Monthly Archives: March 2025

Problem #\(1\): What is the Landau-Ginzburg free energy functional \(F[m]\) for the Ising model? Solution #\(1\): It is defined implicitly through: \[e^{-\beta F[m]}=\sum_{\{\sigma_i\}\to_{\text{c.g.}}m(\textbf x)}e^{-\beta E_{\{\sigma_i\}}}\] where \(E_{\sigma_i}=-E_{\text{ext}}\sum_{i=1}^N\sigma_i-E_{\text{int}}\sum_{\langle i,j\rangle}\sigma_i\sigma_j\) is the energy of a given spin microstate \(\{\sigma_i\}\) and “\(\text{c.g.}\)” is … Continue reading →

Problem #\(1\): Define the Poincaré group. Solution #\(1\): In words, the Poincaré group is the isometry group of Minkowski spacetime \(\textbf R^{1,3}\). Mathematically, it is the semidirect product \(\textbf R^{1,3}⋊O(1,3)\) of the normal subgroup \(\textbf R^{1,3}\) of spacetime translations with … Continue reading →