LINEAR THEORY OF FLUCTUATIONS 127 



where v, as previously defined, is the number of dimensions of the region in 

 which the process occurs. 



Combining Eqs. (4.2) and (4.3) we get 



/. (0 



ci(r', Oci(r, t + u) = j c,{f, t)c,{r", t) p{\ r - r" |, u) dr". (4.5) 



Now using the fact that 



(0 



c,{r', l)c,{r", t) = cyir', t)cy{r", l) (4.6) 



and using the relation 



c,(r', l)c,{r", I) = ^, 5(r' - r") (4.7) 



iproved in Appendix II, Eq. (4.5) reduces to 



ci(r', t)ci(r, t-\-u) = ^,pi\r - r'\, u). (4.8) 



Introducing the expression (4.8) into Eq. (4.1) we obtain at once the desired 

 result 



'Ri{t)Ry{t + u) = ^, fffir)f{r')p{\ r - r' \, u) dr dr' 



(4.9) 



For the sake of comparison with Eq. (3.23) it is necessary to write (4.9) 

 in terms of the Fourier space-transforms of the pertinent quantities. We 

 write 



/W = ff(.k)e"'' 



where 



Also, we write 



P(| r - r' I, u) = ^^J^, exp [-\r - r' f/4Du] 



= (2^. / e^P [-Duk' + ik- (r - r')] dk. 



After introduction of these expressions into (4.9) a short calculation yields 

 the result 



R,(OR,il + u) = (27r)'' ^, f \f{k) 1%-^"^-' dk (4.10) 



