Solution (source code)

= Solution

<Universality of critical phenomena> means that many microscopic models approach the same long-distance <renormalization-group fixed point>. After repeated coarse-graining, <irrelevant operators> lose influence, whereas a few <relevant operators> describe deviations from criticality. Models with the same spatial dimension, <order parameter> symmetry, interaction range and appropriate relevant controls can therefore have the same <universality class> despite very different microscopic constituents.

The <critical exponents>, appropriately normalized scaling functions and certain amplitude ratios are universal. The transition temperature, individual amplitudes, microscopic length scale and analytic background in the <free energy> are generally nonuniversal. The numerical value assigned to an <order parameter> also depends on normalization. <Landau-Ginzburg theory> expresses this distinction by using a common symmetry-allowed functional form while allowing its coefficients to depend on the material. Computing the true <critical exponents> below the <upper critical dimension> requires retaining fluctuations, rather than merely minimizing that functional.