Advertisement

Communications in Mathematical Physics

, Volume 357, Issue 1, pp 43–60 | Cite as

Non-equilibrium Thermodynamics and Conformal Field Theory

Article

Abstract

We present a model independent, operator algebraic approach to non-equilibrium quantum thermodynamics within the framework of two-dimensional Conformal Field Theory. Two infinite reservoirs in equilibrium at their own temperatures and chemical potentials are put in contact through a defect line, possibly by inserting a probe. As time evolves, the composite system then approaches a non-equilibrium steady state that we describe. In particular, we re-obtain recent formulas of Bernard and Doyon (Ann Henri Poincaré 16:113–161, 2015).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Araki H.: Relative Hamiltonians for faithful normal states of a von Neumann algebra. Pub. R.I.M.S. Kyoto Univ. 9, 165–209 (1973)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Araki H., Haag R., Kastler D., Takesaki M.: Extension of KMS states and chemical potential. Commun. Math. Phys. 53, 97–134 (1977)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bernard D., Doyon B.: Energy flow in non-equilibrium conformal field theory. J. Phys. A Math. Theor. 45, 362001 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bernard D., Doyon B.: Non-equilibrium steady states in Conformal Field Theory. Ann. Henri Poincaré 16, 113–161 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bernard D., Doyon B.: Conformal field theory out of equilibrium: a review. J. Stat. Mech. 2016, 064005 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bischoff, M., Kawahigashi, Y., Longo, R., Rehren, K.-H.: Tensor categories and endomorphisms of von Neumann algebras, SpringerBriefs in Mathematical Physics, vol. 3 (2015)Google Scholar
  7. 7.
    Bischoff M., Kawahigashi Y., Longo R., Rehren K.-H.: Phase boundaries in algebraic conformal QFT. Commun. Math. Phys. 342, 1–45 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bischoff M., Kawahigashi Y., Longo R.: Characterization of 2D rational local conformal nets and its boundary conditions: the maximal case. Documenta Math. 20, 1137–1184 (2015)MathSciNetMATHGoogle Scholar
  9. 9.
    Bisognano J.J., Wichmann E.H.: On the duality condition for quantum fields. J. Math. Phys. 17, 303 (1976)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Bros J., Buchholz D.: Towards a relativistic KMS-condition. Nucl. Phys. B 429, 291–318 (1994)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Brunetti R., Guido D., Longo R.: Modular structure and duality in Conformal Field Theory. Commun. Math. Phys. 156, 201–219 (1993)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Buchholz D., Ojima I., Roos H.: Thermodynamic properties of non-equilibrium states in Quantum Field Theory. Ann. Phys. 297(2), 219–242 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Buchholz D., Mack G., Todorov I.: The current algebra on the circle as a germ of local field theories. Nucl. Phys. B (Proc. Suppl.) 5, 20–56 (1988)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Camassa P., Longo R., Tanimoto Y., Weiner M.: Thermal states in Conformal Field Theory. I. Commun. Math. Phys. 309, 703–735 (2012)ADSCrossRefMATHGoogle Scholar
  15. 15.
    Camassa P., Longo R., Tanimoto Y., Weiner M.: Thermal states in Conformal Field Theory. II. Commun. Math. Phys. 315, 771–802 (2012)ADSCrossRefMATHGoogle Scholar
  16. 16.
    Connes A.: Une classification des facteurs de type III. Ann. Sci. Ec. Norm. Sup. 6, 133–252 (1973)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Haag R.: Local Quantum Physics. Springer, Berlin (1996)CrossRefMATHGoogle Scholar
  18. 18.
    Haag R., Hugenoltz N. M., Winnik M.: On the equilibrium states in quantum statistical mechanics. Commun. Math. Phys. 5, 215–236 (1967)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Kawahigashi Y., Longo R.: Classification of two-dimensional local conformal nets with \({c < 1}\) and 2-cohomology vanishing for tensor categories. Commun. Math. Phys. 244, 63–97 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kawahigashi Y., Longo R., Müger M.: Multi-interval subfactors and modularity of representations in conformal field theory. Commun. Math. Phys. 219, 631–669 (2001)ADSMathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Longo R.: Index of subfactors and statistics of quantum fields I. Commun. Math. Phys. 126, 217–247 (1989)ADSMathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Longo R.: An analogue of the Kac–Wakimoto formula and black hole conditional entropy. Commun. Math. Phys. 186, 451–479 (1997)ADSMathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Longo R.: Notes for a quantum index theorem. Commun. Math. Phys. 222, 45–96 (2001)ADSMathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Longo R.: Conformal subnets and intermediate subfactors. Commun. Math. Phys. 23, 7–30 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Ruelle D.: Natural non-equilibrium states in quantum statistical mechanics. J. Stat. Phys. 98, 57 (2000)CrossRefMATHGoogle Scholar
  26. 26.
    Takesaki, M.: Theory of Operator Algebras. Vol. I, II, III, Springer Encyclopaedia of Mathematical Sciences 124 (2002), 125, 127 (2003)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikUniversität LeipzigLeipzigGermany
  2. 2.Dipartimento di MatematicaUniversità di Roma Tor VergataRomeItaly

Personalised recommendations