The Fermi–Pasta–Ulam (FPU) problem is discussed in connection with its physical relevance, and it is shown how apparently there exist only two possibilities: either the FPU problem is just a curiosity, or it has a fundamental role for the foundations of physics, casting a new light on the relations between classical and quantum mechanics. To this end, a short review is given of the main conceptual proposals that have been advanced. Particular emphasis is given to the perspective of a metaequilibrium scenario, which appears to be the only possible one for the FPU paradox to survive in the physically relevant case of infinitely many particles.
REFERENCES
1.
E.
Fermi
, J.
Pasta
, and S.
Ulam
, “Studies of nonlinear problems
,” in E. Fermi: Note e Memorie (Collected Papers)
(Accademia Nazionale dei Lincei, Roma, and The University of Chicago Press
, Chicago, 1965
), Vol. II
, No. 266, pp. 977
–988
.2.
A.
Carati
, L.
Galgani
, and A.
Giorgilli
, “Dynamical systems and thermodynamics
,” in Encyclopedia of Mathematical Physics
(Elsevier
, Oxford, in press).3.
E.
Fermi
, “Beweiss das ein mechanisches Normalsystem im allgemeinen quasi-ergodisch ist
,” Phys. Z.
24
, 261
(1923
);E.
Fermi
, in Note e Memorie (Collected Papers)
(Accademia Nazionale dei Lincei, Roma, and The University of Chicago Press
, Chicago, 1965
), Vol. I
, No. 11, pp. 79
–87
.4.
G.
Benettin
, G.
Ferrari
, L.
Galgani
, and A.
Giorgilli
, “An extension of the Poincaré–Fermi theorem on the nonexistence of invariant manifolds in nearly integrable Hamiltonian systems
,” Nuovo Cimento Soc. Ital. Fis., B
72
, 137
(1982
);G.
Benettin
, L.
Galgani
, and A.
Giorgilli
, “Poincaré’s non-existence theorem and classical perturbation theory in nearly-integrable Hamiltonian systems
,” in Advances in Nonlinear Dynamics and Stochastic Processes
, edited by R.
Livi
and A.
Politi
(World Scientific
, Singapore, 1985
).5.
A. N.
Kolmogorov
, “Preservation of conditionally periodic movements with small change in the Hamilton function
,” Dokl. Akad. Nauk
98
, 527
(1954
);English translation in
G.
Casati
and G.
Ford
, Lecture Notes in Physics No. 93
(Springer-Verlag
, Berlin, 1979
).See also
G.
Benettin
, L.
Galgani
, A.
Giorgilli
, and J.-M.
Strelcyn
, “A proof of Kolmogorov’s theorem on invariant tori using canonical transformations defined by the Lie method
,” Nuovo Cimento Soc. Ital. Fis., B
79
, 201
(1984
).6.
N. J.
Zabusky
and M. D.
Kruskal
, “Interaction of solitons in a collisionless plasma and the recurrence of initial states
,” Phys. Rev. Lett.
15
, 240
(1965
).7.
F. M.
Izrailev
and B. V.
Chirikov
, “Statistical properties of a nonlinear string
,” Sov. Phys. Dokl.
11
, 30
(1966
).8.
D. I.
Shepelyansky
, “Low-energy chaos in the Fermi–Pasta–Ulam problem
,” Nonlinearity
10
, 1331
(1997
).9.
P.
Bocchieri
, A.
Scotti
, B.
Bearzi
, and A.
Loinger
, “Anharmonic chain with Lennard-Jones interaction
,” Phys. Rev. A
2
, 2013
(1970
).10.
L.
Galgani
and A.
Scotti
, “Planck-like distribution in classical nonlinear mechanics
,” Phys. Rev. Lett.
28
, 1173
(1972
);L.
Galgani
and A.
Scotti
, “Recent progress in classical nonlinear dynamics
,” Riv. Nuovo Cimento
2
, 189
(1972
).11.
C.
Cercignani
, L.
Galgani
, and A.
Scotti
, “Zero-point energy in classical nonlinear mechanics
,” Phys. Lett.
38A
, 403
(1972
).12.
C.
Cercignani
, “On a nonquantum derivation of Planck’s distribution law
,” Found. Phys. Lett.
11
, 189
(1998
).13.
E.
Fucito
, F.
Marchesoni
, E.
Marinari
, G.
Parisi
, L.
Peliti
, S.
Ruffo
, and A.
Vulpiani
, “Approach to equilibrium in a chain of nonlinear oscillators
,” J. Phys. (France)
43
, 707
(1982
);G.
Parisi
, “On the approach to equilibrium of a Hamiltonian chain of anharmonic oscillators
,” Europhys. Lett.
40
, 357
(1997
).See also
B.
Bassetti
, P.
Butera
, M.
Raciti
, and M.
Sparpaglione
, “Complex poles, spatial intermittencies, and energy transfer in a classical nonlinear string
,” Phys. Rev. A
30
, 1033
(1984
).14.
R.
Livi
, M.
Pettini
, S.
Ruffo
, M.
Sparpaglione
, and A.
Vulpiani
, “Equipartition threshold in nonlinear large Hamiltonian systems: The Fermi–Pasta–Ulam model
,” Phys. Rev. A
31
, 1039
(1985
).15.
R.
Livi
, M.
Pettini
, S.
Ruffo
, and A.
Vulpiani
, “Further results on the equipartition threshold in large nonlinear Hamiltonian systems
,” Phys. Rev. A
31
, 2740
(1985
).16.
H.
Kantz
, “Vanishing stability thresholds in the thermodynamic limit of nonintegrable conservative systems
,” Physica D
39
, 322
(1989
);H.
Kantz
, R.
Livi
, and S.
Ruffo
, “Equipartition thresholds in chains of anharmonic oscillators
,” J. Stat. Phys.
76
, 627
(1994
);P.
Poggi
, S.
Ruffo
, and H.
Kantz
, “Shock waves and time scales to reach equipartition in the Fermi–Pasta–Ulam model
,” Phys. Rev. E
52
, 307
(1995
);J.
De Luca
, A. J.
Lichtenberg
, and S.
Ruffo
, “Energy transition and time scale to equipartition in the Fermi–Pasta–Ulam oscillator chain
,” Phys. Rev. E
51
, 2877
(1995
);J.
De Luca
, A. J.
Lichtenberg
, and S.
Ruffo
, “Universal evolution to equipartition in oscillator chains
,” Phys. Rev. E
54
, 2329
(1996
);J.
De Luca
, A. J.
Lichtenberg
, and S.
Ruffo
, “Finite time to equipartition in the thermodynamic limit
,” Phys. Rev. E
60
, 3781
(1999
);J.
De Luca
and A. J.
Lichtenberg
, “Transitions and time scales to equipartition in oscillator chains: low-frequency initial conditions
,” Phys. Rev. E
66
, 026206
(2002
);L.
Casetti
, M.
Cerruti-Sola
, M.
Pettini
, and E. G. D.
Cohen
, “The Fermi–Pasta–Ulam problem revisited: Stochasticity thresholds in nonlinear Hamiltonian systems
,” Phys. Rev. E
55
, 6566
(1997
);L.
Casetti
, M.
Cerruti-Sola
, M.
Modugno
, G.
Pettini
, M.
Pettini
, and R.
Gatto
, “Dynamical and statistical properties of Hamiltonian systems with many degrees of freedom
,” Riv. Nuovo Cimento
22
, 1
(1999
);L.
Caiani
, L.
Casetti
, and M.
Pettini
, “Hamiltonian dynamics of the two-dimensional lattice model
,” J. Phys. A
31
, 3357
(1998
);P. R.
Kramers
, J. A.
Biello
, and Y.
Lvov
, “Stages of energy transfer in the FPU model
,” Discrete Contin. Dyn. Syst., Ser. B
Suppl. 113
, 113
(2003
);P. R.
Kramers
, J. A.
Biello
, and Y.
Lvov
, “Application of weak turbulence theory to FPU model. Dynamical systems and differential equations
,” Discrete Contin. Dyn. Syst., Ser. B
Suppl. 113
, 482
(2003
);M.
Casartelli
and S.
Sello
, “Stochasticity and rate of energy exchanges in nonlinear chains
,” Advances in Nonlinear Dynamics and Stochastic Processes
(World Scientific
, Singapore, 1987
), Vol. II
, pp. 113
–125
;C.
Alabiso
and M.
Casartelli
, “Quasi-harmonicity and power spectra in the FPU model
,” J. Phys. A
33
, 831
(2000
).17.
L.
Casetti
, R.
Livi
, and M.
Pettini
, “Gaussian model for chaotic instability of Hamiltonian flows
,” Phys. Rev. Lett.
74
, 375
(1995
).18.
N. N.
Nekhoroshev
, in Topics in Modern Mathematics: Petrovskii Sem.
No. 5, edited by O. A.
Oleinik
(Consultants Bureau
, New York, 1985
).See also
G.
Benettin
, L.
Galgani
, and A.
Giorgilli
, “A proof of Nekhoroshev’s theorem for the stability times in nearly integrable Hamiltonian systems
,” Celest. Mech.
37
, 1
(1985
);A.
Giorgilli
, “On the problem of stability for near to integrable Hamiltonian systems
,” Proceedings of the International Congress of Mathematicians
, Berlin, 1998
, Vol. III
, Documenta Mathematica, extra volume ICM 1998
, pp. 143
–152
;G.
Benettin
, “The elements of Hamiltonian perturbation theory
,” in Hamiltonian Dynamics and Frequency Analysis (Porquerolles School, 2001)
, edited by D.
Benest
, C.
Froeschlé
, and E.
Lega
(Taylor and Francis
, London, in press).19.
G.
Benettin
, L.
Galgani
, and A.
Giorgilli
, “Boltzmann’s ultraviolet cutoff and Nekhoroshev’s theorem on Arnold diffusion
,” Nature (London)
311
, 444
(1984
).20.
L. D.
Landau
and E.
Teller
, “On the theory of sound dispersion
,” Phys. Z. Sowjetunion
10
, 34
(1936
);also in the
Collected Papers of L. D. Landau
, edited by D.
ter Haar
(Pergamon
, Oxford, 1965
), pp. 147
–153
;G.
Benettin
, L.
Galgani
, and A.
Giorgilli
, “Exponential law for the equipartition times among translational and vibrational degrees of freedom
,” Phys. Lett. A
120
, 23
(1987
);O.
Baldan
and G.
Benettin
, “Classical ‘freezing’ of fast rotations. A numerical test of the Boltzmann–Jeans conjecture
,” J. Stat. Phys.
62
, 201
(1991
);G.
Benettin
, A.
Carati
, and P.
Sempio
, “On the Landau–Teller approximation for the energy exchanges with fast degrees of freedom
,” J. Stat. Phys.
73
, 175
(1993
);G.
Benettin
, A.
Carati
, and G.
Gallavotti
, “A rigorous implementation of the Landau–Teller approximation for adiabatic invariants
,” Nonlinearity
10
, 479
(1997
);G.
Benettin
, P.
Hjorth
, and P.
Sempio
, “Exponentially long equilibrium times in a one-dimensional collisional model of a classical gas
,” J. Stat. Phys.
94
, 871
(1999
);A.
Carati
, L.
Galgani
, and B.
Pozzi
, “Levy flights in the Landau–Teller model of molecular collisions
,” Phys. Rev. Lett.
90
, 010601
(2003
);
[PubMed]
A.
Carati
, L.
Galgani
, and B.
Pozzi
, “The problem of the rate of thermalization, and the relations between classical and quantum mechanics,” in Mathematical Models and Methods for Smart Materials
, edited by M.
Fabrizio
, B.
Lazzari
, and A.
Morro
(World Scientific
, Singapore, 2002
).21.
L.
Berchialla
, L.
Galgani
, and A.
Giorgilli
, “Localization of energy in FPU chains
,” Discrete Contin. Dyn. Syst., Ser. A
11
, 855
(2004
);L.
Berchialla
, A.
Giorgilli
, and S.
Paleari
, “Exponentially long times to equipartition in the thermodynamic limit
,” Phys. Lett. A
321
, 167
(2004
).22.
A.
Ponno
, “Soliton theory and the Fermi–Pasta–Ulam problem in the thermodynamic limit
,” Europhys. Lett.
64
, 606
(2003
).23.
D.
Bambusi
and A.
Ponno
, “KdV equation and energy sharing in FPU
,” Chaos
15
, 015107
(2005
) (this issue).24.
R.
Livi
, M.
Pettini
, S.
Ruffo
, M.
Sparpaglione
, and A.
Vulpiani
, “Relaxation to different stationary states in the Fermi–Pasta–Ulam model
,” Phys. Rev. A
28
, 3544
(1983
).25.
A.
Ponno
, in Proceeding of Cargese Summer School 2003: Chaotic Dynamics and Transport in Classical and Quantum Systems
(Kluwer
, Dordrecht, in press).26.
L.
Galgani
, A.
Giorgilli
, A.
Martinoli
, and S.
Vanzini
, “On the problem of energy equipartition for large systems of the Fermi–Pasta–Ulam type: analytical and numerical estimates
,” Physica D
59
, 334
(1992
).27.
G.
Benettin
, L.
Galgani
, and A.
Giorgilli
, “Realization of holonomic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory, II
,” Commun. Math. Phys.
121
, 557
(1989
);G.
Benettin
, J.
Fröhlich
, and A.
Giorgilli
, “A Nekhoroshev-type theorem for Hamiltonian systems with infinitely many degrees of freedom
,” Commun. Math. Phys.
119
, 95
(1988
).28.
A.
Carati
, “The averaging principle in the thermodynamic limit
” (in preparation).29.
R.
Livi
, M.
Pettini
, S.
Ruffo
, and A.
Vulpiani
, “Chaotic behavior in nonlinear Hamiltonian systems and equilibrium statistical mechanics
,” J. Stat. Phys.
48
, 539
(1987
).30.
M.
Born
, W.
Heisenberg
, and P.
Jordan
, “Zur Quantenmechanik
,” Z. Phys.
35
, 557
(1926
);translated in
B. L.
van der Waerden
, Sources of Quantum Mechanics
(Dover
, New York, 1968
);W.
Heisenberg
, Die Physikalishchen Prinzipien der Quantentheorie
, (V. S.
Hirzel
, Leipzig, 1930
), Chap. V, Sec. VII.31.
A.
Perronace
and A.
Tenenbaum
, “Classical specific heat of an atomic lattice at low temperature, revisited
,” Phys. Rev. E
57
, 100
(1998
);G.
Marcelli
and A.
Tenenbaum
, “Quantumlike short-time behavior of a classical crystal
,” Phys. Rev. E
68
, 041112
(2003
).32.
A.
Carati
and L.
Galgani
, “On the specific heat of the Fermi–Pasta–Ulam systems
,” J. Stat. Phys.
94
, 859
(1999
);A.
Carati
, P.
Cipriani
, and L.
Galgani
, “On the definition of temperature in FPU systems
,” J. Stat. Phys.
115
, 1119
(2004
).33.
S.
Lepri
, R.
Livi
, and A.
Politi
, “Thermal conduction in classical low-dimensional lattices
,” Phys. Rep.
377
, 1
(2003
).34.
35.
36.
G.
Benettin
, “Time-scale for energy equipartition in a 2-dimensional FPU model
,” Chaos
15
, 015108
(2005
) (this issue).37.
A.
Carati
and L.
Galgani
, “Analog of Planck’s formula and effective temperature in classical statistical mechanics far from equilibrium
,” Phys. Rev. E
61
, 4791
(2000
);A.
Carati
and L.
Galgani
, “Einstein’s nonconventional conception of the photon, and the modern theory of dynamical systems,” in Chance in Physics
, Lecture Notes in Physics No. 574
, edited by J.
Bricmont
et al. (Springer-Verlag
, Berlin, 2001
);A.
Carati
and L.
Galgani
, “Planck’s formula and glassy behaviour in classical nonequilibrium statistical mechanics
,” Physica A
280
, 105
(2000
);A.
Carati
and L.
Galgani
, “The theory of dynamical systems and the relations between classical and quantum mechanics
,” Found. Phys.
31
, 69
(2001
);L.
Galgani
, “Relaxation times and the foundations of classical statistical mechanics in the light of modern perturbation theory
,” in Non-Linear Evolution and Chaotic Phenomena
, NATO ASI Series R71B
, edited by G.
Gallavotti
and P. F.
Zweifel
(Plenum
, New York, 1988
), Vol. 176
.© 2005 American Institute of Physics.
2005
American Institute of Physics
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