# Is there an influence of entanglement on statistical mechanics and thermodynamics?

Being not an expert in these fields, I wonder whether the quantum mechanical entanglement, e.g., of electrons in an electron gas, is already taken into account in the statistical mechanics and thermodynamics of such a system and what impact (if any) entanglement has on its physical behavior.

• Hi again, entanglement will not, AFAIK, survive interaction with other particles, the jargon here is decoherence, often caused by unavoidable coupling with the environment, leading to degradation of quantum coherence, For a multipartite quantum system, decoherence leads to degradation of entanglement and, in certain cases, entanglement sudden death. ...So to preserve entanglement, you need to keep the particles involved away from everything else. Hope you get an expert answer though.
– user108787
Sep 18, 2016 at 23:28
• Hi CountTo10, thank you for pointing this out. I thought that as there are constantly new interactions between the particles many will get frequently newly entangled so that there might still be a net effect on the system even if individual entanglements get lost by decoherence. Sep 18, 2016 at 23:46
• I self study, so my opinions and answers are entirely my own interpretation of those of experts, but I think I am correct in this. To check my waffling, use Google on how they actually perform the entanglement experiments. Absolutely, definetely read Matt Strassler's blog, and there a very good book , that you might have already Deep Down Things Bruce Schaumm which I highly recommend.
– user108787
Sep 18, 2016 at 23:59
• Thank you very much, also for your helpful suggestions... Sep 19, 2016 at 0:01
• I don't understand what you mean by entanglement being "taken into account". In quantum statistical mechanics, the system is described by a density matrix, and such a density matrix can describe both pure and entangled states. Why would we need to explicitly "take into account" that the state can be entangled? Sep 19, 2016 at 0:38

There is an entanglement entropy between two systems $A$ and $B$ that is $$S(A,B)~=~S(A)~+~S(B)~-~S(A|B).$$ This is a fine grained entropy where the observer has access to everything about the system. For thermodynamics the thermal entropy drops the $S(A|B)$ which results in a larger entropy. This is because $S(A|B)$ is a conditional entropy for the entanglement, which in a complex system will migrate throughout. The entanglement phase, or equivalently the overlap phase for a superposition, will enter in to a larger number of states and it is not practical to account for this. Coarse graining of the system results in the thermal entropy that is larger than the entanglement entropy
• Yes, and some experiments have been done for small $N$ particle systems along these lines. Thermal entropy is fine grained entropy with entanglement removed by coarse graining. Since thermodynamics involves logarithms these differences are not that large. That is basically what makes classical thermodynamics work well enough. A change in coarse graining of phase space or of how one set up decoherent sets in a quantum situation leads to pretty small deviations. Sep 19, 2016 at 15:11