32
STANLEI’ C‘ORRSIN
’The r:2 and r,, contributions balarice individually between pwrticles crossing : rronl Ihc: I M ~
directions.
k o r ;I siniplc conscrvativc process. in which P changes only because of the
diffusion, a lirst estimate of r,, can be obtained merely by differentiating
Eq. (14) with respect to z. With D = inYI, this approach shows that a
sufficient condition for the negligibility of the P,, term is equivalent to that
for ncgligibility of the r,,, term.
On the other hand, when the evolution of r ( z , t ) is in part due to another
process. such as a chemical reaction, then the negligibility of the r,, term is
not equivalent to a spatial condition. Instead. we require that
(20)
&/rz I B 1, or T I ~ z z f / L
I 4 1.
5. INHOMOGENEITY A N D NONSTATIONARITY
OF THE TRANSPORT MECHANISM
As rcmarked in the preceding section, this simplest model can also be
generalized to allow for both inhomogeneity and nonstationarity of the
randoni transport mechanism, e.g. of the r m s particle speed and of the mean
free path, V = V ( r , 1 ) and I = I(z. r).
An obvious ad hoc first attempt at generalization is to allow variability of
the diffusivily:
(21)
(22)
~ ( z ,
t ) = - ~ ( z ,
firz ,
rf = mzz + D Z r z ,
lWD I Q I ( w r * I ’
where I) = fnVI. To see when this variability of D can be neglected, we
substitute Eq. (21) into Eq. (2). to get a generalized diffusion equation,
so the condition. written in suggestive form, is
(23)
The validity of Eq. (21) as a second approximation is, however, not
obvious. For example, the result of the power series expansions of the three
functions r, V , and I in the steady but inhomogeneous case (see the Appendix) is
to the first order in “correction.” To discover conditions sufficient for the
viability of Eys. (21) and (22), we substitute Eq. (24) into the balance equation. Thcn the generalized diffusion equation is
STANLEI’ C‘ORRSIN
’The r:2 and r,, contributions balarice individually between pwrticles crossing : rronl Ihc: I M ~
directions.
k o r ;I siniplc conscrvativc process. in which P changes only because of the
diffusion, a lirst estimate of r,, can be obtained merely by differentiating
Eq. (14) with respect to z. With D = inYI, this approach shows that a
sufficient condition for the negligibility of the P,, term is equivalent to that
for ncgligibility of the r,,, term.
On the other hand, when the evolution of r ( z , t ) is in part due to another
process. such as a chemical reaction, then the negligibility of the r,, term is
not equivalent to a spatial condition. Instead. we require that
(20)
&/rz I B 1, or T I ~ z z f / L
I 4 1.
5. INHOMOGENEITY A N D NONSTATIONARITY
OF THE TRANSPORT MECHANISM
As rcmarked in the preceding section, this simplest model can also be
generalized to allow for both inhomogeneity and nonstationarity of the
randoni transport mechanism, e.g. of the r m s particle speed and of the mean
free path, V = V ( r , 1 ) and I = I(z. r).
An obvious ad hoc first attempt at generalization is to allow variability of
the diffusivily:
(21)
(22)
~ ( z ,
t ) = - ~ ( z ,
firz ,
rf = mzz + D Z r z ,
lWD I Q I ( w r * I ’
where I) = fnVI. To see when this variability of D can be neglected, we
substitute Eq. (21) into Eq. (2). to get a generalized diffusion equation,
so the condition. written in suggestive form, is
(23)
The validity of Eq. (21) as a second approximation is, however, not
obvious. For example, the result of the power series expansions of the three
functions r, V , and I in the steady but inhomogeneous case (see the Appendix) is
to the first order in “correction.” To discover conditions sufficient for the
viability of Eys. (21) and (22), we substitute Eq. (24) into the balance equation. Thcn the generalized diffusion equation is