Physical Sciences
June 13, 2022

How geometry expresses the Second Law of Thermodynamics

Physicists have long struggled to explain how the inevitable increase in the universe’s entropy can be reconciled with the reversible laws of quantum mechanics. Now, Professor Chris Jeynes at the University of Surrey Ion Beam Centre, UK, believes he has found a solution in geometry. This new geometrical thermodynamics shows how the stability in time of structures as diverse as atomic nuclei, the DNA helix, and spiral galaxies can be explained as a natural result of systems adopting their maximum possible entropy configuration, even as the universe evolves over time.

Both quantum mechanics and thermodynamics are central concepts in modern physics. Yet while quantum mechanics is widely believed to be the fundamental description of how the universe works, the goal of deriving thermodynamics from quantum principles has eluded physicists for many decades.

A central part of this challenge lies in the Second Law of Thermodynamics, which guarantees that no physical process where energy is converted from one form to another can ever be 100% efficient. The process must generate ‘waste heat’ in some form, and the ‘entropy’ of the universe as a whole must increase.

This phenomenon (quantified by ‘entropy’) is often described through the concept of the ‘Arrow of Time’ which flows in only one direction: a video that looks ok when run backwards must be very boring indeed! Strangely, this stands in direct contrast with the quantum picture, where almost all physical processes are known to be ‘time-symmetric’: they behave just the same if time is reversed.

Hubble image of NGC 1566, taken 2nd June 2014 by NASA Goddard Space Flight Center. Parker and Jeynes demonstrate that the stability of spiral galaxies is a consequence of their stable (Maximum Entropy) geometry. Parker & Jeynes (dx.doi.org/10.1038/s41598-019-46765-w; 2019, see Fig. 2).

Despite decades of intense investigation, a solution to this mystery still evades physicists today. Yet through close analysis of real systems and structures in physics, Dr Michael Parker (University of Essex) together with Professor Chris Jeynes (University of Surrey) believe they have finally found a solution.

Two questions to answer
Ultimately, Parker’s theory simultaneously addresses two important questions: firstly, ‘where does the second law come from?’ And secondly, ‘if the basic mechanical laws of nature are reversible, where does the second law’s irreversibility come from?’

So far, attempts to answer these questions have claimed that thermodynamics must ‘emerge’ from more fundamental laws. These claims were made by Ludwig Boltzmann himself: the Austrian physicist who provided the first statistical explanations for the second law. All the same, many physicists aren’t comfortable in believing this ‘emergence’ explanation.

Quantum Mechanics yields a pure measurement, but it’s the thermodynamics that explains it.

Bringing together quantum mechanics and general relativity
Parker and Jeynes provide an alternative explanation in which the second law itself is fundamental. At first glance, this appears to entirely contradict the concept of reversibility in quantum-scale systems. However, in 2022, in collaboration with Professor Wilton Catford (University of Surrey), they showed that even though the alpha particle is absolutely stable, its size can be readily calculated using their new methods of thermodynamics which they call ‘Quantitative Geometrical Thermodynamics’ (QGT). That is, it is the very geometry of the alpha particle that intrinsically embodies the Arrow of Time.

But in 2019, Parker and Jeynes also demonstrated that the stability of spiral galaxies had an alternative explanation independent of the usual ones. It is a direct consequence of a QGT analysis of the (known) entropy of black holes in which their calculations used the example of the central supermassive black hole of the Milky Way. And in 2021 they showed that such black holes have a constant rate of entropy increase (‘entropy production’ is a ‘conserved’ quantity just as energy is).

Thermodynamics seems to have been difficult to reconcile with an increase in entropy.
Artist: C.Evans-Pughe/howandwhy.com

Thus, the new thermodynamics is demonstrably valid across, and applicable to, a vast range of scales: from the galactic to the sub-atomic. It applies equally to black holes (those archetypal general relativistic entities) and alpha particles (which are inescapably quantum mechanical).

Organisation needs a flow of entropy
The second law is strongly related to order – and disorder. Living beings are low-entropy systems and depend on a flow of entropy simply to stay alive. As Jeynes explains, ‘in the context of life on Earth, the Sun is a source of entropy, and the dark night sky is a sink of entropy’. Parker and Jeynes have shown that entropy production is a conserved physical quantity, just as energy is also a conserved physical quantity. Notably, although entropy production necessarily involves dissipation processes, it is still subject to its own conservation law. This is rather shocking for physicists.

When thinking about ‘reversibility’ versus ‘irreversibility’, it may be useful to remember that quantum mechanics provides the mathematical tools we need to handle data from big particle accelerators. Indeed, quantum mechanics was invented to explain the structures of atoms, but rapidly shifted its focus to studying the structures of atomic nuclei. Note that ‘atom’ is a Greek word whose primary meaning used to be ‘indivisible’, but we know today that atoms can be split!

The point is that all big particle accelerator experiments involve high energies. Physicists know that high-energy collisions between nuclei are reversible, that is, there are no ‘dissipation’ processes: if we run the video backwards it looks fine. In fact, physicists frequently use the ‘reverse reaction’ to measure the thing they are interested in. It is when the energy is reduced that the irreversible dissipative processes (such as friction) start to become more important (and all the interesting things happen). All the same, Parker and Jeynes have shown that even in these cases, entropy production is conserved.

Measured and calculated matter radii (in femtometres) are compared. Mass number A (in atomic mass units) is given. Published measurements are referenced in the paper. The calculated values use the Quantitative Geometrical Thermodynamics (QGT) formalism and are a simple function of the number of ‘degrees of freedom’ (DoF) and a scale factor (the proton radius). Measurements include uncertainties given as a last-figure-error. Note that the measurement precision is far better for the stable nuclei 4He and 12C than for the halo nuclei 6He and 8He. Adapted from Tables 1 & 2 of Parker et al (2022) Annalen der Physik, dx.doi.org/10.1002/andp.202100278.

Geometry in the alpha particle
One system where Parker, Jeynes, and Catford have investigated this behaviour is the alpha particle: the nucleus of the helium atom. Containing two protons and two neutrons, this particle is an extremely stable isotope of helium – which never decays into its constituent protons and neutrons.

In their analysis, the researchers showed that this stability is rooted in the alpha particle’s Maximum Entropy geometry. The alpha particle radius obtained by QGT yielded the measured values. ‘Here, we calculated the size of the alpha particle using our QGT formalism, with a precisely correct result,’ Jeynes describes. ‘This depends only on the newly established radius of the proton – from first principles using this single parameter, the measured radius of the alpha particle can be exactly derived!’

Leading on from this result, the team also explored the geometries of two heavier isotopes of helium having ‘halos’ of either two or four neutrons. Each of these nuclei contain an alpha particle ‘core,’ surrounded by a halo of identical neutron pairs. Both the ‘matter radius’ and the ‘charge radius’ of these halo nuclei could also be correctly calculated using QGT, by simply accounting for how the geometry of its centre of mass is displaced from the core by the neutrons in the halo. The agreement with the experimental measurements was astonishing. Likewise, the radii of the nuclei in the ‘helium series’ (including elements such as carbon-12, oxygen-16, silicon-28, and calcium-40) could also all be correctly calculated using QGT.

‘Quantitative geometrical thermodynamics’ (QGT) is the new formalism of this universal thermodynamics.

Spirals on different scales
In parallel with these studies, Parker and Jeynes have also explored the relevance of QGT in two very different yet very common structures, each displaying unique spiral geometries. The first of these are DNA molecules, which convey the genetic information of all life on Earth: from the most primitive bacteria to the cells in our own bodies. Remarkably, all natural DNA molecules have a double-helix structure that is right-handed.

Previously, scientists thought that this was an accidental property of DNA which occurred as life was first emerging, and replicated ever since. But as Jeynes explains, the thermodynamics of the DNA geometry provides a more reliable explanation. ‘We have now demonstrated from QGT that natural DNA must be right-handed,’ he says. ‘This contradicts the usual explanation of the ubiquity of the right-handed form as being only an amplification of an original accident.’

On astronomical scales, spiral galaxies like our own Milky Way are ubiquitous, showing that these spiral structures must remain stable over timescales of billions of years. Again, Parker and Jeynes show that this stability is a natural result of QGT.

The 8He nucleus is modelled in QGT as an alpha particle with a four-neutron shell which is a holomorphic pair of holomorphic neutron pairs. See Parker et al (2022), dx.doi.org/10.1002/andp.202100278; for image see the journal Table of Contents, onlinelibrary.wiley.com/toc/15213889/2022/534/2.

The ‘Arrow of Time’ implicit in the geometry
Ultimately, Parker and Jeynes describe how the geometry of the structures exhibited in each of these widely varied cases is a natural consequence of the QGT formalism. ‘The alpha particle and DNA are both ‘Maximum Entropy’ (very stable structures with zero entropy production), meaning that their entropy is not increasing in time,’ Jeynes explains. ‘That is, the Arrow of Time is not apparent, but it is there nevertheless!’

Spiral galaxies are a different case, since although they have approximately Maximum Entropy (stable) geometries, they also evolve in time (having non-zero entropy production). Considering the dynamics of the supermassive black hole thought to be at the centre of each spiral galaxy and controlling galactic evolution, QGT has shown that the Arrow of Time imposes two fundamental effects: the first is the extremely low rate of mass lost from the black hole by Hawking radiation; and the second is the accumulation of mass (stellar material) by the accretion disk at the galactic centre. The implications on actual galactic evolution haven’t yet been explicitly analysed using QGT, but it is clear that geometrical thermodynamics with its implicit Arrow of Time formalism will have interesting consequences. For example, QGT provides new insights into the long-term stability of such spiral galaxies without recourse to the so-called Dark Matter.

QGT treats the Second Law of Thermodynamics as fundamental. We know that Quantum Mechanics and General Relativity are apparently incompatible with each other as they stand, although each theory has been proved to be astonishingly accurate. But the explanations provided by QGT show that although it may not appear the case at first glance, both Quantum Mechanics and General Relativity are governed by the same laws of thermodynamics operating at all scales of space and time, and which also provide new conservation principles that constrain the allowable evolutionary behaviours. These new conservation principles also imply a family of new ‘symmetries’ (by Noether’s theorem) which underpin the fundamental structure of the universe. QGT offers a better fundamental understanding of how the universe works, and will have immediate and wide-ranging applications in drug development, meteorology, nuclear physics, and cosmology. What the longer-term possibilities will be remains to be seen, but expect to be surprised…

Personal Response

This is a major advance with many applications in widely distinct fields. What are some other interesting (and currently intractable) problems which could be solved using QGT methods?

Both DNA and Buckminsterfullerene (C60) have been proved stable using heavy computational methods of chemistry. But we have proved them stable from QGT simply (and without any numerical computation). These are very simple cases: the problem of DNA folding is of great importance but is currently imperfectly understood. In meteorology, cyclones have the same double spiral shape as spiral galaxies, but cannot be calculated directly. QGT offers the prospect of new approaches to such intractable cases.

The nuclear physics results we report are very simple cases (for QGT) but have puzzled the physicists for decades. These new geometrical methods promise dramatic advances in the knowledge of nuclear structure and stability. In particular, the thermodynamic relationship between stable (reversible) and unstable (displaying irreversibility) nuclei will continue to throw light on routes towards a grand unification theory.

In astrophysics, we have so far treated only black holes and heavily idealised spiral galaxies – very simple cases. But the evolution of real galaxies is a hard problem that we expect QGT to dramatically simplify.

Your research brings together quantum mechanics and general relativity. What potential real-world applications could be developed from this?

This reminds me of Michael Faraday’s famous reply to Disraeli (the then Prime Minister) who was touring his lab and asked him the same question: “What use is it?”; Faraday said something to the effect, “I have no idea Sir, but I guarantee that in twenty years time you’ll be taxing it.” Physicists have been searching for a unified QM-GR for more than two generations now. We have shown a new and entirely unexpected way to obtain it. Look at what physics has achieved in its current ‘monocular’ state (that is, using only energetic ideas): what will become possible when it is unified, and a ‘binocular’ viewpoint (fully using geometric entropy ideas too) becomes available?

And of course, we can already see a variety of ways of making dramatic progress in all the different fields mentioned above.

This feature article was created with the approval of the research team featured. This is a collaborative production, supported by those featured to aid free of charge, global distribution.

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