Many macroscopic dynamical phenomena, for example in hydrodynamics and oscillatory chemical reactions, have been observed to display erratic or random time evolution, in spite of the deterministic character of their dynamics—a phenomenon known as macroscopic chaos1,2,3,4,5. On the other hand, it has been long supposed that the existence of chaotic behaviour in the microscopic motions of atoms and molecules in fluids or solids is responsible for their equilibrium and non-equilibrium properties. But this hypothesis of microscopic chaos has never been verified experimentally. Chaotic behaviour of a system is characterized by the existence of positive Lyapunov exponents, which determine the rate of exponential separation of very close trajectories in the phase space of the system6. Positive Lyapunov exponents indicate that the microscopic dynamics of the system are very sensitive to its initial state, which, in turn, indicates that the dynamics are chaotic; a small change in initial conditions will lead to a large change in the microscopic motion. Here we report direct experimental evidence for microscopic chaos in fluid systems, obtained by the observation of brownian motion of a colloidal particle suspended in water. We find a positive lower bound on the sum of positive Lyapunov exponents of the system composed of the brownian particle and the surrounding fluid.
Subscribe to Journal
Get full journal access for 1 year
only $3.90 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Rent or Buy article
Get time limited or full article access on ReadCube.
All prices are NET prices.
Eckmann, J.-P., Oliffson Kamphorst, S., Ruelle, D. & Ciliberto, S. Liapunov exponents from time series. Phys. Rev. A 34, 4971–4979 (1986).
Roux, J.-C., Simoyi, R. H. & Swinney, H. L. Observation of a strange attractor. Physica D 8, 257–266 (1983).
Wisdom, J., Peale, S. J. & Mignard, F. The chaotic rotation of Hyperion. Icarus 58, 137–152 (1984).
Klavetter, J. J. Rotation of Hyperion. I. Observations. Astron. J. 97, 570–579 (1989).
Laskar, J. Anumerical experiment on the chaotic behaviour of the Solar System. Nature 338, 237–238 (1989).
Eckmann, J.-P. & Ruelle, D. Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656 (1985).
Livi, R., Politi, A. & Ruffo, S. Distribution of characteristic exponents in the thermodynamic limit. J.Phys. A 19, 2033–2040 (1986).
Posch, H. A. & Hoover, W. G. Lyapunov instability of dense Lennard-Jones fluids. Phys. Rev. A 38, 473–482 (1988); Equilibrium and nonequilibrium Lyapunov spectra for dense fluids and solids. Phys. Rev. A 39, 2175–2188 (1989).
Dellago, Ch., Posch, H. A. & Hoover, W. G. Lyapunov instability in a system of hard disks in equilibrium and nonequilibrium steady states. Phys. Rev. E 53, 1485–1501 (1996).
Gaspard, P. Chaos, Scattering Theory, and Statistical Mechanics (Cambridge Univ. Press, (1998)).
Krylov, N. Relaxation processes in statistical systems. Nature 153, 709–710 (1944).
Bunimovich, L. A. & Sinai, Ya. G. Statistical properties of Lorentz gas with periodic configuration of scatterers. Commun. Math. Phys. 78, 479–497 (1981).
Evans, D. J., Cohen, E. G. D. & Morriss, G. P. Viscosity of a simple fluid from its maximal Lyapunov exponents. Phys. Rev. A 42, 5990–5997 (1990).
Gaspard, P. & Nicolis, G. Transport properties, Lyapunov exponents, and entropy per unit time. Phys. Rev. Lett. 65, 1693–1696 (1990).
Dorfman, J. R. & Gaspard, P. Chaotic scattering theory of transport and reaction-rate coefficients. Phys. Rev. E 51, 28–35 (1995).
Ruelle, D. Positivity of entropy production in nonequilbirium statistical mechanics. J. Stat. Phys. 85, 1–23 (1996).
Chernov, N. I., Eyink, G. L., Lebowitz, J. L. & Sinai, Ya. G. Derivation of Ohm's law in a deterministic mechanical model. Phys. Rev. Lett. 70, 2209–2212 (1993).
Gallavotti, G. & Cohen, E. G. D. Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 74, 2694–2697 (1995); Dynamical ensembles in stationary states. J. Stat. Phys. 80, 931–970 (1995).
Gaspard, P. Can we observe microscopic chaos in the laboratory? Adv. Chem. Phys. XCIX, 369–392 (1997).
Gaspard, P. & Wang, X.-J. Noise, chaos, and (ε, τ)-entropy per unit time. Phys. Rep. 235, 291–345 (1993).
Chaitin, G. J. Algorithm Information Theory (Cambridge Univ. Press, (1987)).
Kolmogorov, A. N. Combinatorial foundations of information theory and the calculus of probabilities. Russ. Math. Survey 38, 29–40 (1983).
Grassberger, P. & Procaccia, I. Estimation of the Kolmogorov entropy from a chaotic signal. Phys. Rev. A 28, 2591–2593 (1983).
Cohen, A. & Procaccia, I. Computing the Kolmogorov entropy from time signals of dissipative and conservative dynamical systems. Phys. Rev. A 31, 1872–1882 (1985).
Ott, E. Chaos in Dynamical Systems (Cambridge Univ. Press, (1993)).
Kolmogorov, A. N. On the Shannon theory of information transmission in the case of continuous signals. IRE Trans. Information Theory IT-2, 102–108 (1956).
Berger, T. Information rates of Wiener processes. IEEE Trans. Information Theory IT-16, 134–139 (1970).
Tél, T., Gaspard, P. & Nicolis, G. (eds) Chaos 8, (Focus Issue on Chaos and Irreversibility) (1998).
van Beijeren, H. Transport properties of stochastic Lorentz models. Rev. Mod. Phys. 54, 195–234 (1982).
Grier, D. G. & Murray, C. A. in Structure and Dynamics of Strongly Interacting Colloids and Supramolecular Aggregates in Solution (eds Chen, S. H., Huang, J. S. & Tartaglia, P.) 145 (NATO ASI Ser. C, 369, Kluwer, Boston, (1991)).
Schaertl, W. & Sillescu, H. Dynamics of colloidal hard spheres in thin aqueous suspension layers. J.Colloid Interface Sci. 155, 313–318 (1993).
P.G. was supported by the National Fund for Scientific Research (FNRS Belgium); J.R.D. thanks T. Gilbert for helpful remarks, and the US NSF for support; M.E.B. thanks the Physics Department at the University of Utah for support. This work was supported in part by the IUAP/PAI program of the Belgium Government and by the Banque Nationale de Belgique.
About this article
Cite this article
Gaspard, P., Briggs, M., Francis, M. et al. Experimental evidence for microscopic chaos. Nature 394, 865–868 (1998). https://doi.org/10.1038/29721
Spatiotemporal spread of perturbations in a driven dissipative Duffing chain: An out-of-time-ordered correlator approach
Physical Review E (2020)
Chaos: An Interdisciplinary Journal of Nonlinear Science (2020)
Rapid onset of molecular friction in liquids bridging between the atomistic and hydrodynamic pictures
Communications Physics (2020)
Mathematical Methods in the Applied Sciences (2020)
Optics Express (2020)