Ergodicity of One-dimensional Systems Coupled to the Logistic Thermostat
Tapias Diego 1, Bravetti Alessandro 2, Sanders David 3,4
1 Departamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma de México
Ciudad Universitaria, Ciudad de México 04510, México
E-mail: diego.tapias@nucleares.unam.mx2 Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México,
Ciudad Universitaria, Ciudad de México 04510, México
E-mail: alessandro.bravetti@iimas.unam.mx3 Departamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma de México
Ciudad Universitaria, Ciudad de México 04510, México
E-mail: dpsanders@ciencias.unam.mx4 Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology
77 Massachusetts Avenue, Cambridge, MA 02139, USA
Received:
Received: 04 December 2016; revised: 10 January 2017; accepted: 11 January 2017; published online: 31 January 2017
DOI: 10.12921/cmst.2016.0000061
Abstract:
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.
Key words:
ergodicity, Gibbs’ ensemble, Ian Snook Prize, logistic thermostat
References:
[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 sta-
tistical 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, Generaliza-
tion 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 ther-
mostats, Physical Review E81(3), 036705 (2010).
[11] Wm.G. Hoover, C.G. Hoover, J. Clinton Sprott, Nonequilib-
rium 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 Ergod-
icity 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 generat-
ing 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 canon-
ical measure for nonconservative systems from contact ge-
ometry, 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 dy-
namics simulations, In Advanced computer simulation, pages
105-149 Springer, 2005.
[20] P.K. Patra, B. Bhattacharya, An ergodic configurational ther-
mostat 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 stochas-
tic 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 mini-
mum 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 dy-
namical 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 sys-
tems, In Dynamics reported, pages 130-202 Springer, 1995.
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.
Key words:
ergodicity, Gibbs’ ensemble, Ian Snook Prize, logistic thermostat
References:
[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 sta-
tistical 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, Generaliza-
tion 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 ther-
mostats, Physical Review E81(3), 036705 (2010).
[11] Wm.G. Hoover, C.G. Hoover, J. Clinton Sprott, Nonequilib-
rium 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 Ergod-
icity 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 generat-
ing 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 canon-
ical measure for nonconservative systems from contact ge-
ometry, 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 dy-
namics simulations, In Advanced computer simulation, pages
105-149 Springer, 2005.
[20] P.K. Patra, B. Bhattacharya, An ergodic configurational ther-
mostat 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 stochas-
tic 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 mini-
mum 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 dy-
namical 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 sys-
tems, In Dynamics reported, pages 130-202 Springer, 1995.
