Peeter Joot's (OLD) Blog.

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Posts Tagged ‘occupation numbers’

An updated compilation of notes, for ‘PHY452H1S Basic Statistical Mechanics’, Taught by Prof. Arun Paramekanti

Posted by peeterjoot on March 27, 2013

Here’s my second update of my notes compilation for this course, including all of the following:

March 27, 2013 Fermi gas

March 26, 2013 Fermi gas thermodynamics

March 26, 2013 Fermi gas thermodynamics

March 23, 2013 Relativisitic generalization of statistical mechanics

March 21, 2013 Kittel Zipper problem

March 18, 2013 Pathria chapter 4 diatomic molecule problem

March 17, 2013 Gibbs sum for a two level system

March 16, 2013 open system variance of N

March 16, 2013 probability forms of entropy

March 14, 2013 Grand Canonical/Fermion-Bosons

March 13, 2013 Quantum anharmonic oscillator

March 12, 2013 Grand canonical ensemble

March 11, 2013 Heat capacity of perturbed harmonic oscillator

March 10, 2013 Langevin small approximation

March 10, 2013 Addition of two one half spins

March 10, 2013 Midterm II reflection

March 07, 2013 Thermodynamic identities

March 06, 2013 Temperature

March 05, 2013 Interacting spin

plus everything detailed in the description of my first update and before.

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PHY452H1S Basic Statistical Mechanics. Lecture 14: Grand canonical ensemble. Taught by Prof. Arun Paramekanti

Posted by peeterjoot on March 13, 2013

[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]

Disclaimer

Peeter’s lecture notes from class. May not be entirely coherent.

This lecture had a large amount of spoken content not captured in these notes. Reference to section 4 [1] was made for additional details.

Grand canonical ensemble

Fig 1.1: Ensemble pictures

We are now going to allow particles to move to and from the system and the reservoir. The total number of states in the system is

\begin{aligned}\Omega_tot (N, V, E) =\sum_{N_S, E_S} \Omega_S(N_S, V_S, E_S)\Omega_R(N - N_S, V_R, E - E_S),\end{aligned} \hspace{\stretch{1}}(1.2.1)

so for N_S \ll N, and E_S \ll E, we have

\begin{aligned}\Omega_R &= \exp\left( \frac{1}{{k_{\mathrm{B}}}} S_R(N- N_S, V_R, E - E_S) \right) \\ &\approx \exp\left( \frac{1}{{k_{\mathrm{B}}}} S_R(N, V_R, E) - \frac{N_S}{k_{\mathrm{B}}} \left({\partial {S_R}}/{\partial {N}}\right)_{{V, E}} - \frac{E_S}{k_{\mathrm{B}}} \left({\partial {S_R}}/{\partial {E}}\right)_{{N, V}}  \right) \\ &\propto \Omega_S(N_S, V_S, E_S)e^{-\frac{\mu}{k_{\mathrm{B}} T} N_S}e^{-\frac{E_S}{k_{\mathrm{B}} T} },\end{aligned} \hspace{\stretch{1}}(1.2.2)

where the chemical potential \index{chemical potential} and temperature \index{temperature} are defined respectively as

\begin{aligned}\frac{\mu}{T} = -\left({\partial {S_R}}/{\partial {N}}\right)_{{V,E}}\end{aligned} \hspace{\stretch{1}}(1.0.3a)

\begin{aligned}\frac{1}{T} = \left({\partial {S_R}}/{\partial {E}}\right)_{{N,V}}.\end{aligned} \hspace{\stretch{1}}(1.0.3b)

\begin{aligned}\mathcal{P} \propto e^{\frac{\mu}{k_{\mathrm{B}} T} N_S}e^{-\frac{E_S}{k_{\mathrm{B}} T} }.\end{aligned} \hspace{\stretch{1}}(1.0.4)

With \{c\} as the set of all possible configuration pairs \{N_S, E_S\}, we define the grand partition function

\begin{aligned}Z_{\mathrm{G}} = \sum_{\{c\}}e^{\frac{\mu}{k_{\mathrm{B}} T} N_S}e^{-\frac{E_S}{k_{\mathrm{B}} T} }.\end{aligned} \hspace{\stretch{1}}(1.0.5)

So that the probability of finding a given state with energy and particle numbers \{E_S, N_S\} is

\begin{aligned}\mathcal{P}(E_S, N_S) = \frac{e^{\frac{\mu}{k_{\mathrm{B}} T} N_S}e^{-\frac{E_S}{k_{\mathrm{B}} T} }}{Z_{\mathrm{G}}}.\end{aligned} \hspace{\stretch{1}}(1.0.6)

For a classical system we have

\begin{aligned}\{ c \} \rightarrow \{ x \} \{ p \},\end{aligned} \hspace{\stretch{1}}(1.0.7)

whereas in a quantum content we have

\begin{aligned}\{ c \} \rightarrow \text{eigenstate}.\end{aligned} \hspace{\stretch{1}}(1.0.8)

\begin{aligned}Z_{\mathrm{G}}^{\mathrm{\mathrm{\mathrm{QM}}}} = {\text{Tr}}_{\{\text{energy}, N\}} \left( e^{ -\beta (\hat{H} - \mu \hat{N} } \right).\end{aligned} \hspace{\stretch{1}}(1.0.9)

We want to do this because the calculation of the number of states

\begin{aligned}\int_{\{ x \} \{ p \}} \delta\left( \frac{p_1^2}{2m} + \frac{p_2^2}{2m} + \cdots + m g x_1 + m g x_2 + \cdots \right),\end{aligned} \hspace{\stretch{1}}(1.0.10)

can quickly become intractable. We want to go to the canonical ensemble was because the partition function

\begin{aligned}Z_c = \int_{\{ x \} \{ p \}}e^{-\beta \left( \frac{p_1^2}{2m} + \frac{p_2^2}{2m} + \cdots + m g x_1 + m g x_2 + \cdots \right)},\end{aligned} \hspace{\stretch{1}}(1.0.11)

yields the same results, but can be much easier to compute. We have a similar reason to go to the grand canonical ensemble, because this computation, once we allow the number of particles to vary also becomes very hard.

We are now going to define a notion of equilibrium so that it includes

  1. All forces are equal (mechanical equilibrium)
  2. Temperatures are equal (no net heat flow)
  3. Chemical potentials are equal (no net particle flow)

We’ll isolate a subsystem, containing a large number of particles fig. 1.2.

Fig 1.2: A subsystem to and from which particle motion is allowed

When we think about Fermions we have to respect the “Pauli exclusion” principle \index{Pauli exclusion principle}.

Suppose we have just a one dimensional Fermion system for some potential as in fig. 1.3.

Fig 1.3: Energy level filling in a quantum system

For every momentum k there are two possible occupation numbers n_k \in \{0, 1\}

our partition function is

\begin{aligned}Z_c = \sum_{n_k,\sum_k n_k = N} e^{-\beta \sum_k \epsilon_k n_k}.\end{aligned} \hspace{\stretch{1}}(1.0.12)

We’d find that this calculation with this \sum_k n_k = N constraint becomes essentially impossible.

We’ll see that relaxing this constraint will allow this calculation to become tractable.

References

[1] RK Pathria. Statistical mechanics. Butterworth Heinemann, Oxford, UK, 1996.

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