Hostname: page-component-5d59c44645-n6p7q Total loading time: 0 Render date: 2024-02-22T05:57:03.825Z Has data issue: false hasContentIssue false

The discrete versus continuous controversy in physics

Published online by Cambridge University Press:  01 April 2007

ANNICK LESNE*
Affiliation:
LPTMC UMR7600, Université Pierre et Marie Curie-Paris 6, 4 place Jussieu, F-75252 Paris And: Institut des Hautes Études Scientifiques, 35 route de Chartres, F-91440 Bures-sur-Yvette, France Email: lesne@ihes.fr

Extract

This paper presents a sample of the deep and multiple interplay between discrete and continuous behaviours and the corresponding modellings in physics. The aim of this overview is to show that discrete and continuous features coexist in any natural phenomenon, depending on the scales of observation. Accordingly, different models, either discrete or continuous in time, space, phase space or conjugate space can be considered. Some caveats about their limits of validity and their interrelationships (discretisation and continuous limits) are pointed out. Difficulties and gaps arising from the singular nature of continuous limits and from the information loss accompanying discretisation are discussed.

Type
Paper
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arnold, L. (1980) On the consistency of the mathematical models of chemical reactions. In: Haken, H. (ed.) Dynamics of synergetic systems, Springer-Verlag 107118.Google Scholar
Auger, P. and Roussarie, R. (1994) Complex ecological models with simple dynamics: From individuals to populations. Acta Biotheoretica 42 111136.Google Scholar
Badii, R. and Politi, A. (1999) Complexity. Hierarchical structures and scaling in physics, Cambridge University Press.Google Scholar
Bailly, F. and Longo, G. (2004) Causalités et symétries dans les sciences de la nature. Le continu et le discret mathématiques. In: Joinet, J. B. (ed.) Logique et interaction: pour une géométrie de la cognition, Presses Universitaires de la Sorbonne, Paris.Google Scholar
Baker, G. L. and Gollub, J. B. (1996) Chaotic dynamics: An introduction, 2nd edition, Cambridge University Press.Google Scholar
Balian, R. (2004) Entropy, a protean concept. In: Dalibard, J., Duplantier, B. and Rivasseau, V. (eds.) Poincaré Seminar 2003, Birkhaüser.Google Scholar
Bensoussan, A., Lions, J. L. and Papanicolaou, G. (1978) Asymptotic analysis for periodic structures, North Holland.Google Scholar
Berry, M. (2001) Chaos and the semiclassical limit of quantum mechanics (is the moon there when somebody looks?). In: Russell, R. J., Clayton, P., Wegter-McNelly, K. and Polkinghorne, J. (eds.) Quantum mechanics: Scientific perspectives on Divine Action, Vatican Observatory – CTNS Publications 4154.Google Scholar
Boffetta, G., Cencini, M., Falcioni, M. and Vulpiani, A. (2002) Predictability: a way to characterize complexity. Phys. Rep. 356 367474.Google Scholar
Canny, J. F. (1986) A computational approch to edge detection. IEEE Transactions on Pattern Analysis and Machine Intelligence 8 679714.Google Scholar
Cantor, G. (1883) Über unendliche, lineare Punktlannigfaltigkeiten. Mathematische Annalen 21 545591.Google Scholar
Cardy, J. (2004) Field theory and nonequilibrium statistical mechanics. Lecture notes available online at http://www.thphys.physics.ox.ac.uk/users/JohnCardy/home.html.Google Scholar
Castiglione, P., Falcioni, M., Lesne, A. and Vulpiani, A. (2007) Chaos and coarse-grainings in non equilibrium statistical mechanics, Springer-Verlag (to appear).Google Scholar
Chernov, N. and Lebowitz, J. L. (1997) Stationary nonequilibrium states in boundary driven Hamiltonian systems: shear flow. J. of Stat. Phys. 86 953990.Google Scholar
Chopard, B. and Droz, M. (1998) Cellular automata modeling of physical systems, Cambridge University Press.Google Scholar
Cross, M. C. and Hohenberg, P. C. (1993) Pattern formation outside of equilibrium. Revs. Mod. Phys. 65 8511112.Google Scholar
Dettmann, C. P. and Morriss, G. P. (1996) Proof of Lyapunov exponent pairing for systems at constant kinetic energy. Phys. Rev. E 53 R5545R5548.Google Scholar
Diener, F. and Diener, M. (eds.) (1995) Nonstandard analysis in practice, Springer-Verlag.Google Scholar
Dorfman, J. R. (1999) An introduction to chaos in nonequilibrium statistical mechanics, Cambridge University Press.Google Scholar
Droz, M. and Pekalski, A. (2004) Population dynamics with or without evolution: a physicist's approach. Physica A 336 8492.Google Scholar
Eckmann, J. P. (1981) Roads to turbulence in dissipative dynamical systems. Revs. Mod. Phys. 53 643654.Google Scholar
Eckmann, J. P. and Ruelle, D. (1985) Ergodic theory of chaos and strange attractors. Revs. Mod. Phys. 57 617656.Google Scholar
Ernst, M. H. (2000) Kinetic theory of granular fluids: hard and soft inelastic spheres. In: Karkheck, J. (ed.) Proc. NATO ASI on dynamics: models and kinetic methods for non-equilibrium many body systems, Kluwer 239266.Google Scholar
Falcioni, M., Vulpiani, A., Mantica, G. and Pigolotti, S. (2003) Coarse-grained probabilistic automata mimicking chaotic systems. Phys. Rev. Lett. 91 044101.Google Scholar
Frisch, U. (1995) Turbulence: The legacy of A. N. Kolmogorov, Cambridge University Press.Google Scholar
Gardiner, C. W. (1983) Handbook of stochastic methods, Springer-Verlag.Google Scholar
Gaspard, P. (2004a) Maps. In: Scott, A. (ed.) Encyclopedia of Nonlinear Science, Taylor and Francis, London.Google Scholar
Gaspard, P. (2004b) Quantum theory. In: Scott, A. (ed.) Encyclopedia of Nonlinear Science, Taylor and Francis, London.Google Scholar
Gaspard, P. and Dorfman, J. R. (1995) Chaotic scattering theory, thermodynamic formalism and transport coefficients. Phys. Rev. E 52 35253552.Google Scholar
Gaspard, P. and Wang, X. J. (1993) Noise, chaos, and (τ, ε)-entropy per unit time. Phys. Rep. 235 321373.Google Scholar
Ghil, M. and Mullhaupt, A. (l985) Boolean delay equations: periodic and aperiodic solutions. J. Stat. Phys. 41 l25l74.Google Scholar
Givon, D., Kupferman, R. and Stuart, A. (2004) Extracting macroscopic dynamics: model problems and algorithms. Nonlinearity 17 R55R127.Google Scholar
Gonze, D., Halloy, J and Gaspard, P. (2002) Biochemical clocks and molecular noise: Theoretical study of robustness factors. J. Chem. Phys. 116 1099711010.Google Scholar
Gouyet, J. J. (1996) Physics and fractal structures, Springer-Verlag.Google Scholar
Gruber, C., Pache, S. and Lesne, A. (2004) The Second Law of thermodynamics and the piston problem. J. Stat. Phys 117 739772.Google Scholar
Guckenheimer, J. and Holmes, P. (1983) Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Springer-Verlag.Google Scholar
Gutzwiller, M. C. (1990) Chaos in classical and quantum mechanics, Springer-Verlag.Google Scholar
Hänggi, P., Talkner, P. and Borkovec, M. (1990) Reaction-rate theory: fifty years after Kramers. Revs. Mod. Phys. 62 251341.Google Scholar
Hausdorff, F. (1919) Dimension und ausseres Mass. Math. Ann. 29 157179.Google Scholar
Hilbert, D. (1891) Über die stetige Abbildung einer Linie auf ein Flächenstück. Mathematische Annalen 38 459460.Google Scholar
Jaynes, E. T. (1989) Papers on probability, statistics and statistical physics, Kluwer.Google Scholar
Kirkpatrick, T. R. and Ernst, M. H. (1991) Kinetic theory for lattice gas cellular automata. Phys. Rev. A 44 80518061.Google Scholar
Kolmogorov, A. N. and Tikhomirov, V. M. (1959) ε-entropy and ε-capacity of sets in functional space. Russian Mathematical Surveys 2 277–364. (Translated in Translations Am. Math. Soc. (1961) 17 277–364; also available in Shiryayev, A. N. (ed.) (1993) in Selected works of A. N. Kolmogorov, Vol. III, Kluwer 86–170.)Google Scholar
Krivine, H. and Lesne, A. (2003) Mathematical puzzle in the analysis of a low-pitched filter. American Journal of Physics 71 3133.Google Scholar
Laguës, M. and Lesne, A. (2003) Invariances d'échelle, Series ‘Échelles’, Belin, Paris.Google Scholar
Landau, L. D. and Lifschitz, E. M. (1984a) Theory of elasticity, Pergamon Press, Oxford.Google Scholar
Landau, L. D. and Lifschitz, E. M. (1984b) Hydrodynamics, Pergamon Press, Oxford.Google Scholar
Landau, L. D. and Lifschitz, E. M. (1984c) Electromagnetism of continuous media, Pergamon Press, Oxford.Google Scholar
Lebowitz, J. L. (1993) Boltzmann's entropy and time's arrow. Physics Today 46 3238.Google Scholar
Lesne, A. (1998) Renormalization methods, Wiley.Google Scholar
Lind, D and Marcus, B. (1995) An introduction to symbolic dynamics and coding, Cambridge University Press.Google Scholar
Longo, G. (2002) Laplace, Turing and the ‘imitation game’ impossible geometry: randomness, determinism and program's in Turing's test. In: Conference on cognition, meaning and complexity, Univ. Roma II.Google Scholar
Ma, S. K. (1976) Modern theory of critical phenomena, Benjamin.Google Scholar
MacKernan, D. and Nicolis, G. (1994) Generalized Markov coarse-graining and spectral decompositions of chaotic piecewise linear maps. Phys. Rev. E 50 988999.Google Scholar
Mandelbrot, B. (1977a) Fractals: form, chance and dimension, Freeman.Google Scholar
Mandelbrot, B. (1977b) The fractal geometry of Nature, Freeman.Google Scholar
Mielke, A. and Zelik, S. (2004) Infinite-dimensional hyperbolic sets and spatio-temporal chaos in reaction-diffusion systems in Rn (preprint available at http://www.iadm.uni-stuttgart.de/LstAnaMod/Mielke).Google Scholar
Murray, J. D. (2002) Mathematical biology, 3rd edition, Springer-Verlag.Google Scholar
Nicholson, C. (2001) Diffusion and related transport mechanisms in brain tissue. Rep. Prog. Phys. 64 815884.Google Scholar
Nicolis, G. and Gaspard, P. (1994) Toward a probabilistic approach to complex systems. Chaos, Solitons and Fractals 4 4157.Google Scholar
Nyquist, H. (1928) Certain topics in telegraph transmission theory. AIEE Trans. 47 617644.Google Scholar
Parry, W. (1981) Topics in ergodic theory, Cambridge University Press.Google Scholar
Poincaré, H. (1892) Les méthodes nouvelles de la mécanique céleste, Gauthiers-Villars, Paris.Google Scholar
Pollicott, M. and Yuri, M. (1998) Dynamical systems and ergodic theory, Cambridge University Press.Google Scholar
Rezakhanlou, F. (1996) Kinetic limits for a class of interacting particle systems. Probab. Theory Related Fields 104 97146.Google Scholar
Ruelle, D. (1986) Resonances of chaotic dynamical systems. Phys. Rev. Lett. 56 405407.Google Scholar
Ruelle, D. and Takens, F. (1971) On the nature of turbulence. Commun. Maths. Phys. 20 167192; Commun. Maths. Phys. 23343344.Google Scholar
Schnakenberg, J. (1976) Network theory of microscopic and macroscopic behavior of master equation systems. Revs. Mod. Phys. 48 571585.Google Scholar
Shannon, C. E. (1948) A mathematical theory of communication The Bell System Technical Journal 27 479–423 and 623656.Google Scholar
Shannon, C. (1949) Communication in the presence of noise. Proceedings of the IRE 37 10–21. (Reprinted in Proceedings of the IEEE 86 447–457 (1998).)Google Scholar
Stauffer, D. and Aharony, A. (1992) Introduction to percolation theory, Taylor and Francis, London.Google Scholar
Taniguchi, T. and Morriss, G. P. (2002) Stepwise structure of Lyapunov spectra for many-particle systems using a random matrix dynamics Phys. Rev. E 65 056202.Google Scholar
Thomas, R. and Kaufman, M. (2001) Multistationarity, the basis of cell differentiation and memory. I. Structural conditions of multistationarity and other non-trivial behaviour. II. Logical analysis of regulatory networks in terms of feedback circuits. Chaos 11 170195.Google Scholar
Turing, A. M. (1950) Computing machinery and intelligence. Mind 59 433560.Google Scholar
Turing, A. M. (1952) The chemical basis of morphogenesis. Phil. Trans. R. Soc. London B 237 37–72. (Reprinted in Saunders, P. T. (ed.) (1992) Collected works of A. M. Turing, vol. 2, North Holland.)Google Scholar
Van Beijeren, H. and Dorfman, J. R. (1995) Lyapunov exponents and Kolmogorov–Sinai entropy for the Lorentz gas at low densities. Phys. Rev. Lett. 74 13191322.Google Scholar
Vicsek, T. (ed.) (2001) Fluctuations and scaling in biology, Oxford University Press.Google Scholar
Werhl, A. (1978) General properties of entropy. Rev. Mod. Phys. 50 221260.Google Scholar
Zaks, M. and Pikovsky, A. (2003) Dynamics at the border of chaos and order. In: Livi, R. and Vulpiani, A. (eds.) The Kolmogorov legacy in physics 61–82.Google Scholar
Zurek, W. H. and Paz, J. P. (1995) Quantum chaos: a decoherent definition. Physica D 83 300308.Google Scholar