Exercise #05

due date: oral exam.

You can alternatively choose:

  • exercises a) and b)

  • point c) or d)

a) 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.

  • Discuss the mean-field phase transition found in the Ising model in the context of the lattice-gas.

b) After reading chapter II of [R.J. Baxter - Exactly solved models in statistical mechanics-Academic Press (1982)].

  • Solve the Ising one-dimensional model with the transfer matrix method and prove that there is no phase transition at non zero temperature.

  • Discuss the zero temperature limit.

c) Following the instruction here below

http://www.aquila.infn.it/ciuchi/didattica/DOTT/OpenSystems/index.html

try to compile and run the program Oscillators (a fortran compiler is needed).

Once compiled successfully try exercises #1 #2 #3.

d) Derive the Landau theory for the superconducting transition after watching the following recorded lecture:

You can also follow a set of notes:

For reading you can see the book of Linda E. Reichl "A Modern Course in Statistical Physics" chapt. 4 and chapt. 6.11