# Can the second law of thermodynamics be derived from Quantum randomness?

The second law of thermodynamics says that the entropy of an isolated system continuously increases. Can we say that this is due to Quantum mechanics, which continuously increases the information contained in the system by producing random numbers? Is the entropy of a classical system without randomness always constant?

• I'd love to be proven wrong here, but I don't think there is any established way to do this. First of all, the second law is arguably not a fundamental law, but more of a way to state the "obvious" fact that what is more likely to happen will more likely happen. Also, if there is no randomness in the (classical) system, and therefore the system is fully characterised, then it is not even clear how you would define its entropy. If full knowledge is assumed, then by definition there is a single microstate corresponding to the state, and $S=0$ – glS Dec 11 '18 at 10:46
• Going into QM only murks the waters. QM is fully deterministic up until measurements are performed, and there is probably still no consensus as to what exactly constitutes a "measurement", and as to whether a system's state collapses without anyone there to "measure it". To answer the question though, I personally don't think that there is any need to involve QM to derive the second law, as the second law is simply a statement about the statistical behaviour of a partially characterised system. – glS Dec 11 '18 at 10:51

The laws of thermodynamics can be derived (or at least motivated) from statistical physics. To do so, we assume that there is an "underlying theory" describing the microscopic physics of the system. However, we do not know have perfect information about the state of the system (we say that we do not know the "microstate") - we only know some macroscopic properties, the "macrostate".

This derivation is remarkably universal, it does not care much whether the underlying theory is classical or quantum.

Is the entropy of a classical system without randomness always constant?

No, it is not. While the microscopic processes are deterministic, there is a randomness which comes from the fact that we don't have perfect information about the microstate. The second law of Thermodynamics only says that in this situation of imperfect information, the system is extremely likely to assume the most typical macrostate.

If we consider the case where we have perfect knowledge of a classical system, that means that the entropy of the system is zero and it will remain zero. (You could say that entropy measures the amount of missing information.) Note that the same is true for a quantum system, if you have perfect information and don't measure it, its entropy will not change. Only if you measure a quantum state (and don't look at the result of the measurement), you suddenly have less information about the system state than before: its entropy increases.

One of the most catchy Gedankenexperiment of an isolated system with constant entropy is a photon which gets reflected from the surfaces of two perfect mirrors.

At a first glance it looks like the photon gets re-emitted with the same wavelength as it was absorbed. But beside the presumption of such a ideal absorption-emission process there is a second phenomenon, that isn’t negligible: their momentum.

Photons carry a momentum and hitting the mirror, the photon creates a recoil. A recoil means a movement of the mirror or a part of the mirror or at least a deformation of its surface. Such a displacement has two effects:

• The re-emitted photon is with less energy (is redshifted) and
• the mirror gets for example a photonics excitation which lead at least to heat radiation (nothing else as photon emission).

QM was firstly established to describe processes on atomic ranges. And the older introduction of quanta for the electromagnetic radiation is enough to understand the second law of thermodynamics.

Can we say that this is due to Quantum mechanics, which continuously increases the information contained in the system by producing random numbers?

QM describes something, it does not influence things. And the very specific (concrete) description of the photon interaction with the mirrors could be supplemented by a QM description. I prefer - than ever it is possible - concrete description of a single process compared to statistical processes.

Is the entropy of a classical system without randomness always constant?

Interactions in classical system (BTW what is a classical system?) are also interactions of photons. And all photon interactions are random processes. A system has a constant entropy if it is a closed system without any exchange with the surrounding. Such isolated systems simply not exist.