**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

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