Exercise #11

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.

b) Within the mean-field theory for the Lattice Gas, as the previous point is concerned, derive the equation of state

  • plot the pressure as a function of $V/V_0$ where $V_0=N v_0$ is the total excluded volume.

  • Determine $T_c$

c) Apply the linear response to the Ising model

  • Determine (exactly) the relation between non-local susceptibility $\chi_{i,j}=\frac{\partial }{\partial h_j} m_i$ and the corresponding correlation function

  • Evaluate the susceptibility within mean-field

  • Evaluate the correlation function within the same approximation and discuss the result.