Lecture on Debye model for solids

 I just finished preparing my undergraduate lecture on the Debye model for solids. Here are my slides. The course is a third-year course on statistical mechanics. In the previous lecture, they learned about blackbody radiation and derived the Planck spectrum.

(Note. The derivations are not included, as I sketch them by hand on a document viewer. This slows down the pace of the lecture and engages the students as they detect mistakes I make. :) ) 

A few things I find interesting about the lecture content.

The Debye theory is very similar to Planck's theory of blackbody radiation. The main difference is that for Debye there is a high-frequency cutoff for the phonon spectrum. This defines the Debye temperature.

Given its simplicity, the success of Debye theory, qualitatively and quantitatively, is impressive. Arguably, this is because it is an effective theory that is constrained to be consistent with the long-wavelength physics (elasticity theory) and the classical limit (number of modes of lattice vibration).

The beautiful textbook by Schroeder is a bit confusing on page 312. He claims that the Debye temperature lead (88K) is much less than that for diamond (1860 K) because the former is "soft and dense" whereas the latter is "stiff and light".  The speed of sound in diamond is about ten times that in graphite. The mass density is not particularly relevant. However, the number density of atoms (which is what enters the Debye formula) in diamond is actually 5 times larger than lead because of the strong covalent bonding which makes the interatomic distance small.  

A few historical notes I found interesting.  

Einstein's model for solids (1907) was important because it provided significant evidence for quantum theory, i.e., that the energy of lattice vibrations was quantised.

Sommerfeld said that his biggest discovery was his student Peter Debye.

Around 2006, there were moves to "cancel" Debye because of alleged Nazi sympathies. However, because the evidence was ambiguous, these moves faltered.

Reflections of the 2026 Nobel Prizes

 Last week I enjoyed following the Nobel Prize announcements in Physiology or Medicine, Physics, and Chemistry. Reading the popular accounts behind the science is enjoyable, as they are well-written and accessible. Here are a few things I found interesting.

I found the Physics prize a bit surprising. On the one hand, the creativity and vision needed to construct the IceCube neutrino observatory is beautiful and impressive. They have clearly detected high-energy neutrinos, and there is potential for some important astrophysical discoveries. On the other hand, the key word is potential. This prize seems in contrast to earlier prizes for gravitational wave detectors (2017) and for observations of cosmic neutrinos (2002, 2015). For example, the latter solved fundamental issues in our understanding of neutrinos. But perhaps I am missing something.

A fun cultural side to the Physiology or Medicine prize to Karl Deisseroth is that there is a beautiful novel, Transcendent Kingdom, that centres on the life of a graduate student who works in a fictional lab at Stanford based on his. On the science side, a key aspect was the discovery of the protein channelrhodopsin. The essential photophysics is that the protein contains a chromophore, all-trans retinal, that upon photoexcitation undergoes a conformational change (isomerisation) to a 13-cis configuration. This is similar to the photophysics responsible for vision. In both cases, the conformational change of the chromophore triggers a conformational change of the whole protein and an electrical signal.

The figure is taken from here.

The chromophore conformational changes are associated with conical intersections between the potential energy surfaces of the ground and excited states. The essential physics can be understood in terms of a two-site Hubbard model.

The Chemistry Prize was for the development of new synthesis methods to selectively make molecules of definite chirality. The popular science background from the Nobel foundation mentions three important conditions for chiral synthesis that were stated in a short paper in 1953, by Charles Frank, a condensed matter theorist. He was a pretty amazing scientist. He showed in 1951 how topological defects determine the growth rates and strength of crystals and wrote an influential review on snow crystals. In 1958  Frank wrote down a free energy density for nematic liquid crystals, without reference to Landau, and long before Anderson emphasised the importance of generalised rigidity in condensed matter. (Albeit, and perhaps more remarkably, Frank's paper freely builds on a 1933 paper by Carl Wilhelm Oseen.

Aside. There is a fascinating and fundamental question about why chiral molecules do not spontaneously quantum tunnel backwards and forwards between the two possible chiralities. Briefly, this is because of quantum decoherence due to the environment. This question often comes up in discussions about emergence in chemistry.

Was there anything that you thought was particularly interesting about any of these prizes?

Nobel Prize Predictions for 2026

This week Nobel Prizes will be announced. Predicting the awardees is a fun annual exercise. It is also good to reflect on what has been achieved, including outside our own areas, and big advances from the past we may now take for granted.

Before writing this I looked at suggestions from readers of Doug Natelson's blog, Nanoscale views, an article in Physics World, predictions from Clarivate based on citations, and recent recipients of the Wolf Prize.

Not surprisingly, this post is not that different from last year's.

Although we know little about how the process actually works or the explicit criteria used, I have a few speculative suggestions and observations.

1. The Wolf Prize is often a precursor. According to Wikipedia, for physics, "from the 26 prizes awarded between 1978 and 2010, fourteen winners have gone on to win the Nobel Prize, five of those in the following year." For chemistry, 13 awardees have subsequently won a Nobel.

2. Every now and then, the Swedes surprise us. Sometimes this may be because we now take for granted something discovered thirty or forty years ago.

3. Every few years, the physics committee seems to go for something technological, sometimes arguably outside physics, perhaps to remind people how important physics is to modern technology and other areas of science.

4. They seem to spread the awards around between different areas of physics.

5. Theory only gets awards when it has led to well-established experimental observations and new experimental fields. Brilliant theoretical discoveries that motivate large research enterprises (more theory and unsuccessful or ambiguous experimental searches) are not good enough. This is why predictions based on citation numbers may be misleading. High citations (particularly for a specific paper) are usually a necessary, but not sufficient, condition for a prize.

6. Once an award has been made on one topic, it is unlikely that there will be another award for a long time, if ever, on that same topic. In other words, there is a high bar for a second award.

7. I don't think the logic is to pick an important topic and then choose who should get the prize for the topic. This approach works against topics where many researchers independently made contributions that were all important. The awardee needs to be a standout who won't be a debatable choice.

8. The prize can go to at most 3 individuals and does not go to groups. This works against the large collaborations in particle physics and cosmology that produce significant work.

What do you think of these principles?

For some of the above reasons, I discuss below why I am sceptical about some specific predictions.

My top prediction for physics is Metamaterials with negative refractive index, going to John Pendry (theory) and David Smith (experiment). I know little about this topic.

Is it just a matter of time before twisted bilayer graphene wins a prize? This might go to Allan MacDonald (theory) and Pablo Jarillo-Herrero (experiment). They recently received a Wolf Prize. One thing that convinced me of the importance of this discovery was a preprint on moiré WSe2 with beautiful phase diagrams such as this one.

The level of control is truly amazing. Helpful background is the recent Physics Today article by Bernevig and Efetov.

This is big enough to overcome 6. and the earlier prize for graphene.

The prediction of Berry and Aharonov for topological phases in quantum mechanics is reasonable, except for questions about historical precursors in optics and quantum chemistry.

The prediction of topological insulators goes against 6. and the award to Haldane in 2016.

Predictions of a prize for quantum algorithms (Shor, Deutsch, Brassard, Bennett), conformal field theories, and braiding statistics go against 5. 

Chemistry 

I don't know enough chemistry to make meaningful predictions. On the other hand, in 2019 I correctly predicted John Goodenough for lithium batteries.  I do like the prediction (last year) from Clarivate for Biomolecular condensates (Brangwynne, Hyman, and Rosen). I discussed them briefly in my review article on emergence.

What do you think about my "principles"?

What are your predictions?

What are the assumptions of science?

 Science makes assumptions about the nature of the world and the nature of humanity. We are so used to these ideas that we may overlook that they are assumptions. Before the emergence of modern science, people did not necessarily believe these things were true. 

Existence of regularity and order

There are patterns and regularity in nature, even though at first glance this may not appear to be so. Nature is not completely random, chaotic, or unpredictable. Not every object or event is different from every other object or event. Some things are similar. Finding and describing this regularity is a challenge that requires persistent effort. Scientists assume the patterns are there and eventually they will discover them.

Existence of laws of nature

The order and regularity can be encoded in laws that describe diverse systems and situations. Often, the laws can be succinctly stated and may be written in a simple mathematical form.

Uniformity (universality)

 Many laws are believed to apply everywhere and for all time. For example, the laws of physics apply on earth and in outer space. In contrast, Aristotle claimed there were different principles involved in the motion of objects on earth and in the “heavens.” Things may change, but there are also laws describing how they change. Even though some laws have a limited range of applicability, that range can often be explicitly defined.

Intelligibility and rationality

Although humans have finite and fallible minds, they have the ability to discover and understand the laws of nature.

Objectivity

All observations and their analysis, interpretation, and communication involve human subjects. Yet, it is assumed that with due diligence, the results can be independent of human subjectivity.

Induction

Deduction is the rational process of making specific assumptions and then using the rules of logic to draw a conclusion. In contrast, induction is the process of moving from particular examples to making universal statements. For example, every day of my life, I have noted that the sun has risen. Everyone I ask agrees that this is true in their lives as well. Historical records confirm this to be true. And there are solid scientific reasons that verify this reality. From these observations, I induce the general proposition that for the rest of my life, and probably for much longer, the sun will continue to rise each day. Science is built on induction. Scientists make specific observations about the natural world, and from these observations they hypothesise a general proposition to hold for a wide range of circumstances beyond the initial observation.

Induction is not logically justified. Nevertheless, the success of science testifies to the value of inductive reasoning. But induction can fail when data is limited or difficult to obtain, mistakes are made in gathering or interpreting the data, or prejudice blinds someone when deciding what data to accept or reject. A famous example of the failure of induction was the existence of black swans.  In seventeenth-century Europe, this was considered an impossibility as all observed swans were white. However, in 1697 a Dutch expedition to Western Australia discovered black swans. The philosopher John Stuart Mill (1806-1873) wrote: “No amount of observations of white swans can allow the inference that all swans are white, but the observation of a single black swan is sufficient to refute that conclusion.”

On the one hand, the success of science suggests that making these assumptions is reasonable. On the other hand, why do we assume they will continue to be true, even when we have experiences, individually and collectively, where we fail to find patterns or make sense of things? In other words, we have empirical evidence that is consistent with the assumptions not always being true.

Max Planck stated that, “Anybody who has been seriously engaged in scientific work of any kind realizes that over the entrance to the gates of the temple of science are written the words: 'Ye must have faith.'” 

Do you agree these are assumptions? Would you restate any of them? Can you think of other things that scientists assume?

For my blog Google Analytics has gone weird

I occasionally look at Google Analytics for my blog in the slim hope it might tell me something about the level and nature of interest (or lack thereof) from my audience.

Until a few months ago, I thought the data was somewhat meaningful and believable. It gives data on the number of views for the whole blog and for individual posts. It also provides information for specific countries.

Until a few months ago, there were (supposedly) about one thousand page views per day. A typical post would attract a few hundred views within a week of posting and sometimes up to a few thousand after a few years.

However, the graph below shows how daily traffic increased dramatically at the beginning of August, rising to several hundred thousand views per day!

[Apologies for the poor quality of the screenshot... click on it for higher clarity]

Of course, at first I thought that this was exciting and flattering. However, I quickly realised this was ridiculous. This can be seen by looking at the breakdown by country.

Of the 10 million views from the past 3 months, 1.1 million came from Brazil, and about 300 K each from Ukraine, Bangladesh, Saudi Arabia, and Venezuela. Somehow, I don't think there are that many people in those countries interested in my blog!

Perhaps more importantly, if the traffic was real, it should translate into an increased volume of blog comments, personal emails, speaking invitations, enquiries from potential students... I have seen none of that.

The traffic must be driven by some sort of bots, troll farms, or AI site scraping. I asked Google Gemini whether AI could have increased my traffic. It responded

your traffic surge is a direct result of leveraging the content creation boom and capturing high-intent organic traffic ahead of your competitors [1].When ChatGPT launched, it created an immediate divide between blogs that adapted quickly and those that ignored the technology.

This is wrong, as I completely ignored the technology! I have never used it to write content. 

It also provided me with an unrequested "Step-by-step plan to protect my gains." [Shameless AI self-promotion].

A bit more data. The number of views of individual posts has not changed dramatically and still seems reasonable. Overall traffic on my other blog has also increased, but not in such ridiculous and unbelievable ways.

Overall, this now makes me sceptical about whether any of the Google Analytics data is meaningful. It also makes me more doubtful about the massive traffic claimed by some prominent bloggers.

This is all weird. Do you know of other bloggers with similar experience?

Lecture on degenerate Fermi gases at low temperatures

Here are the slides for an undergraduate lecture I gave today. The slides also include the derivations I gave by hand on the document viewer.

For historical background, I found a book chapter from Out of the Crystal Maze, interesting and helpful.

Some fascinating context concerning Arnold Sommerfeld. Over a period of 32 years, he built up an amazing school of theoretical physics in Munich. He mentored seven future Nobel laureates. However, sadly, when he retired in 1934 from a Chair in Theoretical Physics, it took four years to replace him because of interference from the Nazi Party. Even worse, in the end, Wilhelm Muller, a Nazi loyalist, was appointed. According to Wikipedia

"Müller, an aerodynamicist, had not been thought of as a theoretical physicist before this time, and opposed the "new" theoretical physics promoted by scientists such as Albert Einstein. His appointment is seen by historians as political, and during his tenure he would teach only classical physics"

What is so amazing about the degeneracy pressure for Fermi liquids?

 I just finished preparing my undergraduate lecture on ideal Fermi gases at zero temperature. Here are my slides. 

(Note. The derivations are not included, as I sketch them by hand on a document viewer.) (Aside. Personally, I would prefer to do derivations on the whiteboard, but this is viewed unfavourably as the video recording of the lecture will not capture that.)

A few things I find amazing about the lecture content.

Degeneracy pressure is a macroscopic quantum effect, since it vanishes in the limit Planck's constant goes to zero.

Degeneracy pressure is relevant to astrophysics. I enjoyed learning more about how it prevents the collapse of white dwarfs and neutron stars. For the first time (to my memory) I derived the Chandrasekhar limit for white dwarfs. I also enjoyed seeing how this limit is essentially the same physics as the Tolman-Oppenheimer-Volkoff limit for neutron stars.

An earlier post discussed some things I find amazing about Chandrasekhar's expression for the upper.

A few historical notes I found interesting.

It is the 100th anniversary of Fermi-Dirac statistics and the distribution function. It is impressive how Ralph Fowler immediately saw how the degeneracy pressure could explain the stability of white dwarfs. He was the PhD supervisor of Dirac, Chandrasekhar, and Mott. He was also the first to define the zeroth law of thermodynamics.

Chandrasekhar's limit was not accepted for a long time because of the opposition of Arthur Eddington. In hindsight, some consider this was a setback for astrophysics, particularly interest in the possibility of black holes, for decades. This opposition to the profound new ideas of a young PhD student by a legendary senior figure reminded me of Bardeen's opposition to Josephson. However, in that case, within a few years, experiments showed that the establishment figure was wrong.