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Is our mathematics natural? The case of equilibrium statistical mechanics
Author:
David Ruelle
Journal:
Bull. Amer. Math. Soc. 19 (1988), 259-268
MSC (1985):
Primary 00A25; Secondary 82A05
MathSciNet review:
940484
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References |
Similar Articles |
Additional Information
- 1.
P. Cartier, La pratique—et les pratiques—des mathématiques, Encyclopédie Philosophique Universelle, Presses Univ. de France, Paris, 1988 (to appear).
- 2.
Eugene
P. Wigner, The unreasonable effectiveness of mathematics in the
natural sciences [Comm. Pure Appl. Math. 13 (1960), 1–14; Zbl 102,
7], Mathematical analysis of physical systems, Van Nostrand Reinhold,
New York, 1985, pp. 1–14. MR
824292
- 3.
David
Ruelle, Statistical mechanics: Rigorous results, W. A.
Benjamin, Inc., New York-Amsterdam, 1969. MR 0289084
(44 #6279)
- 4.
Rufus
Bowen, Equilibrium states and the ergodic theory of Anosov
diffeomorphisms, Lecture Notes in Mathematics, Vol. 470,
Springer-Verlag, Berlin, 1975. MR 0442989
(56 #1364)
- 5.
David
Ruelle, Thermodynamic formalism, Encyclopedia of Mathematics
and its Applications, vol. 5, Addison-Wesley Publishing Co., Reading,
Mass., 1978. The mathematical structures of classical equilibrium
statistical mechanics; With a foreword by Giovanni Gallavotti and
Gian-Carlo Rota. MR 511655
(80g:82017)
- 6.
Robert
B. Israel, Convexity in the theory of lattice gases, Princeton
University Press, Princeton, N.J., 1979. Princeton Series in Physics; With
an introduction by Arthur S. Wightman. MR 517873
(80i:82002)
- 7.
Ya.
G. Sinaĭ, Theory of phase transitions: rigorous
results, International Series in Natural Philosophy, vol. 108,
Pergamon Press, Oxford, 1982. Translated from the Russian by J. Fritz, A.
Krámli, P. Major and D. Szász. MR 691854
(84k:82002)
- 8.
D.
Ruelle, Correlation functionals, J. Mathematical Phys.
6 (1965), 201–220. MR 0201174
(34 #1058)
- 9.
O. E. Lanford, Entropy and equilibrium states in classical statistical mechanics, Statistical Mechanics and Mathematical Problems, Lecture Notes in Physics, vol. 20, Springer-Verlag, Berlin, 1973, pp. 1-113.
- 10.
T.
D. Lee and C.
N. Yang, Statistical theory of equations of state and phase
transitions. II. Lattice gas and Ising model, Physical Rev. (2)
87 (1952), 410–419. MR 0053029
(14,711c)
- 11.
Ja.
G. Sinaĭ, Markov partitions and U-diffeomorphisms,
Funkcional. Anal. i Priložen 2 (1968), no. 1,
64–89 (Russian). MR 0233038
(38 #1361)
- 12.
Ja.
G. Sinaĭ, Construction of Markov partitionings,
Funkcional. Anal. i Priložen. 2 (1968), no. 3,
70–80 (Loose errata) (Russian). MR 0250352
(40 #3591)
- 13.
Ja.
G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat.
Nauk 27 (1972), no. 4(166), 21–64 (Russian). MR 0399421
(53 #3265)
- 14.
Rufus
Bowen, Markov partitions for Axiom 𝐴 diffeomorphisms,
Amer. J. Math. 92 (1970), 725–747. MR 0277003
(43 #2740)
- 15.
Rufus
Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math.
95 (1973), 429–460. MR 0339281
(49 #4041)
- 16.
William
Parry and Mark
Pollicott, An analogue of the prime number theorem for closed
orbits of Axiom A flows, Ann. of Math. (2) 118
(1983), no. 3, 573–591. MR 727704
(85i:58105), http://dx.doi.org/10.2307/2006982
- 17.
Huzihiro
Araki, Some contact points of mathematics and physics, Acta
Appl. Math. 1 (1983), no. 1, 5–15. MR 702044
(85a:46032), http://dx.doi.org/10.1007/BF02433839
- 18.
R.
Haag, N.
M. Hugenholtz, and M.
Winnink, On the equilibrium states in quantum statistical
mechanics, Comm. Math. Phys. 5 (1967), 215–236.
MR
0219283 (36 #2366)
- 19.
A.
Connes, H.
Narnhofer, and W.
Thirring, Dynamical entropy of 𝐶* algebras and von Neumann
algebras, Comm. Math. Phys. 112 (1987), no. 4,
691–719. MR
910587 (89b:46078)
- 20.
A. Connes, The dynamical entropy of quantum systems, preprint.
- 21.
James
Glimm and Arthur
Jaffe, Quantum physics, Springer-Verlag, New York, 1981. A
functional integral point of view. MR 628000
(83c:81001)
- 22.
Taro
Asano, Lee-Yang theorem and the Griffiths inequality for the
anisotropic Heisenberg ferromagnet, Phys. Rev. Lett.
24 (1970), 1409–1411. MR 0261884
(41 #6494)
- 23.
Taro
Asano, Theorems on the partition functions of the Heisenberg
ferromagnets., J. Phys. Soc. Japan 29 (1970),
350–359. MR 0280127
(43 #5848)
- 24.
Masuo
Suzuki and Michael
E. Fisher, Zeros of the partition function for the Heisenberg,
ferroelectric, and general Ising models, J. Mathematical Phys.
12 (1971), 235–246. MR 0275824
(43 #1577)
- 25.
David
Ruelle, Extension of the Lee-Yang circle theorem, Phys. Rev.
Lett. 26 (1971), 303–304. MR 0292441
(45 #1527)
- 1.
- P. Cartier, La pratique—et les pratiques—des mathématiques, Encyclopédie Philosophique Universelle, Presses Univ. de France, Paris, 1988 (to appear).
- 2.
- E. Wigner, The unreasonable effectiveness of mathematics in the natural sciences, Comm. Pure Appl. Math. 13 (1960), 1-14. MR 824292
- 3.
- D. Ruelle, Statistical mechanics. Rigorous results, Benjamin, New York, 1969. MR 289084
- 4.
- R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math., vol. 470, Springer-Verlag, Berlin, 1975. MR 442989
- 5.
- D. Ruelle, Thermodynamic formalism, Encyclopedia of Math. and its Appl., vol. 5, Addison-Wesley, Reading, Mass., 1978. MR 511655
- 6.
- R. Israel, Convexity in the theory of lattice gases, Princeton Univ. Press, Princeton, N. J., 1979. MR 517873
- 7.
- Ia. G. Sinai, Theory of phase transitions: Rigorous results, Pergamon, Oxford, 1982. MR 691854
- 8.
- D. Ruelle, Correlation functionals, J. Math. Phys. 6 (1965), 201-220. MR 201174
- 9.
- O. E. Lanford, Entropy and equilibrium states in classical statistical mechanics, Statistical Mechanics and Mathematical Problems, Lecture Notes in Physics, vol. 20, Springer-Verlag, Berlin, 1973, pp. 1-113.
- 10.
- T. D. Lee and C. N. Yang, Statistical theory of equations of state and phase transitions. II, Lattice gas and Ising model, Phys. Rev. 87 (1952), 410-419. MR 53029
- 11.
- la. G. Sinai, Markov partitions and C-diffeomorphisms, Funktsional Anal. i Prilozhen 2 No. 1 (1968), 64-89; English transl., Functional Anal. Appl. 2 (1968), 61-82. MR 233038
- 12.
- la. G. Sinai, Construction of Markov partitions, Funktsional Anal. i Prilozhen 2 No. 3 (1968), 70-80; English transl., Functional Anal. Appl. 2 (1968), 245-253. MR 250352
- 13.
- la. G. Sinai, Gibbs measures in ergodic theory, Uspekhi Mat. Nauk 27 No. 4 (1972), 21-64; English transl., Russian Math. Surveys 27 No. 4 (1972), 21-69. MR 399421
- 14.
- R. Bowen, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math. 92 (1970), 725-747. MR 277003
- 15.
- R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math. 95 (1973), 429-459. MR 339281
- 16.
- W. Parry and M. Pollicott, An analogue of the prime number theorem for closed orbits of Axiom A flows, Ann. of Math. (2) 118 (1983), 573-591. MR 727704
- 17.
- H. Araki, Some contact points of mathematics and physics, Invited talk, First Pan-African Congress of Mathematicians, Rabat, 1976 (Zi F, Bielefeld, preprint). MR 702044
- 18.
- R. Haag, N. M. Hugenholtz and M. Winnink, On the equilibrium states in quantum statistical mechanics, Comm. Math. Phys. 5 (1967), 215-236. MR 219283
- 19.
- A. Connes, H. Narnhofer and W. Thirring, Dynamical entropy of C* algebras and von Neumann algebras, Comm. Math. Phys. 112 (1987), 691-719. MR 910587
- 20.
- A. Connes, The dynamical entropy of quantum systems, preprint.
- 21.
- J. Glimm and A. Jaffe, Quantum physics: a functional integral point of view, Springer-Verlag, New York, 1981. MR 628000
- 22.
- T. Asano, Lee-Yang theorem and the Griffiths inequality for the anisotropic Heisenberg ferromagnet, Phys. Rev. Letters 24 (1970), 1409-1411. MR 261884
- 23.
- T. Asano, Theorems on the partition functions of the Heisenberg ferromagnets, J. Phys. Soc. Japan 29 (1970), 350-359. MR 280127
- 24.
- M. Suzuki and M. E. Fisher, Zeros of the partition function for the Heisenberg, ferroelectric, and general Ising models, J. Math. Phys. 12 (1971), 235-246. MR 275824
- 25.
- D. Ruelle, Extension of the Lee-Yang circle theorem, Phys. Rev. Letters 26 (1971), 303-304. MR 292441
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0273-0979-1988-15634-0
PII:
S 0273-0979(1988)15634-0
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