Scaling and universality in statistical physics

Abstract

The twin concepts of Scaling and Universality have played an important role in the description of statistical systems. Hydrodynamics contains many applications of scaling including descriptions of the behavior of boundary layers (Prandtl, Blasius) and of the fluctuating velocity in turbulent flow (Kolmogorov, Heisenberg, Onsager). Phenomenological theories of behavior near critical points of phase transitions made extensive use of both scaling, to define the size of various fluctuations, and universality to say that changes in the model would not change the answers. These two ideas were combined via the statement that elimination of degrees of freedom and a concomitant scale transformation left the answers quite unchanged. In Wilson's hands, this mode of thinking led to the renormalization group approach to critical phenomena. Subsequently, Feigenbaum showed how scaling, universality, and renormalization group ideas could be applied to dynamical systems. Specifically, this approach enabled us to see how chaos first arises in those systems in which but a few degrees of freedom are excited. In parallel Libchaber developed experiments aimed at understanding the onset of chaos, the results of which were subsequently used to show that Feigenbaum's universal behavior was in fact realized in honest-to-goodness hydrodynamical systems. More recently, Gemunu Gunaratne, Mogens Jensen, and Itamar Procaccia have indicated that they believe that a different (and weaker) universality might hold for the fully chaotic behavior of low dimensional dynamical systems. Dynamically generated situations often seem to show kinds of scaling and universality quite different from that seen in critical phenomena. A technical difference which seems to arise in these intrinsically dynamical processes is that instead of having a denumerable list of different critical quantities, each with their critical index, instead there is continuum of critical indices. This so-called multifractal behavior may nonetheless show some kinds of universality. And indeed this might be the kind of scaling and universality shown by those hydrodynamical systems in which many degrees of freedom are excited.