A Brief History of Entropy pt. 3 – Disorder

Still, something was just not right.

Dalton’s atomic theory was ridiculed by the Academics who refused to believe in the existence of such tiny particles, impossible to disintegrate and impossible to observe. Or are they?

A Checkmate to Caloric Fluid

What is heat, exactly? In 1777 Antoine Lavoisier proposed the idea, based on his experimental results, that heat was indeed a self-repelling fluid. The caloric flows from warmer to colder bodies and its amount is constant throughout the Universe. It cannot be destroyed nor created. At that time there was a debate over whether the cold was a fluid as well or not, but I can assure you that it is not for his theory that Lavoisier was executed in 1794 during the Reign of Terror. After all, Sadi Carnot worked out his theorems on the efficiency of thermodynamic cycles assuming that the caloric theory was right. But there is a but. A large one.

After much work, physicists agreed on the principle of conservation of energy. What about the caloric? Benjamin Thompson, also known as Count Rumford, an American-born English physicist, inventor and many other things performed a series of study on this matter. Being particularly at ease with ammunitions as a counter-revolutionary (indeed, he was a loyalist during the American Revolutionary War), he noticed that the heat produced by friction while producing cannon balls was unlimited for all practical purpouses. This meant that it was not a conserved quantity as previously believed. Some years later, in the middle of the 19th century, the combined work of James Prescott Joule and Rudolf Clausius concluded that heat was actually a form of energy. Heat and mechanical work are in fact equivalent and can be treated on the same footing. The caloric theory was intrinsically flawed, but it took almost 50 years to get rid of it.

A Disordered Mind

Finally, the idea of heat as an effect of mechanical work poured into scientific papers. In Treatise on the Mechanical Theory of Heat, the American engineer Richard Sears McCulloh stated very clearly that

the mechanical theory of heat, sometimes called thermo-dynamics, is that branch of science which treats of the phenomena of heat as effects of motion and position.

The idea was sound, but proving it with mathematical rigour was a completely different matter. Anyway, in 1859, James Clerk Maxwell, a young and brilliant Scottish physicist, after reading Clausis’ papers put forward a formula for the evaluation of molecular velocities, which gives the proportion of molecules having a certain velocity in a specific range. This is still considered one of the most important formulae ever conceived. In fact, not only Maxwell provided a way to relate the number of molecules, the volume and the pressure of a gas, but he even introduced statistics into classical mechanics. In fact, given the ridiculous large number of particles, it is too hard to follow the evolution of the system completely. For instance, one mililiter of water contains about 3×10²² molecules! Instead, a small number of quantities, calculated as an average of some properties of the ensemble, describe the macroscopic behaviour of the gas as a whole. These are usually pressure, volume, temperature and the number of moles. An aleatory component never entered any law of physics before.

In 1871, the Austrian physicist Ludwig Boltzmann generalized Maxwell’s formula into the now-called Maxwell-Boltzmann distribution. Boltzmann was a sad genius. He got his PhD in 1866 when he was only 23 years old with a thesis on the kinetic theory of gases, under the supervision of Joseph Stefan. Nowadays, with their Stefan-Boltzmann law we can calculate the temperature of stars. As a professor, Boltzmann supervised the work of Walther Nerst, who won the Nobel Prize later in 1920.

If heat seemed obscure to most of the physicists of that time, entropy was (and partly is) still a mythological beast. Probably, one of the most important contribution of Boltzmann is in fact his explanation of entropy. At the turn of the century, he proposed a statistical interpretation of this quantity. Bear in mind that around 1900 the debate about the existence of atoms was still very fierce, let alone the kinetic theory of gases. One of the main opponents of this theory was in fact Ernst Mach, a very tough guy in the history of philosophy of science. Boltzmann’s idea was to use statistical mechanics to describe the average properties of a system, namely a gas or a solid. When observed, a system is found in a certain configuration, called state. Now, many slightly different states might share on average the same values of pressure and temperature. If W is the number of the number of such states in which the system could be found, then the entropy is given by:

S = k_{B}\ln W

The constant of proportionality kB is now called Boltzmann constant and in the kinetic theory of gases it relates the temperature to the average velocity of the molecules:

\frac{1}{2}m<v^{2}> = \frac{3}{2}k_{B}T

With this definition of entropy, the Second Principle of Thermodynamics can be rewritten as:

\Delta S' = \int \frac{dQ}{k_{B}T}

This deeper formulation, proposed for the first time in 1877, will be of magnificent importance for the communication technology during the second half of the XXth century.

So why does entropy increase over time? It is largely more probable to find the system in a disordered state rather than in an ordered one. The enormous number of collisions tends to shift the system towards one of such states. Since the number of disordered state configurations much is larger than ordered ones, entropy (ln W) tends to increase on average. Finding all the molecules in one corner of the box is not forbidden, is just incredibly improbable!

Unfortunately, only few people supported Boltzmann’s theory. In particular, Ernst Mach fiercely opposed himself to the idea of molecules: basically, it was not economic one. In fact, in order to explain the properties of a gas, one had to suppose the existence of an incredible number of entities which are impossible to observe!

In 1906 Boltzmann fell to pieces, his mental stability collapsed and he committed suicide during a summer vacation in Duino, near Trieste (at the time still part of the Austro-Hungarian Empire). To some extent, Boltzmann anticipated the tragic ending of his Nation and of his Times. His now-celebrated formula for entropy was engraved on his tombstone in Vienna.

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