# Prigogine's theorem of minimum entropy production (Pentcho Valev)

Bryant Fujimoto fujimoto at u.washington.edu
Tue Nov 4 15:50:58 EST 1997

```<RUMYM at BGEARN.ACAD.BG> writes:

>Maybe it is not very suitable to present so special problems on this
>forum. On the other hand, nonequilibrium thermodynamics is regarded as
>very promising for the future theoretical biology. I addressed this
>message to you, in the btk group, a few months ago. You were tired and did
>not answer, but Herbert Sauro replied and confirmed my conclusion. As you
>can see, the problem is still existing, and while nobody should care about
>my personal problems, everybody should care about scientific ones.

>That was my message:
>Bryant, I do not find your answers satisfactory. Of course, I do not blame
>you - nonequilibrium thermo is so complicated and abstruse that maybe one
>can only use some final results - verifying all mathematical technicalities
>is too difficult. Still I think that any theory - even the most respectable
>one - should be verified in detail from time to time. That is what I am
>going to do - I hope nobody would find this obtruding - maybe I will even
>   Let me describe an interesting paradox I have recently found in textbooks.
>Maybe I am wrong, maybe the textbooks are wrong, but still the problem should
>somehow be settled.
>   Consider the following process (the notation is somewhat strange, due to
>e-mail restrictions):

>   S1  -v1->  S2  -v2->  S3                             /1/

>S1 is converted into S2 (at a rate v1), and S2 is converted into S3 (at a rate
>v2).

>   v1 = v2 = v                                          /2/

>The affinity is

>   A = mu(S1) - mu(S3)                                  /3/

>Also, it is said in textbooks that, in a steady state,

>  J = LA = dC(S2)/dt = 0                                /4/

>where C(S2) is the concentration of S2.
>   The condition /4/ is essential for the derivation of Prigogine's
>theorem of minimum entropy production:

>   B = JA = LA^2                                        /5/

>where B = T(dSi/dt) is the entropy production. Therefore,

>   dB/dA = 2LA = 2J = 0                                 /6/

>i.e. the conditions of steady state and minimum entropy production are
>equivalent.
>   In my opinion, /4/ is wrong. The flow, J, corresponding to A, is not
>dC(S2)/dt, but

>   J = LA = v                                           /7/

The problem is that the flux (or flow) is defined to be the
rate of change of a thermodynamic variable, in this case the
concentration.  Therefore dC(S2)/dt is the definition of J and /4/
is correct for steady state.  Somebody decided that a good name for
expressions like dC(S2)/dt would be flux or flow.  So whenever the
flux or flow is called for in this theory, that is what you must
use.  If you insert anything else, as you want to do, you will
of course get nonsense.

Bryant Fujimoto
fujimoto at u.washington.edu

[rest deleted]

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