American Institute of Mathematical Sciences

Advanced
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.

[1]

Pierre-Emmanuel Jabin. A review of the mean field limits for Vlasov equations. Kinetic and Related Models, 2014, 7 (4) : 661-711. doi: 10.3934/krm.2014.7.661
[2]

Patrick Gerard, Christophe Pallard. A mean-field toy model for resonant transport. Kinetic and Related Models, 2010, 3 (2) : 299-309. doi: 10.3934/krm.2010.3.299
[3]

Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080
[4]

Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic and Related Models, 2021, 14 (3) : 429-468. doi: 10.3934/krm.2021011
[5]

Marco Cirant, Diogo A. Gomes, Edgard A. Pimentel, Héctor Sánchez-Morgado. On some singular mean-field games. Journal of Dynamics and Games, 2021, 8 (4) : 445-465. doi: 10.3934/jdg.2021006
[6]

Charles Bordenave, David R. McDonald, Alexandre Proutière. A particle system in interaction with a rapidly varying environment: Mean field limits and applications. Networks and Heterogeneous Media, 2010, 5 (1) : 31-62. doi: 10.3934/nhm.2010.5.31
[7]

Illés Horváth, Kristóf Attila Horváth, Péter Kovács, Miklós Telek. Mean-field analysis of a scaling MAC radio protocol. Journal of Industrial and Management Optimization, 2021, 17 (1) : 279-297. doi: 10.3934/jimo.2019111
[8]

Diogo A. Gomes, Gabriel E. Pires, Héctor Sánchez-Morgado. A-priori estimates for stationary mean-field games. Networks and Heterogeneous Media, 2012, 7 (2) : 303-314. doi: 10.3934/nhm.2012.7.303
[9]

Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Mean-field backward stochastic Volterra integral equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (7) : 1929-1967. doi: 10.3934/dcdsb.2013.18.1929
[10]

Gerasimenko Viktor. Heisenberg picture of quantum kinetic evolution in mean-field limit. Kinetic and Related Models, 2011, 4 (1) : 385-399. doi: 10.3934/krm.2011.4.385
[11]

Franco Flandoli, Enrico Priola, Giovanni Zanco. A mean-field model with discontinuous coefficients for neurons with spatial interaction. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3037-3067. doi: 10.3934/dcds.2019126
[12]

Seung-Yeal Ha, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. Uniform stability and mean-field limit for the augmented Kuramoto model. Networks and Heterogeneous Media, 2018, 13 (2) : 297-322. doi: 10.3934/nhm.2018013
[13]

Michael Herty, Mattia Zanella. Performance bounds for the mean-field limit of constrained dynamics. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 2023-2043. doi: 10.3934/dcds.2017086
[14]

Zhenghong Qiu, Jianhui Huang, Tinghan Xie. Linear-Quadratic-Gaussian mean-field controls of social optima. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021047
[15]

René Aïd, Roxana Dumitrescu, Peter Tankov. The entry and exit game in the electricity markets: A mean-field game approach. Journal of Dynamics and Games, 2021, 8 (4) : 331-358. doi: 10.3934/jdg.2021012
[16]

Nastassia Pouradier Duteil. Mean-field limit of collective dynamics with time-varying weights. Networks and Heterogeneous Media, 2022, 17 (2) : 129-161. doi: 10.3934/nhm.2022001
[17]

Tigran Bakaryan, Rita Ferreira, Diogo Gomes. A potential approach for planning mean-field games in one dimension. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2147-2187. doi: 10.3934/cpaa.2022054
[18]

Michael Herty, Torsten Trimborn, Giuseppe Visconti. Mean-field and kinetic descriptions of neural differential equations. Foundations of Data Science, 2022, 4 (2) : 271-298. doi: 10.3934/fods.2022007
[19]

Matthew Rosenzweig. The mean-field limit of the Lieb-Liniger model. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 3005-3037. doi: 10.3934/dcds.2022006
[20]

Ikuo Arizono, Yasuhiko Takemoto. Statistical mechanics approach for steady-state analysis in M/M/s queueing system with balking. Journal of Industrial and Management Optimization, 2022, 18 (1) : 25-44. doi: 10.3934/jimo.2020141

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (188)
  • HTML views (120)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy