Can entropy of a system decrease if we wait long enough?

Question

A ball rests on a smooth surface. The ball's particles are in constant motion. So are the particles of the floor. Some of the ball's particles collide with the floor's particles and transfer kinetic energy. But overall, the kinetic energy of the floor and the ball is constant. They are at thermal equilibrium.

But, according to probability, there is an infinitesimally small chance, but a chance nonetheless, that the particles of the floor align in such a manner that they bump all the particles in the ball just the right way, so that it bounces of the ground.

Notice, the law of conservation of energy is not violated here. The temperature of the floor will decrease in doing so. But the second law of thermodynamics is violated. Entropy is decreasing. If the ball were smaller (much smaller), the probability of this happening would be much larger.

How is this possible?

Answer 1

You have to realize that depending on the dimensions of the variables under consideration the laws of physics change.

When the dimensions are of order of $\hbar$, the Planck constant, it is the framework of quantum mechanics, and this is the underlying framework from which all other physics frameworks emerge. Quantum mechanics sees elementary particles, molecules and structures less than nanometers, although its results can manifest macroscopically, as in the crystal lattice order, or in superconductors and superfluids which obey large dimensions quantum mechanical equations. In addition to quantum mechanics at the particle level special relativity is also the law of nature.

Nature and physics which describes nature mathematically have no discontinuities. As dimensions grow statistical mechanics is the framework to describe ensembles of particles/atoms/molecules, in the beginning quantum statistical mechanics emerges as the next framework and then classical statistical mechanics where the velocities are such the Newtonian mechanics emerges from special relativity.

When instead of looking at the particulate nature of matter, numbers become very large macroscopically, order of $10^{23}$ particle per mole, thermodynamics emerges as a framework to describe the macroscopic behaviors. The laws of thermodynamics were discovered long before statistical mechanics. They seemed absolute and were for this reason called "laws" because they were the postulates on which the mathematics of thermodynamics was developed. But there is continuity, and in the framework on which thermodynamics emerges, statistical mechanics, the law of entropy is shown to be probabilistic, and holds with very very high probability as long as the numbers are of order $10^{23}$.

A common use of statistical mechanics is in explaining the thermodynamic behavior of large systems. Microscopic mechanical laws do not contain concepts such as temperature, heat, or entropy, however statistical mechanics shows how these concepts arise from the natural uncertainty that arises about the state of a system when that system is prepared in practice. The benefit of using statistical mechanics is that it provides exact methods to connect thermodynamic quantities (such as heat capacity) to microscopic behaviour, whereas in classical thermodynamics the only available option would be to just measure and tabulate such quantities for various materials. Statistical mechanics also makes it possible to extend the laws of thermodynamics to cases which are not considered in classical thermodynamics, for example microscopic systems and other mechanical systems with few degrees of freedom.

One can define entropy within statistical mechanics considering the probabilities of the microsystems contained in the macrosystem, for example the Gibbs entropy:

$$S=-k_B\sum_ip_i\ln p_i$$

The quantity $k_{B}$ is a physical constant known as Boltzmann's constant, which, like the entropy, has units of heat capacity. The logarithm is dimensionless.

So what is a law in one scale of physics is a value dependent on probabilities derived from the microstates. It is very very improbable for the atoms of the floor to align and kick the ball.

Answer 2

Short answer yes (but we have to wait a very long time)

But entropy is a real thing and the second law cannot be consistently violated. The recent paper by Dr. Hawking on black holes (and what happens at the event horizon) along with his previous debates with Leonard Susskind (The black hole war: Book by Susskind, Lecture by Susskind, Wikipedia) are all trying to explain (among other things) what happens to entropy as it enters a black hole. As it may be possible for entropy to decline slightly over very long periods of time it is not possible for entropy to be consistently reduced in a physical process such as entering a black hole.

If we wait a very very long time we will eventually see in an isolated area entropy locally decreasing before increasing again. However for an appreciable decrease in entropy we would almost certainly have to wait so long that entropy has already maxed out at which point the entropy arrow of time is meaningless. See Boltzmann Brain:

A Boltzmann brain is a hypothesized self aware entity which arises due to random fluctuations out of a state of chaos. The idea is named for the physicist Ludwig Boltzmann (1844–1906), who advanced an idea that the known universe arose as a random fluctuation in the process of inflation, similar to a process through which Boltzmann brains might arise.

The problem being that if the universe lives for an infinitely long time there should be an infinite number of Boltzmann Brains exactly like your own compared to only one you have. This paper claims that the problem can be resolved by things like the acceleration of the expansion of the universe.