Introduction to entropy - Revision history

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2014-12-03T22:50:11Z

Reverting possible vandalism by 72.45.167.106 to version by AnomieBOT. False positive? Report it. Thanks, ClueBot NG. (2050819) (Bot)

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	~~In [[mathematical physics]]~~, ~~the '''primon gas''' or '''free Riemann gas''' is a [[toy model]] illustrating in a simple way some correspondences between [[number theory]]~~ and ~~ideas in [[quantum field theory]] and [[dynamical systems]]. It is a quantum field theory of a set of non-interacting particles, the '''primons''';~~ it ~~is called a [[gas]] or a ''free model'' because the particles are non-interacting. The idea of the primon gas was independently discovered by~~		Hi there, I am Alyson Pomerleau and I believe it seems quite good when you say it. Her family lives in Ohio. What me and my family members love is performing ballet but I've been using on new issues recently. Office supervising is what she does for a living.<br><br>Look into my web-site :: real psychics ([http://chorokdeul.co.kr/index.php?document_srl=324263&mid=customer21 chorokdeul.co.kr])
	~~Donald Spector<ref>D. Spector, Supersymmetry and the Möbius Inversion Function, Communications in Mathemtical Physics 127 (1990) pp. 239–252.</ref> and [[Bernard Julia]]~~.~~<ref>Bernard L. Julia, Statistical theory of numbers,~~ in ~~Number Theory and Physics, eds~~. ~~J. M. Luck, P. Moussa,~~ and ~~M. Waldschmidt, Springer Proceedings in~~ '~~'Physics'', Vol~~. ~~'''47''', Springer-Verlag, Berlin, 1990, pp. 276–293.</ref> Later works by Bakas and~~
	~~Bowick<ref>I. Bakas and M.J. Bowick, Curiosities of Arithmetic Gases, J. Math. Phys. 32 (1991) p. 1881</ref>~~
	~~and Spector <ref>D. Spector, Duality, Partial Supersymmetry, and Arithmetic Number Theory, J. Math. Phys. 39 (1998) pp. 1919–1927</ref> explored the connection of such systems to~~
	~~string theory.~~

	~~==The model==~~
	~~Consider~~ a ~~simple [[quantum Hamiltonian]] ''H'' having [[eigenstate]]s <math>\|p\rangle</math> labelled by the [[prime number]]s ''p'', and having energies proportional to log ''p''~~. ~~That is,~~

	~~:<math>H\|p\rangle = E_p \|p\rangle</math>~~

	~~with~~

	:<~~math~~>~~E_p=E_0 \log p \,~~ <~~/math~~>

	~~The [[second quantization\|second-quantized]] version of this Hamiltonian converts states~~ into ~~particles, the '''primons'''. A multi-particle state is given by the numbers <math>k_p</math> of primons in the single~~-~~particle states <math>p</math>~~:

	:~~<math>\|n\rangle = \|k_2, k_3, k_5, k_7, k_{11}, \ldots, k_p, \ldots\rangle</math>~~

	~~This corresponds to the factorization of <math>n</math> into primes:~~

	~~:<math>n = 2^{k_2} \cdot 3^{k_3} \cdot 5^{k_5} \cdot 7^{k_7} \cdot 11^{k_{11}} \cdots p^{k_p} \cdots</math>~~

	~~The labelling by the integer ''n'' is unique, since every number has a unique factorization into primes.~~

	~~The energy of such a multi-particle state is clearly~~

	~~:<math>E(n) = \sum_p k_p E_p = E_0 \cdot \sum_p k_p \log p = E_0 \log n</math>~~

	~~The statistical mechanics [[partition function~~ (~~mathematics)\|partition function]] ''Z'' is given by the~~ [~~[Riemann zeta function]]:~~

	~~:<math>Z(T)~~ :~~= \sum_{n=1}^\infty \exp \left(\frac{-E(n)}{k_B T}\right) = \sum_{n=1}^\infty \exp \left(\frac{-E_0 \log n}{k_B T}\right) = \sum_{n=1}^\infty \frac{1}{n^s} = \zeta (s) </math>~~

	~~with ''s'' = ''E''<sub>0<~~/~~sub>~~/~~''k''<sub>B</sub>''T'' where ''k''<sub>B<~~/~~sub> is [[Boltzmann's constant]] and ''T'' is the absolute [[temperature]]~~. ~~The divergence of the zeta function at ''s'' ~~=&~~nbsp;1 corresponds to the divergence of the partition function at a [[Hagedorn temperature]] of ''T''<sub>H</sub> ~~=~~ ''E''<sub>0</sub>/''k''<sub>B</sub>~~.

	~~==The supersymmetric model==~~
	The above second-quantized model takes the particles to be [[boson]]s. If the particles are taken to be [[fermion]]s, then the [[Pauli exclusion principle]] prohibits multi-particle states which include squares of primes. By the [[spin-statistics theorem]], field states with an even number of particles are bosons, while those with an odd number of particles are fermions. ~~The fermion operator [[(-1)^F\|(−1)<sup>F</sup>]] has a very concrete realization in this model as the [[Möbius function~~]~~] <math>\mu(n~~)~~</math>, in that the Möbius function is positive for bosons, negative for fermions, and zero on exclusion-principle-prohibited states.~~

	~~==More complex models==~~
	The connections between number theory and quantum field theory can be somewhat further extended into connections between [[topological field theory]] and [[K-theory]], where, corresponding to the example above, the [[spectrum of a ring]] takes the role of the spectrum of energy eigenvalues, the [[prime ideal]]s take the role of the prime numbers, the [[group representation]]s take the role of integers, [[group character]]s taking the place the [[Dirichlet character]]s, and so on.

	~~==References==~~
	~~<references/>~~
	* [[John Baez]], [http://math.ucr.edu/home/baez/week199.html This Week's Finds in Mathematical Physics, Week 199]

	~~[[Category:Number theory]]~~
	~~[[Category:Quantum field theory]]~~

en>Vegaswikian: Disambiguated: phase change → phase transition; Unlinked: Work

2014-01-12T21:10:43Z

Disambiguated: phase change → phase transition; Unlinked: Work

← Older revision		Revision as of 23:10, 12 January 2014
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	~~Wilber Berryhill~~ is the ~~name his mothers~~ and ~~fathers gave him~~ and ~~he completely digs that name~~. I'~~ve usually loved residing in Kentucky but now I~~'~~m considering other options~~. ~~To play lacross is some thing~~ I ~~truly enjoy doing~~. ~~He works as a bookkeeper~~.<br><br>~~Here is my website~~ ... [~~http~~://~~gcjcteam~~.~~org~~/~~index~~.~~php?mid~~=~~etc_video~~&~~document_srl~~=~~696611~~&~~sort_index~~=~~regdate~~&~~order_type~~=~~desc love psychic readings~~]		In [[mathematical physics]], the '''primon gas''' or '''free Riemann gas''' is a [[toy model]] illustrating in a simple way some correspondences between [[number theory]] and ideas in [[quantum field theory]] and [[dynamical systems]]. It is a quantum field theory of a set of non-interacting particles, the '''primons'''; it is called a [[gas]] or a ''free model'' because the particles are non-interacting. The idea of the primon gas was independently discovered by
			Donald Spector<ref>D. Spector, Supersymmetry and the Möbius Inversion Function, Communications in Mathemtical Physics 127 (1990) pp. 239–252.</ref> and [[Bernard Julia]].<ref>Bernard L. Julia, Statistical theory of numbers, in Number Theory and Physics, eds. J. M. Luck, P. Moussa, and M. Waldschmidt, Springer Proceedings in ''Physics'', Vol. '''47''', Springer-Verlag, Berlin, 1990, pp. 276–293.</ref> Later works by Bakas and
			Bowick<ref>I. Bakas and M.J. Bowick, Curiosities of Arithmetic Gases, J. Math. Phys. 32 (1991) p. 1881</ref>
			and Spector <ref>D. Spector, Duality, Partial Supersymmetry, and Arithmetic Number Theory, J. Math. Phys. 39 (1998) pp. 1919–1927</ref> explored the connection of such systems to
			string theory.

			==The model==
			Consider a simple [[quantum Hamiltonian]] ''H'' having [[eigenstate]]s <math>\|p\rangle</math> labelled by the [[prime number]]s ''p'', and having energies proportional to log ''p''. That is,

			:<math>H\|p\rangle = E_p \|p\rangle</math>

			with

			:<math>E_p=E_0 \log p \, </math>

			The [[second quantization\|second-quantized]] version of this Hamiltonian converts states into particles, the '''primons'''. A multi-particle state is given by the numbers <math>k_p</math> of primons in the single-particle states <math>p</math>:

			:<math>\|n\rangle = \|k_2, k_3, k_5, k_7, k_{11}, \ldots, k_p, \ldots\rangle</math>

			This corresponds to the factorization of <math>n</math> into primes:

			:<math>n = 2^{k_2} \cdot 3^{k_3} \cdot 5^{k_5} \cdot 7^{k_7} \cdot 11^{k_{11}} \cdots p^{k_p} \cdots</math>

			The labelling by the integer ''n'' is unique, since every number has a unique factorization into primes.

			The energy of such a multi-particle state is clearly

			:<math>E(n) = \sum_p k_p E_p = E_0 \cdot \sum_p k_p \log p = E_0 \log n</math>

			The statistical mechanics [[partition function (mathematics)\|partition function]] ''Z'' is given by the [[Riemann zeta function]]:

			:<math>Z(T) := \sum_{n=1}^\infty \exp \left(\frac{-E(n)}{k_B T}\right) = \sum_{n=1}^\infty \exp \left(\frac{-E_0 \log n}{k_B T}\right) = \sum_{n=1}^\infty \frac{1}{n^s} = \zeta (s) </math>

			with ''s'' = ''E''<sub>0</sub>/''k''<sub>B</sub>''T'' where ''k''<sub>B</sub> is [[Boltzmann's constant]] and ''T'' is the absolute [[temperature]]. The divergence of the zeta function at ''s'' = 1 corresponds to the divergence of the partition function at a [[Hagedorn temperature]] of ''T''<sub>H</sub> = ''E''<sub>0</sub>/''k''<sub>B</sub>.

			==The supersymmetric model==
			The above second-quantized model takes the particles to be [[boson]]s. If the particles are taken to be [[fermion]]s, then the [[Pauli exclusion principle]] prohibits multi-particle states which include squares of primes. By the [[spin-statistics theorem]], field states with an even number of particles are bosons, while those with an odd number of particles are fermions. The fermion operator [[(-1)^F\|(−1)<sup>F</sup>]] has a very concrete realization in this model as the [[Möbius function]] <math>\mu(n)</math>, in that the Möbius function is positive for bosons, negative for fermions, and zero on exclusion-principle-prohibited states.

			==More complex models==
			The connections between number theory and quantum field theory can be somewhat further extended into connections between [[topological field theory]] and [[K-theory]], where, corresponding to the example above, the [[spectrum of a ring]] takes the role of the spectrum of energy eigenvalues, the [[prime ideal]]s take the role of the prime numbers, the [[group representation]]s take the role of integers, [[group character]]s taking the place the [[Dirichlet character]]s, and so on.

			==References==
			<references/>
			* [[John Baez]], [http://math.ucr.edu/home/baez/week199.html This Week's Finds in Mathematical Physics, Week 199]

			[[Category:Number theory]]
			[[Category:Quantum field theory]]

en>Zaereth: Changed photo, so I'm also updating the caption here to describe the insets.

2012-07-30T21:59:15Z

Changed photo, so I'm also updating the caption here to describe the insets.

New page

Wilber Berryhill is the name his mothers and fathers gave him and he completely digs that name. I've usually loved residing in Kentucky but now I'm considering other options. To play lacross is some thing I truly enjoy doing. He works as a bookkeeper.<br><br>Here is my website ... [http://gcjcteam.org/index.php?mid=etc_video&document_srl=696611&sort_index=regdate&order_type=desc love psychic readings]