Symmetry, Chaos and Temperature in the One-dimensional Lattice φ4 Theory
Research and Education Center for Natural Sciences
and Hiyoshi Departament of Physics
Keio University, Yokohama 223–8521, Japan
E-mail: ken@phys-h.keio.ac.jp
Received:
Received: 31 December 2017; accepted: 05 April 2018; published online: 30 May 2018
DOI: 10.12921/cmst.2017.0000055
Abstract:
The symmetries of the minimal φ4 theory on the lattice are systematically analyzed. We find that symmetry can restrict trajectories to subspaces, while their motions are still chaotic. The chaotic dynamics of autonomous Hamiltonian systems are discussed in relation to the thermodynamic laws. Possibilities of configurations with non-equal ideal gas temperatures in the steady state in Hamiltonian systems, are investigated, and examples of small systems in which the ideal gas temperatures are different within the system are found. The pairing of local (finite-time) Lyapunov exponents are analyzed, and their dependence on various factors, such as the energy of the system, the characteristics of the initial conditions are studied and discussed. We find that for the φ4 theory, higher energies lead to faster pairing times. We also find that symmetries can impede the pairing of local Lyapunov exponents and the convergence of Lyapunov exponents.
Key words:
Lyapunov exponents, second law of thermodynamics, symmetries in chaotic dynamics
References:
[1] R.C. Hilborn, Chaos and nonlinear dynamics”, Oxford Univer-
sity Press (New York, 1994).
[2] M. Tabor, Chaos and Integrability in Nonlinear Dynamics, John
Wiley & Sons (New York, 1989).
[3] W.G. Hoover, C.G. Hoover, Simulation and Control of Chaotic
Nonequilibrium Systems, World Scientific Publishing Company
(Singapore, 2015), and references therein.
[4] Wm. G. Hoover, C. G. Hoover, Instantaneous Pairing of Lya-
punov Exponents in Chaotic Hamiltonian Dynamics and the
2017 Ian Snook Prizes, Computational Methods in Science and
Technology 23, 73 (2017).
[5] K. Aoki, D. Kusnezov, Bulk properties of anharmonic chains in
strong thermal gradients: non-equilibrium φ4 theory, Phys. Lett.
A 265, 250 (2000); K. Aoki, D. Kusnezov, Non-Equilibrium
Statistical Mechanics of Classical Lattice φ4 Field Theory, Ann.
Phys. 295, 50 (2002).
[6] B. Hu, B. Li, H. Zhao, Heat conduction in one-dimensional
nonintegrable systems, Phys. Rev. E 61, 3828 (2000).
[7] W.G. Hoover, K. Aoki, Order and chaos in the one-dimensional
φ4 model: N -dependence and the Second Law of Thermody-
namics, Commun. Nonlinear. Sci. Numer. Simulat. 49, 192
(2017).
[8] Y. Ohnuki, S. Kamefuchi, Some General Properties of Para-
Fermi Field Theory, Phys. Rev. 170, 1279 (1968).
[9] F. Wilczek, Quantum Mechanics of Fractional-Spin Particles,
Phys. Rev. Lett. 49, 957 (1982).
[10] L.Dixon, J.A.Harvey, C.Vafa, E.Witten, Strings on Orbifolds,
Nuc. Phys. B 261, 678 (1985).
[11] G. ’t Hooft, A Property of Electric and Magnetic Flux in Non-
abelian Gauge Theories, Nucl. Phys. B 153, 141 (1979).
[12] R. Rosenberg, On Nonlinear Vibrations of Systems with Many
Degrees of Freedom, Adv. Appl. Mech., 9, 155 (1966).
[13] S.W. Shaw, C. Pierre, Non-linear normal modes and invariant
manifolds, Journal of Sound and Vibration 150, 170 (1991),
Normal Modes for Non-Linear Vibratory Systems, ibid. 164, 85
(1993).
[14] K.W. Sandusky, J.B. Page, Interrelation between the stability of
extended normal modes and the existence of intrinsic localized
modes in nonlinear lattices with realistic potentials, Phys. Rev.
B 50, 866 (1994).
[15] S. Flach, Tangent bifurcation of band edge plane waves, dynam-
ical symmetry breaking and vibrational localization, Physica D
91, 223 (1996).
[16] A.F. Vakakis, Non-linear normal modes (NNMs) and their ap-
plications in vibration theory: an overview, Mech. Sys. Sig.
Proc. 11, 3 (1997); Y.V. Mikhlin and K.V. Avramov, Nonlin-
ear Normal Modes for Vibrating Mechanical Systems. Review
of Theoretical Developments, Appl. Mech. Rev 63, 060802
(2011).
[17] T. Bountis, G. Chechin and V. Sakhnenko, Discrete symmetry
and stability in hamiltonian dynamics, Int. J. Bif. Chaos 21,
1539 (2011).
[18] K. Aoki, Stable and unstable periodic orbits in the one-
dimensional lattice φ4 theory, Phys. Rev. E94, 042209 (2016).
[19] S. Nosé, J. Chem. Phys. 81, 511 (1984); A unified formula-
tion of the constant temperature molecular dynamics methods,
Mol. Phys. 52, 255 (1984).
[20] W. G. Hoover, Canonical dynamics: Equilibrium phase-space
distributions, Phys. Rev. A 31,1695 (1985).
[21] L. D. Landau, E. M. Lifshitz, Statistical Physics, Butterworth-
Heinemann (Burlington, 1980).
[22] J. Delhommelle and D. J. Evans, Configurational temperature
thermostat for fluids undergoing shear flow: applicationliquid chlorine, Mol. Phys. 99, 1825 (2001).
[23] K.P. Travis, C. Braga, Configurational temperature control for
atomic and molecular systems, J. Chem. Phys. 128, 014111
(2008).
[24] F. Reif, Fundamentals of Statistical and Thermal Physics, Wave-
land Press (Long Grove, 2009).
[25] K. Aoki, D. Kusnezov, Violations of Local Equilibrium and
Linear Response in Classical Lattice Systems, Phys. Lett. A309,
377 (2003).
[26] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Nu-
merical Recipes: The Art of Scientific Computing, 3rd Edition,
Cambridge University Press (New York, 2007).
[27] Boost libraries, http://www.boost.org/.
[28] I. Shimada, T. Nagashima, A Numerical Approach to Ergodic
Problem of Dissipative Dynamical Systems, Prog. Theor. Phys.
61, 1605 (1979).
[29] G. Benettin, L. Galgani, A. Giorgilli and J. Strelcyn, Lyapunov
Characteristic Exponents for smooth dynamical systems and for
hamiltonian systems; a method for computing all of them. Part
1: Theory Meccanica 15, 9 (1980); Lyapunov Characteristic
Exponents for smooth dynamical systems and for hamiltonian
systems; A method for computing all of them. Part 2: Numerical
application ibid. 15, 21 (1980).
[30] H.A. Posch, W.G. Hoover, Lyapunov instability of dense
Lennard-Jones fluids, Phys. Rev. A38, 473 (1988).
[31] We thank the referee for the insight on this point.
[32] W.G. Hoover, private communication.
[33] V. Oseledets, A multiplicative ergodic theorem. Characteristic
Lyapunov, exponents of dynamical systems, Tr. Mosk. Mat. Obs.
19, 179 (1968).
[34] D. Ruelle, Ergodic theory of differentiable dynamical systems,
Publ. Math. IHES 50, 275 (1979).
[35] F. Ginelli, H. Chaté, R. Livi, A. Politi, Covariant Lyapunov
vectors, J. Phys. A: Math. Theor. 46, 254005 (2013).
The symmetries of the minimal φ4 theory on the lattice are systematically analyzed. We find that symmetry can restrict trajectories to subspaces, while their motions are still chaotic. The chaotic dynamics of autonomous Hamiltonian systems are discussed in relation to the thermodynamic laws. Possibilities of configurations with non-equal ideal gas temperatures in the steady state in Hamiltonian systems, are investigated, and examples of small systems in which the ideal gas temperatures are different within the system are found. The pairing of local (finite-time) Lyapunov exponents are analyzed, and their dependence on various factors, such as the energy of the system, the characteristics of the initial conditions are studied and discussed. We find that for the φ4 theory, higher energies lead to faster pairing times. We also find that symmetries can impede the pairing of local Lyapunov exponents and the convergence of Lyapunov exponents.
Key words:
Lyapunov exponents, second law of thermodynamics, symmetries in chaotic dynamics
References:
[1] R.C. Hilborn, Chaos and nonlinear dynamics”, Oxford Univer-
sity Press (New York, 1994).
[2] M. Tabor, Chaos and Integrability in Nonlinear Dynamics, John
Wiley & Sons (New York, 1989).
[3] W.G. Hoover, C.G. Hoover, Simulation and Control of Chaotic
Nonequilibrium Systems, World Scientific Publishing Company
(Singapore, 2015), and references therein.
[4] Wm. G. Hoover, C. G. Hoover, Instantaneous Pairing of Lya-
punov Exponents in Chaotic Hamiltonian Dynamics and the
2017 Ian Snook Prizes, Computational Methods in Science and
Technology 23, 73 (2017).
[5] K. Aoki, D. Kusnezov, Bulk properties of anharmonic chains in
strong thermal gradients: non-equilibrium φ4 theory, Phys. Lett.
A 265, 250 (2000); K. Aoki, D. Kusnezov, Non-Equilibrium
Statistical Mechanics of Classical Lattice φ4 Field Theory, Ann.
Phys. 295, 50 (2002).
[6] B. Hu, B. Li, H. Zhao, Heat conduction in one-dimensional
nonintegrable systems, Phys. Rev. E 61, 3828 (2000).
[7] W.G. Hoover, K. Aoki, Order and chaos in the one-dimensional
φ4 model: N -dependence and the Second Law of Thermody-
namics, Commun. Nonlinear. Sci. Numer. Simulat. 49, 192
(2017).
[8] Y. Ohnuki, S. Kamefuchi, Some General Properties of Para-
Fermi Field Theory, Phys. Rev. 170, 1279 (1968).
[9] F. Wilczek, Quantum Mechanics of Fractional-Spin Particles,
Phys. Rev. Lett. 49, 957 (1982).
[10] L.Dixon, J.A.Harvey, C.Vafa, E.Witten, Strings on Orbifolds,
Nuc. Phys. B 261, 678 (1985).
[11] G. ’t Hooft, A Property of Electric and Magnetic Flux in Non-
abelian Gauge Theories, Nucl. Phys. B 153, 141 (1979).
[12] R. Rosenberg, On Nonlinear Vibrations of Systems with Many
Degrees of Freedom, Adv. Appl. Mech., 9, 155 (1966).
[13] S.W. Shaw, C. Pierre, Non-linear normal modes and invariant
manifolds, Journal of Sound and Vibration 150, 170 (1991),
Normal Modes for Non-Linear Vibratory Systems, ibid. 164, 85
(1993).
[14] K.W. Sandusky, J.B. Page, Interrelation between the stability of
extended normal modes and the existence of intrinsic localized
modes in nonlinear lattices with realistic potentials, Phys. Rev.
B 50, 866 (1994).
[15] S. Flach, Tangent bifurcation of band edge plane waves, dynam-
ical symmetry breaking and vibrational localization, Physica D
91, 223 (1996).
[16] A.F. Vakakis, Non-linear normal modes (NNMs) and their ap-
plications in vibration theory: an overview, Mech. Sys. Sig.
Proc. 11, 3 (1997); Y.V. Mikhlin and K.V. Avramov, Nonlin-
ear Normal Modes for Vibrating Mechanical Systems. Review
of Theoretical Developments, Appl. Mech. Rev 63, 060802
(2011).
[17] T. Bountis, G. Chechin and V. Sakhnenko, Discrete symmetry
and stability in hamiltonian dynamics, Int. J. Bif. Chaos 21,
1539 (2011).
[18] K. Aoki, Stable and unstable periodic orbits in the one-
dimensional lattice φ4 theory, Phys. Rev. E94, 042209 (2016).
[19] S. Nosé, J. Chem. Phys. 81, 511 (1984); A unified formula-
tion of the constant temperature molecular dynamics methods,
Mol. Phys. 52, 255 (1984).
[20] W. G. Hoover, Canonical dynamics: Equilibrium phase-space
distributions, Phys. Rev. A 31,1695 (1985).
[21] L. D. Landau, E. M. Lifshitz, Statistical Physics, Butterworth-
Heinemann (Burlington, 1980).
[22] J. Delhommelle and D. J. Evans, Configurational temperature
thermostat for fluids undergoing shear flow: applicationliquid chlorine, Mol. Phys. 99, 1825 (2001).
[23] K.P. Travis, C. Braga, Configurational temperature control for
atomic and molecular systems, J. Chem. Phys. 128, 014111
(2008).
[24] F. Reif, Fundamentals of Statistical and Thermal Physics, Wave-
land Press (Long Grove, 2009).
[25] K. Aoki, D. Kusnezov, Violations of Local Equilibrium and
Linear Response in Classical Lattice Systems, Phys. Lett. A309,
377 (2003).
[26] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Nu-
merical Recipes: The Art of Scientific Computing, 3rd Edition,
Cambridge University Press (New York, 2007).
[27] Boost libraries, http://www.boost.org/.
[28] I. Shimada, T. Nagashima, A Numerical Approach to Ergodic
Problem of Dissipative Dynamical Systems, Prog. Theor. Phys.
61, 1605 (1979).
[29] G. Benettin, L. Galgani, A. Giorgilli and J. Strelcyn, Lyapunov
Characteristic Exponents for smooth dynamical systems and for
hamiltonian systems; a method for computing all of them. Part
1: Theory Meccanica 15, 9 (1980); Lyapunov Characteristic
Exponents for smooth dynamical systems and for hamiltonian
systems; A method for computing all of them. Part 2: Numerical
application ibid. 15, 21 (1980).
[30] H.A. Posch, W.G. Hoover, Lyapunov instability of dense
Lennard-Jones fluids, Phys. Rev. A38, 473 (1988).
[31] We thank the referee for the insight on this point.
[32] W.G. Hoover, private communication.
[33] V. Oseledets, A multiplicative ergodic theorem. Characteristic
Lyapunov, exponents of dynamical systems, Tr. Mosk. Mat. Obs.
19, 179 (1968).
[34] D. Ruelle, Ergodic theory of differentiable dynamical systems,
Publ. Math. IHES 50, 275 (1979).
[35] F. Ginelli, H. Chaté, R. Livi, A. Politi, Covariant Lyapunov
vectors, J. Phys. A: Math. Theor. 46, 254005 (2013).
