Understanding MinEP and MaxEP

In this posting, I offer an electrical circuits analogy that may help
to explain (my current understanding of) the relationship between
Prigogine's theorem of minimum entropy production (MinEP) and Paltrige's
hypothesis of maximum entropy production (MaxEP). This posting deals
with physics, not biology. Personally, I am doubtful that this subject
has much relevance to biology, but other people seem to think differently,
so here we are.

[use fixed width font for diagrams]
Consider the standard textbook example of a simple electrical circuit
with several resistors in series. Suppressing the return path connecting
the current source to the current sink, the circuit might look like this:

High _______Resistor________Resistor_________Resistor_____ Low
Voltage 1 ohm | 3 ohm | 2 ohm Voltage
| |
___ ___
___ ___
| |

Those things hanging off the bottom of the diagram are capacitors -
ignore them for now.

The high and low voltage levels are boundary conditions, or as I
like to call them "exogenous" variables. The two (as yet undetermined)
voltage levels between the resistors are "endogenous variables". They
are not fixed by the problem statement, but they may be fixed by the
physics.

Now it turns out that there is a unique solution to this problem
(of determining the values of the two endogenous variables). The
current across each of the three resistors must be the same. This
is a "conservation law" constraint that forces a unique solution.
But now let us consider those capacitors. They have the ability,
at least for a short time, to "soak up" current. So, now, with
the "conservation law" constraint relaxed, our problem has many
possible instantaneous solutions. The values of the two intermediate
variables can be pretty much anything.

However, the system will eventually reach a steady state, and in that
steady state, the currents will again be equal across all three
resistors. Is there anything special about this steady state solution
which differentiates it from the enormous variety of non steady state
instantaneous solutions? Yes there is. It is the solution in which
the total power dissipated by the three resistors is minimal.

Of course, we haven't even mentioned temperature, so we probably
shouldn't talk about entropy. But still, minimum power dissipation
is pretty close to the same thing as minimum entropy production.
A set of resistors in series is a pretty good example of Prigogine's
theorem of minimal entropy production.

Notice that Prigogine's theorem doesn't tell us anything we couldn't
have figured out for ourselves from Kirchoff's laws. As an optimization
principle it is not particularly useful. All of the alternative
sub-optimal solutions which this theorem excludes are not even
steady state solutions. All that Prigogine's theorem tells us is
that the unique steady state solution dissipates less power than
any non steady state solution.

Hmmm. If resistors in series gives us MinEP, does resistors in
parallel give us MaxEP? Lets try it. This time, instead of
capacitors, I am going to add some switches to the basic circuit.
____Switch_______Resistor_______
| |
| |
Current___|___Switch_______Resistor_______|____Current
Source | | Sink
| |
|___Switch_______Resistor_______|

Also, instead of a voltage source (a battery) I am going to
use a constant source of current, and let the voltage drop
across the circuit to float.

Start with only one switch closed. A certain amount of power
will be dissipated through one resistor. Now close a second
switch. Power dissipated through the original resistor falls,
and so does total power dissipation. But the perceptive
reader will notice that I seem to have cheated. It I had
used an ordinary battery, rather than a constant current source,
then when I closed the switch, power in the first resistor is
unchanged but TOTAL power dissipation increases. This is
interesting.

It appears to be the case that the more "channels" you have
available for dissipating power, the more power that will be
dissipated. This seems pretty obvious, but is it ALWAYS true?
Well, I'm pretty sure you can prove it true if all of the
electrical components are 'linear' (resistors that obey Ohm's
law, for example). But I'm not sure that there are any
guarantees if you include non-linear components.

But, in any case, we are not really discussing "maximization"
here. What we need to do is to determine whether the steady
state solution to this circuit with all switches closed
will dissipate more power than any non steady state solution.
The endogenous variables would be the branch currents. Do
these currents adjust themselves so that the (unique) steady
state currents are the ones that dissipate the most power?
Well, they can't. In the circuit shown, the voltage drop
across all three resistors must be the same. So there is no
way to vary the endogenous variables so as to see where there
is a maximum or minimum. The trick to get around this problem
is to use a constant current source as I showed originally, but
to replace all of those switches by inductors. That way we
retain the constraint that the three branch currents must sum
to a constant, but we permit the voltage drops across the
resistors to vary. (There may also be an instantaneous voltage
drop or even a voltage increase across the coils, depending on
the history of the current flows.) But, when the circuit finally
settles down to a steady state, there will be no voltage drop
across the coils, and the power dissipation will again be at a
**minimum**.

Perhaps MaxEP only applies if the circuit is more complicated.
Well, as I will say below, there is a sense in which this is true.
However, any complicated circuit must be heirarchically built up
out of simpler circuits, and MinEP seems to apply locally in each
sub-circuit. When you connect together a bunch of simple circuits
to create a complex circuit, and each of the simple circuits
"minimizes" power dissipation, then the composite system must also
"minimize" power dissipation. Right?

So when does MaxEP apply? My current understanding is that it
would apply only to circuits in which there are many non-linear
components, and in which there are multiple possible steady
states. That is, the system has bifurcations - it can switch
into either of two states depending on contingencies. If the
system has a huge number of steady states it may well be the
case that power dissipation is different in each of them. So
which of the available steady states will be chosen? Well,
as I understand Dewar, you can't say for sure, but it is
more likely to be one with high power dissipation than one
with low power dissipation. At least so long as someone
hasn't "cooked" the example.

In Prigogine's MinEP, you are noticing that the unique steady
state has less EP than any nearby non steady state. In the
case of Dewar/Paltrige MaxEP, you are noticing that the steady
state actually chosen has more EP than other available steady
states.

The variables which you treat as the "free" variables are different
in the two cases. You really do minimize over some variables
and maximize over others.

So, is any of this relevant to biology? Well, I'm pretty sure
Prigogine's MinEP applies to a variety of situations in
biochemistry. One example might be the steady state concentrations
of intermediate metabolites along the near-equilibrium part of
a metabolic pathway. But I think that MaxEP just does not
apply. As I said, MaxEP usually works unless someone has
"cooked" the example. But Nature/NS is an excellent cook.
MaxEP will apply to biology only if Nature allows it to.
And a "law" that only applies if it is permitted to apply
(by Someone with Her own priorities) is really not a "law"
at all.


.

Relevant Pages

  • Re: Understanding MinEP and MaxEP
    ... > Prigogine's theorem of minimum entropy production and Paltrige's ... > the current source to the current sink, the circuit might look like this: ... > steady state, the currents will again be equal across all three ... > and so does total power dissipation. ...
    (sci.bio.evolution)
  • Re: transient analysis of linear system
    ... Anyway the circuit is shown above. ... the user but it must be the steady state result. ... The point of all this is to 'see' if the resistors change ... ohms, giving a time constant of 4 seconds, not 4000 seconds. ...
    (sci.electronics.design)
  • Re: transient analysis of linear system
    ... Anyway the circuit is shown above. ... Clearly in steady state it's just a voltage divider of the difference of Vx and Vy. ... The point of all this is to 'see' if the resistors change through the "fog" caused the time varying sources. ... Looks like you have everything wrong, attempting to measuring a circuit parameter that nature is forcing to be constant, meaning you have to measure *current* to detect the resistor changes, the voltage measurements will barely move by ppm and be undiscernible from drift. ...
    (sci.electronics.design)
  • Re: transient analysis of linear system
    ... Anyway the circuit is shown above. ... Clearly in steady state it's just a voltage divider of the difference of Vx and Vy. ... The point of all this is to 'see' if the resistors change through the "fog" caused the time varying sources. ... As the resistor fluctuate at a rate nearly instantaneous relative to the circuit time constants, all voltages remain unchanged, and charge will be circulated through the resistors to maintain those node voltages constant. ...
    (sci.electronics.design)
  • Re: transient analysis of linear system
    ... Anyway the circuit is shown above. ... the user but it must be the steady state result. ... The point of all this is to 'see' if the resistors change ...
    (sci.electronics.design)
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