Thermodynamics as stochastics
Preview of Chapter 6 of my book, titled “Atmospheric thermodynamics deduced by stochastics”
Throughout my academic career, I developed the habit of taking on the responsibility of teaching those courses that no other professors wanted to teach. This minimized the friction, which otherwise abounded in my university. For example, back in 1998, my colleague Nikos Mamassis and I started a course on hydrometeorology for the newly created Postgraduate Programme on Water Science and Technology. No other professor wanted it, but soon after it became popular among students. Consequently, Nikos and I gave up teaching it, as other professors were willing to take it on.
Thanks to this course, I got to know entropy, which I tried to understand in depth. I’ve been doing research on entropy for more than 20 years. Initially, I understood the principle of maximum entropy as a powerful explanatory tool for natural behaviours which I identified, particularly when exploring hydrological data sets. I used this tool for explanation in an induction framework based on data. I published a couple of papers about this theme in 2005.1
Soon after, I produced another paper2 explaining the intermittency in rainfall by means of entropy, which in this case is related to the probability of rainfall occurrence at a certain time scale. By maximizing the average entropy over all scales, I was able to find the probability of occurrence at any time scale. In the model calibration, I used the probabilities of rainfall occurrence at only two time scales, while for all other scales I predicted them by the multiscale entropy maximization framework. The agreement with data was impressive, as can be seen in Figure 6.22, reproduced from that study and included in Chapter 6 of my new book.
My next step was to use entropy in a deductive setting to model the presence of moisture in the atmosphere. I produced two papers on this3, the first using classical thermodynamics and the second using stochastics. In both cases the resulting equation was the same, and more accurate than existing analytical expressions, as seen in the figure below. Amusingly, the first paper was rejected by the journal Entropy and the second was published by this same journal. The initially rejected paper now has more than 220 citations (on Google Scholar).
Meanwhile I “composed” a “Hymn to Entropy”4, which I presented in the IUGG conference in Melbourne. This included several mathematical derivations, some of which might have contained errors.
By now I have reworked (multiple times) all these derivations, and many more, and I present in the new Chapter 6 of my new book “Stochastics as Physics” a complete synthesis that satisfies me. Perhaps it still contains errors, which I hope to correct with the readers’ help. I note though that Grok did not find any error.
The chapter is purely based on stochastics. Using stochastics, it derives the full set of thermodynamic principles—of course without premising them. Here is the pdf of the first draft that contains chapters 1-6, placeholders for the next chapters, and the full set of references.
The chapter is completely based on deductive reasoning, and the results agree with experimental values, whenever they exist. The table below shows an example.
I am concluding this post by quoting below the first (6.1) and the last (6.26) sections of the chapter.
6.1 Premises
In the modern period and up to the 19th century, heat was interpreted as the flow of the caloric fluid, assumed to be a weightless substance flowing from hotter to colder bodies, passing through pores of matter. In the mid-19th century this view was replaced by a mechanical theory of heat. This is marked by the introduction of the entropy concept in mechanical terms, and the development of thermodynamic principles (Zeroth, First, Second and Third Law; see section 6.26). As already mentioned in section 1.1, near the end of the 19th century, Boltzmann explained entropy in statistical terms. Despite the subsequent development of statistical thermophysics, still the dominant perception of thermodynamics remains mechanistic and deterministic, and in some respects still caloric.
In this chapter we highlight the fact that all thermodynamic laws in gases, with focus on the atmosphere, are stochastic laws. We deduce these laws using stochastics—and in particular the principle of maximum entropy—and without using the thermodynamic principles at all. We show that thermodynamic laws are stochastic laws—typically relationships of expectations of stochastic variables. We derive these laws by maximizing entropy, i.e. uncertainty, at a microscopic (molecular) level. Interestingly, though, they turn out to macroscopically express near certainties and are commonly misinterpreted as deterministic laws. The explanation of near certainty relies on these two facts:
Typical thermodynamic systems are composed of hugely many identical elements: Na = 6.022 × 1023 per mole of material (Na is the Avogadro’s number).
The random motion of each of the system elements makes their state practically independent of the others’.
As a consequence, a stochastic variable x expressing a macroscopic state of 1 mole of material, will have a coefficient of variation std[x]⁄E[x] ~ 1⁄√Na= 1.3 × 10–12 (see also section 1.4). The fact that the macroscopic variability is practically zero should not mislead us to interpret the laws in deterministic terms.
We stress that here we do not assume the thermodynamic laws as first principles, nor do we dispute them. Rather, we derive them from conservation laws plus the principle of maximum entropy. These are our first principles.
6.26 Principles of thermodynamics
In the above presentation we have followed the Aristotelian suggestion for parsimony (section 1.3), i.e., to use “fewer postulates or hypotheses or propositions”. Namely, to derive the basic results of atmospheric thermodynamics in Chapter 6 we have only used (a) Newton’s laws, (b) the principles of conservation of mass, momentum and energy, and (c) the stochastic principle of maximum entropy. We did not use the so-called four principles (or laws) of thermodynamics. Nor did we use other assumptions typically postulated in statistical physics, such as the equiprobability of microstates (e.g., Moore, 2002) or the principle of equipartition of energy into the available degrees of freedom (often referred to as a postulate). Rather, we saw that all these are results of entropy maximization.
For the completeness of the presentation, in this section we summarize some information about the four laws of thermodynamics and discuss their relationship with our results.
The Zeroth Law states that if two systems are in thermal equilibrium with a third system, then they are in thermal equilibrium with each other. This law defines the notion of thermal equilibrium. In turn, this is necessary to define temperature as two systems that are in equilibrium have the same temperature.
In our framework this law is unnecessary. Two systems are in equilibrium if they are put in contact and the entropy of the compound system has been maximized. Besides, the temperature is defined through equation (6.50). As we have seen, equation (6.99) implies that two systems put in contact, in which entropy has been maximized, will have the same temperature. This is a consequence of entropy maximization and does not presuppose an axiomatic introduction of the Zeroth Law.
Moreover, the usefulness of the law is problematic. As we have seen, there appear gradients in temperature (see sections 6.19 and 6.24) and therefore, different layers of the atmosphere, however thin, do not have the same temperature, even though they are in contact. This does not violate the law per se, because the gradients imply non-equilibrium conditions, but reduces its usefulness as most natural systems are not in equilibrium.
The First Law is related to the conservation of energy. We have already used several times the principle of conservation of energy, according to which energy can be neither created nor destroyed but can only change forms. In classical thermodynamics, this principle is typically stated as: the heat supplied to a system (δQ) equals the increase in internal energy of the system (dE) plus the work done by the system (δW). This version of the principle of conservation of energy has also been produced by our framework in section 6.11.
In classical thermodynamics, the Second Law states that in the process of reaching a thermodynamic equilibrium, the total entropy of a system increases, or at least does not decrease. In this respect, the Second Law is none other than the principle of maximum entropy applied to thermodynamic systems. Starting from any condition and approaching the equilibrium, the entropy change dΦ* of the total system cannot be negative.
Finally, the Third Law of thermodynamics states that the entropy of a system approaches zero as the temperature approaches absolute zero. Here we did not need this law at all. Yet, we were able to replicate it in Digression 6.B within a negligible difference for the value of energy in which the entropy becomes zero. In our framework the entropy-zeroing energy is not zero but a super-tiny quantity, smaller than the quantum zero-point energy that is imposed by Heisenberg’s uncertainty principle. Thus, our stochastic framework resulted in a version of the Third Law that is in better agreement with quantum physics than the classical version which implies that zero energy is in principle feasible.
Your comments will be most welcome
as they help me improve the material I am presenting.
D. Koutsoyiannis, Uncertainty, entropy, scaling and hydrological stochastics, 1, Marginal distributional properties of hydrological processes and state scaling, Hydrological Sciences Journal, 50 (3), 381–404, doi:10.1623/hysj.50.3.381.65031, 2005.
D. Koutsoyiannis, Uncertainty, entropy, scaling and hydrological stochastics, 2, Time dependence of hydrological processes and time scaling, Hydrological Sciences Journal, 50 (3), 405–426, doi:10.1623/hysj.50.3.405.65028, 2005.
D. Koutsoyiannis, Clausius-Clapeyron equation and saturation vapour pressure: simple theory reconciled with practice, European Journal of Physics, 33 (2), 295–305, doi:10.1088/0143-0807/33/2/295, 2012.
D. Koutsoyiannis, Entropy: from thermodynamics to hydrology, Entropy, 16 (3), 1287–1314, doi:10.3390/e16031287, 2014.
Demetris, the post is impressive. I feel a great sense of relief. If we reject determinism, that means we no longer need to wonder whether free will exists within it. And, on top of that, Mark Manson’s theory is confirmed. In the end, yes—"everything is f$#k%*", but hope is alive.
(A conflicted Lutheran...)
"...within a negligible difference for the value of energy in which the entropy becomes zero." In the final para.
Would it be possible to explain this a little bit in the text. Difference between what and what?