**Time-Symmetry Breaking in Hamiltonian Mechanics. Part II. A Memoir for Berni Julian Alder [1925–2020]**

Ruby Valley Research Institute

601 Highway Contract 60

Ruby Valley, Nevada 89833

*E-mail: hooverwilliam@yahoo.com

### Received:

Received: 25 September 2020; revised: 8 October 2020; accepted: 9 October 2020; published online: 22 October 2020

### DOI: 10.12921/cmst.2020.0000029

### Abstract:

This memoir honors the late Berni Julian Alder, who inspired both of us with his pioneering development of molecular dynamics. Berni’s work with Tom Wainwright, described in the 1959 Scientific American [1], brought Bill to interview at Livermore in 1962. Hired by Berni, Bill enjoyed over 40 years’ research at the Laboratory. Berni, along with Edward Teller, founded UC’s Department of Applied Science in 1963. Their motivation was to attract bright students to use the laboratory’s unparalleled research facilities. In 1972 Carol was offered a joint LLNL employee-DAS student appointment at Livermore. Bill, thanks to Berni’s efforts, was already a Professor there. Berni’s influence was directly responsible for our physics collaboration and our marriage in 1989. The present work is devoted to two early interests of Berni’s, irreversibility and shockwaves. Berni and Tom studied the irreversibility of Boltzmann’s “H function” in the early 1950s [2]. Berni called shockwaves the “most irreversible” of hydrodynamic processes [3]. Just this past summer, in simulating shockwaves with time-reversible classical mechanics, we found that reversed Runge-Kutta shockwave simulations yielded nonsteady rarefaction waves, not shocks. Intrigued by this unexpected result we studied the exponential Lyapunov instabilities in both wave types. Besides the Runge-Kutta and Leapfrog algorithms, we developed a precisely-reversible manybody algorithm based on trajectory storing, just changing the velocities’ signs to generate the reversed trajectories. Both shocks and rarefactions were precisely reversed. Separate simulations, forward and reversed, provide interesting examples of the Lyapunov-unstable symmetry-breaking models supporting the Second Law of Thermodynamics. We describe promising research directions suggested by this work.

### Key words:

Lyapunov instability, molecular dynamics, rarefaction waves, reversibility, shock waves

### References:

[1] B.J. Alder, T.E. Wainwright,

Molecular Motions, Scientific American201, 113–126 (1959).[2] B.J. Alder, T.E. Wainwright,

Molecular Dynamics by Electronic Computers, [In:]Transport Processes in Statistical Mechanics, the Proceedings of the 27–31 August 1956 Symposium in Brussels, 97–131, edited by I. Prigogine, Interscience, New York (1958).[3] M. Ross, B. Alder,

Shock Compression of Argon II. Nonadditive Repulsive Potential, Journal of Chemical Physics46, 4203–4210 (1967).[4] B.J. Alder, W.G. Hoover, T.E. Wainwright,

Cooperative Motion of Hard Disks Leading to Melting, Physical Review Letters11, 241–243 (1963).[5] W.G. Hoover, B.J. Alder, F.H. Ree,

Dependence of Lattice Gas Properties on Mesh Size, Journal of Chemical Physics41, 3528–3533 (1964).[6] W.G. Hoover, B.J. Alder,

Cell Theories for Hard Particles, Journal of Chemical Physics43, 2361–2367 (1966).[7] W.G. Hoover, B.J. Alder,

Studies in Molecular Dynamics. IV. The Pressure, Collision Rate, and Their Number-Dependence for Hard Disks, Journal of Chemical Physics46, 686–691 (1967).[8] B.J. Alder, W.G. Hoover, D.A. Young,

Studies in Molecular Dynamics. V. High-Density Equation of State and Entropy for Hard Disks and Spheres, Journal of Chemical Physics49, 3688–3696 (1968).[9] B.J. Alder, W.G. Hoover,

Numerical Statistical Mechanics, [In:]Physics of Simple Liquids, 79–113, edited by H.N.V. Temperley, J.S. Rowlinson, G.S. Rushbrooke, North-Holland, Amsterdam (1968).[10] W.G. Hoover, C.G. Hoover,

Time-Symmetry Breaking in Hamiltonian Mechanics, Computational Methods in Science and Technology19, 77–87 (2013).[11] B.L. Holian, W.G. Hoover, H.A. Posch,

Resolution of Loschmidt’s Paradox: The Origin of Irreversible Behavior in Reversible Atomistic Dynamics, Physical Review Letters59, 10–13 (1987).[12] S. Nosé,

A Molecular Dynamics Method for Simulations in the Canonical Ensemble, Molecular Physics52, 255–268 (1984).[13] S. Nosé,

A Unified Formulation of the Constant Temperature Molecular Dynamics Methods, Journal of Chemical Physics81, 511–519 (1984).[14] W.G. Hoover, H.A. Posch, B.L. Holian, M.J. Gillan, M. Mareschal, C.M. Massobrio,

Dissipative Irreversibility from Nosé’s Reversible Mechanics, Molecular Simulation1, 79–86 (I987).[15] B. Moran, W.G. Hoover, S. Bestiale,

Diffusion in a Periodic Lorentz Gas, Journal of Statistical Physics48, 709–726 (1987).[16] B.L. Holian, W.G. Hoover, B. Moran, G.K. Straub,

Shockwave Structure via Nonequilibrium Molecular Dynamics and Navier-Stokes Continuum Mechanics, Physical Review A22, 2798–2808 (1980).[17] R. Courant, K.O. Friedrichs,

Supersonic Flow and Shock Waves, Springer, New York (1948 and 1999).[18] L.D. Landau, E.M. Lifshitz,

Fluid Mechanics, Elsevier, Amsterdam (1959 and 1987).[19] S.D. Stoddard, J. Ford,

Numerical Experiments on the Stochastic Behavior of a Lennard-Jones Gas System, Physical Review A8, 1504–1512 (1973).[20] I. Shimada, T. Nagashima,

A Numerical Approach to Ergodic Problems of Dissipative Dynamical Systems, Progress of Theoretical Physics61, 1605–1616 (1979).[21] G. Benettin, 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. Part 1: Theory, Meccanica15, 9–20 (1980).[22] D. Levesque, L. Verlet,

Molecular Dynamics and Time Reversibility, Journal of Statistical Physics72, 519–537 (1993).[23] Wm.G. Hoover, K. Boercker, H.A. Posch,

Large-System Hydrodynamic Limit for Color Conductivity in Two Dimensions, Physical Review E57, 3911–3916 (1998).[24] Wm.G. Hoover, C.G. Hoover,

Why Instantaneous Values of the ‘Covariant’ Lyapunov Exponents Depend upon the Chosen State-Space Scale, Computational Methods in Science and Technology20, 5–8 (2014).[25] W.G. Hoover,

Canonical Dynamics: Equilibrium Phase-Space Distributions, Physical Review A31, 1695–1697 (1985).[26] W.G. Hoover, C.G. Hoover,

SPAM-Based Recipes for Continuum Simulations, Computing in Science and Engineering3(2), 78–85 (2001).