Ergodicity of One-dimensional Systems Coupled to the Logistic Thermostat

• Autorzy: Tapias D. , Bravetti A. , Sanders D. P.

Identyfikatory

DOI

10.12921/cmst.2016.0000061

• Języki publikacji: EN

Abstrakty

EN

We analyze the ergodicity of three one-dimensional Hamiltonian systems, with harmonic, quartic and Mexican-hat potentials, coupled to the logistic thermostat. As criteria for ergodicity we employ: the independence of the Lyapunov spectrum with respect to initial conditions; the absence of visual “holes” in two-dimensional Poincaré sections; the agreement between the histograms in each variable and the theoretical marginal distributions; and the convergence of the global joint distribution to the theoretical one, as measured by the Hellinger distance. Taking a large number of random initial conditions, for certain parameter values of the thermostat we find no indication of regular trajectories and show that the time distribution converges to the ensemble one for an arbitrarily long trajectory for all the systems considered. Our results thus provide a robust numerical indication that the logistic thermostat can serve as a single one-parameter thermostat for stiff one-dimensional systems.

Słowa kluczowe

EN

logistic thermostat; ergodicity; Gibbs’ ensemble

• Wydawca: Institute of Bioorganic Chemistry Scientific Publishers OWN, Polish Academy of Sciences
• Czasopismo: Computational Methods in Science and Technology
• Rocznik: 2017
• Tom: Vol. 23, No. 1
• Strony: 11--18
• Opis fizyczny: Bibliogr. 34 poz., rys.

Twórcy

autor

Tapias D.

• Departamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma de México Ciudad Universitaria, Ciudad de México 04510, México

autor

Bravetti A.

• Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Ciudad Universitaria, Ciudad de México 04510, México

autor

Sanders D. P.

• Departamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma de México Ciudad Universitaria, Ciudad de México 04510, México
• Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology 77 Massachusetts Avenue, Cambridge, MA 02139, USA

Bibliografia

• [1] S. Nosé, A molecular dynamics method for simulations in the canonical ensemble, Molecular Physics 52, 255-268 (1984).
• [2] W. G. Hoover, Canonical dynamics: equilibrium phase-space distributions, Physical Review A31(3), 1695 (1985).
• [3] C.R. de Oliveira, T. Werlang, Ergodic hypothesis in classical statistical mechanics, Revista Brasileira de Ensino de Física 29(2), 189-201, (2007).
• [4] I. Aleksandr, A. Khinchin, Mathematical foundations of statistical mechanics, Courier Corporation, 1949.
• [5] D. Kusnezov, A. Bulgac,W. Bauer, Canonical ensembles from chaos, Annals of Physics 204(1), 155-185 (1990).
• [6] D. Kusnezov, A. Bulgac, Canonical ensembles from chaos II: Constrained dynamical systems, Annals of Physics 214(1), 180-218 (1992).
• [7] G.J. Martyna, M.L. Klein, M. Tuckerman, Nosé-Hoover chains: The canonical ensemble via continuous dynamics, The Journal of Chemical Physics 97(4), 2635-2643 (1992).
• [8] Wm.G. Hoover, B.L. Holian, Kinetic moments method for the canonical ensemble distribution, Physics Letters A211(5), 253-257 (1996).
• [9] A.C. Brańka, M. Kowalik, K.W. Wojciechowski, Generalization of the Nosé-Hoover approach, The Journal of Chemical Physics 119(4), 1929-1936 (2003).
• [10] A. Sergi, G.S. Ezra, Bulgac-Kusnezov-Nosé-Hoover thermostats, Physical Review E81(3), 036705 (2010).
• [11] Wm.G. Hoover, C.G. Hoover, J. Clinton Sprott, Nonequilibrium systems: hard disks and harmonic oscillators near and far from equilibrium, Molecular Simulation 42(16), 1300-1316 (2016).
• [12] J.D. Ramshaw, General formalism for singly thermostated Hamiltonian dynamics, Physical Review E92(5), 052138 (2015).
• [13] Wm.G. Hoover, C.G. Hoover, Singly-Thermostated Ergodicity in Gibbs’ Canonical Ensemble and the 2016 Ian Snook Prize, arXiv preprint arXiv:1607.04595 (2016).
• [14] I. Fukuda, H. Nakamura, Tsallis dynamics using the Nosé-Hoover approach, Physical Review E65(2), 026105 (2002).
• [15] A. Bravetti and D. Tapias, Thermostat algorithm for generating target ensembles, Physical Review E 93, 022139 (2016).
• [16] D. Tapias, D.P. Sanders, A. Bravetti, Geometric integrator for simulations in the canonical ensemble, The Journal of Chemical Physics 145(8) (2016).
• [17] A. Bravetti and D. Tapias, Liouville’s theorem and the canonical measure for nonconservative systems from contact geometry, Journal of Physics A: Mathematical and Theoretical 48(24), 245001 (2015).
• [18] Wm.G. Hoover, J. Clinton Sprott, C.G. Hoover, Ergodicity of a singly-thermostated harmonic oscillator, Communications in Nonlinear Science and Numerical Simulation 32, 234-240 (2016).
• [19] P.H. Hünenberger, Thermostat algorithms for molecular dynamics simulations, In Advanced computer simulation, pages 105-149 Springer, 2005.
• [20] P.K. Patra, B. Bhattacharya, An ergodic configurational thermostat using selective control of higher order temperatures, The Journal of Chemical Physics 142(19), 194103 (2015).
• [21] P.K. Patra, J. Clinton Sprott, Wm.G. Hoover, C.G. Hoover, Deterministic time-reversible thermostats: chaos, ergodicity, and the zeroth law of thermodynamics, Molecular Physics 113(17-18), 2863-2872 (2015).
• [22] B. Leimkuhler, Generalized Bulgac-Kusnezov methods for sampling of the Gibbs-Boltzmann measure, Physical Review E81(2), 026703 (2010).
• [23] B. Leimkuhler, E. Noorizadeh, F. Theil, A gentle stochastic thermostat for molecular dynamics, Journal of Statistical Physics 135(2), 261-277 (2009).
• [24] P.K. Patra, B. Bhattacharya, Nonergodicity of the Nosé-Hoover chain thermostat in computationally achievable time, Physical Review E90(4), 043304 (2014).
• [25] A. Basu, H. Shioya, C. Park, Statistical inference: the minimum distance approach, CRC Press, 2011.
• [26] Ch. Skokos, The Lyapunov characteristic exponents and their computation, In Dynamics of Small Solar System Bodies and Exoplanets, pages 63-135 Springer, 2010.
• [27] G. Bennetin, L. Galgani, A. Giorgilli, J.M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them, Meccanica 15(9) (1980).
• [28] https://github.com/dapias/ThermostattedDynamics.jl. [29] J.A.G. Roberts, G.R.W. Quispel, Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems, Physics Reports 216(2), 63-177 (1992).
• [30] J.S.W. Lamb, J.A.G. Roberts, Time-reversal symmetry in dynamical systems: a survey, Physica D: Nonlinear Phenomena 112(1), 1-39 (1998).
• [31] H.A. Posch, Wm.G. Hoover, F.J. Vesely, Canonical dynamics of the Nosé oscillator: stability, order, and chaos, Physical Review A33(6), 4253 (1986).
• [32] L.Wang. X.-S. Yang, The invariant tori of knot type and the interlinked invariant tori in the Nosé-Hoover oscillator, The European Physical Journal B88(3), 1-5 (2015).
• [33] L. Wang, X.-S. Yang, A vast amount of various invariant tori in the Nosé-Hoover oscillator, Chaos: An Interdisciplinary Journal of Nonlinear Science 25(12), 123110 (2015).
• [34] C. Liverani, M.P. Wojtkowski, Ergodicity in Hamiltonian systems, In Dynamics reported, pages 130-202 Springer, 1995.

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).