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The equations are valid for any solid, including 2D crystals like graphene. With the use of Moyal\u2019s calculus and its properties, the pseudo-differential operators are expanded up to the second order in <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\hbar $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u0127<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. An energy transport model is obtained by using the moment method with closure relations based on a quantum version of the Maximum Entropy Principle by employing a relaxation time approximation for the production terms of energy and energy flux. An explicit form of the thermal conductivity with quantum correction up to <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\hbar ^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>\u0127<\/mml:mi>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> order is obtained under a long-time scaling for the most relevant phonon branches.<\/jats:p>","DOI":"10.1007\/s00332-023-09993-z","type":"journal-article","created":{"date-parts":[[2023,11,3]],"date-time":"2023-11-03T12:02:33Z","timestamp":1699012953000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Wigner Equations for Phonons Transport and Quantum Heat Flux"],"prefix":"10.1007","volume":"34","author":[{"given":"V. 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