ON THE STATE OF MINIMUM ENTROPY PRODUCTION


C.D. Andriesse


Faculty of Physics and Astronomy, Utrecht University, Princetonplein 5, NL - 3584 CC Utrecht, The Netherlands


August 14, 2002 – made available on the internet: February 20, 2008


Davor Juretić notes that the relations I derived for the case of minimum entropy production [l] pertain to a state without entropy production, and argues that they therefore cannot be used for a treatment of photosynthesis [2]. My claim is that this state is highly relevant. Before giving my arguments, let me define the quantities we talk about, in order to avoid confusion.

The energy conservation law (du/dt) = (∂u/t) + ∇ • j = 0, where u is the energy density, t the time and j = v u the energy flux at velocity v, taken together with the identity (l/T)(∂u/∂t) = (l/T)(∂u/∂s)(∂s/∂t) ≡ (∂s/∂t), where T is the temperature and s the entropy density, gives us the entropy production law (∂s/∂t) = – (1/T) j = j(1/T) – • (j/T). In this expression, the last term is normally neglected, with the argument that (j/T) can have no divergence when the system is closed. When the system is open, • (j/T) can have any value and the equal sign = is wrong, since energy is no more conserved. So, for an open system the entropy production is undetermined.

Only for a closed system we have (∂/∂t)(s/k) = – j/(kT) • (T)/T, where Boltzmann’s constant k is introduced and (1/T) is replaced by – (T)/T2. In this arrangement, we get at the left hand side the increase of the logarithm of the state-probability in a unit volume, and at the right hand side the in-product of a flux of normalized energy ‘particles’ j/(kT), with dimension [m-2 s-1], and a normalized ‘force’ (T)/T, with dimension [m-1], driving this flux. The normalization allows us to apply this result to other ‘particles’ and ‘forces’ which also give rise to entropy production. Adding material particles and diffusion forces (density gradients), we obtain (∂/∂t)(s/k) = – jD • (n)/n – jT • (T)/T, abbreviated as š = – jDXjT . Y, where jD is the flux of the material particles, n their density, and jT stands for the above j/(kT). The last equation has been the starting point of my calculations [l].

The formula I used and its formal derivation have been thoroughly discussed by De Groot and Mazur [3]. They arrive at (dΔS/dt) = Σi Ji Xi – see their page 39 – where the fluxes Ji and the ‘forces’ Xi are defined in an abstract way (S, the entropy, is related to the entropy density s by s = nS), but further on in their book we find applications where Ji appears to have the dimension of a real flux, namely [m-2 s-1], and Xi that of a gradient, namely [m-1]. So, my starting point is in line with the fundamental text on non-equilibrium thermodynamics. Hill’s book [4], to which Juretić refers, gives a seemingly similar formula: T(dS/dt) = J1 X1 + J2 X2 see Hill’s page 39. However, here the ‘fluxes’ J1 and J2 appear to have the dimension of a rate, namely [s-1], and the ‘forces’ X1 and X2 that of an energy, namely [kg m2 s-2], as the preceding pages make clear. So, despite the resemblance, Hill’s formula is not identical to mine, which means that one cannot identify jD with J1 and jT with J2, or X with X1 and Y with X2. Considering the dimensions, this formula does define a change of entropy, but not necessarily the one in accordance with energy conservation. It is not derived or justified.

Having defined the quantities we talk about, let’s turn to the content of my calculation. The minimum of entropy production, found by differentiating š{X,Y} with respect to X and Y and putting the derivatives equal to zero, depends on assumptions about jD{X,Y} and jT{X,Y}. When these dependencies are assumed to be linear, the consequence is that, in this minimum, there is no entropy production (š = 0), as Juretić has noted [2]. When the dependencies are assumed to be slightly non-linear, this is only true in approximation (š ≈ 0) [5]. The coefficients in these linear, or slightly non-linear, relations, such as D (diffusion coefficient) and λ (thermal conductivity) have to be determined in experiments. Transport properties such as D and λ cannot be specified, and determined, in terms of the rates J and energies X.

What can we say about a state where the entropy production is exactly (or approximately) zero? Of course, it can be a dead (or an almost dead) system, without fluxes of matter and energy. But when such fluxes are present, it cannot be called dead. It then describes the dynamic equilibrium whereby a flux of energy produces just as much (or about as much) entropy as is consumed in a flux of matter, for instance by the creation of ordered structures. Wouldn’t this nicely fit to what is going on in plants? Consider the entropy change in the chemical reaction


6 H2O (liquid) + 6 CO2 (gas) → C6H12O6 (solid) + 6 O2 (gas)


where the solid D-glycose is taken as an example of the carbohydrates formed in photosynthesis. In fact, 6 moles of water, of gaseous carbon dioxide and of gaseous oxygen have an entropy of respectively 419, 1282 and 1230 J/K at standard temperature and pressure [6], while under these standard conditions 1 mol of solid D-glycose has an entropy of 213 J/K [7]. The reactants at the left-hand side together have 1701 J/K and the products at the right-hand side have 1443 J/K, which is 358 J/K less. Clearly entropy is consumed. (In a qualitative sense, it remains true if a possibly more appropriate entropy-value of D-glycose were known, namely the one in aqueous solution, since it should be between 213 and 419 J/K.) The above consumption of entropy in the formation of carbohydrates will be precisely, or largely, balanced by the production of entropy in the degradation of photons. (When they are absorbed their temperature is about 6000 K, and when they are emitted that temperature is about 300 K, so that their entropy has become about 20 times larger.)

Let me finally address Juretić’s statement that – (X2 J2)/(X1 J1), which he erroneously identifies with – (jTY)/(jD X), would be the efficiency of free-energy conversion [2,4]. When minimum entropy production implies that the latter ratio is about 1, nothing is stated about the efficiency of energy transfer, but only something about the efficiency of entropy transfer – as is discussed above. The efficiency of energy conversion, I maintain, can only be obtained by multiplying the normalized ‘particle’ fluxes jD and jT with the energy per ‘particle’ and by dividing the two, while the ‘forces’ X and Y have nothing else to do with it than that they are driving these fluxes [8]. We must be grateful to Davor Juretić that his comment has brought to light an important difference in the significance of symbols used in physics and biology.


References:

1. Andriesse, C.D. (2000) On the relation between stellar mass loss and luminosity, Astrophysical Journal 539, 364-365.

2. Juretić, D. (2002) Comment on “Minimum entropy production in photosynthesis”, draft on the internet: http://mapmf.pmfst.hr/~juretic/Juretic-revised-comment.pdf; part of it is published by Juretić, D. and Županović, P. (2003) Photosynthetic models with maximum entropy production in irreversible charge transfer steps, Computational Biology and Chemistry 27, 541-553.

3. De Groot, S.R. and Mazur, P. (1984) Non-equilibrium thermodynamics, Dover, New York.

4. Hill, T.L. (1977) Free energy transduction in biology, Academic Press, New York.

5. Andriesse, C.D., to be published.

6. Cox, J.D., Wagman, D.D. and Medvedev, V.A. (1989) CodataKey values for thermodynamics, Hemisphere Publ. Corp., New York.

7. Daubert, T.E. and Danner, R.P. (1989) Physical and thermodynamic properties of pure chemicals, Hemisphere Publ. Corp., New York.

8. Andriesse, C.D. and Hollestelle, M.J. (2001) Minimum entropy production in photosynthesis, Biophysical Chemistry 90, 249-253.