Merge pull request #701 from bobmyhill/add_relaxation_docs

added relaxation documentation

Author: bobmyhill

docs/background.rst

@@ -6,6 +6,7 @@ Here is a bit of background on the methods used to calculate thermoelastic and t
 
 .. include:: background_thermoelastics.txt
 .. include:: background_thermodynamics.txt
+.. include:: background_relaxation.txt
 .. include:: background_solutions.txt
 .. include:: background_composites.txt
 .. include:: background_equilibration.txt

docs/background_relaxation.txt

@@ -0,0 +1,87 @@
+Calculating Relaxed Thermodynamic Properties
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Burnman can calculate relaxed thermodynamic properties for materials that have
+isochemical degrees of freedom. This is done by minimizing the Gibbs free energy
+of the system with respect to the isochemical variables at each pressure and
+temperature point. If these isochemical variables can vary on timescales
+shorter than the timescales of changes in pressure and temperature,
+then the second derivatives of the Gibbs free energy with respect to these variables
+are affected by the relaxation. 
+
+To calculate these properties, we use
+variational calculus. Let the extent of the isochemical reactions be given by the
+vector :math:`\xi`. Let us also define the partial derivative function
+:math:`F(P,T,X,\xi) = \partial G / \partial \xi`, where :math:`G` is the Gibbs free energy
+of the system. The relaxed state is given by the condition :math:`F(P,T,X,\xi) = 0`.
+For simplicity, let us group the pressure, temperature, and composition variables
+into a single vector :math:`Z = (P,T,X)` (we aren't actually interested in the composition,
+but keep it here for completeness). Finally, we define a function
+:math:`\phi(Z) = \xi`, which gives the extent of the isochemical reactions
+at equilibrium for given pressure, temperature, and composition.
+Assuming that the reaction variables are always at equilibrium, we have:
+.. math::
+    F = F(Z, \phi(Z)) = \frac{\partial G}{\partial \xi} (Z, \phi(Z))
+
+Taking the total derivative of this expression with respect to :math:`Z`, we have:
+.. math::
+    \frac{d F}{d Z} = \frac{\partial F}{\partial Z} + \frac{\partial F}{\partial \xi}
+    \frac{\partial \phi}{\partial Z}
+
+Now, since :math:`F(Z, \phi(Z)) = 0` for all :math:`Z`, we also have :math:`d F / d Z = 0`, 
+and therefore:
+.. math::
+     \frac{\partial F}{\partial \xi}
+    \frac{\partial \phi}{\partial Z} = - \frac{\partial F}{\partial Z}
+    
+Note that the matrix :math:`\partial F / \partial \xi` is the Hessian matrix of second derivatives
+of the Gibbs free energy with respect to the isochemical variables:
+.. math::
+
+    \frac{\partial F}{\partial \xi} = \frac{\partial^2 G}{\partial \xi^2}
+
+It is therefore symmetric. Let us define the pseudo-inverse of this matrix as
+:math:`R`, so that :math:`R \partial F / \partial \xi = I`, where :math:`I` is the identity matrix.
+Then we have:
+.. math::
+    \frac{\partial \phi}{\partial Z} = - R \frac{\partial F}{\partial Z}
+
+
+Finally, define a function :math:`g(Z) = G(Z, \phi(Z))`, which gives the Gibbs free energy
+of the system at equilibrium. Using the chain rule, we can write the total derivative of this function with respect to :math:`Z` as:
+.. math::
+    \frac{d g}{d Z} = \frac{\partial G}{\partial Z} + \frac{\partial G}{\partial \xi}
+    \frac{\partial \phi}{\partial Z}
+
+However, since :math:`F = \partial G / \partial \xi = 0`, the second term on the right-hand side
+vanishes, and we have:
+.. math::
+    \frac{d g}{d Z} = \frac{\partial G(Z, \phi(Z))}{\partial Z}
+
+Taking the second derivative of :math:`g` with respect to :math:`Z`, we have:
+.. math::
+    \frac{d^2 g}{d Z^2} = \frac{\partial^2 G}{\partial Z^2} +
+    \frac{\partial^2 G}{\partial Z \partial \xi} \frac{\partial \phi}{\partial Z}
+
+
+Now, substituting in our expression for :math:`\partial \phi / \partial Z`, we have:
+.. math::
+    \frac{d^2 g}{d Z^2} = \frac{\partial^2 G}{\partial Z^2} -
+    \frac{\partial^2 G}{\partial Z \partial \xi} R
+    \frac{\partial F}{\partial Z}
+
+or, equivalently:
+.. math::
+    \frac{d^2 g}{d Z^2} = \frac{\partial^2 G}{\partial Z^2} -
+    \frac{\partial^2 G}{\partial Z \partial \xi} 
+    R
+    \frac{\partial^2 G}{\partial \xi \partial Z}
+
+This expression gives the relaxed second derivatives of the Gibbs free energy
+with respect to pressure, temperature, and composition. These second derivatives
+can then be used to calculate the relaxed thermodynamic properties of the system,
+such as the bulk modulus, thermal expansivity, and heat capacity:
+.. math::
+    K_{P,relaxed} = -V \left(\frac{\partial^2 g}{\partial P^2}\right)^{-1}, \quad
+    \alpha_{relaxed} = \frac{1}{V} \frac{\partial^2 g}{\partial P \partial T}, \quad
+    C_{P,relaxed} = - T \frac{\partial^2 g}{\partial T^2}