Diffusion of a passive impurity in a random velocity field

The method of characteristic functional is used for statistical description of the impurity concentration distribution in the random velocity field. A closed set of equations for a mean concentration response to a change of the external source density and for a pair correlation of the concentration fluctuations is obtained. 1. Statement of the problem The processes of a diffusion in a random velocity field are described by the equation           (0) , , , , , , = 0 L C r t u r t C r t r t C u           (1.1) Here   , C r t is the impurity concentration at the space-time point { , } r t , (0) = t L     ; is the molecular diffusivity; ( , ) r t  is the density of a impurity sources, and   , u r t  is the random velocity field given statistically and described by a normal distribution with a given pair correlation function       1 1 2 2 1 1 2 2 , , = , ; , u r t u r t D r t r t     Next, let us introduce a digital notation for the space-time point 1 1 { , } r t according to the condition 1 1 { , } 1 r t  , 1 1 d1 = d d r t  and we will imply an integrating over repeated digital notation and a summing over the components of vector functions (the Einstein rule). In this notation equation (1.1) takes the form               (0) 1,2 2 1| 2,3 2 3 1 = , , = 0 L C V u C C u         (1.2)           (0) (1) (1) (3) 1,2 1 2 , 1| 2,3 1 2 1 3 t L V                   2. Characteristic functional and its representation The concentration (1) C is a random field and the problem of its statistical description in terms of the statistical moments of various orders in various space-time points arises. To do this, one has to know the density distribution of a probability to find a given realization of the concentration field at a given source density   , P C  that can be defined as a mean value of the   functional, which is a functional analog of the   function generalized to the case of field description.   , = (1) (1) P C C C         (2.1) 1 APhM2016 IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 815 (2017) 012014 doi:10.1088/1742-6596/815/1/012014 International Conference on Recent Trends in Physics 2016 (ICRTP2016) IOP Publishing Journal of Physics: Conference Series 755 (2016) 011001 doi:10.1088/1742-6596/755/1/011001 Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd here (1) C is the realization of the field concentration at the point 1 , which is a solution to the equation [ , , ] = 0 C u    at given realizations of the velocity field u and the source density  , the averaging is carried out over the realization of concentration field, which is reduced to averaging over the velocity field realizations u . Using a functional analog of the formula well-known in the theory of generalized functions 0 [ ( )] = ( ) | ( ) |, x x x x       where 0 x is the solution to the equation 0 = ) (x  , one gets       , , (1) (1) = , , (1) C u C C C u C              (2.2) The calculation of the Jacobian of functional transformation [ , , ] / C u C     shows that the result turns out to be independent of the concentration due to a linear dependence of [ , , ] C u   on C as well as on the random quantity u . Since, the Jacobian in equation (2.2) may be included in a redefinition of the probabilistic measure and hence one can take the value of Jacobian in equation (2.2) be equal to unity. The characteristic functional  is a functional Fourier-transform of the probability density [ , ] P C  [ , ] = d[ ] [ , ]exp{ (1) (1)} = d[ ] { [ , , ]} exp{ (1) (1)} C P C i C C C u i C                 (2.3) Next we apply the formula of the   functional representation, which is a functional analog of the   function representation in the form of the Fourier-series expansion { [ , , ]} = d[ ]exp{ (1) [ (1), (1), (1)]} C u i C u           (2.4) As the result we obtain a representation of the characteristic functional [ , ]    in the form of a functional integral over two fields  and C . (0) [ , ] = d[ ] d[ ]exp{ [ (1) (1) (1) (1,2) (2) (1) (1)]} C i C L C                        exp 1 1| 2,3 2 3 i V u C     (2.5) Using the formula for calculating mean value in the case of a central normal distribution           exp 1 1| 2,3 3 2 = i V C u                1 = exp 1 1 | 2,3 3 1 1 | 2 ,3 (3 ) 2, 2 2 V C V C D                   (2.6) we obtain   (0) [ , ] = d[ ] d[ ]exp [ (1) (1) (1) (1,2) (2) (1) (1)] 1 2 (1,1 ;3,3 ) (1) (1 ) (3) (3 ) C i C L C A C C                      (2.7) (1,1 ;3,3 ) = (1|2,3) (1 |2 ,3 ) (2,2 ) A V V D          2 APhM2016 IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 815 (2017) 012014 doi:10.1088/1742-6596/815/1/012014