http://www.sklogwiki.org/SklogWiki/api.php?action=feedcontributions&user=Vql&feedformat=atomSklogWiki - User contributions [en]2024-03-28T13:16:33ZUser contributionsMediaWiki 1.41.0http://www.sklogwiki.org/SklogWiki/index.php?title=Temperature&diff=5972Temperature2008-02-19T22:50:58Z<p>Vql: /* Kinetic temperature */</p>
<hr />
<div>{{Stub-general}}<br />
__NOTOC__<br />
The '''temperature''' of a system in [[classical thermodynamics]] is intimately related to the [[zeroth law of thermodynamics]]; two systems having to have the same temperature if they are to be in thermal equilibrium (i.e. there is no net [[heat]] flow between them).<br />
However, it is most useful to have a temperature scale.<br />
By making use of the [[Equation of State: Ideal Gas |ideal gas law]] one can define an absolute temperature<br />
<br />
:<math>T = \frac{pV}{Nk_B}</math><br />
<br />
however, perhaps a better definition of temperature is <br />
<br />
:<math>\frac{1}{T(E,V,N)} = \left. \frac{\partial S}{\partial E}\right\vert_{V,N}</math><br />
==Units==<br />
Temperature has the SI units of ''kelvin'' (K) (named in honour of [[William Thomson]]) The kelvin is the fraction 1/273.16 of the thermodynamic temperature of the [[triple point]] of [[water]].<br />
====External links====<br />
*[http://physics.nist.gov/cuu/Units/kelvin.html NIST reference page]<br />
<br />
==Kinetic temperature==<br />
:<math>T = \frac{2}{3} \frac{1}{k_B} \overline {\left(\frac{1}{2}m_i v_i^2\right)}</math><br />
<br />
where <math>k_B</math> is the [[Boltzmann constant]]. The kinematic temperature so defined is related to the equipartition theorem; for more details, see [http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 Configuration integral].<br />
<br />
==Configurational temperature==<br />
*[http://dx.doi.org/10.1063/1.480995 András Baranyai "On the configurational temperature of simple fluids", Journal of Chemical Physics '''112''' pp. 3964-3966 (2000)]<br />
==Non-equilibrium temperature==<br />
*[http://dx.doi.org/10.1063/1.2743032 Alexander V. Popov and Rigoberto Hernandez "Ontology of temperature in nonequilibrium systems", Journal of Chemical Physics '''126''' 244506 (2007)]<br />
==Inverse temperature==<br />
It is frequently convenient to define a so-called ''inverse'' temperature, <math>\beta</math>, such that<br />
<br />
:<math>\beta := \frac{1}{k_BT}</math><br />
==References==<br />
#William Thomson "On an Absolute Thermometric Scale, founded on Carnot's Theory of the Motive Power of Heat, and calculated from the Results of Regnault's Experiments on the Pressure and Latent Heat of Steam", Philosophical Magazine '''October''' pp. (1848)<br />
#[http://dx.doi.org/10.1088/0026-1394/27/1/002 H. Preston-Thomas "The International Temperature Scale of 1990 (ITS-90)", Metrologia '''27''' pp. 3-10 (1990)]<br />
#[http://dx.doi.org/10.1088/0026-1394/27/2/010 H. Preston-Thomas "ERRATUM: The International Temperature Scale of 1990 (ITS-90)", Metrologia '''27''' p. 107 (1990)]<br />
[[category: Classical thermodynamics]]<br />
[[category: statistical mechanics]]<br />
[[category: Non-equilibrium thermodynamics]]</div>Vqlhttp://www.sklogwiki.org/SklogWiki/index.php?title=1-dimensional_hard_rods&diff=57291-dimensional hard rods2008-02-12T13:07:13Z<p>Vql: /* Canonical Ensemble: Configuration Integral */</p>
<hr />
<div>Hard Rods, 1-dimensional system with [[hard sphere model | hard sphere]] interactions. The statistical mechanics of this system can be solved exactly (see Ref. 1).<br />
== Canonical Ensemble: Configuration Integral ==<br />
<br />
Consider a system of length <math> \left. L \right. </math> defined in the range <math> \left[ 0, L \right] </math>.<br />
<br />
Our aim is to compute the [[partition function]] of a system of <math> \left. N \right. </math> hard rods of length <math> \left. \sigma \right. </math>.<br />
<br />
Model:<br />
<br />
* External Potential; the whole length of the rod must be inside the range:<br />
<br />
: <math> V_{0}(x_i) = \left\{ \begin{array}{lll} 0 & ; & \sigma/2 < x < L - \sigma/2 \\<br />
\infty &; & {\rm elsewhere}. \end{array} \right. </math><br />
<br />
* [[Intermolecular pair potential]]:<br />
<br />
: <math> \Phi (x_i,x_j) = \left\{ \begin{array}{lll} 0 & ; & |x_i-x_j| > \sigma \\<br />
\infty &; & |x_i-x_j| < \sigma \end{array} \right. </math><br />
<br />
where <math> \left. x_k \right. </math> is the position of the center of the k-th rod.<br />
<br />
Consider that the particles are ordered according to their label: <math> x_0 < x_1 < x_2 < \cdots < x_{N-1} </math>; <br />
taking into account the pair potential we can write the canonical partition function <br />
([http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 configuration integral]) <br />
of a system of <math> N </math> particles as:<br />
<br />
: <math><br />
\frac{ Z \left( N,L \right)}{N!} = \int_{\sigma/2}^{L+\sigma/2-N\sigma} d x_0 <br />
\int_{x_0+\sigma}^{L+\sigma/2-N\sigma+\sigma} d x_1 \cdots <br />
\int_{x_{i-1}+\sigma}^{L+\sigma/2-N\sigma+i \sigma} d x_i \cdots <br />
\int_{x_{N-2}+\sigma}^{L+\sigma/2-N\sigma+(N-1)\sigma} d x_{N-1}.<br />
</math><br />
<br />
Variable change: <math> \left. \omega_k = x_k - \left(k+\frac{1}{2}\right) \sigma \right. </math> ; we get:<br />
<br />
:<math><br />
\frac{ Z \left( N,L \right)}{N!} = \int_{0}^{L-N\sigma} d \omega_0 <br />
\int_{\omega_0}^{L-N\sigma} d \omega_1 \cdots <br />
\int_{\omega_{i-1}}^{L-N\sigma} d \omega_i \cdots <br />
\int_{\omega_{N-2}}^{L-N\sigma} d \omega_{N-1}.<br />
</math><br />
<br />
Therefore:<br />
:<math><br />
\frac{ Z \left( N,L \right)}{N!} = \frac{ (L-N\sigma )^{N} }{N!}.<br />
</math><br />
<br />
: <math><br />
Q(N,L) = \frac{ (L-N \sigma )^N}{\Lambda^N N!}.<br />
</math><br />
<br />
== Thermodynamics ==<br />
<br />
[[Helmholtz energy function]]<br />
: <math> \left. A(N,L,T) = - k_B T \log Q \right. </math><br />
<br />
In the thermodynamic limit (i.e. <math> N \rightarrow \infty; L \rightarrow \infty</math> with <math> \rho = \frac{N}{L} </math>, remaining finite):<br />
<br />
:<math> A \left( N,L,T \right) = N k_B T \left[ \log \left( \frac{ N \Lambda} { L - N \sigma }\right) - 1 \right]. </math><br />
<br />
== Equation of state ==<br />
<br />
From the basic thermodynamics, the [[pressure]] [''linear tension in this case''] <math> \left. p \right. </math> can<br />
be written as:<br />
<br />
:<math><br />
p = - \left( \frac{ \partial A}{\partial L} \right)_{N,T} = \frac{ N k_B T}{L - N \sigma};<br />
</math><br />
<br />
:<math><br />
Z = \frac{p L}{N k_B T} = \frac{1}{ 1 - \eta}, <br />
</math><br />
<br />
where <math> \eta \equiv \frac{ N \sigma}{L} </math>; is the fraction of ''volume'' (length) occupied by the rods.<br />
<br />
==References==<br />
#[http://dx.doi.org/10.1103/PhysRev.50.955 Lewi Tonks "The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres", Physical Review '''50''' pp. 955- (1936)]<br />
#[http://dx.doi.org/10.1016/0031-8914(49)90059-2 L. van Hove "Quelques Propriétés Générales De L'intégrale De Configuration D'un Système De Particules Avec Interaction", Physica, '''15''' pp. 951-961 (1949)]<br />
#[http://dx.doi.org/10.1016/0031-8914(50)90072-3 L. van Hove, "Sur L'intégrale de Configuration Pour Les Systèmes De Particules À Une Dimension", Physica, '''16''' pp. 137-143 (1950)]<br />
<br />
[[Category:Models]]<br />
[[Category:Statistical mechanics]]</div>Vqlhttp://www.sklogwiki.org/SklogWiki/index.php?title=User_talk:Carl_McBride&diff=5446User talk:Carl McBride2008-01-25T02:34:07Z<p>Vql: </p>
<hr />
<div>This is the discussion page for the user [[User:Carl McBride |Carl McBride]]. To add a new 'comment' to this page simply click<br />
on the '''+''' symbol found in the tabs at the top of the page.<br />
==VQWiki==<br />
Hola Carl, it is good to get to know you. I stumbled by chance on <br />
your wiki for statistical mechanics, and just added some links, <br />
which you can follow to see my wiki, publications, and contact info. <br />
Any comments would be welcome. Loc [[User:Vql|Vql]] 18:42, 16 January 2008 (CET)<br />
:Dear [[User:Vql|Loc]], thank you very much for the link to your (comprehensive) page concerning the [http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 configurational integral] on [http://clesm.mae.ufl.edu/wiki.pub/index.php/Main_Page VQWiki]. <br />
:Concerning SklogWiki I should like to make a minor distinction; although I am the founder/administrator for SklogWiki, it is not ''my'' wiki; it is for ''everyone'' who shares our interest in stat. mech., thermodynamics, and computer simulation :-D <br />
:All the best, --[[User:Carl McBride | <b><FONT COLOR="#8B3A3A">Carl McBride</FONT></b>]] ([[User_talk:Carl_McBride |talk]]) 18:55, 16 January 2008 (CET)<br />
==Strength of Sklogwiki==<br />
Hola Carl, the strength of Sklogwiki is in the reference to <br />
up-to-date journal articles, even though some Sklogwiki articles need <br />
to be written and/or completed to some extent. It is important to <br />
continue maintain this strength that clearly distinguishes <br />
Sklogwiki from Wikipedia. I linked to some Sklogwiki articles <br />
in my article, and mentioned the above strength of Sklogwiki.<br />
Take care, Loc [[User:Vql|Vql]] 18:44, 19 January 2008 (CET)<br />
:Dear [[User:Vql|Loc]], thank you very much for your comments and links to SklogWiki. I totally agree with your perspective regarding SklogWiki. I personally feel that the placement of SklogWiki with the most potential is between the standard text book on one side, and refereed research articles on the other. SklogWiki is about to complete its first year soon, and most of the work so far has been in setting up the general framework and structure of the Wiki. Now that this is in place, the focus will shift to 'filling out' the stub pages. Any contributions that you can make to such stub pages would obviously be most appreciated. All the best, --[[User:Carl McBride | <b><FONT COLOR="#8B3A3A">Carl McBride</FONT></b>]] ([[User_talk:Carl_McBride |talk]]) 11:14, 21 January 2008 (CET)<br />
<br />
<hr><br />
Hola Carl, here are some ideas that would make Sklogwiki more<br />
visible, different from, but complementing other wikis (e.g.,<br />
Wikipedia, Citizendium), in addition to maintaining the existing<br />
strength of Sklogwiki already mentioned above. It is not necessary<br />
to repeat what other wikis have been doing; it is better to<br />
complement these wikis with something of <i>superior quality</i><br />
where applicable. In other words, develop of niche for Sklogwiki that<br />
distinguishes it from the other wikis.<br />
<br />
To attract contributors to Sklogwiki, it is important to remove<br />
the many pitfalls that beset Wikipedia. For these pitfalls, <br />
many of which were the reason for the existence of Citizendium,<br />
see the very informative Wikipedia article<br />
[http://en.wikipedia.org/wiki/Criticism_of_Wikipedia Criticism of Wikipedia].<br />
<br />
Specifically, what I have in mind is to make Sklogwiki a venue that<br />
academics, particularly university professors and researchers,<br />
would be interested in publishing <i>their</i> articles (which<br />
would not fit in a research journal, such as their lecture notes,<br />
opinion, etc.). <br />
<br />
* explicit authorship: It is an important incentive for academics to own their articles by having their names listed in the byline of their articles.<br />
<br />
* free market of ideas: Allow multiple articles on the same subject by different authors. Sometimes articles on the same subject could have conflicting ideas and opinions; let the readers judge. There are plenty of examples in science where reasonable people would disagree with each other. Let all ideas and opinions on the same subject have equal chance to be expressed by the author(s). An example would be an article by an author on his/her method, which would be critiqued by another author in a different, but parallel article on the same subject.<br />
<br />
* have a range of copyrights (from the most restrictive to the least restrictive) available so author(s) could select selected by the author(s) of each article. Some authors may prefer to have their articles fully copyrighted with all rights reserved; some other authors would select a less restrictive copyright such as the GNU-type copyleft. To this end, one possibility to protect the copyright of the author(s) is to have the most restrictive copyright for the site, and then let each article have its own copyright, which may be less restrictive. By default, it would be the most restrictive copyright that covers all articles.<br />
<br />
* possibility to restrict the editing of an article as decided by the author(s). For example, the author(s) of an article could decide not to have other users modify their work without their knowledge. Some other authors could be open for collaboration. Several issues could be thought of.<br />
** Identity of contributors to an existing article having explicit author(s) in the byline: All contributors to such an existing article should have their identity and credentials revealed; they should not be anonymous users. Such article is like a house in a bucolic village where people don't lock their door, but it does not mean than their house is open for vandalism by anonymous users with unknown credentials. Contributors should be courteous to inform the author(s) of their modifications.<br />
** Listing of co-authors: If a contributor made significant contribution to an existing article, then such contributor could be listed at a co-author, with the agreement of the existing author(s). In case of disagreement, the contributor can take out his/her contribution to create a separate and parallel article on the same subject. This situation is possible since several articles on the same subject are allowed; see above.<br />
<br />
* authors could post their articles in Sklogwiki as well as in other venues (e.g., on the own web site, etc.) in parallel, i.e., there is no restriction where the authors could post their articles.<br />
<br />
* invite well-known authors to contribute: Once the above rules are in place, there is an incentive from academics to contribute. See for example the Stanford Encyclopedia of Philosophy. It is then possible to invite well-known and well-respected researchers to contribute their articles to the site. Some names come to mind: Evans and Searle, Jarzynski, Crooks, Cohen, etc.<br />
<br />
There may be more that can be discussed. The above is a start.<br />
Take care. Loc [[User:Vql|Vql]] 03:34, 25 January 2008 (CET)</div>Vqlhttp://www.sklogwiki.org/SklogWiki/index.php?title=User:Vql&diff=5433User:Vql2008-01-23T14:42:11Z<p>Vql: New page: [http://clesm.mae.ufl.edu/~vql Loc Vu-Quoc]</p>
<hr />
<div>[http://clesm.mae.ufl.edu/~vql Loc Vu-Quoc]</div>Vqlhttp://www.sklogwiki.org/SklogWiki/index.php?title=Ideal_gas_partition_function&diff=5423Ideal gas partition function2008-01-22T17:22:16Z<p>Vql: </p>
<hr />
<div>The [[canonical ensemble]] [[partition function]], ''Q'',<br />
for a system of ''N'' identical particles each of mass ''m'' is given by<br />
<br />
:<math>Q_{NVT}=\frac{1}{N!}\frac{1}{h^{3N}}\int\int d{\mathbf p}^N d{\mathbf r}^N \exp \left[ - \frac{H({\mathbf p}^N,{\mathbf r}^N)}{k_B T}\right]</math><br />
<br />
where ''h'' is [[Planck's constant]], ''T'' is the [[temperature]] and <math>k_B</math> is the [[Boltzmann constant]]. When the particles are distinguishable then the factor ''N!'' disappears. <math>H(p^N, r^N)</math> is the [[Hamiltonian]]<br />
corresponding to the total energy of the system.<br />
''H'' is a function of the ''3N'' positions and ''3N'' momenta of the particles in the system.<br />
The Hamiltonian can be written as the sum of the kinetic and the potential energies of the system as follows<br />
<br />
:<math>H({\mathbf p}^N, {\mathbf r}^N)= \sum_{i=1}^N \frac{|{\mathbf p}_i |^2}{2m} + {\mathcal V}({\mathbf r}^N)</math><br />
<br />
Thus we have <br />
<br />
:<math>Q_{NVT}=\frac{1}{N!}\frac{1}{h^{3N}}\int d{\mathbf p}^N \exp \left[ - \frac{|{\mathbf p}_i |^2}{2mk_B T}\right]<br />
\int d{\mathbf r}^N \exp \left[ - \frac{{\mathcal V}({\mathbf r}^N)} {k_B T}\right]</math><br />
<br />
This separation is only possible if <math>{\mathcal V}({\mathbf r}^N)</math> is independent of velocity (as is generally the case).<br />
The momentum integral can be solved analytically:<br />
<br />
:<math>\int d{\mathbf p}^N \exp \left[ - \frac{|{\mathbf p} |^2}{2mk_B T}\right]=(2 \pi m k_B T)^{3N/2}</math><br />
<br />
Thus we have <br />
<br />
:<math>Q_{NVT}=\frac{1}{N!} \frac{1}{h^{3N}} \left( 2 \pi m k_B T\right)^{3N/2}<br />
\int d{\mathbf r}^N \exp \left[ - \frac{{\mathcal V}({\mathbf r}^N)} {k_B T}\right]</math><br />
<br />
<br />
The integral over positions is known as the <br />
[[#configintegral|''configuration integral'']], <br />
<math>Z_{NVT}</math> (from the German ''Zustandssumme'' meaning "sum over states")<br />
<br />
:<math>Z_{NVT}= \int d{\mathbf r}^N \exp \left[ - \frac{{\mathcal V}({\mathbf r}^N)} {k_B T}\right]</math><br />
<br />
In an [[ideal gas]] there are no interactions between particles so <math>{\mathcal V}({\mathbf r}^N)=0</math>.<br />
Thus <math>\exp(-{\mathcal V}({\mathbf r}^N)/k_B T)=1</math> for every gas particle.<br />
The integral of 1 over the coordinates of each atom is equal to the volume so for ''N'' particles<br />
the ''configuration integral'' is given by <math>V^N</math> where ''V'' is the volume.<br />
Thus we have <br />
<br />
:<math>Q_{NVT}=\frac{V^N}{N!}\left( \frac{2 \pi m k_B T}{h^2}\right)^{3N/2}</math><br />
<br />
If we define the [[de Broglie thermal wavelength]] as <math>\Lambda</math><br />
where<br />
<br />
:<math>\Lambda = \sqrt{h^2 / 2 \pi m k_B T}</math><br />
<br />
one arrives at<br />
<br />
:<math>Q_{NVT}=\frac{1}{N!} \left( \frac{V}{\Lambda^{3}}\right)^N = \frac{q^N}{N!}</math> <br />
where <br />
:<math>q= \frac{V}{\Lambda^{3}}</math><br />
is the single particle translational partition function.<br />
<br />
Thus one can now write the partition function for a real system can be built up from<br />
the contribution of the ideal system (the momenta) and a contribution due to<br />
particle interactions, ''i.e.''<br />
<br />
:<math>Q_{NVT}=Q_{NVT}^{\rm ideal} ~Q_{NVT}^{\rm excess}</math><br />
<br />
==References==<br />
* <span id="configintegral"></span> [http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 Configuration integral (statistical mechanics)]<br />
<br />
[[Category:Ideal gas]]<br />
[[Category:Statistical mechanics]]</div>Vqlhttp://www.sklogwiki.org/SklogWiki/index.php?title=User_talk:Carl_McBride&diff=5394User talk:Carl McBride2008-01-19T17:44:15Z<p>Vql: </p>
<hr />
<div>This is the discussion page for the user [[User:Carl McBride |Carl McBride]]. To add a new section to this page simply click<br />
on the '''+''' symbol found in the tabs at the top of the page.<br />
<br />
Hola Carl, it is good to get to know you. I stumbled by chance on <br />
your wiki for statistical mechanics, and just added some links, <br />
which you can follow to see my wiki, publications, and contact info. <br />
Any comments would be welcome. Loc [[User:Vql|Vql]] 18:42, 16 January 2008 (CET)<br />
:Dear [[User:Vql|Loc]], thank you very much for the link to your (comprehensive) page concerning the [http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 configurational integral] on [http://clesm.mae.ufl.edu/wiki.pub/index.php/Main_Page VQWiki]. <br />
:Concerning SklogWiki I should like to make a minor distinction; although I am the founder/administrator for SklogWiki, it is not ''my'' wiki; it is for ''everyone'' who shares our interest in stat. mech., thermodynamics, and computer simulation :-D <br />
:All the best, --[[User:Carl McBride | <b><FONT COLOR="#8B3A3A">Carl McBride</FONT></b>]] ([[User_talk:Carl_McBride |talk]]) 18:55, 16 January 2008 (CET)<br />
<br />
Hola Carl, the strength of Sklogwiki is in the reference to <br />
up-to-date journal articles, even though some Sklogwiki articles need <br />
to be written and/or completed to some extent. It is important to <br />
continue maintain this strength that clearly distinguishes <br />
Sklogwiki from Wikipedia. I linked to some Sklogwiki articles <br />
in my article, and mentioned the above strength of Sklogwiki.<br />
Take care, Loc [[User:Vql|Vql]] 18:44, 19 January 2008 (CET)</div>Vqlhttp://www.sklogwiki.org/SklogWiki/index.php?title=De_Broglie_thermal_wavelength&diff=5392De Broglie thermal wavelength2008-01-16T19:16:08Z<p>Vql: </p>
<hr />
<div>The '''de Broglie thermal wavelength''' is defined as <br />
<br />
:<math>\Lambda= \sqrt{\frac{h^2}{2\pi mk_BT}}</math><br />
<br />
where<br />
<br />
* ''h'' is the [[Planck constant]]<br />
* ''m'' is the mass <br />
* <math>k_B</math> is the [[Boltzmann constant]]<br />
* ''T'' is the [[temperature]]. <br />
<br />
A detailed derivation of the above expression can be found in<br />
[http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 Configuration integral].<br />
<br />
==References==<br />
#[http://www.ensmp.fr/aflb/LDB-oeuvres/De_Broglie_Kracklauer.htm Louis-Victor de Broglie "On the Theory of Quanta" Thesis (1925)]<br />
#[http://dx.doi.org/10.1088/0143-0807/21/6/314 Zijun Yan, "General thermal wavelength and its applications", Eur. J. Phys. '''21''' pp. 625-631 (2000)]</div>Vqlhttp://www.sklogwiki.org/SklogWiki/index.php?title=User_talk:Carl_McBride&diff=5390User talk:Carl McBride2008-01-16T17:42:36Z<p>Vql: </p>
<hr />
<div>This is the discussion page for the user [[User:Carl McBride |Carl McBride]]. To add a new section to this page simply click<br />
on the '''+''' symbol found in the tabs at the top of the page.<br />
<br />
Hola Carl, it is good to get to know you. I stumbled by chance on <br />
your wiki for statistical mechanics, and just added some links, <br />
which you can follow to see my wiki, publications, and contact info. <br />
Any comments would be welcome. Loc [[User:Vql|Vql]] 18:42, 16 January 2008 (CET)</div>Vqlhttp://www.sklogwiki.org/SklogWiki/index.php?title=Partition_function&diff=5389Partition function2008-01-16T17:34:21Z<p>Vql: </p>
<hr />
<div>The '''partition function''' of a system is given by<br />
<br />
:<math> \left. Z \right.= {\mathrm {Tr}} \{ e^{-\beta H} \}</math><br />
<br />
where ''H'' is the [[Hamiltonian]]. The symbol ''Z'' is from the German ''Zustandssumme'' meaning "sum over states". The [[canonical ensemble]] partition function of a system in contact with a thermal bath<br />
at temperature <math>T</math> is the normalization constant of the [[Boltzmann distribution]]<br />
function, and therefore its expression is given by<br />
<br />
:<math>Z(T)=\int \Omega(E)\exp(-E/k_BT)\,dE</math>,<br />
<br />
where <math>\Omega(E)</math> is the [[density of states]] with energy <math>E</math> and <math>k_B</math><br />
the [[Boltzmann constant]]. <br />
<br />
In classical statistical mechanics, there is a close connection <br />
between the partition function and the <br />
[http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 configuration integral], <br />
which has played an important role in many applications <br />
(e.g., drug design).<br />
<br />
==Helmholtz energy function==<br />
The partition function of a system is related to the [[Helmholtz energy function]] through the formula<br />
<br />
:<math>\left.A\right.=-k_BT\log Z.</math><br />
<br />
This connection can be derived from the fact that <math>k_B\log\Omega(E)</math> is the<br />
[[entropy]] of a system with total energy <math>E</math>. This is an [[Extensive properties | extensive magnitude]] in the<br />
sense that, for large systems (i.e. in the [[thermodynamic limit]], when the number of<br />
particles <math>N\to\infty</math><br />
or the volume <math>V\to\infty</math>), it is proportional to <math>N</math> or <math>V</math>.<br />
In other words, if we assume <math>N</math> large, then<br />
<br />
:<math>\left.k_B\right. \log\Omega(E)=Ns(e),</math><br />
<br />
where <math>s(e)</math> is the entropy per particle in the [[thermodynamic limit]], which is<br />
a function of the energy per particle <math>e=E/N</math>. We can<br />
therefore write<br />
<br />
:<math>\left.Z(T)\right.=N\int \exp\{N(s(e)-e/T)/k_B\}\,de.</math><br />
<br />
Since <math>N</math> is large, this integral can be performed through [[steepest descent]],<br />
and we obtain<br />
<br />
:<math>\left.Z(T)\right.=N\exp\{N(s(e_0)-e_0/k_BT)\}</math>,<br />
<br />
where <math>e_0</math> is the value that maximizes the argument in the exponential; in other<br />
words, the solution to<br />
<br />
:<math>\left.s'(e_0)\right.=1/T.</math><br />
<br />
This is the thermodynamic formula for the inverse temperature provided <math>e_0</math> is<br />
the mean energy per particle of the system. On the other hand, the argument in the exponential<br />
is<br />
<br />
:<math>\frac{1}{k_BT}(TS(E_0)-E_0)=-\frac{A}{k_BT}</math><br />
<br />
the thermodynamic definition of the [[Helmholtz energy function]]. Thus, when <math>N</math> is large,<br />
<br />
:<math>\left.A\right.=-k_BT\log Z(T).</math><br />
==Connection with thermodynamics==<br />
We have the aforementioned [[Helmholtz energy function]],<br />
<br />
:<math>\left.A\right.=-k_BT\log Z(T)</math><br />
<br />
we also have the [[internal energy]], which is given by<br />
<br />
:<math>U=k_B T^{2} \left. \frac{\partial \log Z(T)}{\partial T} \right\vert_{N,V}</math><br />
<br />
and the pressure, which is given by<br />
<br />
:<math>p=k_B T \left. \frac{\partial \log Z(T)}{\partial V} \right\vert_{N,T}</math>.<br />
<br />
These equations provide a link between [[Classical thermodynamics | classical thermodynamics]] and <br />
[[Statistical mechanics | statistical mechanics]]<br />
==See also==<br />
*[[Ideal gas partition function]]<br />
[[category:classical thermodynamics]]<br />
[[category:statistical mechanics]]</div>Vqlhttp://www.sklogwiki.org/SklogWiki/index.php?title=Chemical_potential&diff=5387Chemical potential2008-01-16T17:27:41Z<p>Vql: /* Statistical mechanics */</p>
<hr />
<div>==Classical thermodynamics==<br />
Definition:<br />
<br />
:<math>\mu=\left. \frac{\partial G}{\partial N}\right\vert_{T,p} = \left. \frac{\partial A}{\partial N}\right\vert_{T,V}</math><br />
<br />
where <math>G</math> is the [[Gibbs energy function]], leading to <br />
<br />
:<math>\mu=\frac{A}{Nk_B T} + \frac{pV}{Nk_BT}</math><br />
<br />
where <math>A</math> is the [[Helmholtz energy function]], <math>k_B</math><br />
is the [[Boltzmann constant]], <math>p</math> is the pressure, <math>T</math> is the temperature and <math>V</math><br />
is the volume.<br />
<br />
==Statistical mechanics==<br />
The chemical potential is the derivative of the [[Helmholtz energy function]] with respect to the <br />
number of particles<br />
<br />
:<math>\mu= \left. \frac{\partial A}{\partial N}\right\vert_{T,V}=\frac{\partial (-k_B T \ln Z_N)}{\partial N} = -\frac{3}{2} k_BT \ln \left(\frac{2\pi m k_BT}{h^2}\right) + \frac{\partial \ln Q_N}{\partial N}</math><br />
where <math>Z_N</math> is the [[partition function]] for a fluid of <math>N</math><br />
identical particles<br />
:<math>Z_N= \left( \frac{2\pi m k_BT}{h^2} \right)^{3N/2} Q_N</math><br />
and <math>Q_N</math> is the <br />
[http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 configurational integral]<br />
:<math>Q_N = \frac{1}{N!} \int ... \int \exp (-U_N/k_B T) dr_1...dr_N</math><br />
<br />
==Kirkwood charging formula==<br />
See Ref. 2<br />
<br />
:<math>\beta \mu_{\rm ex} = \rho \int_0^1 d\lambda \int \frac{\partial \beta \Phi_{12} (r,\lambda)}{\partial \lambda} {\rm g}(r,\lambda) dr</math><br />
<br />
where <math>\Phi_{12}(r)</math> is the [[intermolecular pair potential]] and <math>{\rm g}(r)</math> is the [[Pair distribution function | pair correlation function]].<br />
==See also==<br />
*[[Ideal gas: Chemical potential]]<br />
==References==<br />
#[http://dx.doi.org/10.1007/s10955-005-8067-x T. A. Kaplan "The Chemical Potential", Journal of Statistical Physics '''122''' pp. 1237-1260 (2006)]<br />
#[http://dx.doi.org/10.1063/1.1749657 John G. Kirkwood "Statistical Mechanics of Fluid Mixtures", Journal of Chemical Physics '''3''' pp. 300-313 (1935)]<br />
#[http://dx.doi.org/10.1119/1.17844 G. Cook and R. H. Dickerson "Understanding the chemical potential", American Journal of Physics '''63''' pp. 737-742 (1995)]<br />
[[category:classical thermodynamics]]<br />
[[category:statistical mechanics]]</div>Vql