Exercise #4
due date: December 4th
a) In the mean file solution of the Ising model plot the isotherms $h(m)$ in the tree cases $T>T_c$,$T=T_c$,$T<T_c$. Plot the mean field estimate for the free energy $F(T,h)$ as a function of $h$ in the tree cases $T>T_c$,$T=T_c$,$T<T_c$ and discuss the results. I advice to use gnuplot which allows to define parametric functions.
b) Prove the isomorphism between the Ising model in the Canonincal Ensemble and the Lattice Gas model in the Grand-Canonical ensemble i.e. prove that $H_{LG}-\mu <N>$ maps onto $H_{Ising}$ up to some additive constants. After the solution of the previous point you will be able to plot the isotherms in the pressure-volume space for the lattice gas and discuss the results.
In the following you can alternatively choose c1) or c2).
c1) Read the notes about the Kac ring model:
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which set of variables describes a microscopic state?
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which set of variables describes the macroscopic state?
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write a code to calculate the number of black and white point (you can also do it considering a small size say N=4 and doing the evolution by hand…)
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compare the output of the code with the "molecular-chaos" solution given in the notes and discuss the results.
c2) Read the notes about the Logistic map:
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write a code to calculate the map evolution and the Lyapunov exponent.
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discuss the results in the parameters space described in the note discussing the local stability and the Lyapunov exponent.
d) Read the chapter 3 of the textbook (Kerson Huang, "Statistical Mechanics", Second Edition, Wiley). Try to answer to one of the problems of the chapter at your convenience. If you dare you can try exercise 3.5 and for that it would be useful to read this paper