The isothermal isobaric ensemble

The isothermal isobaric ensemble

The isothermal isobaric ensemble is a statistical model that can be used to describe the behavior of a system with a constant temperature, a constant number of atoms and a constant pressure. If we have a Hamiltonian, $H(x,p)$, for the system the isothermal-isobaric partition function can be calculated using: $$ Z_i = \sum_i e^{-\beta H(x_i,p_i) } e^{-\beta P V(x_i,p_i) } $$ The sum here runs over all the microstates in the system and is replaced by an integral if you have a continuous rather than a discrete state space. $\beta$, meanwhile, is equal to one divided by the temperature, $T$, multiplied by Boltzmann's constant, $k_B$. That is to say $\beta = \frac{1}{k_B T}$. The probabibility of being with an individual microstate in the isothermal-isobaric ensemble is given by: $$ p_i = \frac{ e^{-\beta H(x_i,p_i)} e^{-\beta P V(x_i,p_i) } }{ Z_i } $$ We can thus calculate the average volume of the system by computing an emsemble average using: $$ \langle V \rangle = \sum_i V(x_i,p_i) p_i $$ or by using the analytical result below: $$ \langle V \rangle = -\frac{ \partial \ln Z_i }{\partial (\beta P) } $$ There is a similar result that we can use to calculate the isothermal compressibility: $$ \kappa_T = -\frac{1}{V} \left( \frac{\partial V}{\partial P} \right)_T = \frac{k_B T}{\langle V \rangle} \left\langle ( V - \langle V \rangle )^2 \right\rangle = \frac{1}{\langle V \rangle} \left( \frac{ \partial^2 \ln Z }{\partial (\beta P)^2 } \right) $$

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