söndag 18 juni 2017

Repulsive forces as the cause of heat diffusion

Recently some thoughts have crossed my mind about our general understanding of the fundamentals of thermodynamics, especially concerning its alleged foundation in mechanics at the microscopic level. These thoughts are modest in scope but the more papers I read on the subject it seems to me that this particular aspect is rarely mentioned, if it is mentioned at all. Therefore I would like to give it some attention here. The classical formulation of the second law of thermodynamics goes something along the lines

Heat flow spontaneously from higher to lower temperature

This is an empirical observation dating back to the time when the atomic theory of matter was not established science. Hence, at that time, it was a somewhat bold to say that the spread of heat in space from higher to lower temperature could in fact be explained by inter-molecular forces at the microscopic level. One problem, pointed out by Poincaré and others, was the time-reversibility of the standard laws of mechanics. If, at any moment, you were to reverse the velocity vectors of every molecule in the system, then the process would start to run backwards resulting in the opposite process of heat going instead from lower to higher temperture. This might not seem as that great of a difficulty to overcome, since an argument could be put forward that such an event is so unlikely to occur that it is never seen in reality. This probabilistic argument was to become the starting point for statistical thermodynamics and subsequently statistical mechanics, and it has prevailed to this day.

A somewhat more controversial branch of early thermodynamics is the never ending story about the ever increasing entropy. Entropy has its origins in an apparently innocuous looking term arrived at in the analysis of the Carnot cycle

dS = dQ/T

which during the course of an entire Carnot cycle can be shown to increase, assuming the usual form of the second law of thermodynamics. This observation, allied with the probabilistic explanation for heat spread, sparked an intense quest for the microscopic foundation of this mysterious new quantity called entropy. At this stage people were no longer talking simply about the spread of heat in physical space but instead more generally of the increase in entropy in the 6N-dimensional phase space of the molecules taking part in the process. A mechanical explanation for the increase in entropy was never found though, the Gibbs entropy can in fact be shown to be constant under Newtonian dynamics. However, the narrative has prevailed to the present day, and still we here stories of the irreversibility of the smashing of glass and the like as caused by an increase in some entity called entropy.

In any case, I would now like to take some steps back and look at our initial intuitive understanding of the spread of heat and matter in space. 

Imagine an ensmble of atoms and molecules confined to some space as ilustrated above. Thermodynamics tells us that heat should spread from regions with higher temperature to lower, likewise we expect molecules to spread from denser regions to less dense regions. Based on our experience from observing the behaviour of billiard balls in a snooker game it certainly makes sense to envisage these processes as the result of microscopic mechanics, however, I would like you to think carefully about what tacit assumptions you might be making of the molecular interactions taking place. One assumption that comes to my mind is the following:

We assume that there are repulsive forces acting between the molecules

Imagine what the situation would be like if we replaced the repulsive forces by attractive forces, like for example the gravitational force. In that case our intuition would no longer dictate that heat and matter spread in space, but rather the opposite. You agree?

Ok so what?, you might say, it is indeed plausible that there are repulsive forces acting between the electron shells of each atom, that makes sense doesn't it, so what is the problem? Nothing really, except that this simple assumption about the forces being repulsive is hardly ever mentioned. What insights can we gain from this? Well, consider for example the entropy discussion earlier. For many years people tried to explain the increase in entropy from mechanical considerations alone, but if no assumption about the repulsive nature of the forces is injected into the analysis, no wonder why they didn't succeed. If the forces were instead attractive you wouldn't expect any such thing as increase in "disorder".

Furthermore, in atmospheric and cosmological thermodynamics we have seen that the classical picture of thermodynamics probably breaks down at some level. Could that be because we have not properly taken into account the effect of attractive forces such as gravity? Who knows, but I just wanted to make this little point, repulsive forces are the cause of diffusion, remember that. That's all.

Further reading:

Boltzmann’s H-theorem, its limitations, and the birth of (fully) statistical mechanics