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February  2021, 14(1): 77-88. doi: 10.3934/krm.2020049

On two properties of the Fisher information

Nicolas Rougerie

Université Grenoble-Alpes & CNRS, LPMMC (UMR 5493), B.P. 166, F-38042 Grenoble, France

Received  January 2020 Revised  August 2020 Published  February 2021 Early access  November 2020

Alternative proofs for the superadditivity and the affinity (in the large system limit) of the usual and some fractional Fisher informations of a probability density of many variables are provided. They are consequences of the fact that such informations can be interpreted as quantum kinetic energies.

 

Erratum: The month information has been corrected from January to February. We apologize for any inconvenience this may cause.

Keywords: Fisher information, de Finetti's theorem, statistical mechanics, mean-field limits.
Mathematics Subject Classification: Primary: 82Bxx, 35Qxx; Secondary: 49Sxx.
Citation: Nicolas Rougerie. On two properties of the Fisher information. Kinetic and Related Models, 2021, 14 (1) : 77-88. doi: 10.3934/krm.2020049
References:
[1]

J. Bourgain, H. Brézis and P. Mironescu, Another look at Sobolev spaces, in Optimal control and Partial Differential equations, IOS Press, (2001), 439–455.

[2]

J. Bourgain, H. Brezis and P. Mironescu, Limiting embedding theorems for $W^{s,p}$ when $s\uparrow 1$ and applications, J. Anal. Math., 87 (2002), 77-101.  doi: 10.1007/BF02868470.

[3]

E. A. Carlen, Superadditivity of Fisher's information and logarithmic Sobolev inequalities, J. Funct. Anal., 101 (1991), 194-211.  doi: 10.1016/0022-1236(91)90155-X.

[4]

G. F. dell'Antonio, On the limits of sequences of normal states, Comm. Pure Appl. Math., 20 (1967), 413-429.  doi: 10.1002/cpa.3160200209.

[5]

N. Fournier, M. Hauray and S. Mischler, Propagation of chaos for the 2d viscous vortex model, J. Eur. Math. Soc., 16 (2014), 1423-1466.  doi: 10.4171/JEMS/465.

[6]

F. Golse, On the dynamics of large particle systems in the mean field limit, arXiv: 1301.5494, (2013)., Lecture notes for a course at the NDNS+ Applied Dynamical Systems Summer School "Macroscopic and large scale phenomena", Universiteit Twente, Enschede (The Netherlands).

[7]

M. Hauray, Limite de Champ Moyen et Propagation du Chaos Pour des Systèmes de Particules, Limites Gyro-cinétique et Quasi-neutre Pour Les Plasmas., Habilitation thesis, 2014.

[8]

M. Hauray and S. Mischler, On Kac's chaos and related problems, J. Func. Anal., 266 (2014), 6055-6157.  doi: 10.1016/j.jfa.2014.02.030.

[9]

E. Hewitt and L. J. Savage, Symmetric measures on Cartesian products, Trans. Amer. Math. Soc., 80 (1955), 470-501.  doi: 10.1090/S0002-9947-1955-0076206-8.

[10]

M. Hoffmann-Ostenhof and T. Hoffmann-Ostenhof, Schrödinger inequalities and asymptotic behavior of the electron density of atoms and molecules, Phys. Rev. A, 16 (1977), 1782-1785.  doi: 10.1103/PhysRevA.16.1782.

[11]

R. L. Hudson and G. R. Moody, Locally normal symmetric states and an analogue of de Finetti's theorem, Z. Wahrscheinlichkeitstheor. und Verw. Gebiete, 33 (1975/76), 343-351.  doi: 10.1007/BF00534784.

[12]

M. K.-H. Kiessling, The Hartree limit of Born's ensemble for the ground state of a bosonic atom or ion, J. Math. Phys., 53 (2012), 095223, 21 pp. doi: 10.1063/1.4752475.

[13]

M. Lewin, Mean-Field limit of Bose systems: Rigorous results, arXiv: 1510.04407, Proceedings of the International Congress of Mathematical Physics, 2015

[14]

M. Lewin, P. T. Nam and N. Rougerie, Derivation of Hartree's theory for generic mean-field Bose systems, Adv. Math., 254 (2014), 570-621.  doi: 10.1016/j.aim.2013.12.010.

[15]

E. H. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2nd ed., 2001. doi: 10.1090/gsm/014.

[16]

W. Masja and J. Nagel, Über äquivalente normierung der anisotropen Funktionalraüme $H ^{\mu} ( { {\mathbb R} } ^n)$, Beiträge zur Analysis, 12 (1978), 7-17. 

[17]

V. Maz'ya and T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Func. Anal., 195 (2002), 230-238.  doi: 10.1006/jfan.2002.3955.

[18]

D. W. Robinson and D. Ruelle, Mean entropy of states in classical statistical mechanics, Commun. Math. Phys., 5 (1967), 288-300.  doi: 10.1007/BF01646480.

[19]

N. Rougerie, De Finetti theorems, mean-field limits and Bose-Einstein condensation, arXiv: 1506.05263, 2014. LMU lecture notes.

[20]

——, Théorèmes de De Finetti, Limites de Champ Moyen et Condensation de Bose-Einstein, Les cours Peccot, Spartacus IDH, Paris, 2016., Cours Peccot, Collège de France : février-mars 2014.

[21]

S. Salem, Propagation of chaos for fractional Keller Segel equations in diffusion dominated and fair competition cases, Journal de Mathématiques Pures et Appliquées, 132 (2019), 79-132. doi: 10.1016/j.matpur.2019.04.011.

[22]

S. Salem, Propagation of chaos for the Boltzmann equation with moderately soft potentials, arXiv: 1910.01883, 2019.

[23]

R. Schatten, Norm Ideals of Completely Continuous Operators, vol. 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete, Folge, 1960.

[24] B. Simon, Trace Ideals and Their Applications, vol. 35 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1979. 
[25]

G. Toscani, The fractional Fisher information and the central limit theorem for stable laws, Ric. Mat., 65 (2016), 71-91.  doi: 10.1007/s11587-015-0253-9.

[26]

G. Toscani, The information-theoretic meaning of Gagliardo-Nirenberg type inequalities, Rend. Lincei Mat. Appl., 30 (2019), 237-253.  doi: 10.4171/RLM/845.

[27]

G. Toscani, Score functions, generalized relative Fisher information and applications, Ricerche mat., 66 (2017) 15–26. doi: 10.1007/s11587-016-0281-0.

show all references

References:
[1]

J. Bourgain, H. Brézis and P. Mironescu, Another look at Sobolev spaces, in Optimal control and Partial Differential equations, IOS Press, (2001), 439–455.

[2]

J. Bourgain, H. Brezis and P. Mironescu, Limiting embedding theorems for $W^{s,p}$ when $s\uparrow 1$ and applications, J. Anal. Math., 87 (2002), 77-101.  doi: 10.1007/BF02868470.

[3]

E. A. Carlen, Superadditivity of Fisher's information and logarithmic Sobolev inequalities, J. Funct. Anal., 101 (1991), 194-211.  doi: 10.1016/0022-1236(91)90155-X.

[4]

G. F. dell'Antonio, On the limits of sequences of normal states, Comm. Pure Appl. Math., 20 (1967), 413-429.  doi: 10.1002/cpa.3160200209.

[5]

N. Fournier, M. Hauray and S. Mischler, Propagation of chaos for the 2d viscous vortex model, J. Eur. Math. Soc., 16 (2014), 1423-1466.  doi: 10.4171/JEMS/465.

[6]

F. Golse, On the dynamics of large particle systems in the mean field limit, arXiv: 1301.5494, (2013)., Lecture notes for a course at the NDNS+ Applied Dynamical Systems Summer School "Macroscopic and large scale phenomena", Universiteit Twente, Enschede (The Netherlands).

[7]

M. Hauray, Limite de Champ Moyen et Propagation du Chaos Pour des Systèmes de Particules, Limites Gyro-cinétique et Quasi-neutre Pour Les Plasmas., Habilitation thesis, 2014.

[8]

M. Hauray and S. Mischler, On Kac's chaos and related problems, J. Func. Anal., 266 (2014), 6055-6157.  doi: 10.1016/j.jfa.2014.02.030.

[9]

E. Hewitt and L. J. Savage, Symmetric measures on Cartesian products, Trans. Amer. Math. Soc., 80 (1955), 470-501.  doi: 10.1090/S0002-9947-1955-0076206-8.

[10]

M. Hoffmann-Ostenhof and T. Hoffmann-Ostenhof, Schrödinger inequalities and asymptotic behavior of the electron density of atoms and molecules, Phys. Rev. A, 16 (1977), 1782-1785.  doi: 10.1103/PhysRevA.16.1782.

[11]

R. L. Hudson and G. R. Moody, Locally normal symmetric states and an analogue of de Finetti's theorem, Z. Wahrscheinlichkeitstheor. und Verw. Gebiete, 33 (1975/76), 343-351.  doi: 10.1007/BF00534784.

[12]

M. K.-H. Kiessling, The Hartree limit of Born's ensemble for the ground state of a bosonic atom or ion, J. Math. Phys., 53 (2012), 095223, 21 pp. doi: 10.1063/1.4752475.

[13]

M. Lewin, Mean-Field limit of Bose systems: Rigorous results, arXiv: 1510.04407, Proceedings of the International Congress of Mathematical Physics, 2015

[14]

M. Lewin, P. T. Nam and N. Rougerie, Derivation of Hartree's theory for generic mean-field Bose systems, Adv. Math., 254 (2014), 570-621.  doi: 10.1016/j.aim.2013.12.010.

[15]

E. H. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2nd ed., 2001. doi: 10.1090/gsm/014.

[16]

W. Masja and J. Nagel, Über äquivalente normierung der anisotropen Funktionalraüme $H ^{\mu} ( { {\mathbb R} } ^n)$, Beiträge zur Analysis, 12 (1978), 7-17. 

[17]

V. Maz'ya and T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Func. Anal., 195 (2002), 230-238.  doi: 10.1006/jfan.2002.3955.

[18]

D. W. Robinson and D. Ruelle, Mean entropy of states in classical statistical mechanics, Commun. Math. Phys., 5 (1967), 288-300.  doi: 10.1007/BF01646480.

[19]

N. Rougerie, De Finetti theorems, mean-field limits and Bose-Einstein condensation, arXiv: 1506.05263, 2014. LMU lecture notes.

[20]

——, Théorèmes de De Finetti, Limites de Champ Moyen et Condensation de Bose-Einstein, Les cours Peccot, Spartacus IDH, Paris, 2016., Cours Peccot, Collège de France : février-mars 2014.

[21]

S. Salem, Propagation of chaos for fractional Keller Segel equations in diffusion dominated and fair competition cases, Journal de Mathématiques Pures et Appliquées, 132 (2019), 79-132. doi: 10.1016/j.matpur.2019.04.011.

[22]

S. Salem, Propagation of chaos for the Boltzmann equation with moderately soft potentials, arXiv: 1910.01883, 2019.

[23]

R. Schatten, Norm Ideals of Completely Continuous Operators, vol. 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete, Folge, 1960.

[24] B. Simon, Trace Ideals and Their Applications, vol. 35 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1979. 
[25]

G. Toscani, The fractional Fisher information and the central limit theorem for stable laws, Ric. Mat., 65 (2016), 71-91.  doi: 10.1007/s11587-015-0253-9.

[26]

G. Toscani, The information-theoretic meaning of Gagliardo-Nirenberg type inequalities, Rend. Lincei Mat. Appl., 30 (2019), 237-253.  doi: 10.4171/RLM/845.

[27]

G. Toscani, Score functions, generalized relative Fisher information and applications, Ricerche mat., 66 (2017) 15–26. doi: 10.1007/s11587-016-0281-0.

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