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Communications in Mathematical Physics

, Volume 331, Issue 3, pp 887–926 | Cite as

An Analog of the 2-Wasserstein Metric in Non-Commutative Probability Under Which the Fermionic Fokker–Planck Equation is Gradient Flow for the Entropy

  • Eric A. Carlen
  • Jan Maas
Article

Abstract

Let \({\mathfrak{C}}\) denote the Clifford algebra over \({\mathbb{R}^n}\), which is the von Neumann algebra generated by n self-adjoint operators Q j , j = 1,…,n satisfying the canonical anticommutation relations, Q i Q j  + Q j Q i =  2δ ij I, and let τ denote the normalized trace on \({\mathfrak{C}}\). This algebra arises in quantum mechanics as the algebra of observables generated by n fermionic degrees of freedom. Let \({\mathfrak{P}}\) denote the set of all positive operators \({\rho\in\mathfrak{C}}\) such that τ(ρ) = 1; these are the non-commutative analogs of probability densities in the non-commutative probability space \({(\mathfrak{C},\tau)}\). The fermionic Fokker–Planck equation is a quantum-mechanical analog of the classical Fokker–Planck equation with which it has much in common, such as the same optimal hypercontractivity properties. In this paper we construct a Riemannian metric on \({\mathfrak{P}}\) that we show to be a natural analog of the classical 2-Wasserstein metric, and we show that, in analogy with the classical case, the fermionic Fokker–Planck equation is gradient flow in this metric for the relative entropy with respect to the ground state. We derive a number of consequences of this, such as a sharp Talagrand inequality for this metric, and we prove a number of results pertaining to this metric. Several open problems are raised.

Keywords

Relative Entropy Clifford Algebra Planck Equation Logarithmic Sobolev Inequality Geodesic Convexity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient flows in metric spaces and in the space of probability measures. In: Lectures in Mathematics ETH Zürich. 2nd ed. Birkhäuser Verlag, Basel (2008)Google Scholar
  2. 2.
    Ambrosio, L., Gigli, N., Savaré, G.: Calculus and heat flow in metric measure spaces and applications to spaces with ricci bounds from below. Invent. Math. 195(2), 289–391 (2014)Google Scholar
  3. 3.
    Benamou J.-D., Brenier Y.: A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numer. Math. 84, 375–393 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Biane P., Voiculescu D.: A free probability analogue of the Wasserstein metric on the trace-state space. Geom. Funct. Anal. 11, 1125–1138 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Bogoliubov N.N.: Phys. Abh. SU. 1, 229 (1962)Google Scholar
  6. 6.
    Carlen E.A.: Trace inequalities and quantum entropy: an introductory course. Entropy and the quantum. Contemp. Math. 529, 73–140 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Carlen E.A., Gangbo W.: Constrained steepest descent in the 2-Wasserstein metric. Arch. Ration. Mech. Anal. 172, 21–64 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Carlen E.A., Gangbo W.: Solution of a model Boltzmann equation via steepest descent in the 2-Wasserstein metric. Ann. Math. 157(3), 1–40 (2003)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Carlen E.A., Lieb E.H.: Optimal hypercontractivity for Fermi Fields and related non-commutative integration inequalities. Commun. Math. Phys. 155, 27–46 (1993)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Carlen E.A., Lieb E.H.: Brascamp–Lieb inequalities for non-commutatative integration. Documenta Math. 13, 553–584 (2008)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Carrillo J.A., McCann R.J., Villani C.: Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Ration. Mech. Anal. 179, 217–263 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Chow, S.-N., Huang, W., Li, Y., Zhou, H.: Fokker–Planck equations for a free energy functional or Markov process on a graph. Arch. Ration. Mech. Anal. 203(3), 969–1008 (2012)Google Scholar
  13. 13.
    Daneri S., Savaré G.: Eulerian calculus for the displacement convexity in the Wasserstein distance. SIAM J. Math. Anal. 40(3), 1104–1122 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Dyson F.J., Lieb E.H., Simon B.: Phase transitions in quantum spin systems with isotropic and nonisotropic interactions. J. Stat. Phys. 18(4), 335–383 (1978)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Erbar M.: The heat equation on manifolds as a gradient flow in the Wasserstein space. Ann. Inst. Henri Poincaré Probab. Stat. 46, 1–23 (2010)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Erbar M., Maas J.: Ricci curvature of finite Markov chains via convexity of the entropy. Arch. Ration. Mech. Anal. 206, 997–1038 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Fang S., Shao J., Sturm K.-Th.: Wasserstein space over the Wiener space. Probab. Theory Relat. Fields 146, 535–565 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Gigli, N., Kuwada, K., Ohta, S.-i.: Heat flow on Alexandrov spaces. Commun. Pure Appl. Math. 66(3), 307–331 (2013)Google Scholar
  19. 19.
    Gross L.: Existence and uniqueness of physical ground states. J. Funct. Anal. 10, 52–109 (1972)zbMATHCrossRefGoogle Scholar
  20. 20.
    Gross L.: Hypercontractivity and logarithmic Sobolev inequalities for the Clifford–Dirichlet form. Duke Math. J. 42, 383–396 (1975)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Jordan R., Kinderlehrer D., Otto F.: The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29, 1–17 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Kubo R.: Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems. J. Phys. Soc. Jpn. 12, 570–586 (1957)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Lieb E.H.: Convex trace functions and the Wigner–Yanase–Dyson conjecture. Adv. Math. 11, 267–288 (1973)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Maas J.: Gradient flows of the entropy for finite Markov chains. J. Funct. Anal. 261, 2250–2292 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    McCann R.J.: A convexity principle for interacting gases. Adv. Math. 128, 153–179 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Mielke A.: A gradient structure for reaction–diffusion systems and for energy–drift–diffusion systems. Nonlinearity 24, 1329–1346 (2011)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Mielke, A.: Geodesic convexity of the relative entropy in reversible Markov chains. Calc. Var. Partial Differ. Equ. 48(1–2), 1–31 (2013)Google Scholar
  28. 28.
    Mori H.: Transport, collective motion, and Brownian motion. Progr. Theor. Phys. 33, 423–455 (1965)ADSzbMATHCrossRefGoogle Scholar
  29. 29.
    Nelson E.: The free Markoff field. J. Funct. Anal. 12, 211–227 (1973)zbMATHCrossRefGoogle Scholar
  30. 30.
    Ohta S.-I., Sturm K.-Th.: Heat flow on Finsler manifolds. Commun. Pure Appl. Math. 62, 1386–1433 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Otto F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26, 101–174 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Otto F., Westdickenberg M.: Eulerian calculus for the contraction in the Wasserstein distance. SIAM J. Math. Anal. 37(4), 1227–1255 (2005)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Otto F., Villani C.: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173, 361–400 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Segal I.E.: A non-commutative extension of abstract integration. Ann. Math. 57, 401–457 (1953)zbMATHCrossRefGoogle Scholar
  35. 35.
    Segal I.E.: Tensor algebras over Hilbert spaces II. Ann. Math. 63, 160–175 (1956)zbMATHCrossRefGoogle Scholar
  36. 36.
    Segal I.E.: Algebraic integration theory. Bull. Am. Math. Soc. 71, 419–489 (1965)zbMATHCrossRefGoogle Scholar
  37. 37.
    Talagrand M.: Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6, 587–600 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Villani, C.: Topics in optimal transportation, vol. 58. In: Graduate Studies in Mathematics. American Mathematical Society, Providence (2003)Google Scholar
  39. 39.
    Villani, C.: Optimal transport, old and new, vol. 338. Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (2009)Google Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Department of Mathematics, Hill CenterRutgers UniversityPiscatawayUSA
  2. 2.Institute for Applied MathematicsUniversity of BonnBonnGermany

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