Anomalous scaling in passive scalar advection from anomalous exponents in polymer partition function

The problem of turbulence in liquids is addressed by a recent theory of Brownian relativity. It relates the dynamics of the liquid molecules to the static properties of macromolecules dispersed in a polymer solution. As such, features of the random molecular motion in a turbulent fluid should also be recovered from a (simpler) statistical mechanics analysis of chain configurations. We took advantage of a covariant Vineyard-like approximation, formerly derived to get the van Hove's distribution functions in a dilute polymer solution. It points out an extended Gaussian statistics for correlated systems and a condition of probability conservation, exploited here to deal with a turbulently advected passive scalar by the partition function of a polymer network of given topology. Working with star polymers, a semi-heuristic closed-form expression for the anomalous scaling of passive structure exponents is derived. It agrees with earlier numerical simulations, encouraging the search of turbulent patterns “within” static chain configurations alone.