Date: December 17, 2020

Topic: The canonical (NVT) ensemble

Recall

Which extensive variables are fixed in the canonical ensemble

Give a formula for the canonical partition function

Give the probability for being in a particular microstate in the canonical ensemble

Which is the appropriate thermodynamic potential to use when considering the behaviour of the canonical ensemble

Describe two ways in which one can calculate the ensemble average of the energy

Explain how the heat capacity can be calculated

Definition

In statistical mechanics when we develop models for closed systems that cannot do PV work we use the canonical ensemble. Such systems have constant volume and a constant number of atoms. The energy, however, has to be calculated as an ensemble average. Inserting these requirements into the generalised partition function gives:

$\frac{\textrm{d}S}{k_B} = -\lambda^{(1)} \left\langle \frac{\partial E}{\partial N} \right\rangle \textrm{d}N - \lambda^{(1)} \left\langle \frac{\partial E}{\partial V} \right\rangle \textrm{d}V + \lambda^{(1)} \textrm{d}\langle E \rangle$

Remembering that we can also calculate $\textrm{d}S$ by combining the first and second laws of thermodynamics we arrive at:

$\textrm{d}S = \frac{P}{T} \textrm{d}V - \frac{\mu}{T} \textrm{d}N + \frac{1}{T}\textrm{d}E$

Equating coefficients of $\textrm{d}E$ gives us $\lambda^{(1)} =\frac{1}{k_B T} = \beta$

https://www.youtube.com/watch?v=SbZXEyeP4wk&t=3s&ab_channel=GarethTribello

Partition function

The canonical partition function is equal to:

<aside> 💡 $Z_c(N,V,T) = \sum_i e^{-\beta E_i}$

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where the sum here runs over all microstates that have volume $V$ and number of atoms $N$.

Probability of being in a microstate

The probability of being in any microstate with volume $V$ and number of atoms $N$ is:

<aside> 💡 $p_i = \frac{ e^{-\beta E_i}}{Z_c(N,V,T)}$

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Ensemble averages

By inserting the expression above into $\sum_i p_i$ and differentiating it with respect to $\beta$ it is possible to show that:

<aside> 💡 $\langle E \rangle = \sum_i E_i p_i = -\left( \frac{\partial \ln Z_c(N,V,T)}{\partial \beta} \right)$

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and that:

$\langle (E - \langle E \rangle)^2 \rangle = \left( \frac{\partial^2 \ln Z_c(N,V,T)}{\partial \beta^2} \right)$

Thermodynamic potential

In addition, we can relate the logarithm of the canonical partition function to the Helmholtz free energy using:

<aside> 💡 $F(N,V,T) = - k_B T \log Z_c(N,V,T)$

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Response functions

Last but not least we can show that the constant volume heat capacity (a response function) is related to the ensemble average of the fluctuations in the internal energy via:

$C_v = \frac{1}{k_B T^2} \langle (E - \langle E \rangle)^2 \rangle$

https://www.youtube.com/watch?v=KiLr6ErYCKE&ab_channel=GarethTribello

<aside> 📌 SUMMARY: Closed systems are examined using the canonical ensemble in which the volume and number of atoms are assumed fixed. The canonical partition function is connected to the Helmholtz free energy. The internal energy is calculated as an ensemble average.

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