  We present a simple one-dimensional model with molecular interactions
favouring the formation of clusters with a defined optimal size. Increasing the
density, at low temperature, the system goes from a nearly-ideal gas of
independent molecules to a system with most of the molecules in optimal
clusters, in a way that resembles the formation of micelles in a dilution of
amphiphilic molecules, at the critical micellar concentration. Our model is
simple enough to have an exact solution, but it contains some basic features of
more realistic descriptions of amphiphilic systems: molecular excluded volume
and molecular attractions which are saturated at the optimal cluster. The
comparison between the exact results and the mean field density functional
approximation suggests new approaches to study the more complex and realistic
models of micelle formation; in particular it addresses the long-standing
controversy surrounding separation of internal degrees of freedom in the
formulation of cluster association phenomena.
