126 BELL SYSTEM TECHNICAL JOURNAL 



Markoff processes. ^^ We set out directly to calculate the self-covariance for 

 Ri{l) which, by Eq. (2.4), is given by 



R,{l)R,{t + u) = fffir')f{r)c,{r't)c,{r',t + u) dr' dr. (4.1) 



Thus the problem is now reduced to the calculation of Ci(r', t)ci{r, t + u). 

 The quantity Ci{r' , t)ci{r, t + u) is calculated in two steps. First we find 



Ci{t, t -\- u) , the average value of Ci at the point r at the time / + u 

 with the restriction that Ci is known at every point r' with certainty to be 

 Ci(r', /) at the time / (assuming, of course, that u > 0). Then we find that 

 the required self-covariance for d, is given by multiplying the above 



(t+u) 



Ci(r, t -\- u) by Ci{r', i) and averaging over-all values of Ci{r, t) at 



time /; thus: 



ci(r', /)ci(r, / + w)"-^"' = ci(r', Oci(r, t -f- u). (4.2) 



.{t+u) 



Now we assume that C\{r, / + m) is related to c{r\ t) by an integral 



equation, analogous to the Smoluchowski equation, as follows: 



(<+«) /• 



ci(r, t -\- u) = j p(\r - r' \, u)ci(r', t) dr'. (4.3) 



In the case that c represents a concentration as in the diffusion of particles, 

 p(| r — r' 1, ii) dr is the probability that a particle be in the j^-dimensional 

 volume element dr at time / + u when it is known with certainty to be at 

 r' a time /. Now the number of particles in dr' at r' at time / is evidently 

 [c + Ci{r' , t)] dr'; consequently, the probable number of particles in dr at 

 time / + u which were in dr' at time / is p(| r — r' |, u)[c + Ci{r', t)] dr dr' . 

 Integration over dr' gives the total probable number 



{l+u) 



(c -\- Ci(r, t -\- u) ) dr of particles in dr equal to 



( / p(| r — r' I, u)[c + Ci(r', /)] dr' j dr which reduces to 



( c + I p(\r — r' \, u)ci(r', I) dr' I dr. Division by dr and subtraction of f 



from both sides of the equality leads directly to Eq. (4.3). For the case of 

 heat flow in crystal lattices the above picture can be used approximately if 

 one uses the concept of phonons.-" For a diffusional process p(| r — r' |, w) 

 is the normalized singular solution of the diffusion equation-^; thus 



p{\r-r'\,u) = ^^, exp [- 1 r - r' \'/4Du] (4.4) 



i» ix)c cit. 



20 J. Weigle, Experientia, 1, 99-103 (1945). 



2' Chandrasekhar, Rea. Mod. Phys. 15, 1 (1943). 



