M. Chertkov et al., EXACT FIELD-THEORETICAL DESCRIPTION OF PASSIVE SCALAR CONVECTION IN AN N-DIMENSIONAL LONG-RANGE VELOCITY-FIELD, Physics letters. A, 192(5-6), 1994, pp. 435-443
We describe a new functional integral method for the computation of av
erages containing chronological exponentials of random matrices of arb
itrary dimension. We apply these results to the rigorous study of the
statistics of a passive scalar advected by a large-scale N-dimensional
flow. In the delta-correlated case the statistics of the rate of line
stretching appears to be exactly Gaussian at all times and we explici
tly compute the dependence of the mean value and variance of the stret
ching rate on the space dimension N. The probability distribution func
tion of the passive scalar is also exactly computed. Further applicati
ons of our functional integral method are suggested.