Peeter Joot's (OLD) Blog.

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Posts Tagged ‘variance’

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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open system variance of N

Posted by peeterjoot on March 16, 2013

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Question: Variance of N in open system ([1] pr 3.14)

Show that for an open system

\begin{aligned}\text{var}(N) = \frac{1}{{\beta}} \left({\partial {\bar{N}}}/{\partial {\mu}}\right)_{{V, T}}.\end{aligned} \hspace{\stretch{1}}(1.0.1)

Answer

In terms of the grand partition function, we find the (scaled) average number of particles

\begin{aligned}\frac{\partial {}}{\partial {\mu}} \ln Z_{\mathrm{G}} &= \frac{\partial {}}{\partial {\mu}} \ln \sum_{r,s} e^{\beta \mu N_r - \beta E_s} \\ &= \frac{1}{{Z_{\mathrm{G}}}} \sum_{r,s} \beta N_r e^{\beta \mu N_r - \beta E_s} \\ &= \beta \bar{N}.\end{aligned} \hspace{\stretch{1}}(1.0.2)

Our second derivative provides us a scaled variance

\begin{aligned}\frac{\partial^2 {{}}}{\partial {{\mu}}^2} \ln Z_{\mathrm{G}} &= \frac{\partial {}}{\partial {\mu}} \left( \frac{1}{{Z_{\mathrm{G}}}} \sum_{r,s} \beta N_r e^{\beta \mu N_r - \beta E_s}  \right) \\ &= \frac{1}{{Z_{\mathrm{G}}}} \sum_{r,s} (\beta N_r)^2 e^{\beta \mu N_r - \beta E_s}-\frac{1}{{Z_{\mathrm{G}}^2}} \left( \sum_{r,s} \beta N_r e^{\beta \mu N_r - \beta E_s} \right)^2 \\ &= \beta^2 \left( \bar{N^2} - {\bar{N}}^2  \right)\end{aligned} \hspace{\stretch{1}}(1.0.3)

Together this gives us the desired result

\begin{aligned}\text{var}(N) &= \frac{1}{{\beta^2}}\frac{\partial {}}{\partial {\mu}} \left( \beta \bar{N}  \right) \\ &= \frac{1}{{\beta}}\frac{\partial {\bar{N}}}{\partial {\mu}}.\end{aligned} \hspace{\stretch{1}}(1.0.4)

References

[1] E.A. Jackson. Equilibrium statistical mechanics. Dover Pubns, 2000.

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

Posted by peeterjoot on March 3, 2013

In A compilation of notes, so far, for ‘PHY452H1S Basic Statistical Mechanics’ I posted a link this compilation of statistical mechanics course notes.

That compilation now all of the following too (no further updates will be made to any of these) :

February 28, 2013 Rotation of diatomic molecules

February 28, 2013 Helmholtz free energy

February 26, 2013 Statistical and thermodynamic connection

February 24, 2013 Ideal gas

February 16, 2013 One dimensional well problem from Pathria chapter II

February 15, 2013 1D pendulum problem in phase space

February 14, 2013 Continuing review of thermodynamics

February 13, 2013 Lightning review of thermodynamics

February 11, 2013 Cartesian to spherical change of variables in 3d phase space

February 10, 2013 n SHO particle phase space volume

February 10, 2013 Change of variables in 2d phase space

February 10, 2013 Some problems from Kittel chapter 3

February 07, 2013 Midterm review, thermodynamics

February 06, 2013 Limit of unfair coin distribution, the hard way

February 05, 2013 Ideal gas and SHO phase space volume calculations

February 03, 2013 One dimensional random walk

February 02, 2013 1D SHO phase space

February 02, 2013 Application of the central limit theorem to a product of random vars

January 31, 2013 Liouville’s theorem questions on density and current

January 30, 2013 State counting

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Application of the central limit theorem to a product of random vars

Posted by peeterjoot on February 1, 2013

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Our midterm had a question asking what the central limit theorem said about a product of random variables. Say, Y = X_1 X_2 \cdots X_N, where the random variables X_k had mean and variance \mu and \sigma^2 respectively. My answer was to state that the Central limit theorem didn’t apply since it was for a sum of independent and identical random variables. I also stated the theorem and said what it said of such summed random variables.

Wondering if this was really all the question required, I went looking to see if there was in fact some way to apply the central limit theorem to such a product and found http://math.stackexchange.com/q/82133. The central limit theorem can be applied to the logarithm of such a product (provided all the random variables are strictly positive)

For example, if we write

\begin{aligned}Z = \ln Y = \sum_{k = 1}^N \ln X_k,\end{aligned} \hspace{\stretch{1}}(1.0.1)

now we have something that the central limit theorem can be applied to. It will be interesting to see if this is the answer that the midterm was looking for. It is one that wasn’t obvious enough for me to think of it at the time. In fact, it’s also not something that we can even state a precise central limit theorem result for, because we don’t have enough information to state the mean and variance of the logarithm of the random vars X_k. For example, if the random vars are continuous, we have

\begin{aligned}\left\langle{{\ln X}}\right\rangle = \int \rho(X) \ln X dX.\end{aligned} \hspace{\stretch{1}}(1.0.2)

Conceivably, if we knew all the moments of X we could expand the logarithm in Taylor series. In fact we need more than that. If we suppose that 0 < X < 2 \mu, so that \left\lvert {X/\mu - 1} \right\rvert \le 1, we can write

\begin{aligned}\ln X &= \ln \mu + (X - \mu) \\ &= \ln \mu + \ln \left( { 1 + \left(\frac{X}{\mu} - 1\right) } \right) \\ &= \ln \mu + \sum_{k = 1}^{\infty} (-1)^{k+1} \frac{\left( {\frac{X}{\mu} -1} \right)^k}{k}.\end{aligned} \hspace{\stretch{1}}(1.0.3)

With such a bounding for the random variable X we’d have

\begin{aligned}\left\langle{{\ln X}}\right\rangle = \ln \mu + \sum_{k = 1}^{\infty} \frac{(-1)^{k+1}}{k} \left\langle{{\left( {\frac{X}{\mu} -1} \right)^k}}\right\rangle\end{aligned} \hspace{\stretch{1}}(1.0.4)

We need all the higher order moments of X/\mu - 1 (or equivalently all the moments of X), and can’t just assume that \left\langle{{\ln X}}\right\rangle = \ln \mu.

Suppose instead that we just assume that it is possible to find the mean and variance of the logarithm of the random variables X_k, say

\begin{subequations}

\begin{aligned}\mu_{\mathrm{ln}} = \left\langle{{\ln X}}\right\rangle\end{aligned} \hspace{\stretch{1}}(1.0.5a)

\begin{aligned}\sigma_{\mathrm{ln}}^2 = \left\langle{{(\ln X)^2}}\right\rangle - \left\langle{{\ln X}}\right\rangle^2.\end{aligned} \hspace{\stretch{1}}(1.0.5b)

\end{subequations}

Now we can state that for large N the random variable Z has a distribution approximated by

\begin{aligned}\rho(Z) = \frac{1}{{\sigma_{\mathrm{ln}} \sqrt{2 \pi N}}} \exp\left( - \frac{ (\ln X - N \mu_{\mathrm{ln}})^2}{2 N \sigma_{\mathrm{ln}}^2} \right).\end{aligned} \hspace{\stretch{1}}(1.0.6)

Given that, we can say that the random variable Y = X_1 X_2 \cdots X_N, is the exponential of random variable with the distribution given approximately (for large N) by 1.0.6.

It will be interesting to see if this is the answer that we were asked to state. I’m guessing not. If it was, then a lot more cleverness than I had was expected.

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