by Vitaly Khvorostyanov
Some replies to and comments on the remarks by WebHubTelescope and others, written in response to the blog post Thermodynamics, Kinetics and Microphysics of Clouds.
Einstein used to say: “The physics is a drama of ideas”. A good example here is Bose-Einstein (B-E) statistics. When Satyendra Nath Bose wrote a short paper in 1924 on his statistics and submitted to the “Philosophical Magazine”, the reviewer wrote very negative review, and the journal rejected the paper. Perhaps, it would never be published, but fortunately, Bose was a fighter. He sent the paper to Einstein with request to publish it in “Zeitschrift für Physik”. Einstein examined the paper, organized a positive review, translated himself from the English into the German, wrote his own paper on this topic, but with another derivation, and asked the journal to publish both papers together. So, they were published and we have since then Bose-Einstein statistics. This example shows how difficult sometimes is the destiny of research, which becomes later commonly accepted. It seems, our attempt to slightly generalize the nucleation rate in classical nucleation theory with B-E statistics also meets counteraction from some audiences.
Most comments on Curry’s blog on the book by Khvorostyanov and Curry are positive, constructive and friendly, we greatly appreciate them. The most critical is a series of comments by WebHubTelescope (WHUT) aka Paul Pukite, who criticized application of Bose-Einstein (B-E) statistics for nucleation rates at low temperatures. This initiated a substantial discussion, which took much time of busy people. After this discussion, Paul Pukite wrote a 1-star comment at Amazon site.
To clarify the situation and to respond to the criticisms by WHUT and Paul Pukite, we must emphasize at once the following. Even the Table of Contents of the book shows that the basis of our consideration is the Boltzmann statistics traditionally used in classical nucleation theory (CNT) and having the form exp( -E/kT). The title of section 8.2.1. is “Application of the Boltzmann Statistics”. The title of section 8.3.1. is “Nucleation rates with the Boltzmann distribution”. Here, the nucleation rates and crystal concentrations are derived with the Boltzmann distribution for the polydisperse aerosol and are used in all calculations of homogeneous nucleation in this book. Possible extensions of CNT with B-E statistics are only briefly outlined in 2 short subsections, 8.2.3 and 8.3.2 (half a page each), where B-E statistics is considered as a possible candidate for generalization at low T (the reason for this is explained below). However, we emphasize that these sections 8.2.3 and 8.3.2 with Bose-Einstein statistics are NOT used in any calculations in the book. All calculations of the homogeneous nucleation processes in this book, all parcel model simulations, and analytical parameterizations for the cloud and climate models in this book are based on Boltzmann statistics described in Sections 8.2, 8.3.1. All calculations and parameterizations of heterogeneous nucleation and deposition in Chapters 9 and 10 are also based ONLY on the Boltzmann distribution. Consideration of deliquescence and efflorescence in Chapter 11 is also based on the Boltzmann distribution. Bose-Einstein statistics in never used in this book, beyond brief mentions in sections 8.2.3 and 8.3.2! Dan Hughes confirms this in the comments to the previous thread by searching using the Kindle version of the book.
Now, it should be clear to all who have been following this on Judith’s blog that the criticisms by WebHub and Paul Pukite are inconsistent and irrelevant to this book.
This discussion initiated by WebHub and Paul Pukite took a lot of time and distracted the community from discussion of the things that are really important and new and are synthesized in this book. For several decades, ice nucleation was parameterized based on empirical data as functions either of temperature T or saturation ratio Sw, the attempts to combine these dependencies were rare and not successful. The authors of this book derived for the first time the expressions for the critical radius rcr and critical energy DFcr of the ice germs with simultaneous analytical dependencies on the temperature, saturation ratio, external pressure, and finite size of the freezing particle. Using these equations along with nucleation rates from 8.2, 8.3.1 (based on the Boltzmann distribution), this allowed:
- to suggest the first simple analytical (not empirically based) equations for ice nucleation;
- to perform much more precise calculations of ice nucleation at varying T and Sw;
- to derive many previous empirical parameterizations (seemingly unrelated) from CNT and to express their parameters via physical constants and aerosol parameters; after all, these seemingly unrelated parameterizations appeared to be the close relatives of one family;
- to find thermodynamic constraints for the previous empirical parameterizations; some of them appeared to be in conflict with thermodynamics in some domains of their applications;
- to find for the first time the analytical expressions for the critical temperatures and humidities of freezing;
- to find a simple quantitative relation between the solution and pressure effects in nucleation;
- to find a separable analytical representation of the nucleation rates and crystal concentrations by T and Sw;
- to integrate this separable representation by time and to obtain ready parameterizations for the large- scale models (GCMs) as a substep process;
- to find the diffusion and kinetic limits of this parameterization, which allows to study effects of pollution on microphysics and optics of crystalline and mixed-phase clouds;
All of this is described in detail in Chapters 8, 9, and 10, and all this was done with Boltzmann distribution, not with B-E statistics.
And now, after this general clarification, we will respond in detail to the criticisms by WHUT. To give precise replies point-by-point, we use the style as in replies to the referee in the journal: we cite WebHub’s statement and then our reply follows.
1) WHUT: “So right there in Section 8.2.3 is a bad description of statistical mechanics.”
Reply. This is a wrong and misleading statement. There is no description of statistical mechanics in sections 8.2.3 and 8.3.2 – not at all; all the derivations and description of the 5 main statistics are given in Chapter 3. At the beginning of Chapter 8, in sections 8.2.1 and 8.2.2, the traditional derivations of the classical nucleation theory (CNT) are given for the nucleation rates based on the Boltzmann statistics. In these short sections 8.2.3 and 8.3.2, (half a page each) are simply given equations of the classical nucleation theory (CNT), with suggested possible modification by using Bose-Einstein (B-E) statistics instead of the traditionally used Boltzmann statistics. These are just briefly outlined as a subject for further possible verification in the experiment or theory, but are NOT used in any calculations in this book.
Why did we come to this idea? We worked together for almost 2 decades on ice nucleation, and our experience here is reflected by papers cited in the book by Khvorostyanov and Curry. Many hundreds numerical experiments and comparisons with experimental data were performed with various sets of parameters. Unfortunately, despite almost 90 years of developments of CNT since 1920s, the values of parameters are still uncertain. This is especially true for the surface tension s. It is known that surface tension decreases with decreasing temperature, the rate of decrease is different in different sources, but it is known that at low temperatures (can be around Tlim of -70 to -100 C), s becomes very low, even for pure water without surfactants, and may become even negative in some extrapolations to the low T, which prevents calculations at these T. Since the critical energy of activation DFcr ~ s3, the enery DFcr also becomes very small (or negative), and eventually can become comparable to or smaller than kT, so that DFcr ~ kT or DFcr < kT. However, Boltzmann statistics requires a condition DFcr >> kT. Thus, we come to a dead end: calculations of nucleation become impossible below Tlim with Boltzmann statistics although many cirrus clouds may form at these T (especially in the tropics), polar stratospheric clouds, playing important role in ozone depletion, noctilucent and mesospheric clouds form at even lower T (see Chapter 4 in the book).
So, we have a situation when clouds form at very cold temperatures, but we cannot calculate crystal nucleation rate at these T because of inapplicability of the Boltzmann statistics and cannot simulate these clouds. This situation is not characteristic of only our work, but is a common problem for many researchers who try to deal with low T.
What to do? Wait for another 50-90 years when the new data on s, latent heat, etc, will become available at low T? But there is no hope that any reliable data will be available in the near future, because experiments are usually limited to -38 to -40 C and are difficult or impossible at lower T (“No Man’s Land”, see Chapter 4), and the available theories of water at low T are still incomplete, proceed slowly, and does not provide reliable values of s. The only way to solve the problem now was to try another statistics, valid at low T. A possible candidate is the Bose-Einstein statistics, valid to very low T, even close to 0 K. It is more general than the Boltzmann statistics and includes it as a particular case at sufficiently high T. We just outlined it, but did not develop in detail and did not use in this book. This is a suggestion that can be verified or rejected by any researcher who has appropriate experimental or theoretical tools at very low T. But the use of B-E statistics at low T cannot be rejected until it is proven that it leads to the wrong results. To our knowledge, such data at very low T are absent. Therefore, criticisms by WHUT and Paul Pukite is pure speculation based on nothing.
2) WHUT: “Bose-Einstein stats are more general than Maxwell-Boltzmann stats”
Reply. This is again a misleading statement, WHUT mixes water with oil. The Maxwell and Boltzmann statistics are substantially different (see Chapter 3). The Maxwell statistics is formulated in terms of velocities and used usually in cloud physics for evaluation of the kinetic vapor fluxes around a growing drop or crystal (Chapter 5). The Boltzmann statistics is formulated in terms of the energies and is used here for evaluation of the nucleation rates and nucleated crystal concentrations (Chapters 8, 9, 10, 11).
3) WHUT: “Yes, Bose-Einstein stats are more general than Maxwell-Boltzmann stats, but it doesn’t apply to the kinds of particles you are talking about. Instead, in the case of Boltzmann should be if anything a simplification of Fermi-Dirac statistics which forbids two particles from occupying the same state.”
Reply: According to the Correspondence Principle, we always have a right to replace in some equation a more particular expressions with a more general expression that includes the previous expression as a particular case, if the necessary conditions for such generalization are satisfied. Of course, application of B-E statistics should be done with caution. There are 3 major requirements here: a) the particles should behave as an ideal gas with weak interactions; b) the particles should have integer spin; c) as the B-E statistics was derived for sufficiently simple, or “elementary” particles, the particles should be “elementary enough”. Consider these conditions in more detail.
a) The water vapor molecules are considered in CNT as an ideal gas, thus, the first requirement is satisfied.
b) We hypothesized applicability of B-E statistics to the water vapor molecules because they have integer spin and can be considered as some analog of bosons, for which B-E is applicable. Application of B-E statistics requires that particles have integer spin. Responses to WHUT were given on Curry’s blog on 4-5-6-7 September by Carrick, Pekka Perilla, Judith, Tomas Milanovic and several other people, with corresponding references, both from the theory and optical and other measurements (including the paper with rigorous quantum mechanical calculations by Konukhov et al., the link was given by Carrick). Water vapor can be considered as an ideal gas and represents usually a mixture of “ortho-water” and “para-water” in the ratio 3:1, although this ratio can vary at various conditions. In ortho-water”, the hydrogen spins are parallel, so that the molecular spin is s= 1 h (h is the Planck function). In “para-water”, where the hydrogen spins are anti-parallel, the spin of the molecule is zero, s = 0 h (see e.g., Born, 1963; Landau and Lifshitz, Quantum Mechanics, Wikipedia, etc). Thus, in both cases, the spin of the molecule is integer, and they should behave as bosons, therefore B-E statistics is applicable to them. But Fermi-Dirac (F-D) statistics, as WHUT hypothesized without any justification, is inapplicable.
c) “Elementary” particles. One could think that an “elementary particle” is simpler than a water molecule consisting of 3 atoms. Note first that derivation of the B-E or F-D statistics is rather simple (see Landau and Lifshitz, v.5, Statistical physics, or Chapter 3 here) and does not consider internal structure of the particles. Then, how elementary are “elementary particles”? The present day Standard Theory of elementary particles counts more than hundred particles, described by the symmetry group SU3xSU6xU1. In particular, there are more than 64 mesons that have integer spin and are bosons (most famous are pi-mesons, elementary quanta of the field of strong interactions in atomic nuclei, analogous to photons in electromagnetic interactions, Higgs boson for gravitation, etc), and many particles with half-integer spin, which are fermions, like proton, neutron (building atomic nuclei), and many similar barions. These “elementary particles” are composed of 6 types of quarks, which have spin s= 1/2 h, electrical charge of 2/3 or -1/3, and many other quantum numbers (charm, strangeness, color, etc.). For example, proton and neutron consist of 3 quarks, therefore have spin s = 1/2 and are fermions. A meson consists of a quark and antiquark, therefore their spin is integer and they are bosons. Besides, each such “elementary” particle may include an unlimited number of virtual “sea” quarks, antiquarks and gluons (particles with spin s= 1 h and zero mass), which help to “glue” these quarks together. Therefore, each “elementary” particle consists of many or of unlimited “more elementary” sub-particles. Until about 1964, the elementary particles were considered as really elementary. In 1964, Gell-Mann and Zweig suggested that the elementary particles are not so elementary, but are composed of quarks. Gradually, and especially around 1980, an elementary particle’s status as indeed elementary—an ultimate constituent of substance—was mostly discarded, and they are considered now as consisting of many sub-particles.
However, neither Bose-Einstein statistics nor Fermi-Dirac statistics were changed, they are still applied in their old forms to these “elementary” but very complicated particles, composed of many sub-particles, as these B-E and F-D statistics were applied before. Now, if to compare just described structure of an “elementary particle”, a huge monster consisting of many sub-particles, and our poor small beautiful water molecule, consisting of only 3 particles (atoms), in some sense, the water molecule may look simpler, and can be considered as an “elementary particle”. Thus, all requirements a), b) and c) are satisfied and we can apply to water molecules B-E statistics. Why not? Therefore, we have chosen B-E statistics as a possible natural generalization for CNT for the boson-like water molecules.
4) WHUT: “There is a phenomenon known as Bose-Einstein condensate which describes atoms (not photons) taking on Bose-Einstein statistics, but that requires special laboratory conditions using Helium and at very low temperatures. If you think that is happening with ice crystals or water droplets in the atmosphere, I don’t know what to say but stand incredulous. These particles don’t have integer spin, despite what you say in the book.”
Reply. Bose-Einstein condensate has nothing in common with real condensation. As was emphasized by Landau and Lifshitz (v. 5, Statistical Physics, 1958a, section 59, Degenerate Bose Gas, footnote at the end): “The phenomenon of accumulation of the particles in the state with lowest energy e =0, is often called “Bose-Einstein condensation”. However, we can talk here only about “condensation” in the momentum space, “and no any real condensation does not occur, of course”. That was exact citation.
Meanwhile, in our book, we consider real condensation and deposition. WHUT does not appear to know the difference between the formal term “Bose-Einstein condensation” with no condensation, and the real condensation processes.
5) WHUT: “If you indeed think that B-E statistics applies to condensation and deposition, you really should place a citation in your book. Good luck finding one!”
Reply. In the literature on the atmospheric physics and nucleation, we did not find any use of B-E statistics for evaluation of nucleation rates — Boltzmann statistics is always used. It is clear why: since the very beginning of development of CNT in 1920’s, CNT was applied to the processes at not very low T, when DFcr >> kT and Boltzmann statistics is valid. Only during the last few decades, the studies of very cold clouds began, when this condition may become invalid and generalizations may be required. We also use Boltzmann statistics through this book in all calculations. The problem why we turned to B-E was explained above: at T below some -70 to -100 C, the existing parameterizations of surface tension, and of critical energy give very low or even negative values, and Boltzmann statistics becomes inapplicable. If such a citation existed, we, of course, would use the existing methods with or without B-E, and would not “invent a bicycle”. Thus, if in the future, B-E statistics will appear to be valid for nucleation at low T, the first reference will be this book. If somebody can supply us with such reference, we would appreciate it.
6) WHUT, September 4, 2014 at 2:15 pm. “Bose-Einstein statistics only applies to particles like photons. I can see what you are trying to do — torturing an equation to extract a prefactor. But it is based on the completely wrong premise, which is applying Bose-Einstein statistics to matter incorrectly. Water particles don’t have integer spin.”
Reply. a) Bose-Einstein statistics applies not only to photons, but to all particles with integer spin, i. e., to all bosons. As mentioned above, there are more than 64 types of mesons (pi-mesons of nuclear forces, Higgs boson present everywhere and giving inertia to all bodies; K-mesons, etc., gluons with spin s =1h). B-E statistics is applied to all 64 mesons and to all other bosons.
b) “I can see what you are trying to do — torturing an equation to extract a prefactor.”
Reply. Did WHUT actually read the book? We are not “torturing equations” to extract a prefactor in sections 8.2.3 and 8.3.2 with B-E statistics, there are not any prefactors in these sections with B-E. The prefactors for the nucleation rate are derived earlier in section 8.2.3 for deposition, and in section 8.3.1 for freezing, both are based on Boltzmann statistics.
c) “Water particles don’t have integer spin.”
Reply. It was explained above (and earlier in the blog by Carrick, Pekka Pirilla, Judith, Tomas Milanovic and several other people) that both ortho-water and para-water have integer spin of 1 and 0 respectively, both are integer, and, as Pekka Pirilla wrote: “normal water molecules are bosons, not fermions”.
7) WHUT: Paul Pukite (aka WHUT) wrote a review of the book on amazon.com, called it “Easy to find glaring errors”, claimed that the book “creates an activation dependence that trends as kT, where T=Temperature, instead of the accepted Arrhenius-rate law exp(-E/kT)” and warned climate modelers that the use of the results from this book may be harmful for climate models, they give wrong temperature dependence due to use of B-E statistics, modelers “should beware” of that danger.
Reply. As was explained above, possible generalizations with B-E at very low T are only briefly outlined in the book, but never used. All calculations in the whole book were made with Boltzmann statistics, exp(-E/kT) (which Pukite erroneously prescribed to Arrhenius), but with new equations of Khvorostyanov and Curry for the critical radii and energies as described above. The ice nucleation rates are not described by the “Arrhenius-rate law exp(-E/kT)”. Numerous theoretical and experimental works described in this book (including the authors’ works), show that the temperature dependence of ice nucleation rates is substantially different from the simple exponential law, exp(-E/kT), cited by Pukite, is much more complicated, and this temperature dependence is different under different conditions in clouds. Thus, this comment of Paul Pukite is beyond misleading, both for potential readers, and for Amazon.
Summary: We appreciate all of the discussion of the book, but are very sorry to see the discussion hijacked by a focus on a fairly inconsequential few paragraphs in the book about B-E statistics. We look forward to your further comments, questions and critiques.
JC comment: This is a guest post, please keep your comments on topic and relevant.
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