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63. Forwards or Backwards?

Loschmidt’s Paradox

The Question:

Scene 1: On a foggy day, Xavier drives his truck towards Quincey’s car and, without slowing down, crashes into it. Luckily, nobody is hurt but the damage is great. Naturally, Qunicey sues Xavier. He has the good fortune that a nearby surveillance camera had filmed the accident, and though the day was foggy and the background fuzzy, one can clearly see the truck crashing into the car.

Come the day in court and Xavier presents the video. Surely, it would be an open and shut case. But no, Quincey disputes the series of events. It was not he who crashed into Xavier’s car but the other way around: Xavier crashed into his truck. To prove his version of events, he too presents the video. Only, he plays it in the other direction.

The judge is at a loss. From the video clip she cannot determine which is the guilty party.

Scene 2: Zoe kicks a ball into the air towards Yvette. It rises on a parabolic path and descends towards Yvette’s foot. The nearby surveillance camera recorded the scene. With the clip run backwards, the ball rises from Yvette’s foot, describes a parabola and lands on Zoe’s foot. Depending on how the clip is run, either one could be the kicker or the kickee.

Scene 3: Cecily pours blue ink into a bottle of water. The ink disperses until the bottle’s content is a pale blue liquid. Ah yes, the surveillance camera… The video is played backwards and, like magic, the liquid in the bottle clears and the ink gathers back into the inkpot.



The Paradox:

Scene 1: yes. Scene 2: yes. Scene 3: Nooo!

In 1872, the Austrian physicist Ludwig Boltzman, described the properties of physical systems that are composed of a very large number of particles, in particular gasses and liquids that are composed of molecules. His main thesis was that as such systems evolve, they become more and more disordered. As time advances, their ‘entropy’ (the degree of unorderliness) always increases. This is what’s known as the ‘Second Law of Thermodynamics’.

Hence, according to Boltzmann and the Second Law, the ink molecules and the water molecules intermix to such a degree that they can never again separate into two distinct liquids.

Not so, claimed his compatriot, the scientist Josef Loschmidt. According to Newton’s laws of motion, if at some point in time, the velocities and directions of all particles are reversed, an evolving system of interacting particles will go through the exact same motions, but backwards. This is called ‘time reversal symmetry’.

It is due to time reversal symmetry that the video clips of Xavier’s truck and Qunicey’s car, and of Zoe’s and Yvette’s football, can be run backwards without anybody noticing anything amiss. And it is due to the Second Law of Thermodynamics that when Cecily’s video clip is run backwards, it is immediately obvious that it is not realistic.

So, a good argument can be made for the Second Law of Thermodynamics and a good argument can be made for time-reversal symmetry. But the two are incompatible. A paradox!



Johann Josef Loschmidt (1821-1895) and his younger colleague Ludwig Boltzmann (1844-1906), both Austrians, both teaching at the university of Vienna, were good friends. But while Boltzmann gained fame and is nowadays a household name, at least among physicists, Loschmidt, due to his shy and self-effacing nature, is largely a forgotten genius.

Born to a poor peasant family, Loschmidt hated work on the farm, so much so that his family considered him useless for anything but academic studies. The penniless student started out in Vienna, first as a janitor then as a school teacher, eventually becoming university professor. A slim, 51-page booklet in which he described molecular structures, and which was the first to give a graphic representation of molecules, has been called ‘the masterpiece of the century in organic chemistry’.[1]



According to Loschmidt, if the ink-and-water-system evolves for, say, a minute, and is then run backwards, it would, after another minute, end up exactly where it started: water and ink separated. That, again according to Loschmidt, is the inescapable consequence of time reversal symmetry, as implied by Newton’s Laws. And to achieve this result, all one has to do is reverse the velocity vectors of the molecules.

Hmmm…There are about 33,500,000,000,000,000,000,000,000 molecules in a liter of water. That’s why Loschmidt aptly called his set-up a Gedankenexperiment (thought experiment). Obviously, it can’t be done in reality. So, classical Newtonian mechanics with its time reversal symmetry does not hold for systems with myriad particles, like gasses and liquids.

Hence, a different approach is required to deal with Scene 3. It was Boltzmann who invented the appropriate tool box for gasses and liquids: not classical mechanics but thermodynamics. True, molecules bump into each other, and interact with each other, according to Newton’s laws. But when the number of these interactions becomes humongous, different things begin to happen. Thus, while themodynamics is derived from classical dynamics on a microscopic level, on the macroscopic level it is governed by statistical laws. The upshot of all this is that such systems cannot be reversed back to their initial states.


Technical supplement:

One way to reconcile the paradoxical contradiction between Newton’s time reversal symmetry on the one hand, and thermodynamics on the other, is to regard the Second Law of Thermodynamics not as an actual physical law but rather as an overwhelming probability. This was apparently Boltzmann’s answer to Loschmidt: his theory on gasses and liquids was to be considered a statistical statement, which must be true in the overwhelming majority of scenarios. In that sense, it can be considered more of an axiom than a physical law.


Apart from its central statement – that entropy, i.e., disorder, must increase over time – the Second Law of Thermodynamics has another extremely important consequence. The Law, call it an overwhelming probability or a law, implies that a perpetuum mobile, a machine that runs forever without addition of energy is impossible…or overwhelmingly improbable.


© George Szpiro, 2019





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