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A microscopic understanding of the thermodynamic entropy in quantum systemshas been a mystery ever since the invention of quantum mechanics. In classicalphysics, this entropy is believed to be the logarithm of the volume of phasespace accessible to an isolated system [1]. There is no quantum mechanicalanalog to this. Instead, Von Neumann's hypothesis for the entropy [2] is mostwidely used. However this gives zero for systems with a known wave function,that is a pure state. This is because it measures the lack of information aboutthe system rather than the flow of heat as obtained from thermodynamicexperiments. Many arguments attempt to sidestep these issues by considering thesystem of interest coupled to a large external one, unlike the classical casewhere Boltzmann's approach for isolated systems is far more satisfactory. Withnew experimental techniques, probing the quantum nature of thermalization isnow possible [3, 4]. Here, using recent advances in our understanding ofquantum thermalization [5-10] we show how to obtain the entropy as is measuredfrom thermodynamic experiments, solely from the self-entanglement of thewavefunction, and find strong numerical evidence that the two are in agreementfor non-integrable systems. It is striking that this entropy, which is closelyrelated to the concept of heat, and generally thought of as microscopic chaoticmotion, can be determined for systems in energy eigenstates which arestationary in time and therefore not chaotic, but instead have a very complexspatial dependence.

The microscopic origin of thermodynamic entropy in isolated systems