Dear Alan Forrester,

I agree with you on that nature has both digital and analog features. I think that some issues in your Essay need commenting, but I will only comment in some few.

You write: "The second law of thermodynamics states that entropy always increases". One finds this kind of statements in introductory treatises (e.g. general physics or physical chemistry textbooks),...

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Dear Alan Forrester,

I agree with you on that nature has both digital and analog features. I think that some issues in your Essay need commenting, but I will only comment in some few.

You write: "The second law of thermodynamics states that entropy always increases". One finds this kind of statements in introductory treatises (e.g. general physics or physical chemistry textbooks), but are not accurate and can be misleading. In fact, they are in basis of the recent misguided attacks to biological evolution becoming from creationists.

Evolution is in complete agreement with the second law! Prigogine won a Nobel Prize for his work on the extension of classical thermodynamics and specially for his introduction of concept of dissipative structures. Living systems are dissipative structures.

The second law says (in modern notation) that (d_iS >= 0). The equality (dS = d_iS) is valid for a crystal at equilibrium but is not valid for dissipative structures. The second law of thermodynamics is perfectly compatible with a decrease in entropy as the observed in evolving living systems.

You continue: "Entropy can be understood in terms of free energy." Here and thereafter you are really referring to Helmholtz free energy (which is different from Gibbs free energy). It is true that one can derive free energies from entropy, using Legendre transformations, but free energies (both Helmholtz and Gibbs) are thermodynamic potentials only in special cases and cannot substitute to entropy in the general case. For this reason modern treatments of thermodynamics are built around the central role played by entropy. For instance the modern thermodynamic theory of fluctuations is based in entropy instead of free energies or enthalpies as in the old thermodynamic theory of fluctuations, which has a limited validity.

You say that "if E remains constant, F decreases as S increases", but again this is not at all true. If one starts from the Legendre transformation (F = E - TS). At constant T and E, one obtains (dF = - TdS). Now if, and only if, (dS = d_iS), then one obtains (dF =< 0). The fact that T must be also constant is the reason which F is a thermodynamic potential only for systems at constant T. Moreover, the constraint (dE = 0) can be relaxed. In fact, textbooks often emphasize that F is a thermodynamic potential, dF =< 0, for systems at constant temperature T and volume V, but they do not say at constant energy, because this is not a requirement for being a potential.

Finally, let me remark that the entropy of a quantum system is

S = - k Tr rho ln rho

This can written as S = -k H where H is given by your equation (10). H is that we call the H-function (a generalization of Boltzmann H-function) not entropy as you say. The H-function decreases when entropy increases (H-theorem).

You write that "The expectation values of observables of S_1 are now correlated with those of S_2 and they contain information about one another that they did not contain before the interaction". You state that this is the basis for the second law. This looks pretty much as a rather old idea formulated by Prigogine in the 60s. He dubbed it the "flow of correlations". He even offered the following social analogy in his best-seller books (e.g. in "The End of Certainty"): Two persons meet in the street and talk; after meeting, each one of them continue walking independently and remember the talk with the other, whereas the inverse process never happens.

He and his group have been working in giving a rigorous mathematical foundation for this idea. One of the results that they found recently (in the last 90s) is that an irreversible process of that kind cannot be described by an unitary evolution. They have introduced a more general non-unitary evolution in their recent extension of quantum mechanics to 'large' systems with Poincaré resonances; with quantum mechanics recovered as a special case, of course.

There was a recent International Solvay Conference where they and other groups discussed irreversible extensions of quantum mechanics (e.g. see the generalized quantum mechanics proposed by particle physicist Sudarshan and coworkers). The results were published in Adv. Chem. Phys. Volume 99.

You write "The present contains records of the past but the past doesn't contain records of the present and it is this asymmetry that makes the present later than the past". I find this a bit misleading. First, for reversible processes the past contains "records of the present" because the evolution is unitary. Consider the motion of the Moon around Earth. At a very good approximation, this is a reversible motion (dissipation is practically zero) and if you know the position of the Moon at present you can compute what was his position in the past with an excellent precision. Second, the inverse of the process A(past) --> B(present) is not B(present) --> A(past) as you seem to believe, but B(present) --> A(future).

I consider that the origin of irreversibility is related to the existence of bifurcations. Note, however, that outside a bifurcation point the evolution is completely reversible.

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