{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T02:27:33Z","timestamp":1777602453938,"version":"3.51.4"},"reference-count":72,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2017,4,2]],"date-time":"2017-04-02T00:00:00Z","timestamp":1491091200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We consider the generic model of a finite-size quantum electron system connected to two (temperature and particle) reservoirs. The quantum open system is driven out of equilibrium by the presence of both potential temperature and chemical differences between the two reservoirs. The nonequilibrium (NE) thermodynamical properties of such a quantum open system are studied for the steady state regime. In such a regime, the corresponding NE density matrix is built on the so-called generalised Gibbs ensembles. From different expressions of the NE density matrix, we can identify the terms related to the entropy production in the system. We show, for a simple model, that the entropy production rate is always a positive quantity. Alternative expressions for the entropy production are also obtained from the Gibbs\u2013von Neumann conventional formula and discussed in detail. Our results corroborate and expand earlier works found in the literature.<\/jats:p>","DOI":"10.3390\/e19040158","type":"journal-article","created":{"date-parts":[[2017,4,3]],"date-time":"2017-04-03T13:32:30Z","timestamp":1491226350000},"page":"158","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Nonequilibrium Thermodynamics and Steady State Density Matrix for Quantum Open Systems"],"prefix":"10.3390","volume":"19","author":[{"given":"Herv\u00e9","family":"Ness","sequence":"first","affiliation":[{"name":"Department of Physics, Faculty of Natural and Mathematical Sciences, King\u2019s College London, Strand, London WC2R 2LS, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,4,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1227","DOI":"10.1063\/1.523789","article-title":"Entropy production for quantum dynamical semigroups","volume":"19","author":"Spohn","year":"1978","journal-title":"J. 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Matter"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"372001","DOI":"10.1088\/1751-8113\/46\/37\/372001","article-title":"Time-reversal symmetry and fluctuation relations in non-equilibrium quantum steady states","volume":"46","author":"Bernard","year":"2013","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"398","DOI":"10.1103\/PhysRev.91.398","article-title":"The Formal Theory of Scattering","volume":"91","author":"Goldberger","year":"1953","journal-title":"Phys. Rev."},{"key":"ref_42","unstructured":"Akhiezer, A.I., and Peletminskii, S.V. (1981). Methods of Statistical Physics, Pergamon Press."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Bohm, A. (1993). Quantum Mechanics: Foundations and Applications, Springer.","DOI":"10.1007\/978-1-4612-4352-6"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"5341","DOI":"10.1088\/0305-4470\/34\/26\/305","article-title":"Moeller operators and Lippmann\u2013Schwinger equations for steplike potentials","volume":"34","author":"Baute","year":"2001","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_45","unstructured":"It is known from quantum transport theory [70] that, in the steady state, the currents at the L\/C and C\/R interfaces are equal to each other (up to the proper sign convention). Hence any linear combination of the type J = aJL \u2212 bJR (with a + b = 1) can be used. It is then possible to transfrom Equation (6) into a form similar to Equation (10), i.e., \u03c1NE \u221d \n          \n            \n              \n\t\t      exp\n                      \n                        \u2212\n                        \n                          \u03b2\n                          \u0303\n                        \n                        \n                          (\n                          \n                            H\n                            \n                              L\n                              +\n                              R\n                            \n                            \n                              (\n                              +\n                              )\n                            \n                          \n                          \u2212\n                          \n                            \u03bc\n                            \u0303\n                          \n                          \n                            N\n                            \n                              L\n                              +\n                              R\n                            \n                            \n                              (\n                              +\n                              )\n                            \n                          \n                          )\n                        \n                        +\n                        \n                          \u0394\n                          \u03bc\n                        \n                        \n                          Q\n                          \n                            (\n                            +\n                            )\n                          \n                        \n                        \u2212\n\t\t\t\t\t\t \u0394\n\t\t\t\t\t\t\u03b2\n\t\t\t\t\t\t\n                          E\n                          \n                            (\n                            +\n                            )\n                          \n                        \n                      \n\t\t        \n            \n          \n         by using any linear combination of the type Q = aNL \u2212 (1 \u2212 a)NR and E = aHL \u2212 (1 \u2212 a)HR with a \u2208 [0, 1]. Simple mathematical manipulations show that \u0394\u03bc = \u03b2L\u03bcL \u2212 \u03b2R\u03bcR and \u0394\u03b2 = \u03b2L \u2212 \u03b2R, as in Equation (10), and that \n          \n            \n              \n\t\t    \n                  \u03b2\n                   \u0303\n                       \n\t\t          \n            \n          \n         = (1 \u2212 a)\u03b2L + a\u03b2R and \n          \n            \n              \n\t\t    \n                  \u03b2\n                   \u0303\n                       \n\t\t\t\t\t\n                            \u03bc\n                            \u0303\n                          \n\t\t          \n            \n          \n         = (1 \u2212 a)\u03b2L\u03bcL + a\u03b2R\u03bcR. Obviously, for a = 1\/2, one recovers the expected Equation (10)."},{"key":"ref_46","unstructured":"In Appendix A of [34], we have demonstrated the intertwining relation H\u03a9(+) = \u03a9(+)H0. Following the same steps, we can also show that N\u03a9(+) = \u03a9(+)N when one assumes that the total number of electron operator N commutes with the Hamiltonian H. Hence we have not only [H\u03b1,N\u03b2] = 0 but also [H,\u2211\u03b2 N\u03b2] = 0. In other words, the total number of electron is conserved and [W, NL + NC + NR] = 0, implying to all the electrons coming out (in) the reservoirs are going in (out) the central region C."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"235116","DOI":"10.1103\/PhysRevB.81.235116","article-title":"Correlated Initial Condition for an Embedded Process by Time Partitioning","volume":"81","year":"2010","journal-title":"Phys. Rev. B"},{"key":"ref_48","unstructured":"Such a fact may not appear as a rigorous mathematical derivation, however it is known that the steady state does not dependent of the initial choice of \u03c10. Furthermore, one could note that another splitting of the total Hamiltonian H can be used to reduce the complexity of the expressions for the NE density matrix. In an earlier work [34], we considered splitting the Hamiltonian H into H0 + W where H0 is only H0 = HL + HR, hence the initial density matrix is only the direct product \u03c10 = \u03c1L \u2297 \u03c1R, and the expected form of the NE density matrix is obtained. A difference however occurs in the construction of the Y operators given in Equation (13). In the present work, the operator W does not include HC while it does in [34]. In the calculation of the Gibbs-like entropy in the central region, one deals with products of terms including \u03c1(\n          \n            \n              \n\t\t\t  \n\t\t\t  H\n\t\t\t  C\n\t\t\t  +\n\t\t\t  \n\t\t      \n            \n          \n        )ln\u03c1C. The asymptotic operator \u03c1(\n          \n            \n              \n\t\t\t  \n\t\t\t  H\n\t\t\t  C\n\t\t\t  +\n\t\t\t  \n\t\t      \n            \n          \n        ) can be expanded in a series of \n          \n            \n              \n\t\t\t  \n\t\t\t  \u03c1\n\t\t\t  C\n\t\t\t  n\n\t\t\t  \n\t\t      \n            \n          \n         from the series expansion of Moeller operators. Hence leading to a series of terms in \n          \n            \n              \n\t\t\t  \n\t\t\t  \u03c1\n\t\t\t  C\n\t\t\t  n\n\t\t\t  \n\t\t      \n            \n          \n        ln\u03c1C. By considering that initially the central region (of finite size) is fully isolate, there cannot be any partial occupation of the electronic levels, and hence the terms \n          \n            \n              \n\t\t\t  \n\t\t\t  \u03c1\n\t\t\t  C\n\t\t\t  n\n\t\t\t  \n\t\t      \n            \n          \n        ln\u03c1C expressed in the basis set of the central region will lead to the evaluation of either 1nln1 or 0nln0. This obviously leads to a zero contribution to the entropy, and therefore the terms in \u03c1(\n          \n            \n              \n\t\t\t  \n\t\t\t  H\n\t\t\t  C\n\t\t\t  +\n\t\t\t  \n\t\t      \n            \n          \n        ) in the density matrix can be ignored."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"022121","DOI":"10.1103\/PhysRevE.88.022121","article-title":"Nonequilibrium density matrix for quantum transport: Hershfield approach as a McLennan Zubarev form of the statistical operator","volume":"88","author":"Ness","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1016\/j.matpr.2015.05.066","article-title":"Nonlinear heat conduction in Coulomb-blockaded quantum dots","volume":"2","author":"Sierra","year":"2015","journal-title":"Mater. 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In [12] only the assumed standard definition for heat flux is used and hence Equation (15) follows automatically. Finally, only weak coupling regime was considered in [14]."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"125319","DOI":"10.1103\/PhysRevB.73.125319","article-title":"Quantum simulation of many-body effects in steady-state nonequilibrium: Electron-phonon coupling in quantum dots","volume":"73","author":"Han","year":"2006","journal-title":"Phys. Rev. B"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"125122","DOI":"10.1103\/PhysRevB.75.125122","article-title":"Mapping of strongly correlated steady-state nonequilibrium system to an effective equilibrium","volume":"75","author":"Han","year":"2007","journal-title":"Phys. Rev. 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