Mathematics of thermodynamics is being rewritten after 200 years

The theory of thermodynamics has helped us understand devices like engines for more than 200 years, but its mathematical foundations have always been a little too imprecise. Now, researchers are putting the famous theory on a firmer footing, with mathematics more commonly used to describe quantum fields.

Among all branches of physics, thermodynamics is among the easiest to connect to everyday life. This is because its development was driven in part by engineers looking to understand and maximise the efficiency of heat engines, which are idealised devices that model a wide range of familiar technologies, including car engines and refrigerators.

But although thermodynamics is a very successful theory, it has historically lacked mathematical rigour, says Bryan Roberts at the London School of Economics and Political Science. He has set out to rebuild it based on mathematical ideas that draw on geometry and quantum field theory, a notable departure from how thermodynamics has long been understood and taught.

Central to Roberts’s remaking of thermodynamics is the concept of “gauge theory”, which typically deals with properties of objects that aren’t directly observable or manipulable.

A simplified example, involving marbles rolling along a surface, helps to explain the approach. The marbles appear identical, but each has a different colour hidden at its centre.

In a gauge theory, there would be a mathematical space – the “observable” space – defined by numbers that capture the marbles’ motion, and another space – the “bundle” space – that can be formulated to contain information about each marble’s internal colour.

These two mathematical spaces are deeply connected, such that the observable space is the projection of the unobservable bundle space. Roberts says this is a little like shining a light on an object. Even if you were, for some reason, unable to look at the object directly, you could discern some of its properties simply by studying its shadow.

He thinks that it makes sense to use this approach to study thermodynamics because it, too, involves both accessible and inaccessible quantities.

“There’s kind of two levels to thermodynamics,” he says. “There’s the accessible level, things that you can extract work out of, because you can kind of grab onto them and move them around – like the piston that goes in and out of an engine.” And there is a less accessible level: the heat that is generated or lost in a system, which can’t really be manipulated as directly. Roberts defines this as a hidden contribution to energy.

This isn’t a distinction that carries mathematical weight in traditional thermodynamics. There, “work” and “heat” are placed on an equal footing and their sum accounts for changes in an object’s total energy. For Roberts, however, the hiddenness of the heat component of energy leads him to map thermodynamics onto the structure of a gauge theory, placing it in the bundle space.

He says that taking this approach provides an opportunity to take what has already been proved about gauge theory in other areas of physics, and use it to gain a deeper understanding of thermodynamics.

For instance, temperature and entropy – two fundamental quantities in thermodynamics – can be defined in terms of a specific projection from the bundle space onto the observable space. Roberts says that this is a more geometrical definition of entropy than many previous ones, which makes it easier to apply to general systems, from engines to black holes.

Additionally, gauge structure has been connected to experiments when it comes to the quantum theory of electromagnetic fields, and Roberts anticipates something similar might happen with thermodynamics. Specifically, he says that preliminary experiments with certain molecular junctions hint at a thermodynamic version of the Aharonov-Bohm effect, a famous experiment in which a charged particle seems to experience a hidden magnetic field.

Roberts presented the work at the Foundations of Physics conference in Irvine, California, on 16 June.

Lucas Céleri at the Federal University of Goiás in Brazil says that Roberts’s idea is beautiful and complementary to ongoing efforts to understand thermodynamics in the quantum realm as a gauge theory as well.

When applied to quantum objects, thermodynamics becomes even less well-defined and clear, says Céleri. “I’m worried about quantum thermodynamics because there are so many definitions of heat and work, for example. So, if you can put this in a rigorous mathematical theory, then maybe we can formulate a [more] consistent and a unique understanding,” he says.

He and his colleagues have been working to do so by turning to gauge theory, which Céleri says has so far been successful in reproducing some standard quantum thermodynamic results.

One big challenge for both quantum and classical thermodynamics going forward will be combining it with Albert Einstein’s theory of special relativity, but here too the mathematics of gauges might be better suited than more traditional approaches, says Céleri.