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Topic: Derivation of the partition function

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	PlanetMath: integer partition
		Conventionally, parts of a partition are written from the largest to the smallest.
		The dual partition is the partition obtained by reflecting the Young diagram along the main diagonal.
		(Combinatorics :: Enumerative combinatorics :: Partitions of integers)
	planetmath.org /encyclopedia/Partition2.html (169 words)

	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.
		A 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) (1645 words)

	Derivation of the partition function: Definition and Links by Encyclopedian.com
		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.
		The partition function provides a measure of the total number of energetic states that are accessible at a particular temperature and can be related to many different thermodynamic properties (see Statistical Mechanics).
	www.encyclopedian.com /de/Derivation-of-the-partition-function.html (1251 words)

	Systematic derivation of partition functions for ligand binding to two-dimensional lattices -- Wang and Di Cera ...
		Systematic derivation of partition functions for ligand binding to two-dimensional lattices
		The partition function for this lattice is evidently
		are the partition functions of a linear lattice of
	www.pnas.org /cgi/content/full/93/23/12953 (3138 words)

	Function Decomposition Algorithm
		The resulting functions G and H have to be consistent with F. For the purpose of decomposition, a set of features X is partitioned to two disjoint subsets A and B, referred to as a free and bound set, respectively.
		However, for each function F there are many possible partitions, each one yielding a different feature c and a different pair of functions G and H. In feature transformation, it is important to identify partitions that lead to simple but useful new features.
		Partition matrix is thus a formal construct used for explanation or analysis of the decomposition algorithm.
	www.ailab.si /blaz/hint/algorithm.htm (936 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 partition function is a function of, firstly, the temperature T; and, secondly, the microstate energies E1, E2, E3, etc. The microstate energies are determined by other thermodynamic variables, such as the number of particles and the volume, as well as microscopic quantities like the mass of the constituent particles.
		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)

	June 2000 : Derivation, tolerance and validity
		We open by informally sketching a model of the declaration, the notion of tolerance, and the kinds of derivations with respect to which validations might be successfully conducted.
		Without saying what can count as a derivation (other than to note that every type is a derivation of itself), we extend the definition of validity to accommodate derivation.
		The intent of derivation is to control the circumstances under which a tag might occur at a node where another kind of tag is expected.
	www.infoloom.com /gcaconfs/WEB/paris2000/S06-02.HTM (2338 words)

	FUNCTION - Online Information article about FUNCTION
		In the case of irrational functions, or rational f unctions which are not integral, new variables are introduced in such a way as to make the equations contain rational integral terms only.
		The rule for a function of a function is not stated explicitly but is illustrated by 'examples in which new variables are introduced, in much the same way as in Newton's Methodus fluxionum.
		In connexion with the problem of maxima and minima, it is noted that the differential of y is positive or negative according as y increases or decreases when x increases, and the discrimination of maxima from minima depends upon the sign of ddy, the differential of dy.
	encyclopedia.jrank.org /FRA_GAE/FUNCTION.html (8431 words)

	More on Statistical Mechanics
		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.artilifes.com /statistical-mechanics.htm (1368 words)

	Definition of a Line Integral
		The integral is then defined as the limit of the sequence of values Sn when the size of the partitions (usually taken to be the size of the largest piece) goes to zero.
		Where the domain is now the curve C and the function is a function that assigns a scalar (number) to each point on the curve.
		The function f needs to be defined at least on the curve C but it is often known in a much larger part of the space that contains C. Here is an example.
	omega.albany.edu:8008 /calc3/line-integrals-dir/define-m2h.html (749 words)

	Statistical thermodynamics - Chemistry (Site not responding. Last check: 2007-11-03)
		Ensembles In exploring the Canonical Partition function, our focus in the Boltzmann distribution changes from that of how molcules are distributed between their individual quantum states to that of the distribution of energy between the whole series of possible energies for the COMPLETE system containing all molecules.
		In terms of derivations, the derivation for Internal energy and Pressureare remarkibly similar.
		The derivation for Heat capacity uses the derivation for Internal energy.
	chemistry.tidalswan.com /index.php?title=Statistical_thermodynamics (843 words)

	CS267: Notes for Lectures 11 and 12, Feb 20 and 22, 1996
		The partition on the left only has 6 edges crossing processor boundaries, whereas the partition on the right crosses ten, so assuming each edge has the same weight, the left partitioning requires less communication and so is superior.
		The best way to partition for parallelizing the external force computation is to partition by particle number, or simply assigning an equal number of particles to each processor by particle number, independent of how their physical locations may change.
		A better way to partition for nearby-forces is to partition the physical space in which the particles reside, which each processor being responsible for the particles that happen to reside in the part of space it owns.
	www.cs.berkeley.edu /~demmel/cs267/lecture16/lecture16.html (5528 words)

	User Defined RDML Built-In Function (Site not responding. Last check: 2007-11-03)
		If an RDML Built-In Function has to be made available to a partition other than that where it has been compiled, it must be accessible in the library list for that other job.
		The Built-In Function is passed two arguments :a working list of packed 11,2 numbers and optionally whether to return those that are above or below the average of the numbers, defaulting to be above.
		Because the function is heavyusage, its evaluation logic is conditioned by ensuring that it is not a shutdown call.
	www.lansa.com /support/tips/t0085.htm (1526 words)

	PlanetMath: partition function
		is defined to be the number of partitions of the integer
		This function grows very quickly, as we see in the following theorem due to Hardy and Ramanujan.
		This is version 6 of partition function, born on 2006-06-09, modified 2006-09-30.
	planetmath.org /encyclopedia/PartitionFunction2.html (138 words)

	Statistical mechanics: the Riemann zeta function interpreted as a partition function
		In the theory of the distribution of primes, the fundamental object is the Riemann zeta function.
		The derivation is based on the use of modular arithmetic to solve an isomorphic combinatoric problem, that of describing the global organization of the sequence of all ordered placements of N indistinguishable balls into M distinguishable non-empty bins or boxes.
		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.
	secamlocal.ex.ac.uk /~mwatkins/zeta/physics2.htm (8404 words)

	Big-Oh for Recursive Functions: Recurrence Relations
		Note that the last line is derived by seeing a pattern --- this is the Eureka/leap of faith/practice with generalizing mathematical patterns part of the problem.
		In the worst case the partition function is malicious and partitions the vector poorly every time it's called.
		This means that the partition function will divide the vector into two roughly equal pieces each time it's called.
	www.cs.duke.edu /~ola/ap/recurrence.html (1255 words)

	CygnaCom Solutions SEL: PC Guardian: Encryption Plus® for Hard Disks 7.0
		The data written to and read from the partition or disk is respectively encrypted and decrypted on the fly as required driven by operating system use of the storage device.
		It is recommended that all partitions on all disks be encrypted with EPHD to minimize risk of swap files and other application and operating system generated temporary files, being stored on an unprotected disk.
		The EC KRK is derived by first negotiating a key with Elliptic Curve Diffie-Hellman (ECDH) encryption using the corporate administrator and the local administrator's public ECDH keys, and then deriving the EC KRK from the negotiated key and the username with the KDF2 key derivation function.
	www.cygnacom.com /labs/pfSEL0144a.htm (1512 words)

	CM: NetLogo ProbLab: Partition Permutation Distribution: Riddle #1
		So the permutation 1-4 has a value of 0 (no 2's in it), the permutation 3-2 has a value of 1 (one 2 in it), and the permutation 2-1-2 has a value of 2 (because there are two 2's).
		Now, there are exactly six different partition permutations totaling 5 with a single '2' in them: 1-1-1-2, 1-1-2-1, 1-2-1-1, 2-1-1-1, 3-2, and 2-3.
		The first exlanation was the General Case approach, in which we considered the situations where the target-total is one more, two more and three more than the number for which we are looking.
	ccl.northwestern.edu /curriculum/ProbLab/PartitionRiddle1.htm (904 words)

	The partition function and pressure estimators
		The partition function for the isotropic isothermal-isobaric ensemble (NPT) is given by [39, 5]
		The primitive pressure estimator can be manipulated to produce another pressure estimator in the same manner as the path integral virial kinetic energy estimator is obtained from the primitive kinetic energy estimator [13, 6].
		Finally, the derivation assumes the ``size'' of the pseudomolecules is small compared to the volume (i.e.
	homepages.nyu.edu /~mt33/PINPT/node7.html (496 words)

	Cerius2 Forcefield Based Simulations - Free Energy
		Thus, the ratio of the partition functions can be computed from an ensemble average of the energy difference between a reference state and a perturbed state.
		The first step in any free energy calculation is parameterizing the energy function to provide a continuous change between the thermodynamic states that are being compared.
		Because a function of state (the free energy) is being integrated, the path between the end points is arbitrary.
	www.chem.cmu.edu /courses/09-560/docs/msi/ffbsim/6_FreeEnergy.html (4912 words)

	Arithmetic, Numeration, Number Theory - Numericana
		A partition of n is a collection of positive integers (not necessarily distinct) whose sum equals n.
		If f is an arithmetic function, a function F, called the sum-function of f, is defined by letting F(n) be the sum of the terms f(d) for all divisors d of n.
		A multiplicative function which is zero for squares of primes, and higher powers of prime numbers, is thus the Dirichlet inverse of a totally multiplicative function.
	home.att.net /~numericana/answer/numbers.htm (7607 words)

	Variation in Partition Coefficient, Kd, Values \| Radiation Protection Program \| US EPA (Site not responding. Last check: 2007-11-03)
		parameter derived from ion-exchange literature and its "empiricized" use in contaminant transport codes, and (3) the explicit and implicit assumptions underlying the use of the K
		This is the third volume in the series that describes: (1) the conceptualization, measurement, and use of the partition coefficient parameter; and (2) the geochemical aqueous solution and sorbent properties that are most important in controlling adsorption/retardation behavior of selected contaminants.
		Generic or default partition coefficient values found in the literature can result in significant errors when used to predict the impacts of contaminant migration or site-remediation options.
	www.epa.gov /radiation/cleanup/partition.htm (742 words)

	Classical Statistical Mechanics
		1] discusses a derivation of the ``quasi-classical'' representation of the canonical partition function,
		is the Hamiltonian function which computes the energy of a point in phase space.
		The quantity in the left-hand braces is the ideal gas partition function, because it corresponds to the case when the potential
	www.pages.drexel.edu /~cfa22/msim/node7.html (345 words)

	Quantum Physics: Problem set #3, 2003
		Include in your discussion the derivation of how you find the ground state wave function of the SHO, including normalization, by an operator method.
		A partition is removed so that the well suddenly doubles its size to a new width 2L.
		Derive the matrix that you need to diagonalize for l = 2, but don't attempt to solve this equation unless you have access to a good algebraic matrix solver.
	venables.asu.edu /quant/prob033.html (935 words)

	Simpson's Rule (Site not responding. Last check: 2007-11-03)
		Since this function is unique, this must be the quadratic function with which we approximate f(x) on
		is a function that interpolates all of the points (n,(n-1)!) for positive integers n.
		Note that for x > 1 this function is defined for all s in [0,1] except at s = 1.
	www.mathcs.emory.edu /ccs/ccs215/integral/node4.html (757 words)

	Derivation, tolerance and validity
		We investigate the general mechanism of type derivations in XML schemas1, of which the restrictions and extensions of the proposed schema standard (in [XSD1] and [XSD2]) are but special cases.
		In effect we are proposing that we make applications responsible for telling the validator what classes of transformations lead to substitutable derivations from the application's perspective.
		The former declares an element that permits2 the substitution of a type derived by any sequence of element or attribute extensions.
	www.gca.org /papers/xmleurope2000/papers/s06-02.html (2015 words)

	Formal derivation of the virial expansion
		The expression for the grand canonical partition function
		In the derivation of expression (58) we took into account that the Mayer
		It concerns the convergence of the fugacity expansion of the grand canonical partition function (56), our starting point.
	www.uni-konstanz.de /FuF/Physik/Jaeckle/lecture/StatMech/node3.html (305 words)

	Dyslexia (Site not responding. Last check: 2007-11-03)
		Research shows intense training in dyslexics helps them use the part of their brain to take over the limited functionality the left part.
		These flaws affect connectivity and functionality of cortex in critical areas related to sound visual processing.
		Whether today's models of dyslexia are correct not the main lesson of dyslexia is minor genetic changes affecting the layering of cortex in a minor area of the may impose inborn limitations on the overall function.
	www.freeglossary.com /Dyslexia (2220 words)

	Derivation of the discretized path integral
		The partition function evaluated as a trace in the coordinate basis is
		When expressed in this way, the partition function, for a finite value of P, is isomorphic to a classical configuration integral for a P-particle system, that is a cyclic chain of particles, with harmonic nearest neighbor interactions and interacting with an external potential U(x)/P.
		Thus, for finite (if large) P the partition function in the discretized path integral representation can be treated as any ordinary classical configuration integral.
	www.nyu.edu /classes/tuckerman/stat.mech/lectures/lecture_14/node1.html (474 words)

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