Clas Blomberg - Physics of life-Elsevier Science (2007)

Chapter 14. Thermodynamics formalism and examples 137
If the fluxes and forces are related by linear relations as in (14.50), then the two derivatives in
(14.43) are the same, and we have the relation:
dP
dt
dP d P
x
j
2 2
dt dt
(14.54)
We will go further and show a general inequality in a particular example: heat conduction.
In this case, the energy flux is given by the heat flux:
u
n
t
We use the relation u c V T, and write this in the form:
J q
nc
V
T
t
J
q
(14.55)
For the local and total entropy productions, we have
⎛
s J
1 ⎞
⎛
q P Jq
1 ⎞
⎝⎜
T ⎠⎟
∫
⎝⎜
T ⎠⎟
dV
V
(14.56)
The “force” (X)-derivative concerns the time derivative of the temperature gradient:
dX
P
dt
∫
V
J
q
⎛ 1 ⎞
t ⎝⎜
T ⎠⎟
dV
“V” represents the volume of the system, and dV the volume differential. The order of the
derivatives can be changed, and one can make a partial integration:
dX
P
dt
∫
A
⎛ 1⎞
⎛ 1⎞
Jq
nˆ
dA
J
q
t ⎝⎜
T ⎠⎟
∫
t ⎝⎜
T ⎠⎟
dV
V
The first integral on the RHS is over the boundary surface. We may assume a boundary
relation such that this is zero. For the second one, we use eq. (14.55), which means that
( 1/T
) ⎛ 1 ⎞⎛
T
⎞ (
J )
q
t
⎝⎜
T ( ) .
2
⎠⎟
⎝⎜
t ⎠⎟
nc
V