The renormalization group and optimization of entropy

We illustrate the possible connection that exists between the extremal prop erties of entropy expressions and the renormalization group (RG) approach w hen applied to systems with scaling symmetry. We consider three examples: ( 1) Gaussian fixed-point criticality in a fluid or in the capillary-wave mod el or an interface; (2) Levy-like random walks with self-similar cluster fo rmation; and (3) long-ranged bond percolation. In all cases we find a decre asing entropy function that becomes minimum under an appropriate constraint at the fixed point. We use an equivalence between random-walk distribution s and order-parameter pair correlations in a simple fluid or magnet to stud y how the dimensional anomaly at criticality relates to walks with long-tai led distributions.