The amazing world of physics seems paradoxical in its very essence. Every so often it is fun and stimulating to be reminded of the physical paradoxes and let them blow the circuits of one’s brain. This article will undertake to show the differences between Zermelo’s paradox and Loschmidt’s Paradox as well as travel over into some other paradoxical results of 20th century physics.
The first paradox was named after Zermelo. He was a famous axiomatizer of set theory. This paradox involves two seemingly contradictory aspects of physics. One has to do with Poincare recurrence. The other has to do with the second law of thermodynamics.
Let’s say that you have a system governed by Newton’s mechanical laws and also that energy is conserved(e.g. the universe). The system has some initial conditions and evolves from there. Poincare’s recurrence states that the system will return arbitrarily close to its initial conditions.
This universe also seems to obey the laws of thermodynamics, the principle of the conservation of energy (1st law) as mentioned above, and the 2nd law, which has to do with an isolated system always increasing its entropy.
Now we run up against a problem. When a system is at a point away from its initial conditions, it has higher entropy due to the second law, but Poincare’s recurrence states that a system can return arbitrarily close to its initial conditions. These two statements together entail a reduction of entropy if a system returns because points reached away from the initial conditions have higher entropies than points closer to the initial conditions. Thus, it seems like the laws of Newton and the laws of thermodynamics are at odds.
Zermelo apparently subscribed to the vitalism that was in the air during those times and was a strong supporter of the the second law of thermodynamics. Something had to give and for him it was the Newtonian mechanical view of the universe. The contradiction above led him to declare that the mechanical view of the universe was bankrupt.
Surprisingly, it seems like there is no universal agreement on the solution to this problem yet. People have put forward Boltzmann’s work and statistical mechanics. Under this view, entropy is basically the maximization of the number of ways to arrange units of energy and particles or the like. You can show using the statistical mechanics approach that one can reproduce the macro approach of thermodynamics.
Why is this statistical approach relevant? As the number of ways of arranging things increases, some probabilities become almost certainties. You’d basically see a spike on the graph. So, it’s highly probable that entropy will continue to increase, but there are minute probabilities that the universe could fall back into one of the low entropy states near the initial state of the universe. How the universe got into such a low entropy state has also been a problem in cosmology. Anyway, there are more advanced solutions offered, but this is all I know of this particular paradox.
The second paradox has to do with the fact that newton’s laws are time reversal invariant. This basically means that you can run things backwards and forwards and not beable to notice the difference. There is a symmetry. So, if this is the case that all particles are governed by these time reversal invariant laws, why do we seem to only experience time in one direction? Why do we never see a broken egg come back together?
Some have wanted to connect the “arrow of time” with the second law of thermodynamics because it also has the unidirectional character i.e. entropy always increasing. With the statistical approach, we see that it is actually possible, but highly improbable that an egg really come back together after being broken. Some also have suggested that this is related to cpt violation. It has to do with a certain kind of particle imbalance that might also have to do with the problem of why we find more matter than antimatter in our universe.
Traveling into other paradoxical results, let’s say you have a particle in a box. In classical mechanics, you assume you can find the particle anywhere in the box and at any continuous energy level. If the particle does not have enough energy, it’s not coming out. It will just bounce back and forth against the walls.
It’s very different in quantum mechanics. You find that you can find the particle only in certain regions with a certain probability of being in those regions. You even find that there are some regions where the particle has zero probability of being in that region. This is amazing because you’re thinking of a particle bouncing back and forth here, yet there are gaps in its line of movement. What?
This makes a lot more sense when we think of waves, and we can because we have found that particles have a wave nature and also that light has a particle nature rather than just a wave nature. This is the famous wave-particle duality. Anyway, waves have nodes and these nodal regions correspond to the regions where the particle has zero probability of being.
You also find that energy is discretized; the particle cannot be found at certain energies. Speaking of energy, you find a weird phenomenon in quantum mechanics where a particle can escape its box despite not having enough energy classically to do so. The common example is that of going through a wall, leaving the wall intact. This is called tunneling. There are small probabilities that the particle will come out of the box in this manner. Using the box as a model, we can explain radiation as resulting from these escaped particles having small probabilities of escaping the nucleus. Modern electronics also are based on quantum phenomena like this.
The accepted physics is so wacky, let alone the speculation! When you get to relativity, you get the ideas of time dilation, lorentz contraction, gravity warping spacetime by the presence of matter and energy, time slowing down due the presence of gravitational fields, black holes, time travel speculation, teleportation speculation, multiverse speculation,etc…