Conical coordinates - Revision history

en>Jonesey95: Fixing citation DOI or LCCN parameter error using AutoEd

2014-08-25T03:37:52Z

Fixing citation DOI or LCCN parameter error using AutoEd

← Older revision		Revision as of 05:37, 25 August 2014
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	~~{{Unreferenced\|date=December 2009}}~~		The title of the writer is Luther. What she loves performing is to perform croquet but she hasn't made a dime with it. Bookkeeping has been his day job for a whilst. Years ago we moved to Arizona but my spouse desires us to transfer.<br><br>Also visit my web site; extended car warranty [[http://vendorportal.citypower.co.za/Activity-Feed/My-Profile/UserId/18066 try this web-site]]
	~~In [[statistical mechanics]],~~ the '~~''Rushbrooke inequality'''<!-- who is Rushbrooke? --> relates the [[critical exponent]]s of~~ a ~~[[magnetic]] system which exhibits~~ a ~~first-order [[phase transition]] in the [[thermodynamic limit]] for non-zero [[temperature]] ''T''~~.

	~~Since the [[Helmholtz free energy]] is [[extensive quantity\|extensive]], the normalization~~ to ~~free energy per site is given as~~

	:<~~math~~> ~~f = -kT \lim_{N \rightarrow \infty} \frac{1}{N}\log Z_N~~ <~~/math~~>

	~~The magnetization ''M'' per~~ site ~~in the [~~[~~thermodynamic limit]], depending on the external~~ [~~[magnetic field]] ''H'' and temperature ''T'' is given by~~

	:~~<math> M(T,H) \ \stackrel{\mathrm{def}}{=}\ \lim_{N \rightarrow \infty} \frac{1}{N} \left( \sum_i \sigma_i \right) = - \left( \frac{\partial f}{\partial H} \right)_T <~~/~~math>~~

	~~where <math> \sigma_i <~~/~~math> is the spin at the i-th site, and the [[magnetic susceptibility]] and [[specific heat]] at constant temperature and field are given by, respectively~~

	~~:<math> \chi_T(T,H) = \left( \frac{\partial M}{\partial H} \right)_T </math>~~

	~~and~~

	~~:<math> c_H = -T \left( \frac{\partial^2 f}{\partial T^2} \right)_H~~. </~~math>~~

	~~==Definitions==~~
	The critical exponents <math> \alpha, \alpha', \beta, \gamma, \gamma' </math> and <math> \delta </math> are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows

	~~:<math> M(t,0) \simeq (~~-~~t)^{\beta}\mbox{ for }t \uparrow 0 <~~/~~math>~~

	~~<!-~~- ~~extra line for legibility -->~~

	~~:<math> M(0,H) \simeq \|H\|^{1~~/ ~~\delta} \operatorname{sign}(H)\mbox{ for }H \rightarrow 0 <~~/~~math>~~

	<!-~~- extra line for legibility -->~~

	~~:<math> \chi_T(t,0) \simeq \begin{cases}~~
	~~(t)^{-\gamma}, & \textrm{for} \ t \downarrow 0 \\~~
	~~(-t)^{-\gamma'}, & \textrm{for} \ t \uparrow 0 \end{cases}~~
	~~</math>~~

	~~<!-- extra line for legibility -->~~

	~~:<math> c_H(t,0) \simeq \begin{cases}~~
	~~(t)^{-\alpha} & \textrm{for} \ t \downarrow 0 \\~~
	~~(-t)^{-\alpha'} & \textrm{for} \ t \uparrow 0 \end{cases}~~
	~~</math>~~

	~~where~~

	~~:<math> t \ \stackrel{\mathrm{def}}{=}\ \frac{T-T_c}{T_c}</math>~~

	~~measures the temperature relative to the [[critical point (thermodynamics)\|critical point]].~~

	~~==Derivation==~~
	~~For the magnetic analogue of the [[Maxwell relations]] for the [[response function]]s, the relation~~

	~~:<math> \chi_T (c_H -c_M) = T \left( \frac{\partial M}{\partial T} \right)_H^2 </math>~~

	~~follows, and with thermodynamic stability requiring that <math> c_h, c_M\mbox{ and }\chi_T \geq 0 </math>, one has~~

	~~:<math> c_H \geq \frac{T}{\chi_T} \left( \frac{\partial M}{\partial T} \right)_H^2 </math>~~

	~~which, under the conditions <math> H=0, t<0</math> and the definition of the critical exponents gives~~

	~~:<math> (-t)^{-\alpha'} \geq \mathrm{constant}\cdot(-t)^{\gamma'}(-t)^{2(\beta-1)} </math>~~

	~~which gives the '''Rushbrooke inequality'''~~

	~~:<math> \alpha' + 2\beta + \gamma' \geq 2. </math>~~

	~~Remarkably, in experiment and in exactly solved models, the inequality actually holds as an equality.~~

	~~{{DEFAULTSORT:Rushbrooke Inequality}}~~
	~~[[Category:Critical phenomena]]~~
	~~[[Category:Statistical mechanics~~]]

en>Addbot: Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q5161034

2013-03-15T13:27:09Z

Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q5161034

← Older revision		Revision as of 15:27, 15 March 2013
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	~~Hello buddy~~. ~~Let me introduce myself. I am Ron but I don~~'~~t like when people use my full name. Delaware has usually been my living place~~ and ~~will~~ by ~~no means transfer. Climbing~~ is ~~what love doing~~. ~~Interviewing is what she does but quickly she~~'~~ll be on her own.~~<br><br>~~Also visit my web blog ... extended auto warranty~~ (~~[http~~://~~product~~-~~dev~~.~~Younetco.com~~/~~xuannth~~/~~fox_ios1~~/~~index.php?do~~=/~~profile~~-~~700~~/~~info~~/ ~~Suggested Web page~~])		{{Unreferenced\|date=December 2009}}
			In [[statistical mechanics]], the '''Rushbrooke inequality'''<!-- who is Rushbrooke? --> relates the [[critical exponent]]s of a [[magnetic]] system which exhibits a first-order [[phase transition]] in the [[thermodynamic limit]] for non-zero [[temperature]] ''T''.

			Since the [[Helmholtz free energy]] is [[extensive quantity\|extensive]], the normalization to free energy per site is given as

			:<math> f = -kT \lim_{N \rightarrow \infty} \frac{1}{N}\log Z_N </math>

			The magnetization ''M'' per site in the [[thermodynamic limit]], depending on the external [[magnetic field]] ''H'' and temperature ''T'' is given by

			:<math> M(T,H) \ \stackrel{\mathrm{def}}{=}\ \lim_{N \rightarrow \infty} \frac{1}{N} \left( \sum_i \sigma_i \right) = - \left( \frac{\partial f}{\partial H} \right)_T </math>

			where <math> \sigma_i </math> is the spin at the i-th site, and the [[magnetic susceptibility]] and [[specific heat]] at constant temperature and field are given by, respectively

			:<math> \chi_T(T,H) = \left( \frac{\partial M}{\partial H} \right)_T </math>

			and

			:<math> c_H = -T \left( \frac{\partial^2 f}{\partial T^2} \right)_H. </math>

			==Definitions==
			The critical exponents <math> \alpha, \alpha', \beta, \gamma, \gamma' </math> and <math> \delta </math> are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows

			:<math> M(t,0) \simeq (-t)^{\beta}\mbox{ for }t \uparrow 0 </math>

			<!-- extra line for legibility -->

			:<math> M(0,H) \simeq \|H\|^{1/ \delta} \operatorname{sign}(H)\mbox{ for }H \rightarrow 0 </math>

			<!-- extra line for legibility -->

			:<math> \chi_T(t,0) \simeq \begin{cases}
			(t)^{-\gamma}, & \textrm{for} \ t \downarrow 0 \\
			(-t)^{-\gamma'}, & \textrm{for} \ t \uparrow 0 \end{cases}
			</math>

			<!-- extra line for legibility -->

			:<math> c_H(t,0) \simeq \begin{cases}
			(t)^{-\alpha} & \textrm{for} \ t \downarrow 0 \\
			(-t)^{-\alpha'} & \textrm{for} \ t \uparrow 0 \end{cases}
			</math>

			where

			:<math> t \ \stackrel{\mathrm{def}}{=}\ \frac{T-T_c}{T_c}</math>

			measures the temperature relative to the [[critical point (thermodynamics)\|critical point]].

			==Derivation==
			For the magnetic analogue of the [[Maxwell relations]] for the [[response function]]s, the relation

			:<math> \chi_T (c_H -c_M) = T \left( \frac{\partial M}{\partial T} \right)_H^2 </math>

			follows, and with thermodynamic stability requiring that <math> c_h, c_M\mbox{ and }\chi_T \geq 0 </math>, one has

			:<math> c_H \geq \frac{T}{\chi_T} \left( \frac{\partial M}{\partial T} \right)_H^2 </math>

			which, under the conditions <math> H=0, t<0</math> and the definition of the critical exponents gives

			:<math> (-t)^{-\alpha'} \geq \mathrm{constant}\cdot(-t)^{\gamma'}(-t)^{2(\beta-1)} </math>

			which gives the '''Rushbrooke inequality'''

			:<math> \alpha' + 2\beta + \gamma' \geq 2. </math>

			Remarkably, in experiment and in exactly solved models, the inequality actually holds as an equality.

			{{DEFAULTSORT:Rushbrooke Inequality}}
			[[Category:Critical phenomena]]
			[[Category:Statistical mechanics]]

en>AManWithNoPlan: update lccn to newer template

2012-06-14T04:32:47Z

update lccn to newer template

New page

Hello buddy. Let me introduce myself. I am Ron but I don't like when people use my full name. Delaware has usually been my living place and will by no means transfer. Climbing is what love doing. Interviewing is what she does but quickly she'll be on her own.<br><br>Also visit my web blog ... extended auto warranty ([http://product-dev.Younetco.com/xuannth/fox_ios1/index.php?do=/profile-700/info/ Suggested Web page])