Information geometry in vapour-liquid equilibrium

Journal of Physics A: Mathematical and Theoretical

Volumn 42, Issue 2, 2009, Pages

  Brody, Dorje C ^a   Hook, Daniel W ^a  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 64549100174     PISSN: 17518113     EISSN: 17518121     Source Type: Journal    
DOI: 10.1088/1751-8113/42/2/023001     Document Type: Article

References (92)

1. Amari S 1982 Differential geometry of curved exponential families-curvatures and information loss Ann. Stat. 10 357-85
2. Amari S 1985 Differential-Geometrical Methods in Statistics (Lecture Notes in Statistics vol 28) (New York: Springer)
3. Amari S, Barndorff-Nielsen O E, Kass R E, Lauritzen S L and Rao C R 1987 Differential Geometry in Statistical Inference (Institute of Mathematical Statistics Lecture Notes. Monograph Series vol 10) (Hayward, CA: Institute of Mathematical Statistics)
4. Amari S and Nagaoka H 2000 Methods of Information Geometry (AMS Translations of Mathematical Monograph vol 191) (Oxford: Oxford University Press)
5. Arwini K A and T Dodson C J 2008 Information Geometry (Lecture Notes in Mathematics vol 1953) (Berlin: Springer)
6. Atkinson C and S Mitchell A F 1981 Rao's distance measure Sankhy 43 345-65
7. Barndorff-Nielsen O E and Gill R D 2000 Fisher information in quantum statistics J. Phys. A: Math. Gen. 33 4481-90
8. Bhattacharyya A 1943 On a measure of divergence between two statistical populations defined by their probability distributions Bull. Calcutta Math. Soc. 35 99-109
9. Bhattacharyya A 1946 On a measure of divergence between two multinomial populations Sankhy 7 401-6
10. Brody D C 2000 Differential renormalisation flow in random lattice gauge theories Phys. Lett. B 485 422-8
11. Brody D C and Hughston L P 1998 Statistical geometry in quantum mechanics Proc. R. Soc. Lond. A 454 2445-75
12. Brody D C and Hughston L P 1999 Geometrisation of statistical mechanics Proc. R. Soc. Lond. A 455 1683-715
13. Brody D C and Hughston L P 2001 Geometric quantum mechanics J. Geom. Phys. 677 1-35
14. Brody D C and Ritz A 1998 On the symmetry of real-space renormalisation Nucl. Phys. B 522 588-604
15. Brody D C and Ritz A 2003 Information geometry of finite Ising models J. Geom. Phys. 47 207-20
16. Brody D C and Rivier N 1995 Geometrical aspects of statistical mechanics Phys. Rev. E 51 1006-11
17. Brody E J 1971 An elementary proof of the Gaussian dichotomy theorem Z. Wahrscheinlichkeitstheor. Verwandte Geb. 20 217-26
18. Brody E J 1987 Applications of the Kakutani metric to real-space renormalization Phys. Rev. Lett. 58 179-82
19. Burbea J 1986 Informative geometry of probability spaces Expositiones Math. 4 347-78
20. Burbea J, Oller J M and Reverter F 2002 Some remarks on the information geometry of the Gamma distribution Commun. Stat. Theory Methods 31 1959-75
21. Burbea J and Rao C R 1982a On the convexity of some divergence measures based on entropy functions IEEE Trans. Inf. Theory IT-28 489-95
22. Burbea J and Rao C R 1982b On the convexity of higher order Jensen differences based on entropy functions IEEE Trans. Inf. Theory IT-28 961-3
23. Burbea J and Rao C R 1984 Differential metrics in probability spaces Probab. Math. Stat. 3 241-58
24. Bures D 1969 An extension of Kakutani's theorem on infinite product measures to the tensor product of semifinite w*-algebras Trans. Am. Math. Soc. 135 199-212
25. Cena A and Pistone G 2007 Exponential statistical manifold Ann. Inst. Stat. Math. 59 27-56
26. encov N N 1982 Statistical Decision Rules and Optimal Inference (Translations of Mathematical Monographs vol 53) (Providence, RI: American Mathematical Society) (Originally published as 1972.)
27. Crooks G E 2007 Measuring thermodynamic length Phys. Rev. Lett. 99 100602
28. Dawid A P 1975 Discussion on Professor Efron's paper Ann. Stat. 3 1231-4
29. Diósi L, Forgács G, Lukács B and Frisch H L 1984 Metricization of thermodynamic-state space and the renormalization group Phys. Rev. A 29 3343-5
30. Diósi L and Lukács B 1986 Spatial correlations in diluted gases from the viewpoint of the metric of the thermodynamic state space J. Chem. Phys. 84 5081-4
31. Diósi L, Lukács B and Rácz A 1989 Mapping the van der Waals state space J. Chem. Phys. 91 3061-7
32. Dolan B P 1998 Geometry and thermodynamic fluctuations of the Ising model on a Bethe lattice Proc. R. Soc. Lond. A 454 2655-65
33. Efron B 1975 Defining the curvature of a statistical problem (with applications to second order efficiency). With a discussion by C R Rao, D A Pierce, D R Cox, D V Lindley, L LeCam, J K Ghosh, J Pfanzagl, N Keiding, A P Dawid, J Reeds and with a reply by the author Ann. Stat. 3 1189-242
34. Efron B 1978 The geometry of exponential families Ann. Stat. 6 362-76
35. Fisher R A 1925 Theory of statistical estimation Proc. Camb. Phil. Soc. 122 700-25
36. Gibilisco P and Isola T 2007 Uncertainty principle and quantum Fisher information Ann. Inst. Stat. Math. 59 147-59
37. Gibilisco P, Imparato D and Isola T 2007 Uncertainty principle and quantum Fisher information: II J. Math. Phys. 48 072109
38. Grasselli M R and Streater R F 2001 On the uniqueness of the Chentsov metric in quantum information geometry Infinite Dimens. Anal. Quantum Probab. Relat. Top. 4 173-82
39. Havrda J and Charvát F 1967 Quantification method of classification processes Kybernetika 3 30-5
40. Hellinger E 1909 Neue Begründung der Theorie quadratischer Formen von unendlichvielen Veränderlichen J. Reine Angew. Math. 136 210-71
41. Hemmer P C 1964 On the van der Waals theory of the vapour-liquid equilibrium: IV. The pair correlation function and equation of state for long-range forces J. Math. Phys. 5 75-84
42. Hemmer P C, Kac M and Uhlenbeck G E 1964 On the van der Waals theory of the vapour-liquid equilibrium: III. Discussion of the critical region J. Math. Phys. 5 60-74
43. Ingarden R S 1981 Information geometry in functional spaces of classical and quantum finite statistical systems Int. J. Eng. Sci. 19 1609-33
44. Ingarden R S and Tamassy L 1993 On parabolic geometry and irreversible macroscopic time Rep. Math. Phys. 32 11-33
45. Janke W, Johnston D A and C. Malmini R P K 2002 Information geometry of the Ising model on planar random graphs Phys. Rev. E 66 056119
46. Janyszek H 1990 Riemannian geometry and stability of thermodynamical equilibrium systems J. Phys. A: Math. Gen. 23 477-90
47. Janyszek H and Mrugała R 1989a Geometrical structure of the state space in classical stastical and phenomological thermodynamics Rep. Math. Phys. 27 145-59
48. Janyszek H and Mrugała R 1989b Riemannian geometry and the thermodynamics of model magnetic systems Phys. Rev. A 39 6515-23
49. Janyszek H and Mrugała R 1990 Riemannian geometry and the stability of ideal quantum gases J. Phys. A: Math. Gen. 23 467-76
50. Jeffreys H 1946 An invariant form for the prior probability in estimation problems Proc. R. Soc. Lond. A 186 453-61
51. Jenová A and Petz D 2006 Sufficiency in quantum statistical inference: a survey with examples Infinite Dimens. Anal. Quantum Probab. Relat. Top. 9 331-51
52. Johnston D A, Janke W and Kenna R 2003 Information geometry, one, two, three (and four) Acta Phys. Pol. B 34 4923-37
53. Kac M 1959 On the partition function of a one-dimensional gas Phys. Fluids 2 8-12
54. Kac M, Uhlenbeck G E and Hemmer P C 1963 On the van der Waals theory of the vapour-liquid equilibrium: I. Discussion of a one-dimensional model J. Math. Phys. 4 216-28
55. Kakutani S 1948 On equivalence of infinite product measures Ann. Math. 49 214-24
56. Kass R 1989 The geometry of asymptotic inference. With comments and a rejoinder by the author Stat. Sci. 4 188-234
57. Lebowitz J L and Penrose O 1966 Rigorous treatment of the van der Waals-Maxwell theory of liquid-vapour transition J. Math. Phys. 7 98-113
58. Lebowitz J L and Percus J K 1963 Asymptotic behaviour of the radial distribution function J. Math. Phys. 4 248-54
59. Mahalanobis P C 1930 On tests and meassures of groups divergence J. Asiatic Soc. Bengal 26 541-88
60. Mahalanobis P C 1936 On the generalised distance in statistics Proc. Natl. Inst. Sci. India A 2 49-55
61. Maybank S 2005 The Fisher-Rao metric for projective transformations of the line Int. J. Comput. Vis. 63 191-206
62. Mayer J E and Mayer M G 1940 Statistical Mechanics (New York: Wiley)
63. Mijatović M, Veselinović V and Trenevski K 1987 Differential geometry of equilibrium thermodynamics Phys. Rev. A 35 1863-7
64. Mirza B and Mohammadzadeh H 2008 Ruppeiner geometry of anyon gas Phys. Rev. E 78 021127
65. Mrugała R, Nulton J D, Schön J C and Salamon P 1990 Statistical approach to the geometric structure of thermodynamics Phys. Rev. A 41 3156-60
66. Murray M K and Rice J W 1993 Differential Geometry and Statistics (London: Chapman and Hall)
67. O'Connor D and Stephens C R 1993 Geometry, the renormalisation group and gravity Directions in General Relativity Proc. 1993 Int. Symp., MA vol 1 ed B. L. Hu, M P Ryan Jr and C V Vishveshwava (Cambridge: Cambridge University Press)
68. Peter A and Rangarajan A 2006 Shape analysis using the Fisher-Rao Riemannian metric: unifying shape representation and deformation Proc. 3rd IEEE Int. Symp. on Biomedical Imaging: Nano to Macro pp 1164-7
69. Petz D 2002 Covariance and Fisher information in quantum mechanics J. Phys. A: Math. Gen. 35 929-39
70. Petz D and Sudar C 1996 Geometries of quantum states J. Math. Phys. 37 2662-73
71. Rao C R 1945 Information and the accuracy attainable in the estimation of statistical parameters Bull. Calcutta Math. Soc. 37 81-91
72. Rao C R 1947 The problem of classification and distance between two populations Nature 159 30-1
73. Rao C R 1954 On the use and interpretation of distance functions in statistics Bull. Inst. Int. Stat. 34 90-7
74. Rao C R 1984 Convexity properties of entropy functions and analysis of diversity Inequalities in Statistics and Probability Proc. Symp. on Inequalities in Statistics and Probability (Lincoln, Nebraska 1982) (Institute of Mathematical Statistics Lecture Notes Monograph Series vol 5) ed Y L Tong (Hayward, CA: Institute of Mathematical Statistics)
75. Ruppeiner G 1979 Thermodynamics: a Riemannian geometric model Phys. Rev. A 20 1608-13
76. Ruppeiner G 1990 Thermodynamic curvature of a one-dimensional fluid J. Chem. Phys. 92 3700-9
77. Ruppeiner G 1991 Riemannian geometric theory of critical phenomena Phys. Rev. A 44 3583-95
78. Ruppeiner G 1995 Reimannian geometry in thermodynamic fluctuation theory Rev. Mod. Phys. 67 605-59
79. Ruppeiner G 2008 Thermodynamic curvature and phase transitions in Kerr-Newman black holes Phys. Rev. D 78 024016
80. Ruppeiner G and Chance J 1990 Reimannian geometry in thermodynamic fluctuation theory J. Chem. Phys. 92 3700-9
81. Salamon P and Berry R S 1983 Thermodynamic length and dissipated availability Phys. Rev. Lett. 51 1127-30
82. Schlögl F 1985 Thermodynamic metric and stochastic measures Z. Phys. B 59 449-54
83. Streater R F 2004 Duality in quantum information geometry Open Syst. Inf. Dyn. 11 71-7
84. Tonks L 1936 The complete equation of state of one, two, and three-dimensional gases of hard elastic spheres Phys. Rev. 50 955-63
85. Tsallis C 1988 Possible generalization of Boltzmann-Gibbs statistics J. Stat. Phys. 52 479-87
86. Uhlenbeck G E, Hemmer P C and Kac M 1963 On the van der Waals theory of the vapour-liquid equilibrium: II. Discussion of the distribution functions J. Math. Phys. 4 229-47
87. Uhlmann A 1996 Spheres and hemispheres as quantum state space J. Geom. Phys. 18 76-92
88. van Hove L 1950 Sur l'intégral de configuration pour les systèmes de particules à une dimension Physica 16 137-43
89. van Kampen N G 1964 Condensation of a classical gas with long-range attraction Phys. Rev. 135 A362-9
90. Weinhold F 1975 Metric geometry of equilibrium thermodynamics J. Chem. Phys. 63 2479-83
91. Yau S T 1977 Calabi's conjecture and some new results in algebraic geometry Proc. Natl. Acad. Sci. USA 74 1798-9
92. Zanardi P, Giorda P and Cozzini M 2007 Information-theoretic differential geometry of quantum phase transitions Phys. Rev. Lett. 99 100603