A origem microscópica da termodinâmica: o papel da entropia em um sistema quântico

segunda-feira, julho 31, 2017

Vincenzo Alba a,1 and Pasquale Calabrese a  

Author Affiliations

a International School for Advanced Studies, Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, 34136 Trieste, Italy

Edited by Subir Sachdev, Harvard University, Cambridge, MA, and approved June 6, 2017 (received for review March 3, 2017)


Understanding how statistical ensembles arise from the out-of-equilibrium dynamics of isolated pure systems has been a fascinating question since the early days of quantum mechanics. Recently, it has been proposed that the thermodynamic entropy of the long-time statistical ensemble is the stationary entanglement of a large subsystem in an infinite system. Here, we combine this concept with the quasiparticle picture of the entanglement evolution and integrability-based knowledge of the steady state to obtain exact analytical predictions for the time evolution of the entanglement in arbitrary 1D integrable models. These results explicitly show the transformation between the entanglement and thermodynamic entropy during the time evolution. Thus, entanglement is the natural witness for the generalized microcanonical principle underlying relaxation in integrable models.


Entanglement and entropy are key concepts standing at the foundations of quantum and statistical mechanics. Recently, the study of quantum quenches revealed that these concepts are intricately intertwined. Although the unitary time evolution ensuing from a pure state maintains the system at zero entropy, local properties at long times are captured by a statistical ensemble with nonzero thermodynamic entropy, which is the entanglement accumulated during the dynamics. Therefore, understanding the entanglement evolution unveils how thermodynamics emerges in isolated systems. Alas, an exact computation of the entanglement dynamics was available so far only for noninteracting systems, whereas it was deemed unfeasible for interacting ones. Here, we show that the standard quasiparticle picture of the entanglement evolution, complemented with integrability-based knowledge of the steady state and its excitations, leads to a complete understanding of the entanglement dynamics in the space–time scaling limit. We thoroughly check our result for the paradigmatic Heisenberg chain.

entanglement quantum quench integrability thermodynamics


1 To whom correspondence should be addressed. Email: valba@sissa.it.

Author contributions: V.A. and P.C. designed research, performed research, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1703516114/-/DCSupplemental.


Professores, pesquisadores e alunos de universidades públicas e privadas com acesso ao Portal de Periódicos CAPES/MEC podem ler gratuitamente este artigo do PNAS e mais 33.000 publicações científicas.