Logarithmic Sobolev Inequalities for an Ideal Bose Gas

  • Fabio CiprianiEmail author
Part of the Springer INdAM Series book series (SINDAMS, volume 18)


The aim of this work is to derive logarithmic Sobolev inequalities, with respect to the Fock vacuum state and for the second quantized Hamiltonian \(d\varGamma (H^{\varLambda } -\mu \mathbb{I})\) of an ideal Bose gas with Dirichlet boundary conditions in a finite volume Λ, from the free energy variation with respect to a Gibbs temperature state and from the monotonicity of the relative entropy. Hypercontractivity of the semigroup \(e^{-\beta d\varGamma (H^{\varLambda }) }\) is also deduced.


Free energy Gibbs state Hypercontractivity Ideal bose gas Logarithmic sobolev inequality Relative entropy 



This work has been supported by GREFI-GENCO INDAM Italy-CNRS France and MIUR PRIN 2012 Project No 2012TC7588-003.


  1. 1.
    H. Araki, S. Yamagami, On quasi-equivalence of quasifree states of canonical commutation relations. Publ. RIMS Kyoto Univ. 18, 283–338 (1982)MathSciNetzbMATHGoogle Scholar
  2. 2.
    O. Bratteli, D.W. Robinson, “Operator algebras and Quantum Statistical Mechanics 2”, 2nd edn. (Springer, Berlin, Heidelberg, New York, 1996), 517 p.Google Scholar
  3. 3.
    E. Carlen, D. Stroock, An Application of the Bakry-Emery Criterion to Infinite-Dimensional Diffusions. Séminaire de Probabilités, XX, 1984/85, Lecture Notes in Math., vol. 1204 (Springer, Berlin, 1986), pp. 341–348Google Scholar
  4. 4.
    E.B. Davies, Heat Kernels and Spectral Theory, vol. 92 (Cambridge Tracts in Mathematics, Cambridge, 1989)CrossRefzbMATHGoogle Scholar
  5. 5.
    E.B. Davies, B. Simon, Ultracontractivity and the heat kernel for Schroedinger operators and Dirichlet Laplacians. J. Funct. Anal. 59, 335–395 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    L. Gross, Existence and uniqueness of physical ground states. J. Funct. Anal. 10, 59–109 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    L. Gross, Hypercontractivity and logarithmic Sobolev inequalities for the Clifford–Dirichlet form. Duke Math. J. 42, 383–396 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    E. Lieb, The stability of matter. Rev. Modern Phys. 48(4), 553–569 (1976)MathSciNetCrossRefGoogle Scholar
  9. 9.
    E. Nelson, A quartic interaction in two dimension, Mathematical theory of elementary particles (Proceedings of the Conference on the mathematical Theory of Elementary particles held at Hendicott House in Dedham, Mass., September 12–15, 1965), Roe Goodman and Irving E. Segal eds. (1966), 69–73Google Scholar
  10. 10.
    E. Nelson, The free Markov field. J. Funct. Anal. 12, 211–227 (1973)CrossRefzbMATHGoogle Scholar
  11. 11.
    I.E. Segal, Tensor algebras over Hilbert spaces I. Trans. Am. Math. Soc. 81, 106–134 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    I.E. Segal, Distributions in Hilbert space and canonical systems of operators. Trans. Am. Math. Soc. 88, 12–41 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    B. Simon, “The P(Φ) 2 Euclidean (Quantum) Field Theory” (Princeton University Press, Princeton, New Jersey, 1974)Google Scholar
  14. 14.
    B. Simon, R. Hoegh-Krohn, Hypercontractivity semigroups and two dimensional self-coupled Bose fields. J. Funct. Anal. 9, 121–180 (1972)CrossRefzbMATHGoogle Scholar
  15. 15.
    D. Stroock, B. Zegarlinski, The equivalence of Logarithmic Sobolev Inequality and the Dobrushin-Shlosmann mixing condition. Comm. Math. Phys. 144, 303–323 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    H. Umegaki, Conditional expectation in an operator algebra. IV. Entropy and information. Kodai Math. Sem. Rep. 14, 59–85 (1962)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly

Personalised recommendations