|
|
| Line 1: |
Line 1: |
| {{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]]
| |
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.
Also visit my web site; extended car warranty [try this web-site]