Journal of Applied Nonlinear Dynamics
The Thermalization of Quantum Systems
Journal of Applied Nonlinear Dynamics 14(3) (2025) 499--512 | DOI:10.5890/JAND.2025.09.001
L.E. Reichl
Department of Physics, Center for Complex Quantum Systems, University of Texas, Austin, Texas, 78745, USA
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Abstract
The fact that the Earth is becoming hotter poses an existential threat for some life forms on Earth. But what is heat? Surprisingly, the answer to this question is not completely clear. We here explore the history of the quest to understand the microscopic mechanical origins of heat. We also describe what is presently known about the dynamical structure of phase space when heat is present.
References
-
[1]  | Newton, I. (1686), The Principia: The Mathematical Principles of Natural Philosophy, Patristic Publishing: Omaha, NE, USA, 2019: English translation of the Latin language version.
|
-
[2]  | Thompson, B. (1798), Philosophical Transactions of the Royal Society, 102.
|
-
[3]  | Reichl, L.E. (2021), The Transition to Chaos, 3rd Edition, Springer (ISBN 978-3-030-63533-6).
|
-
[4]  | Lighthill, J. (1986), The recently recognized failure of predictability in Newtonian dynamics, Proceedings of the Royal Society of London A, 407, 35-50.
|
-
[5]  | Sinai, Y.G. (1963), Some remarks on the spectral properties of ergodic dynamical systems, Russian Mathematical Surveys, 18, 37.
|
-
[6]  | Voros, A. (1976), Semi-classical approximations, Annales de l’Institut Henri Poincaré, 24, 31-90.
|
-
[7]  | Berry, M.V. (1977), Regular and irregular semiclassical wavefunctions, Journal of Physics A: Mathematical and General, 10, 2083-2091.
|
-
[8]  | Berry, M.V. (1977), Semi-classical mechanics in phase space: A study of Wigner's function, Philosophical Transactions of the Royal Society of London, 287, 237-271.
|
-
[9]  | Srednicki, M. (1994), Chaos and quantum thermalization, Physical Review E, 50, 888-901.
|
-
[10]  | McDonald, S.W. and Kaufman, A.N. (1979), Spectrum and eigenfunctions for a Hamiltonian with stochastic trajectories, Physical Review Letters, 42, 1189-1191.
|
-
[11]  | Casati, G., Vals-Gris, F., and Guarneri, I. (1980), On the connection between quantization of nonintegrable systems and statistical theory of spectra, Lettere al Nuovo Cimento, 28, 279-282.
|
-
[12]  | Bogomolny, E.B. (1988), Smoothed wave functions of chaotic quantum systems, Physica D: Nonlinear Phenomena, 31, 169-189.
|
-
[13]  | Stockman, H.J. and Stein, J. (1990), ``Quantum" chaos in billiards studied by microwave absorption, Physical Review Letters, 64, 2215-2219.
|
-
[14]  | Graf, H.D., Harney, H.L., Lengeler, H., Lewenkopf, C.H., Rangacharyulu, C., Richter, A., Schardt P., and Weidenmuller H.A. (1992), Distribution of eigenmodes in a superconducting stadium billiard with chaotic dynamics, Physical Review Letters, 69, 1296-1299.
|
-
[15]  | Stein J. and Stockman H.J. (1992), Experimental determination of billiard wave functions, Physical Review Letters, 68, 2867-2870.
|
-
[16]  | Luna-Acosta G.A., Na K., Reichl L.E., and Khrokin A. (1996), Band structure and quantum Poincaré sections of a classically chaotic quantum rippled channel, Physical Review E, 53, 3271-3283.
|
-
[17]  | Akguc A. and Reichl L.E. (2000), Conductance and statistical properties of chaotic and integrable electron waveguides, Journal of Statistical Physics, 98, 813-834.
|
-
[18]  | Li, W., Reichl, L.E., and Wu, B. (2002), Quantum chaos in the ripple billiard, Physical Review E, 65, 56220(1-9).
|
-
[19]  | Bunimovich, L.A. (1979), On the ergodic properties of nowhere dispersing billiards, Communications in Mathematical Physics, 65, 295-312.
|
-
[20]  | Ree, S. and Reichl, L.E. (1999), Classical and quantum chaos in a circular billiard with straight cut, Physical Review E, 60, 1607-1615.
|
-
[21]  | Makino, H., Harayama, T., and Aizawa, Y. (2001), Quantum-classical correspondences of the Berry-Robnik parameter through bifurcations in lemon billiard systems, Physical Review E, 63, 58203(1-11).
|
-
[22]  | Redding, B., Cerjan, A., Huang, X., Lee, M.L., Stone, A.D., Choma, M.A., and Cao, H. (2015), Low spatial coherence electrically pumped semiconductor laser for speckle-free full-field imaging, Proceedings of the National Academy of Sciences, 112, 1304-1309.
|
-
[23]  | Reichl, L.E. (2024), Chaos-assisted tunneling, Entropy, 26, 144(1-16).
|
-
[24]  | Kim, M.W., Rim, S., Yi, C.H., and Kim, C.H. (2013), Chaos-assisted tunneling in a deformed microcavity laser, Optics Express, 21, 32508-32515.
|
-
[25]  | Xiao, Y.F., Jiang, X.F., Yang, Q.F., Wang, L., Shi, K., Li, Y., and Gong, Q. (2013), Tunneling-induced transparency in a chaotic microcavity, Laser and Photonics Reviews, 7, L51-L54.
|
-
[26]  | Qian, Y.J., Liu, H., Cao, Q.T., Kullig, J., Rong, K., Qui, C.W., Wiersig J., Gong Q., Chen J., and Xiao Y.F. (2021), Regulated photon transport in chaotic microcavities by tailoring phase space, Physical Review Letters, 127, 273902(1-6).
|
-
[27]  | Wang, S., Liu, Y., Xiao, S., Wang, Z., Fan, Y., Han, J., Ge, L., and Song, Q. (2021), Direct observation of chaotic resonances in optical microcavities, Light: Science $\&$ Applications, 10, 135.
|
-
[28]  | Ermann, L., Vergini, E., and Shepelyanski, D.L. (2015), Dynamical thermalization of Bose-Einstein condensate in Bunimovich stadiums, Europhysics Letters, 111, 1-6.
|
-
[29]  | Ermann, L., Vergini, E., and Shepelyanski, D.L. (2016), Dynamics and thermalization of a Bose-Einstein condensate in a Sinai-oscillator trap, Physical Review A, 94, 1-10, 013618.
|
-
[30]  | Porter, M.D., Barr, A.D., Barr, A.D., and Reichl, L.E. (2017), Chaos in the band structure of a soft Sinai lattice, Physical Review E, 95, 1-11, 052213.
|
-
[31]  | Barr, A.D., Porter, M.D., and Reichl, L.E. (2017), Signatures of chaos in the Brillouin zone, Chaos: An Interdisciplinary Journal of Nonlinear Science, 27, 1-9, 104604.
|
-
[32]  | Kolmogorov, A.N. (1954), The general theory of dynamical systems, Dokl. Akad. Nauk SSSR, 98, 527 (English translation in R. Abraham, Foundations of Mechanics, W.A. Benjamin: New York, 1997, Appendix D).
|
-
[33]  | Moser, J. (1958), New aspects in the theory of stability of Hamiltonian systems, Communications on Pure and Applied Mathematics, 11, 81-114.
|
-
[34]  | Arnold, V.I. (1963), Small denominators and problems on the stability of motions in classical and celestial mechanics, Russian Mathematical Surveys, 18, 91-192.
|
-
[35]  | Escande, D.F. and Doveil, F. (1981), Renormalization method for computing the threshold of the large-scale stochastic instability in two degrees of freedom Hamiltonian systems, Journal of Statistical Physics, 26, 257-284.
|
-
[36]  | Escande, D.F. (1985), Stochasticity in classical Hamiltonian systems: Universal aspects, Physics Reports, 121, 165-261.
|
-
[37]  | Reichl, L.E. (1989), Mechanism for the extension of the wave function in quantum dynamics, Physical Review A, 39, 4817-4827.
|
-
[38]  | Reichl, L.E. and Haoming, L. (1990), Self-similarity in quantum dynamics, Physical Review A, 42, 4543-4561.
|
-
[39]  | Zaslavsky, G.M. (1981), Stochasticity in quantum dynamics, Physics Reports, 80, 157-250.
|
-
[40]  | Nekhoroshev, N.N. (1977), An exponential estimate of the time of stability in nearly integrable Hamiltonian systems, Russian Mathematical Surveys, 32(1), 1-65.
|
-
[41]  | Froeschle, C., Guzzo, M., and Lega, E. (2000), Graphical evolution of the Arnold web: From order to chaos, Science, 289, 2108-2110.
|
-
[42]  | Boretz, Y. and Reichl, L.E. (2016), Arnold diffusion in a driven optical lattice, Physical Review E, 93, 1-13, 032214.
|
-
[43]  | Reichl, L.E. (2025), The effect of resonance on qudit dynamics, Physica B: Condensed Matter, 696, 1-9, 416642.
|