Scaling: another look

2011 October 10
by Daniel Lakeland

I just got a copy of Scaling, a book by G. I. Barenblatt on the use of similarity and dimensional analysis and so called "intermediate asymptotics". The idea is that for many physical systems there is a domain of the dynamic variables in which the solution has become essentially independent of the initial conditions, but not yet in a long-time stable equilibrium solution. So for example you have time t so that t_0 << t and t << t_\infty. In this intermediate range between initial conditions, and final conditions lies a large portion of the stuff we're interested in. Often we can find some result which is true in this region by virtue of the fact that there is an approximate symmetry in the problem. The symmetry holds because we can neglect the dependence on the details of the initial conditions, and we can neglect things that only become dominant at long times.

The description given above doesn't really do the book justice. This is a very thin book and very readable. I recommend it highly, and I'm going to look into his other more comprehensive book as well.

One thing about this book is that the topic expands on ideas I thought about while reading on renormalization group. For example, the idea of a "nonlinear eigenvalue", that is a parameter which controls the self-similarity of a result. This is very similar in principle to the fixed point in a renormalization group solution.

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