# Breaking laws

I have just got back from Liquids 2014, a conference in Lisbon. I thoroughly enjoyed it, and Lisbon was great – sunny, friendly and with great seafood. I will probably write a post or two on the best talks later, but now I want to talk about how I learnt of two scientists trying to reinvent thermodynamics.

Thermodynamics is one of the triumphs of modern science. It allows us to engineer everything from power stations to fridges. A key element of thermodynamics is temperature. We are all familiar with this, and we all know what happens when we leave a hot cup of coffee without drinking it, it slowly cools down and ends up as room temperature. Heat energy leaves the coffee until the coffee is the same temperature as the room.

Now, temperature can be calculated from the perhaps odd-looking expression that involves the number of ways a total energy E can be distributed among the molecules of the coffee or of anything else, called Ω(E). These Ω different ways are called the states of the system, which can be coffee or anything else. The temperature is related to a derivative of this number of states:

T = 1/(∂[lnΩ(E)]/∂E)

This looks a bit horrible but it is not too bad. In words, it means that temperature is one over the rate of change of ln[Ω(E)] with respect to the energy E. ln is the logarithm to th base e = 2.71…. This is all rather abstract but all it really means is that if you add a little bit of energy to a hot thing, then its number of states increases a little, whereas if you add a little bit of energy to a cold thing, then its number of states increases a lot.

So far I have just described the thermodynamics we have been using for more than hundred years. I am happy with this, but two scientists, Dunkel and Hilbert, are not so happy with this. They have decided that the thermodynamics we are using is not sufficiently mathematically rigorous, and published a paper in Nature Physics in which they derive a new, mathematically rigorous, thermodynamics.

Fine, except for one problem: it does not work, in the sense that as far as I can see it does not allow me to conclude that my coffee should cool down until it reaches room temperature, at which point it should stop cooling. I view this problem as a deal breaker.

In the past I have left coffee cups for a long time, and when I have come back they were at room temperature. I want a thermodynamics that agrees with this. The conventional version predicts this, the Dunkel and Hilbert paper appears to be silent on whether my coffee cup should cool down or heat up*.

I have to say I was a bit astonished that Nature Physics published this. Given that we have been using temperature rather successfully for over a hundred years – for example my room thermostat currently reads 26.5 C – you might have thought they would be cautious about publishing a paper redefining it. Apparently not.

I guess the moral of this is that science should remained grounded in observations, not get carried away with mathematical rigour. If you want to know that coffee cools down, just do the experiment, and if a mathematician or theoretical physicist disagrees with our finding, just dunk your finger in it and say ‘Feels like room temperature to me’.

* Strictly speaking this is a bit harsh on Dunkel and Hilbert, in practice their ‘thermodynamics’ should only differ from thermodynamics for very small objects, so really it is only a nanosized coffee cup that could end up, according to them, at a different ‘temperature’, hotter or colder, than the surroundings. See here for a technical criticism of their work by Frenkel and Warren, and here for Dunkel and Hilbert’s reply to that criticism. There is a further criticism by Schneider it al. here. A technical version of my coffee cup experiments is basically Schneider it al.’s point v).