Weight Hessian for ellipsoidal diffusion.
Oi-Oj partial derivatives
~~~~~~~~~~~~~~~~~~~~~~~~~
d2c-2 / / d2dx ddx ddx \ / d2dy ddy ddy \ / d2dz ddz ddz \ \ d2e
------- = 3 | dx**2 | dx ------- + 3 --- . --- | + dy**2 | dy ------- + 3 --- . --- | + dz**2 | dz ------- + 3 --- . --- | | - ------- ,
dOi.dOj \ \ dOi.dOj dOi dOj / \ dOi.dOj dOi dOj / \ dOi.dOj dOi dOj / / dOi.dOj
d2c-1 / d2dz ddz ddz \ / ddy ddz ddz ddy \ / d2dy ddy ddy \
------- = 6 dy**2 | dz ------- + --- . --- | + 12 dy.dz | --- . --- + --- . --- | + 6 dz**2 | dy ------- + --- . --- | ,
dOi.dOj \ dOi.dOj dOi dOj / \ dOi dOj dOi dOj / \ dOi.dOj dOi dOj /
d2c0 / d2dz ddz ddz \ / ddx ddz ddz ddx \ / d2dx ddx ddx \
------- = 6 dx**2 | dz ------- + --- . --- | + 12 dx.dz | --- . --- + --- . --- | + 6 dz**2 | dx ------- + --- . --- | ,
dOi.dOj \ dOi.dOj dOi dOj / \ dOi dOj dOi dOj / \ dOi.dOj dOi dOj /
d2c1 / d2dy ddy ddy \ / ddx ddy ddy ddx \ / d2dx ddx ddx \
------- = 6 dx**2 | dy ------- + --- . --- | + 12 dx.dy | --- . --- + --- . --- | + 6 dy**2 | dx ------- + --- . --- | ,
dOi.dOj \ dOi.dOj dOi dOj / \ dOi dOj dOi dOj / \ dOi.dOj dOi dOj /
d2c2 / / d2dx ddx ddx \ / d2dy ddy ddy \ / d2dz ddz ddz \ \ d2e
------- = 3 | dx**2 | dx ------- + 3 --- . --- | + dy**2 | dy ------- + 3 --- . --- | + dz**2 | dz ------- + 3 --- . --- | | + ------- ,
dOi.dOj \ \ dOi.dOj dOi dOj / \ dOi.dOj dOi dOj / \ dOi.dOj dOi dOj / / dOi.dOj
where
d2e 1 / / / d2dx ddx ddx \ / d2dz ddz ddz \
------- = - | (1 + 3Dr) | dx**2 | dx ------- + 3 --- . --- | + dy**2 | dz ------- + --- . --- |
dOi.dOj R \ \ \ dOi.dOj dOi dOj / \ dOi.dOj dOi dOj /
/ d2dy ddy ddy \ / ddy ddz ddz ddy \ \
+ dz**2 | dy ------- + --- . --- | + 2dy.dz | --- . --- + --- . --- | |
\ dOi.dOj dOi dOj / \ dOi dOj dOi dOj / /
/ / d2dy ddy ddy \ / d2dz ddz ddz \
+ (1 - 3Dr) | dy**2 | dy ------- + 3 --- . --- | + dx**2 | dz ------- + --- . --- |
\ \ dOi.dOj dOi dOj / \ dOi.dOj dOi dOj /
/ d2dx ddx ddx \ / ddx ddz ddz ddx \ \
+ dz**2 | dx ------- + --- . --- | + 2dx.dz | --- . --- + --- . --- | |
\ dOi.dOj dOi dOj / \ dOi dOj dOi dOj / /
/ / d2dz ddz ddz \ / d2dy ddy ddy \
- 2 | dz**2 | dz ------- + 3 --- . --- | + dx**2 | dy ------- + --- . --- |
\ \ dOi.dOj dOi dOj / \ dOi.dOj dOi dOj /
/ d2dx ddx ddx \ / ddx ddy ddy ddx \ \ \
+ dy**2 | dx ------- + --- . --- | + 2dx.dy | --- . --- + --- . --- | | |
\ dOi.dOj dOi dOj / \ dOi dOj dOi dOj / / /
Oi-tm partial derivatives
~~~~~~~~~~~~~~~~~~~~~~~~~
d2c-2
------- = 0,
dOi.dtm
d2c-1
------- = 0,
dOi.dtm
d2c0
------- = 0,
dOi.dtm
d2c1
------- = 0,
dOi.dtm
d2c2
------- = 0.
dOi.dtm
Oi-Da partial derivatives
~~~~~~~~~~~~~~~~~~~~~~~~~
d2c-2
------- = 0,
dOi.dDa
d2c-1
------- = 0,
dOi.dDa
d2c0
------- = 0,
dOi.dDa
d2c1
------- = 0,
dOi.dDa
d2c2
------- = 0.
dOi.dDa
Oi-Dr partial derivatives
~~~~~~~~~~~~~~~~~~~~~~~~~~~
d2c-2 d2e
------- = - 3 -------,
dOi.dDr dOi.dDr
d2c-1
------- = 0,
dOi.dDr
d2c0
------- = 0,
dOi.dDr
d2c1
------- = 0,
dOi.dDr
d2c2 d2e
------- = 3 -------,
dOi.dDr dOi.dDr
where
d2e 1 / / ddx / ddz ddy \ \
------- = ---- | (1 - Dr) | dx**3 --- + dy.dz | dy --- + dz --- | |
dOi.dDr R**3 \ \ dOi \ dOi dOi / /
/ ddy / ddz ddx \ \
- (1 + Dr) | dy**3 --- + dx.dz | dx --- + dz --- | |
\ dOi \ dOi dOi / /
/ ddz / ddy ddx \ \ \
+ 2Dr | dz**3 --- + dx.dy | dx --- + dy --- | | |
\ dOi \ dOi dOi / / /
tm-tm partial derivatives
~~~~~~~~~~~~~~~~~~~~~~~~~
d2c-2
----- = 0,
dtm2
d2c-1
----- = 0,
dtm2
d2c0
---- = 0,
dtm2
d2c1
---- = 0,
dtm2
d2c2
---- = 0.
dtm2
tm-Da partial derivatives
~~~~~~~~~~~~~~~~~~~~~~~~~
d2c-2
------- = 0,
dtm.dDa
d2c-1
------- = 0,
dtm.dDa
d2c0
------- = 0,
dtm.dDa
d2c1
------- = 0,
dtm.dDa
d2c2
------- = 0.
dtm.dDa
tm-Dr partial derivatives
~~~~~~~~~~~~~~~~~~~~~~~~~
d2c-2
------- = 0,
dtm.dDr
d2c-1
------- = 0,
dtm.dDr
d2c0
------- = 0,
dtm.dDr
d2c1
------- = 0,
dtm.dDr
d2c2
------- = 0.
dtm.dDr
Da-Da partial derivatives
~~~~~~~~~~~~~~~~~~~~~~~~~
d2c-2
------ = 0,
dDa**2
d2c-1
------ = 0,
dDa**2
d2c0
------ = 0,
dDa**2
d2c1
------ = 0,
dDa**2
d2c2
------ = 0.
dDa**2
Da-Dr partial derivatives
~~~~~~~~~~~~~~~~~~~~~~~~~
d2c-2
------- = 0,
dDa.dDr
d2c-1
------- = 0,
dDa.dDr
d2c0
------- = 0,
dDa.dDr
d2c1
------- = 0,
dDa.dDr
d2c2
------- = 0.
dDa.dDr
Dr-Dr partial derivatives
~~~~~~~~~~~~~~~~~~~~~~~~~
d2c-2 3 d2e
------ = - - ------,
dDr**2 4 dDr**2
d2c-1
------ = 0,
dDr**2
d2c0
------ = 0,
dDr**2
d2c1
------ = 0,
dDr**2
d2c2 3 d2e
------ = - ------,
dDr**2 4 dDr**2
where
d2e 1 / \
------ = ---- | (6Dr**2 - 9Dr - 1)(dx**4 + 2dy**2.dz**2) + (6Dr**2 + 9Dr - 1)(dy**4 + 2dx**2.dz**2) - 2(6Dr**2 - 1)(ddz*4 + 2dx**2.dy**2) |
dDr**2 R**5 \ /
|