The canonical ensemble

The canonical ensemble

The canonical 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 volume. If we have a Hamiltonian, $H(x,p)$, for the system the canonical partition function can be calculated using: $$ Z_c = \sum_i e^{-\beta H(x_i,p_i) } $$ 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 canonical ensemble is given by: $$ p_i = \frac{ e{-\beta H(x_i,p_i) } }{ Z_c } $$ We can thus calculate the average energy of the system by computing an emsemble average using: $$ \langle E \rangle = \sum_i H(x_i,p_i) p_i $$ or by using the analytical result below: $$ \langle E \rangle = - \frac{ \partial \ln Z }{\partial \beta } $$ There is a similar result that we can use to calculate the heat capacity: $$ C_v = \frac{1}{k_B T^2} \langle ( E - \langle E \rangle )^2 \rangle = \frac{1}{k_B T^2} \left( \frac{ \partial^2 \ln Z }{\partial \beta^2 } \right) $$

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