Editing Gibbs-Duhem integration

CURRENTLY THIS ARTICLE IS UNDER CONSTRUCTION
== History ==
The so-called Gibbs-Duhem Integration referes to a number of methods that couple
molecular simulation techniques with thermodynamic equations in order to draw
phase coexistence lines.

The method was proposed by Kofke (Ref 1-2).

== Basic Features ==

Consider two thermodynamic phases: <math> a </math> and <math> b  </math>,  at thermodynamic equilibrium at certain conditions.
The thermodynamic equilibrium implies:

* Equal temperature in both phases: <math> T = T_{a} = T_{b} </math>, i.e. thermal equilbirum.
* Equal pressure in both phases <math> p = p_{a} = p_{b} </math>, i.e. mechanical equilbrium.
* Equal chemical potentials for the components <math> \mu_i = \mu_{ia} = \mu_{ib} </math>, i.e. ''material'' equilibrium.

In addition if we are dealing with a statistical mechanics model, with certain parameters that we can represent as <math> \lambda </math> , the
model should be the same in both phases.

== Example: phase equilibria of one-component system ==

Notice: The derivation that follows is just a particular route to perform the integration
* Consider that at given conditions of <math> T , p, \lambda </math> two phases of the systems are at equilibrium, this implies:

: <math> \mu_{a} \left( T, p, \lambda \right) = \mu_{b} \left( T, p, \lambda \right) </math>

Given the thermal equilibrium we can also write:

: <math> \beta \mu_{a} \left( \beta, \beta p, \lambda \right) = \beta \mu_{b} \left( \beta, \beta p, \lambda \right) </math>

where
* <math> \beta = 1/k_B T </math>, where <math> k_B </math> is the [[Boltzmann constant]]
When a differential change of the conditions is performed we wil have for any phase:

: <math> d \left( \beta\mu \right) = \left[ \frac{ \partial (\beta \mu) }{\partial \beta} \right]_{\beta p,\lambda} d \beta +
\left[ \frac{ \partial (\beta \mu) }{\partial (\beta p)} \right]_{\beta,\lambda} d (\beta p) + 
\left[ \frac{ \partial (\beta \mu) }{\partial \lambda} \right]_{\beta,\beta p} d \lambda.
</math> 

Taking into account that <math> \mu </math> is the [[Gibbs energy function|Gibbs free energy]] per particle
: <math> d \left( \beta\mu \right) =  \frac{E}{N}  d \beta +  \frac{ V }{N } d (\beta p)  + 
\left[ \frac{ \partial (\beta \mu) }{\partial \lambda} \right]_{\beta,\beta p} d \lambda.
</math> 

Let us use a bar to design quantities divided by the number of particles: e.g. <math> \bar{E} = E/N; \bar{V} = V/N </math>;
and taking into account the definition:

: <math> \bar{L} \equiv \frac{1}{N} \left[ \frac {\partial (\beta \mu )}{\partial \lambda }\right]_{\beta,\beta p} </math>

Again, let us suppose that we have a phase coexistence at a point given by <math>\left[ \beta_0, (\beta p)_0, \lambda_0 \right]</math> and that
we want to modify slightly the conditions. In order to keep the system at the coexistence conditions:

: <math> d \left[ \beta \mu_{a} - \beta \mu_b \right] = 0 </math>

Therefore, to keep the system on the coexistence conditions, the changes in the variables <math> \beta, (\beta p), \lambda </math> are
constrained to fulfill:

<math>  \left( \Delta  \bar{E} \right) d  \beta + \left( \Delta \bar{V} \right) d (\beta p) + \left(\Delta  \bar{L} \right) d \lambda = 0 </math>

whrere for any porperty <math> X </math> we can define: <math> \Delta X \equiv X_a - X_b </math> (i.e. the difference between the values of the property in the phases).
Taking a path with, for instance constante <math> \beta </math>, the coexistence line will  follow the trajectory produced by the solution of the
differential equation:

<math> d(\beta p) = - \frac{ \Delta \bar{L} }{\Delta \bar{V} } d \lambda. </math>

TO BE CONTINUED

== References ==
#[http://dx.doi.org/10.1080/00268979300100881 David A. Kofke,  Gibbs-Duhem integration: a new method for direct evaluation of phase coexistence by molecular simulation, Mol. Phys.  '''78''' , pp 1331 - 1336 (1993)]
#[http://dx.doi.org/10.1063/1.465023 David A. Kofke,  Direct evaluation of phase coexistence by molecular simulation via integration along the saturation line,  J. Chem. Phys. '''98''' ,pp. 4149-4162 (1993) ]