StM Course Notes - Revision history

David: /* Notes */

2016-06-06T02:00:51Z

‎Notes

2014-06-08T19:17:51Z

‎Gallery

2014-06-08T19:16:19Z

‎Gallery

2014-05-24T23:25:45Z

‎Exercises

2014-05-24T23:24:33Z

‎Exercises

2014-05-24T23:22:47Z

‎Gallery

2014-05-24T23:22:01Z

‎Exercises

2014-05-24T23:21:17Z

‎Exercises

2014-05-24T23:19:38Z

‎Exercises

2014-05-24T23:18:05Z

‎Exercises

@@ Line 15: / Line 15: @@
 Potential for Canonical Ensemble:<div class="mw-collapsible mw-collapsed" style="width:750px">
   \[
-F= U-TS = -kTln(Z)
+F= U-TS = -kT\ln(Z)
 \]
 </div>

@@ Line 462: / Line 462: @@
 [http://physwiki.com/w/images/0/04/StM.pdf StM Course Notes] by [http://www.pa.ucla.edu/content/eric-dhoker-lecture-notes Prof. E. D'Hoker] (with bookmarks).
-[http://physwiki.com/w/images/d/db/Ken_Wilson_Nobel_Lecture.pdf Ken Wilson's Nobel Lecture].
+[http://physwiki.com/w/images/d/db/Ken_Wilson_Nobel_Lecture.pdf Nobel Lecture] by Ken Wilson.
 [http://home.basu.ac.ir/~psu/Books/%5BMehran_Kardar%5D_Statistical_Physics_of_Particles%28BookFi.org%29.pdf Statistical Physics of Particles] by Prof. M. Kardar.

← Older revision		Revision as of 19:16, 8 June 2014
Line 465:		Line 465:

	[http://home.basu.ac.ir/~psu/Books/%5BMehran_Kardar%5D_Statistical_Physics_of_Particles%28BookFi.org%29.pdf Statistical Physics of Particles] by Prof. M. Kardar.		[http://home.basu.ac.ir/~psu/Books/%5BMehran_Kardar%5D_Statistical_Physics_of_Particles%28BookFi.org%29.pdf Statistical Physics of Particles] by Prof. M. Kardar.
		+
		+	[http://paginas.fisica.uson.mx/laura.yeomans/tc/YKL.pdf Problems and Solutions for Statistical Mechanics] by Yung-Kuo Lim.

	'''Biographies'''		'''Biographies'''

@@ Line 450: / Line 450: @@
 Given: \[n_i = \frac{1}{e^{\beta (\epsilon_i -\mu)} + \eta}  \\ S= \sum_{i=1}^{N} S_i = \sum_{i=1}^{N} -kG_i \left[n_i \mathrm{ln}(n_i) - \eta(1-\eta n_i)\mathrm{ln}(1-\eta n_i)\right] \\  N = \sum_{i} n_i G_i \text{ and } E =  \sum_{i} \epsilon_i n_i G_i \\  \Phi_{G.-C.} = E- TS- \mu N =-kT\mathrm{ln}(Z) \]
-Find: \[Z_{FD} = \prod_{i} \left( 1 + e^{-\beta \epsilon_i}\right)  \]\[\  Z_{BE} = \prod_{i} \left( \frac{1}{1 + e^{-\beta \epsilon_i}}\right) =\prod_{i} \left(\sum_{n=0}^{\infty} e^{-\beta \epsilon_i n}\right)  \] in the grand canonical ensemble.
+Find: \[Z_{FD} = \prod_{i} \left( 1 + e^{-\beta \epsilon_i}\right)  \]\[\  Z_{BE} = \prod_{i} \left( \frac{1}{1 - e^{-\beta \epsilon_i}}\right) =\prod_{i} \left(\sum_{n=0}^{\infty} e^{-\beta \epsilon_i n}\right)  \] in the grand canonical ensemble.
 <div class="mw-collapsible mw-collapsed" style="width:1000px">
 The method here is to rewrite $S$ such that we get the LHS of $\Phi_{G.-C.}$ proportional to a logarithm of the partition function.

@@ Line 449: / Line 449: @@
 -->
-Given: \[n_i(\epsilon) = \frac{1}{e^{\beta (\epsilon -\mu)} + \eta}  \\ S= \sum_{i=1}^{N} S_i = \sum_{i=1}^{N} -kG_i \left[n_i \mathrm{ln}(n_i) - \eta(1-\eta n_i)\mathrm{ln}(1-\eta n_i)\right] \\  N = \sum_{i} n_i G_i \text{ and } E =  \sum_{i} \epsilon_i n_i G_i \\  \Phi_{G.-C.} = E- TS- \mu N =-kT\mathrm{ln}(Z) \]
+Given: \[n_i = \frac{1}{e^{\beta (\epsilon_i -\mu)} + \eta}  \\ S= \sum_{i=1}^{N} S_i = \sum_{i=1}^{N} -kG_i \left[n_i \mathrm{ln}(n_i) - \eta(1-\eta n_i)\mathrm{ln}(1-\eta n_i)\right] \\  N = \sum_{i} n_i G_i \text{ and } E =  \sum_{i} \epsilon_i n_i G_i \\  \Phi_{G.-C.} = E- TS- \mu N =-kT\mathrm{ln}(Z) \]
 Find: \[Z_{FD} = \prod_{i} \left( 1 + e^{-\beta \epsilon_i}\right)  \]\[\  Z_{BE} = \prod_{i} \left( \frac{1}{1 + e^{-\beta \epsilon_i}}\right) =\prod_{i} \left(\sum_{n=0}^{\infty} e^{-\beta \epsilon_i n}\right)  \] in the grand canonical ensemble.
 <div class="mw-collapsible mw-collapsed" style="width:1000px">

@@ Line 468: / Line 468: @@
 '''Biographies'''
-[http://en.wikipedia.org/wiki/Willard_Gibbs Willard Gibbs]
+[http://en.wikipedia.org/wiki/Willard_Gibbs Willard Gibbs], [http://en.wikipedia.org/wiki/Ludwig_Boltzmann Ludwig Boltzmann], [http://en.wikipedia.org/wiki/Max_Planck Max Planck], [http://en.wikipedia.org/wiki/Enrico_Fermi Enrico Fermi]
-−
 [http://en.wikipedia.org/wiki/Ludwig_Boltzmann Ludwig Boltzmann]
-−
 [http://en.wikipedia.org/wiki/Max_Planck Max Planck]
-−
-[http://en.wikipedia.org/wiki/Enrico_Fermi Enrico Fermi]
-−
-[http://en.wikipedia.org/wiki/Albert_Einstein Albert Einstein]
 '''Quotes'''

@@ Line 442: / Line 442: @@
 ==Exercises==
-Given $n_i(\epsilon) = \frac{1}{e^{\beta \epsilon} + \eta}$ and $S= \sum_i S_i = \sum_i -kG_i \left[n_i \mathrm{ln}(n_i) - \eta(1-\eta n_i)\mathrm{ln}(1-\eta n_i)\right]$ find the partition functions $Z_{FD, BE}$.
+<!--
 Note for Lim #1009: A better solution may be found [http://physwiki.com/w/images/e/ee/CvCp.pdf here] (courtesy of Elisabeth Mills).