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- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book%3A_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07%3A_Mean_Field_Theory_of_Phase_Transitions/7.02%3A_Fluids_Magnets_and_the_Ising_ModelFor the Lennard-Jones system, \(V\ns_{\BR\BR'}=v(\BR-\BR')<0\) is due to the attractive tail of the potential, hence \(J\ns_{\BR\BR'}\) is positive, which prefers alignment of the spins \(\sigma\ns_\B...For the Lennard-Jones system, \(V\ns_{\BR\BR'}=v(\BR-\BR')<0\) is due to the attractive tail of the potential, hence \(J\ns_{\BR\BR'}\) is positive, which prefers alignment of the spins \(\sigma\ns_\BR\) and \(\sigma\ns_{\BR'}\). and \(E\ns_0=\frac{1}{8}Nz\big(\ve\ns_{\ssr{AA}}+\ve\ns_{\ssr{BB}}+2\ve\ns_{\ssr{AB}}\big)\), where \(N\) is the total number of lattice sites and \(z=8\) is the lattice coordination number, which is the number of nearest neighbors of any given site.
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Statistical_Mechanics_(Styer)/09%3A_Strongly_Interacting_Systems_and_Phase_Transitions/9.04%3A_Correlation_Functions_in_the_Ising_ModelJust as the sum of G i over all sites is related to the susceptibility χ, so the integral of g 2 (r) over all space is related to the compressibility κ T . And just as the susceptibility can be integr...Just as the sum of G i over all sites is related to the susceptibility χ, so the integral of g 2 (r) over all space is related to the compressibility κ T . And just as the susceptibility can be integrated twice (carefully, through comparison to the ideal paramagnet) to give the magnetic free energy F(T, H), so the compressibility can be integrated twice (carefully, through comparison to the ideal gas) to give the fluid free energy F(T, V, N).
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book%3A_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06%3A_Classical_Interacting_Systems/6.01%3A_Ising_ModelFor \(Z\), this means the trivial path \(\Gamma=\{\emptyset\}\), while for \(Y\ns_{kl}\) this means finding the shortest length path from \(k\) to \(l\). (If there is no straight line path from \(k\) ...For \(Z\), this means the trivial path \(\Gamma=\{\emptyset\}\), while for \(Y\ns_{kl}\) this means finding the shortest length path from \(k\) to \(l\). (If there is no straight line path from \(k\) to \(l\), there will in general be several such minimizing paths.) Note, however, that the presence of the string between sites \(k\) and \(l\) complicates the analysis of \(g\ns_\Gamma\) for the closed loops, since none of the links of \(\Gamma\) can intersect the string.
