On the statistical mechanics of self-organized profiles

The radial structure of tokamak profiles determined by anomalous transport is elucidated by studying the statistical mechanics of a sand pile automaton for which the toppling conditions depend on local gradient, alone. In this representation, the sand pile dynamics is Markovian, and spatial profiles may be obtained from calculated expectation values of the local gradient. The Markovian structure of the dynamics is exploited to analytically derive a local gradient probability distribution function from a generalized kinetic equation. for homogeneous, weak noise, the calculated expectation value of the gradient is well below the marginally stable state. In the over-driven limit (i.e., strong homogeneous noise), a region of super-critical gradient is shown to form near the bottom of the pile. For the case of localized noise, the mean self-organized profile is always sub-critical. These results are consistent with numerical studies of simple automata. Their relevance to and implications for tokamak confinement are discussed. (C) 1996 American Institute of Physics.