### Abstract

We revisit the equilibrium one-dimensional ϕ^{4} model from the dynamical systems point of view. We find an infinite number of periodic orbits which are computationally stable. At the same time some of the orbits are found to exhibit positive Lyapunov exponents! The periodic orbits confine every particle in a periodic chain to trace out either the same or a mirror-image trajectory in its two-dimensional phase space. These “computationally stable” sets of pairs of single-particle orbits are either symmetric or antisymmetric to the very last computational bit. In such a periodic chain the odd-numbered and even-numbered particles’ coordinates and momenta are either identical or differ only in sign. “Positive Lyapunov exponents” can and do result if an infinitesimal perturbation breaking a perfect two-dimensional antisymmetry is introduced so that the motion expands into a four-dimensional phase space. In that extended space a positive exponent results. We formulate a standard initial condition for the investigation of the microcanonical chaotic number dependence of the model. We speculate on the uniqueness of the model's chaotic sea and on the connection of such collections of deterministic and time-reversible states to the Second Law of Thermodynamics.

Original language | English |
---|---|

Pages (from-to) | 192-201 |

Number of pages | 10 |

Journal | Communications in Nonlinear Science and Numerical Simulation |

Volume | 49 |

DOIs | |

Publication status | Published - 2017 Aug 1 |

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### Keywords

- Chaotic dynamics
- Lyapunov instability
- Molecular dynamics
- Time-Reversible thermostats

### ASJC Scopus subject areas

- Numerical Analysis
- Modelling and Simulation
- Applied Mathematics

### Cite this

**Order and chaos in the one-dimensional ϕ ^{4} model : N-dependence and the Second Law of Thermodynamics.** / Hoover, William Graham; Aoki, Kenichiro.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Order and chaos in the one-dimensional ϕ4 model

T2 - N-dependence and the Second Law of Thermodynamics

AU - Hoover, William Graham

AU - Aoki, Kenichiro

PY - 2017/8/1

Y1 - 2017/8/1

N2 - We revisit the equilibrium one-dimensional ϕ4 model from the dynamical systems point of view. We find an infinite number of periodic orbits which are computationally stable. At the same time some of the orbits are found to exhibit positive Lyapunov exponents! The periodic orbits confine every particle in a periodic chain to trace out either the same or a mirror-image trajectory in its two-dimensional phase space. These “computationally stable” sets of pairs of single-particle orbits are either symmetric or antisymmetric to the very last computational bit. In such a periodic chain the odd-numbered and even-numbered particles’ coordinates and momenta are either identical or differ only in sign. “Positive Lyapunov exponents” can and do result if an infinitesimal perturbation breaking a perfect two-dimensional antisymmetry is introduced so that the motion expands into a four-dimensional phase space. In that extended space a positive exponent results. We formulate a standard initial condition for the investigation of the microcanonical chaotic number dependence of the model. We speculate on the uniqueness of the model's chaotic sea and on the connection of such collections of deterministic and time-reversible states to the Second Law of Thermodynamics.

AB - We revisit the equilibrium one-dimensional ϕ4 model from the dynamical systems point of view. We find an infinite number of periodic orbits which are computationally stable. At the same time some of the orbits are found to exhibit positive Lyapunov exponents! The periodic orbits confine every particle in a periodic chain to trace out either the same or a mirror-image trajectory in its two-dimensional phase space. These “computationally stable” sets of pairs of single-particle orbits are either symmetric or antisymmetric to the very last computational bit. In such a periodic chain the odd-numbered and even-numbered particles’ coordinates and momenta are either identical or differ only in sign. “Positive Lyapunov exponents” can and do result if an infinitesimal perturbation breaking a perfect two-dimensional antisymmetry is introduced so that the motion expands into a four-dimensional phase space. In that extended space a positive exponent results. We formulate a standard initial condition for the investigation of the microcanonical chaotic number dependence of the model. We speculate on the uniqueness of the model's chaotic sea and on the connection of such collections of deterministic and time-reversible states to the Second Law of Thermodynamics.

KW - Chaotic dynamics

KW - Lyapunov instability

KW - Molecular dynamics

KW - Time-Reversible thermostats

UR - http://www.scopus.com/inward/record.url?scp=85013645241&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85013645241&partnerID=8YFLogxK

U2 - 10.1016/j.cnsns.2017.02.006

DO - 10.1016/j.cnsns.2017.02.006

M3 - Article

AN - SCOPUS:85013645241

VL - 49

SP - 192

EP - 201

JO - Communications in Nonlinear Science and Numerical Simulation

JF - Communications in Nonlinear Science and Numerical Simulation

SN - 1007-5704

ER -