The Fundamentals of Thermodynamics – with David Lavis

David, can you give us a definition of what exactly thermodynamics is?

The experience of temperature, of objects being more or less hot, is part of our everyday interaction with the world. Thermodynamics is the branch of physics which describes this aspect of reality.

Mechanical systems change their states when work is done on them; in thermodynamics a change of state is also produced by means of a flow of energy in the form of heat into or out of the system. In this context a change of state which does not involve a flow of heat is called adiabatic. The first law of thermodynamics is simply the extension of the mechanical law of the conservation of energy to include non-adiabatic changes of state.

What led an applied mathematician and a philosopher of science to choose to write a book on thermodynamics?

We sometimes wonder that ourselves. The trouble with thermodynamics is that it is a minefield. There are so many statements made, both in books and on the web, which are either untrue or true only when heavily qualified. (Try asking an AI essay-generating app to state and explain the second law.) What incentivised us to begin this work was reading the papers of Elliott Lieb and Jakob Yngvason.

Their axiomatic-algebraic development of thermodynamics avoids, as we see it, these problems, with all the versions of the second law together with a definition of entropy arising in a natural way from the axioms. With the encouragement of Lieb and Yngvason we have embedded their work into a wider presentation which, among other things, includes the mathematically consistent but physically exotic cases of systems with negative temperature and/or heat capacity.

You have just mentioned the second law. Among the laws of physics why does it seem to have a special place?

Apart from C P Snow’s famous remark that the ability to state the second law of thermodynamics is culturally on a par with having read a work of Shakespeare, the second law of thermodynamics does have some features that make it of special note both in the public imagination and for physicists and philosophers of physics.

Firstly, it has the curious property that it can be expressed in three different forms, each of which is an assertion of the impossibility of certain processes, but which on first sight do not appear to be obviously equivalent.

The Kelvin-Planck version, which takes its name from William Thomson (ennobled as Lord Kelvin) and Max Planck, and the Clausius version, so named for German physicist Rudolf Clausius, both retain the practical feel associated with the origins of thermodynamics. Each refers to a system which cycles between two sources of heat (heat reservoirs), taking energy in the form of heat from one reservoir and depositing it in the other, while at the same time expending or absorbing energy in the form work done on or by the environment.

The Kelvin-Planck version of this setup is a heat engine where the heat is absorbed by the system and work is done on the environment, and the assertion is simply that it is impossible for all the heat absorbed from one reservoir to be utilised as work, some being `wasted’ into the other reservoir. The Clausius version concerns a heat pump with heat being absorbed from the cooler reservoir and deposited in the warmer reservoir. It asserts the impossibility of this being done without the expenditure of work by the environment.

The third version of the second law was formulated later by the Greek-German mathematician Constantin Carthéodory. It is more abstract, asserting in its most simple form, that for any state of a thermodynamic system there are other nearby states which cannot be reached by an adiabatic process, that is just by doing work on the system.

One might suppose that for the theory to be coherent these versions need to be equivalent. But this is by no means obvious; in fact certain additional conditions need to be imposed. Whereas the Carthéodory version is very general, encompassing systems with positive or negative temperature and positive or negative heat capacity, both the Kelvin-Planck and Clausius versions need positive heat capacity and the Kelvin-Planck version also needs the temperature to be positive.

Each of the versions of the Second Law is a consequence of the Lieb-Yngvason axioms, but with the caveats that, as we have indicated, both the Kelvin-Planck and Clausius versions assume the heat capacity to be positive with the Kelvin-Planck version also requiring the temperature to be positive. With these restrictions the three versions can be shown to be equivalent.

But why should this law, involving on the one hand a rather formalized version of a practical device and on the other a seemingly abstruse mathematical result, have the cultural importance ascribed to it by Snow and the hold it appears to have on the public imagination? I think this is to do with entropy. None of the versions of the second law described above mentions entropy. However, entropy along with temperature is a new quantity which arises in thermodynamics, and the law that states that entropy does not decrease in adiabatic processes is often taken as another version of the second law. In fact it a consequence of, rather than a statement of, the second law and as such it can be shown to follow from the Lieb-Yngvason axioms.

It seems that the wider supposed significance of the second law arises here because of two connections. The first is that made between entropy and order, with more entropy implying more disorder, and the second between entropy change and the arrow of time. So if we take together, the second law understood as the law of non-decreasing entropy (often stated as simply `increasing entropy’), and entropy as measuring disorder and put them together with increasing entropy defining the direction of the arrow of time, we have a broad-brush view of a universe getting more disordered.

Of course these inferences are matters of discussion and dispute. The law of non-decreasing entropy is only for adiabatic processes, including the limiting case of processes in isolated systems, order needs defining, and the identification of entropy increase with the arrow of time is a matter of much discussion and dispute. A treatment of order necessarily involves the microscopic level of statistical mechanics, which we have excluded from our book, and a consideration of the arrow of time needs, and has received, extensive book-length treatments. This is not to say that these inferences from the second law are necessarily invalid. But they do need strong foundations which we have attempted to provide.

What do scientists in the field have to learn from the philosophy of physics?

One of the roles of philosophers of physics is to examine the structures of scientific theories. In the context of any particular theory this will involve an examination of the foundations of the subject.

The impetus in physics is to move forward, exploring the consequences of a theory. However, it is illuminating and important to examine foundational questions.

Why should philosophers of physics be interested in thermodynamics?

Thermodynamics is one of the major branches of physics, and according to Einstein is ``the only physical theory of universal content concerning which I am convinced that, within the framework of the applicability of its basic concepts, it will never be overthrown’’. And yet, compared to quantum mechanics and relativity, thermodynamics has attracted less attention from philosophers of physics.

As I have said, our aim is to ameliorate this situation by exposing the philosophically interesting questions that arise in the theory and laying firm foundations on which they can be explored.