Displacement operator - Revision history

en>Liuhd330: modified the typo in the first paragraph

2014-12-03T03:28:16Z

modified the typo in the first paragraph

← Older revision		Revision as of 05:28, 3 December 2014
Line 1:		Line 1:
	~~The '''spherical model''' in [[statistical mechanics]] is a model of [[ferromagnetism]] similar~~ to ~~the [[Ising model]], which was solved in 1952 by T. H. Berlin and [[Mark Kac\|M. Kac]]. It has the remarkable property that when applied~~ to systems of dimension ''d'' greater than four, the [[critical exponents]] that govern the behaviour of the system near the critical point are independent of ''d'' and the geometry of the system. It is one of the few models of ferromagnetism that can be solved exactly in the presence of an external field.		I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. To perform lacross is some thing he would by no means give up. I am an invoicing officer and I'll be promoted quickly. My wife and I reside in Mississippi but now I'm contemplating other choices.<br><br>Here is my blog: cheap psychic readings, [http://appin.co.kr/board_Zqtv22/688025 appin.co.kr],

	~~== Formulation ==~~
	~~The model describes a set of particles on a lattice <math> \mathbb{L} </math> containing ''N'' sites. For each site ''j'' of <math> \mathbb{L} </math>~~, ~~a spin <math> \sigma_j </math> which interacts only with its nearest neighbours~~ and ~~an external field~~ '~~'H''. It differs from the Ising model in that the <math> \sigma_j </math> are no longer restricted to <math> \sigma_j \in \{1,-1\} </math>, but can take~~ all ~~real values, subject to the constraint that~~

	~~:<math> \sum_{j=1}^N \sigma_j^2 = N </math>~~

	~~which in a homogenous system ensures that the average of the square of any spin is one, as in the usual Ising model~~.

	~~The [[partition function (statistical mechanics)\|partition function]] generalizes from that of the [[Ising model]] to~~

	:<math> Z_N = \int_{-\infty}^{\infty} \ldots \int_{-\infty}^{\infty} d\sigma_1 \ldots d\sigma_N exp \left[ K \sum_{\langle jl \rangle} \sigma_j \sigma_l + h \sum_j \sigma_j \right] \delta \left[N - \sum_j \sigma_j^2 \right] </math>

	~~where <math> \delta </math>~~ is the [[Dirac delta function]], <math> \langle jl \rangle</math> are the edges of the lattice, and <math>K=J/kT </math> and <math> h= H/kT </math>, where ''T'' is the temperature of the system, ''k'' is [[Boltzmann's constant]] and ''J'' the coupling constant of the nearest-neighbour interactions.

	Berlin and Kac saw this as an approximation to the usual Ising model, arguing that the <math>\sigma</math>-summation in the Ising model can be viewed as a sum over all corners of an ''N''-dimensional [[hypercube]] in <math>\sigma</math>-space. The becomes an [[Integral\|integration]] over the [[surface (mathematics)\|surface]] of a hypersphere passing through all such corners.

	~~It was rigorously proved~~ by ~~Kac and C. J~~. ~~Thompson that the spherical model is a limiting case of the [[N-vector model]].~~

	~~== Equation of state ==~~
	~~Solving the partition function and using a calculation of the [[Thermodynamic free energy\|free energy]] yields an equation describing the [[magnetization]] ''M'' of the system~~

	~~:<math> 2J(1-M^2) = kTg'(H/2JM) </math>~~

	~~for the function ''g'' defined as~~

	~~:<math> g(z) = (2 \pi)^{-d} \int_0^{2\pi} \ldots \int_0^{2\pi} d\omega_1 \ldots d\omega_d \ ln [z+d -cos \omega_1 - \ldots - cos \omega_d] </math>~~

	~~The [[internal energy]] per site is given by~~

	~~:<math> u =\frac{1}{2} kT - Jd - \frac{1}{2}H(M+M^{-1}) </math>~~

	an ~~exact relation relating internal energy~~ and ~~magnetization.~~

	~~==Critical behaviour==~~
	~~For <math> d \leq 2 </math> the [[critical temperature]] occurs at [[absolute zero]], resulting in no phase transition for the spherical model. For~~ '~~'d'' greater than 2, the spherical model exhibits the typical ferromagnetic behaviour, with a finite [[Curie temperature]] where ferromagnetism ceases~~. ~~The critical behaviour of the spherical model was derived~~ in ~~the completely general circumstances that the dimension~~ '~~'d'' may be a real non-integer dimension~~.

	~~The critical exponents~~ <~~math~~> ~~\alpha, \beta, \gamma~~ <~~/math~~> ~~and <math> \gamma' </math> in the zero-field case which dictate the behaviour of the system close to were derived to be~~

	:~~<math> \alpha = \begin{cases}~~
	~~- \frac{4-d}{d-2} & \ \mathrm{if} \ 2<d<4 \\~~
	~~0 & \ if \ d > 4 \end{cases} </math>~~

	~~:<math> \beta = \frac{1}{2} </math>~~

	~~:<math> \gamma = \begin{cases}~~
	~~\frac{2}{d-2} & \ if \ 2 <d<4 \\~~
	~~1 & if \ d > 4 \end{cases} </math>~~

	~~:<math> \delta = \begin{cases} \frac{d+2}{d-2} & \ if \ 2 < d < 4 \\~~
	~~3 & if \ d > 4 \end{cases}~~
	~~</math>~~

	~~which are independent of the dimension of ''d'' when it is greater than four~~, ~~the dimension being able to take any real value.~~

	~~== References ==~~
	* [~~[Rodney J. Baxter\|R.J. Baxter]], ''Exactly solved models in statistical mechanics'', London, Academic Press, 1982~~

	~~== Further reading ==~~
	*{{cite journal \| last1=Berlin \| first1=T. H. \| last2=Kac \| first2=M. \| title=The spherical model of a ferromagnet \| url=http://~~link~~.~~aps~~.~~org~~/~~abstract~~/~~PR/v86/p821 \| doi=10~~.~~1103/PhysRev.86.821 \| id={{MathSciNet \| id = 0049829}} \| year=1952 \| journal=Physical Rev. (2) \| volume=86 \| pages=821–835\|bibcode = 1952PhRv~~.~~..86..821B }}~~

	~~[[Category:Lattice models]]~~
	~~[[Category:Exactly solvable models]~~]

130.209.203.66: /* Properties */ minor sign fix

2014-02-03T13:42:42Z

Properties: minor sign fix

← Older revision		Revision as of 15:42, 3 February 2014
Line 1:		Line 1:
	~~Hello~~ and ~~welcome~~. ~~My title~~ is ~~Figures Wunder~~. ~~My day occupation is~~ a ~~meter reader~~. ~~One~~ of the ~~extremely best things~~ in the ~~world for me~~ is to ~~do aerobics~~ and ~~now I~~'~~m attempting~~ to ~~earn money with it~~. For ~~many years he~~'~~s been residing~~ in ~~North Dakota~~ and ~~his family enjoys it.~~<br><br>~~my web site~~ :: ~~std home test;~~ [http://~~dynast~~.Co.kr/xe/~~?document_srl~~=~~150618 click through the up coming web page~~],		The '''spherical model''' in [[statistical mechanics]] is a model of [[ferromagnetism]] similar to the [[Ising model]], which was solved in 1952 by T. H. Berlin and [[Mark Kac\|M. Kac]]. It has the remarkable property that when applied to systems of dimension ''d'' greater than four, the [[critical exponents]] that govern the behaviour of the system near the critical point are independent of ''d'' and the geometry of the system. It is one of the few models of ferromagnetism that can be solved exactly in the presence of an external field.

			== Formulation ==
			The model describes a set of particles on a lattice <math> \mathbb{L} </math> containing ''N'' sites. For each site ''j'' of <math> \mathbb{L} </math>, a spin <math> \sigma_j </math> which interacts only with its nearest neighbours and an external field ''H''. It differs from the Ising model in that the <math> \sigma_j </math> are no longer restricted to <math> \sigma_j \in \{1,-1\} </math>, but can take all real values, subject to the constraint that

			:<math> \sum_{j=1}^N \sigma_j^2 = N </math>

			which in a homogenous system ensures that the average of the square of any spin is one, as in the usual Ising model.

			The [[partition function (statistical mechanics)\|partition function]] generalizes from that of the [[Ising model]] to

			:<math> Z_N = \int_{-\infty}^{\infty} \ldots \int_{-\infty}^{\infty} d\sigma_1 \ldots d\sigma_N exp \left[ K \sum_{\langle jl \rangle} \sigma_j \sigma_l + h \sum_j \sigma_j \right] \delta \left[N - \sum_j \sigma_j^2 \right] </math>

			where <math> \delta </math> is the [[Dirac delta function]], <math> \langle jl \rangle</math> are the edges of the lattice, and <math>K=J/kT </math> and <math> h= H/kT </math>, where ''T'' is the temperature of the system, ''k'' is [[Boltzmann's constant]] and ''J'' the coupling constant of the nearest-neighbour interactions.

			Berlin and Kac saw this as an approximation to the usual Ising model, arguing that the <math>\sigma</math>-summation in the Ising model can be viewed as a sum over all corners of an ''N''-dimensional [[hypercube]] in <math>\sigma</math>-space. The becomes an [[Integral\|integration]] over the [[surface (mathematics)\|surface]] of a hypersphere passing through all such corners.

			It was rigorously proved by Kac and C. J. Thompson that the spherical model is a limiting case of the [[N-vector model]].

			== Equation of state ==
			Solving the partition function and using a calculation of the [[Thermodynamic free energy\|free energy]] yields an equation describing the [[magnetization]] ''M'' of the system

			:<math> 2J(1-M^2) = kTg'(H/2JM) </math>

			for the function ''g'' defined as

			:<math> g(z) = (2 \pi)^{-d} \int_0^{2\pi} \ldots \int_0^{2\pi} d\omega_1 \ldots d\omega_d \ ln [z+d -cos \omega_1 - \ldots - cos \omega_d] </math>

			The [[internal energy]] per site is given by

			:<math> u =\frac{1}{2} kT - Jd - \frac{1}{2}H(M+M^{-1}) </math>

			an exact relation relating internal energy and magnetization.

			==Critical behaviour==
			For <math> d \leq 2 </math> the [[critical temperature]] occurs at [[absolute zero]], resulting in no phase transition for the spherical model. For ''d'' greater than 2, the spherical model exhibits the typical ferromagnetic behaviour, with a finite [[Curie temperature]] where ferromagnetism ceases. The critical behaviour of the spherical model was derived in the completely general circumstances that the dimension ''d'' may be a real non-integer dimension.

			The critical exponents <math> \alpha, \beta, \gamma </math> and <math> \gamma' </math> in the zero-field case which dictate the behaviour of the system close to were derived to be

			:<math> \alpha = \begin{cases}
			- \frac{4-d}{d-2} & \ \mathrm{if} \ 2<d<4 \\
			0 & \ if \ d > 4 \end{cases} </math>

			:<math> \beta = \frac{1}{2} </math>

			:<math> \gamma = \begin{cases}
			\frac{2}{d-2} & \ if \ 2 <d<4 \\
			1 & if \ d > 4 \end{cases} </math>

			:<math> \delta = \begin{cases} \frac{d+2}{d-2} & \ if \ 2 < d < 4 \\
			3 & if \ d > 4 \end{cases}
			</math>

			which are independent of the dimension of ''d'' when it is greater than four, the dimension being able to take any real value.

			== References ==
			* [[Rodney J. Baxter\|R.J. Baxter]], ''Exactly solved models in statistical mechanics'', London, Academic Press, 1982

			== Further reading ==
			*{{cite journal \| last1=Berlin \| first1=T. H. \| last2=Kac \| first2=M. \| title=The spherical model of a ferromagnet \| url=http://link.aps.org/abstract/PR/v86/p821 \| doi=10.1103/PhysRev.86.821 \| id={{MathSciNet \| id = 0049829}} \| year=1952 \| journal=Physical Rev. (2) \| volume=86 \| pages=821–835\|bibcode = 1952PhRv...86..821B }}

			[[Category:Lattice models]]
			[[Category:Exactly solvable models]]

134.96.30.100: /* Properties */

2012-07-16T12:40:17Z

Properties

New page

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