# Misc¶

## Extensive and intensive conjugate quantities¶

For a system described by N extensive quantities $$e_k$$ and N intensive quantities $$i_k$$.

The differential increase in energy $$U$$ per unit volume of the system for a variation of $$e_k$$ is:

$dU = \sum_k i_k d e_k$

The Gibbs potential of the system is defined as:

$G = U - \sum_k i_k e_k$

And, the differential increase of Gibbos potential can be expressed as:

$dG = dU - d\sum_k i_k e_k = \sum_k i_k d e_k - \sum e_k d i_k - \sum i_k d e_k = \sum e_k d i_k$

Therefore, the intensive quantities $$i_k$$ can be defined as partial derivatives of the energy $$U$$ with respect to their extensive conjugate quantities $$e_k$$:

$i_k = \frac{\partial U}{\partial e_k}$

The extensive quantities $$e_k$$ can be defined as partial derivatitives of the Gibbos potential $$G$$ with respect to their intensive conjugate quantities $$i_k$$:

$e_k = - \frac{\partial G}{\partial i_k}$