of the late Nobel prize-winner's book. He has consistently held in his books that nature is probabilistic even though his explanation of the 2nd law of thermodynamics, that entropy can only hold constant or increase in an isolated system, has evolved. Much of his motivation seems to have been in sorting out why Boltzman and Gibbs failed to satisfy the science community that their statistical physics explained the 2nd law, due to reversible classical equations and Poincare recurrences. However in order to make his probabilistic argument he may have created a loophole. He points to the Langevin equation as an irreversible equation with noise (friction) and he says Poincare should have connected nonintegrability with irreversibility and most dynamics are nonintegrable. However everyone agrees some (simple) systems are reversible (pendulums etc) so how can all of nature be stochastic? Maybe because the noise terms tend to but never go to zero? However in addressing the arrow of time he suggests gravity which is ignored in thermodynamics as are all interactions; but this explanation is also used by others in deterministic models. So it may never be provable who is right; but if his loophole is real I think there may be a simpler explanation.
Statistical entropy in all of it's variations is an excellent inference tool but it is about an observer's measurements and not underlying properties of the system being measured (frequentist approaches come close but usually have to extrapolate). In this case Poincare recurrence maybe non-physical, a mere statistical fluctuation with no actuality. (Prigogine says it is false because he introduces new microscopic dynamics, I'm just saying it may not arise in reality but only through statistical assumptions which depend on observer uncertainty.) I agree with the explanation at the website secondlaw.com that the thermodynamic explanation of entropy is fundamental as it is a measure of energy diffusion, and not randomness or uncertainty as the tool of statistical entropy would imply. In this way the 2 approaches are not contradictory; the statistical is merely a measurement tool for observers while the thermodynamic is real dynamics requiring no observers (ice melts, water crystalizes etc long before man was around). The current argument in wikipedia that statistical entropy is considered more fundamental because the others can be derived from it is silly; there are many types of subjective entropy measures, the basic frequentist vs Bayesian approaches, there is volume entropy such as for measuring expanding gases, configurational entropy such as for crystals etc; however there is only one thermodynamic entropy, Clausius's dS = <>q/T (for reversible systems; calculations change of course with potential variables of volume, pressure and temperature). If anything this should be viewed as fundamental as it is a direct measurement of the physical movement of heat. One should not confuse information theory and measurement techniques with real underlying dynamics. When some authors say 'entropy is not a property of a system, it is a property of our description of the system' they are referring to statistical entropy measures and not real thermodynamics. As Prigogine says 'irreversability is not just in our minds', that it applies to nonintegral systems identified by Poincare but not the connection.
If Boltzman had accepted that his equation was not fundamental but an inference tool then most of the debates would likely not have arisen, including Prigogine's criticism of an excellent tool that did not deserve to be criticized on that basis. However what he has done is to show mathematically how irreversibility can apply at the microscopic level for nonintegral systems (in agreement with macroscopics) due to non-local persistent interactions but has to be measured statistically at the level of ensembles and not individual trajectories. This is quite a feat even if controversial. Nevertheless the standard entropy calculations apply for equilibrium systems and the arrow of time is still mysterious though possibly linked to gravity as he says. More details of his derivations are provided in his book The End of Certainty but it would have been nice to see some discussion of entropy of non-equilibrium systems for which there is no universal agreement. For instance it is said that 'the rate of change of entropy with time for a nonequilibrium stochastic process is always positive.' [B.C. Bag; the following references are also available on the net with a simple author search.] This might suggest he already solved the problem and gravity is not required? But-
R. Metzler et. al. say 'Prigogine introduced novel microscopic laws which are irreversible with time. One reason for this ongoing discussion is the absence of rigorous mathematical proofs of irreversible properties in the thermodynamic limit...ensemble averages do not give a basic explanation of irreversible properties, since they contain an average over infinitely many trajectories. Ergodic theory does not help either, since it needs time averages over infinitely many trajectories...In this model we introduce a model with deterministic time reversible dynamics which can be analysed in detail...The Poincare return time is known exactly...' However this takes us back to the usual complaints about statistical fluctuations. [Is there a real arrow of time if everything is eventually reversible?]
Castagnino and Lombardi have developed an interesting approach to the question of the arrow of time. [Clearly Prigogine failed by his own admission and his gravity conjecture!] 'In fact time reversal invariant equations can have irreversible solutions. [e.g. the pendulum is time-reversal invariant...however the trajectories...are irreversible...]...The traditional local approach owes its origin to the attempts to reduce thermodynamics to statistical mechanics...[however] only by means of global considerations can all decaying processes be coordinated. This means that the global arrow of time plays the role of the background scenario where we can meaningfully speak of the temporal direction of irreversible processes, and this scenario cannot be built up by means of local theories that only describe phenomena confined in small regions of spacetime...the geometrical approach to the problem of the arrow of time has conceptual priority over the entropic approach, since the geometrical properties of the universe are more basic than its thermodynamic properties.'
Obviously the debate continues. While Prigogine may not have solved the arrow of time, his work on correlations is important as these are assumed away in classical physics but they are critical to life!