-03-46.html (239 words)

	Science News Online - Ivars Peterson's MathLand - 3/22/97
		Indeed, "anytime the number of ways of writing a number as the sum of other numbers arises, the theory of partitions can't be far off," says number theorist George E. Andrews of Pennsylvania State University.
		Recently, Ken Ono of the Institute for Advanced Study in Princeton, N.J., proved some results involving the parity of the partition function and sets of integers known as arithmetic progressions.
		This effort formed the basis of his submission to the 56th Westinghouse Science Talent Search, from which he emerged as the third-place winner when the awards were announced earlier this month.
	www.sciencenews.org /pages/sn_arc97/3_22_97/mathland.htm (828 words)

	Numerical Study of the Entropy Loss of Dimerization and the Folding Thermodynamics of the GCN4 Leucine Zipper -- ...
		Although it is in principle possible in to estimate the ratio of configurational partition functions in Eq.
		) that is accessible to rigid translation of chain 2 of the dimer as a function of temperature.
		9, as a function of the dimensionless temperature.
	www.biophysj.org /cgi/content/full/83/5/2801 (5994 words)

	The Equilibrium Partition Function
		That is, we consider partition functions for every fragment of the original sequence.
		is the partition function for the excluded fragment from j through the origin and back to to i.
		Partition function computations for loop dependent energy rules were first described by McCaskill [11].
	www.bioinfo.rpi.edu /%7Ezukerm/Bio-5495/RNAfold-html/node7.html (367 words)

	Clearing up the market cycle... best Partition Function P Cong... (Site not responding. Last check: 2007-11-06)
		The most simple mathematical and periodic function is a sine one.
		Periodia in one's mathematical form appeals to this function, but it is fold form from value of price row, values of time, characteristic for given row, function sine and parameter of optimization called as Chimerical price.
		This function will accept a graph with ) ( two designated states, and perform the partitioning algorithm until ) ( either no more split is needed or the designated states end up in ) ( different blocks.
	ascot.pl /th/Fourier5/Partition-Function-P-Cong....htm (408 words)

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