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What is liquid? Lyapunov instability reveals symmetry-breaking irreversibilities hidden within Hamilton's many-body equations of motion
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What is liquid? Lyapunov instability reveals symmetry-breaking irreversibilities hidden within Hamilton's many-body equations of motion
Інші назви:
Що таке рiдина? Нестiйкiсть Ляпунова розкриває необоротнiсть порушення симетрiї, яка прихована у багаточастинкових рiвняннях руху Гамiльтона
Інший ID:
DOI:10.5488/CMP.18.13003
arXiv:1405.2485
PACS: 05.10.-a, 05.45.-a
arXiv:1405.2485
PACS: 05.10.-a, 05.45.-a
Посилання:
What is liquid? Lyapunov instability reveals symmetry-breaking irreversibilities hidden within Hamilton's many-body equations of motion / Wm.G. Hoover, C.G. Hoover // Condensed Matter Physics. — 2015. — Т. 18, № 1. — С. 13003:1-13. — Бібліогр.: 22 назв. — англ.
Підтримка:
We thank the anonymous referee for several useful suggestions. We address most of them here, in
order to clarify some of the underlying concepts and details of our work.
The referee wished for evidence as to the Debye-like nature of manybody Lyapunov spectra. This
became clear for manybody simulations during the 1980s. In addition to figure 1 of reference [6], two
nonequilibrium spectra appear as figure 1 in Wm.G. Hoover, “The Statistical Thermodynamics of Steady
States”, Physics Letters A, 1999, 255, 37–41.
That paper also illustrates the fact that low-temperature “stochastic” thermostats, represented by a
drag force −(p/τ), can lead to fractal distributions. This fractal nature, as judged by the Kaplan-Yorke
dimension, occurs because the phase-space offset vectors rotate very rapidly. Vectors linking a reference
trajectory to an orthogonal array of rapidly-rotating satellite trajectories mix the dimensionality loss associated with drag among all phase-space directions.
It needs to be emphasized that the offset vectors are in phase space rather than tangent space. The
insensitivity of our results to the length of these vectors and to the timestep was carefully checked.
The inversion symmetry of the balls and crystallites in figures 3–5 implies that the top of the leftmost
projectile corresponds to the bottom of its rightmost image. This inversion symmetry is particularly noticeable in the “important particles” shown here in figure 3.
As the referee kindly points out, it has been stated that Hamiltonian offset vectors are paired so that
the sum of the first and last local Lyapunov exponents should vanish exactly. Here, we simply state our
finding that this pairing is sometimes violated, not only initially, but also at long times.
The reason why we have used a collective density-dependent attractive potential for these collision
problems is that this reduces the yield strength so that the flow can more easily be observed and analyzed.
Дата:
2015
Переглядів:
494
Завантажень:
254
Анотація:
Typical Hamiltonian liquids display exponential "Lyapunov instability", also called "sensitive dependence on initial conditions". Although Hamilton's equations are thoroughly time-reversible, the forward and backward Lyapunov instabilities can differ, qualitatively. In numerical work, the expected forward/backward pairing of Lyapunov exponents is also occasionally violated. To illustrate, we consider many-body inelastic collisions in two space dimensions. Two mirror-image colliding crystallites can either bounce, or not, giving rise to a single liquid drop, or to several smaller droplets, depending upon the initial kinetic energy and the interparticle forces. The difference between the forward and backward evolutionary instabilities of these problems can be correlated with dissipation and with the Second Law of Thermodynamics. Accordingly, these asymmetric stabilities of Hamilton's equations can provide an "Arrow of Time". We illustrate these facts for two small crystallites colliding so as to make a warm liquid. We use a specially-symmetrized form of Levesque and Verlet's bit-reversible Leapfrog integrator. We analyze trajectories over millions of collisions with several equally-spaced time reversals.
