Time-Symmetry Breaking in Hamiltonian Mechanics. II. A Memoir for Berni Julian Alder [1925-2020]
aa r X i v : . [ phy s i c s . h i s t - ph ] O c t FIGURE LIST :1. Black particles in two-ball collision forward and backward2. Three Geometries for generating shockwaves3. 8192-particle twofold shockwave4+5. Runge-Kutta Shockwave reversal— > Rarefaction6+7. Shock forward and Shock backward important particles 20488+9 Rarefaction forward and backward important particles 204810. Thermodynamics of the 8192 rarefaction wave
Time-Symmetry Breaking in Hamiltonian Mechanics. II.A Memoir for Berni Julian Alder [1925-2020]
William Graham Hoover with Carol Griswold HooverRuby Valley Research Institute601 Highway Contract 60Ruby Valley, Nevada 89833 (Dated: October 16, 2020) bstract This memoir honors the late Berni Julian Alder, who inspired both of us with his pioneeringdevelopment of molecular dynamics. Berni’s work with Tom Wainwright, described in the 1959Scientific American[1], brought Bill to interview at Livermore in 1962. Hired by Berni, Bill en-joyed over 40 years’ research at the Laboratory. Berni, along with Edward Teller, founded UC’sDepartment of Applied Science in 1963. Their motivation was to attract bright students to use thelaboratory’s unparalleled research facilities. In 1972 Carol was offered a joint LLNL employee-DASstudent 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. Berniand Tom studied the irreversibility of Boltzmann’s “H function” in the early 1950s[2]. Berni calledshockwaves the “most irreversible” of hydrodynamic processes[3]. Just this past summer, in sim-ulating shockwaves with time-reversible classical mechanics, we found that reversed Runge-Kuttashockwave simulations yielded nonsteady rarefaction waves, not shocks. Intrigued by this unex-pected result we studied the exponential Lyapunov instabilities in both wave types. Besides theRunge-Kutta and Leapfrog algorithms, we developed a precisely-reversible manybody algorithmbased 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 theSecond Law of Thermodynamics. We describe promising research directions suggested by thiswork.
Keywords: Molecular Dynamics, Reversibility, Lyapunov Instability, Shock Waves, Rarefaction Waves . INTRODUCTION Bill began to work with Berni in the fall of 1962. Over the next six years they publishedsix joint works , including one each with three coauthors: Francis Ree, Tom Wainwright,and Dave Young. All six works were motivated by Berni’s longstanding interest in under-standing melting transitions for disks and spheres. The titles give an idea of their jointresearch: “Cooperative Motion of Hard Disks Leading to Melting” ; “Dependence of LatticeGas Properties on Mesh Size” ; “Cell Theories for Hard Particles” ; “The Pressure, Col-lision Rate, and Their Number Dependence for Hard Disks” ; “High-Density Equation ofState and Entropy for Hard Disks and Spheres” ; and last of all a longer review of theirwork, “Numerical Statistical Mechanics”, pages 79-113 in Physics of Simple Liquids , editedby three of their friends and colleagues: Neville Temperley, John Rowlinson, and GeorgeRushbrooke. These six papers can be found in the chronological publications list on ourwebsite, hooverwilliam.info, under “[ The 1960s ]”.Besides introducing us to his worldwide colleagues Berni passed on cogent research advice:understanding is the goal; words and pictures are vital to understanding; equations, not somuch; clarity of presentation is essential; of the three routes to understanding, formalism,experiment, and computation, at least two of these must be included and compared to makea publication “useful”.Our goal in the present work is to shed more light on the connection of time-reversibleatomistic dynamics to the irreversible Second Law of Thermodynamics. It is an extensionof work with a similar title published in 2013 . Back then, we expressed our motivation:“The goal we pursue here is improved microscopic understanding of the thermo-dynamic irreversibility described by the Second Law of Thermodynamics.”In Section II we sketch three approaches to the irreversibility question: [ 1 ] the H Theorem,[ 2 ] fractal distributions from thermostatted systems, and [ 3 ] time-symmetry breakingthrough Lyapunov instability. Section III describes the example motivating the presentwork, a one-dimensional strong shockwave, simulated with classical manybody moleculardynamics. The shockwave study led automatically to an investigation of rarefaction waves.Sections IV and V detail the Lyapunov instabilities of both processes, shock and rarefaction,in both time directions, “forward” and “backward”. In both cases we develop and applya novel precisely-reversible integration algorithm. Section VI describes the smooth-particle3echnique for connecting the atomistic and continuum descriptions of flow problems, appliedthere to the measurement of longitudinal and transverse temperatures. A summary follows,in Section VII. II. THREE EXPLANATIONS OF DYNAMICAL IRREVERSIBILITY
In 1956 Berni and Tom described several problems in their Brussels presentation “Molec-ular Dynamics by Electronic Computers” . Their evaluation of Boltzmann’s H Function, the19th-century explanation of irreversibility, showed that low-density hard-sphere moleculardynamics and Boltzmann’s equation agreed quite well. In 1987 a second explanation of ir-reversibility from time-reversible dynamics was offered as a consequence of Shuichi Nos´e’sequilibrium thermostat ideas applied to nonequilibrium problems, following the progressof one- or two-dimensional particles through arrays of scatterers. The time-averaged temper-ature was controlled in the one-dimensional case and the instantaneous temperature wasfixed in the two-dimensional case . Both these problems supported a new explanation ofirreversibility. Both generated fractal phase-space distributions with fractional dimension-alities less than that of the phase space. The rarity of nonequilibrium states, coupled withthe exponential instability of the reversed fractal repellor motion, provided an explanationmore general than Boltzmann’s. Rather than dilute gases the fractal description applied toa wide variety of liquid and solid problems .In 2013 we made a third effort to understand irreversibility for manybody Newtoniansystems through a novel measure of Lyapunov instability . This pervasive instability canbe followed by tracking the rate at which two nearby trajectories, the “reference” and the“satellite”, tend to separate, with the distance, but not the direction, between the twotrajectories held fixed. The direction of the reference-to-satellite vector joining the twomanybody trajectories determines which particles contribute most to the instability. Figure1 shows a striking difference between forward and backward analyses of an inelastic collisionof two 400-particle balls . The simulation is purely classical and precisely time-reversible.Forward in time the satellite particles most sensitive to instability (black in Figure 1 ) arethose on the leading edges, those first to take notice of collision. When precisely the sametrajectory is analyzed backward, with the 800-particle ball spontaneously (and completelyunphysically) separating into its two parts, the “important particles” are very different.4ackward in time such particles are mostly in the high-strainrate necking region wherenew surfaces are being created. The forward collision is physically reasonable and can besimulated easily with a variety of integrators and algorithms, all of them leading to similarresults. The reversed process, in which a single ball spontaneously separates into parts, is adifferent story, “irreversible”. It cannot be simulated directly. Instead it can only be studiedby a brute-force numerical reversal of the forward-in-time collision.
III. SHOCKWAVES–THE “MOST IRREVERSIBLE” PROCESSES A comprehensive 1980 study examined the two shockwaves, with velocities ± u s , thatresult when a periodic liquid manybody system is suddenly compressed by two periodicimages of itself. The left image advances rightward at the “piston velocity” + u p < u s .The right image leftward, at − u p , propelling the faster shock with velocity − u s . In thespace of about two atomic diameters the argon liquid being modelled increases in pressureto 400 kilobars and in temperature to about ten thousand kelvins. The density increasesapproximately twofold.Here we consider an alternative mechanism for shock generation, and in two space di-mensions rather than three. See the middle illustration in Figure 2 . We launch a stress-freecold solid against a fixed barrier at speed u = u p . When complete, this process convertsthe initial macroscopic kinetic energy, ( N u / N e . We model the initial cold state with an N -particle triangular lattice,periodic in y . Each particle pair interacts with the short-ranged repulsive pair potential,arbitrarily normalized to unity: φ ( r <
1) = (10 /π )(1 − r ) ; φ ( r >
1) = 0 → Z ∞ πrφ ( r ) dr ≡ . In the present shockwave work N is either 8192 = 32 ×
256 or 2048 = 16 ×
128 so that theaspect ratio ( L x /L y ), with close-packed columns of particles parallel to the y axis, is initially8 q (3 /
4) = 6 . x axis.The initial velocity, 0.97, is selected to shock-compress the cold solid twofold, to a hotfluid state. To break the lattice symmetry we begin with additional thermal velocities corre-sponding to an otherwise negligible temperature of 0.0001. Figure 3 shows the coexistenceof the hot shocked material with the cold stress-free triangular-lattice as modelled with 81925articles. The number density ρ increases from q (4 /
3) to 2 q (4 /
3) and the internal energychange is consistent with the Hugoniot relation for twofold compression from the stress-freezero-energy cold state to a hot shocked state with temperature T H = 0 .
115 : e H − e C ≡ (1 / P H + P C )( v C − v H ) [ Hugoniot Relation ]with e C = 0 and P C = 0 and v H = ( v C / −→ e H = (1 / P H ( v C /
2) = (1 / P H v H . so that e H = P H (0 . /
2) = 0 . . / → P H = 2 . . To derive the Hugoniot relation imagine the cold zero-energy zero-pressure crystal movingrightward at speed (0.97/2) and stagnating to match the velocity of a leftmoving wall atvelocity ( − . / / . / per particle.Evidently the resulting internal energy e H (the energy exclusive of the macroscopic motion) isidentical to the per-particle work done by the crystal in the compression process, ( P H v H /
2) =(0 . / simulations show that the structures of such strong shockwavesare steady and accurately one-dimensional, with a shockwidth on the order of two particlediameters. In the shock-based coordinate system ( fixed on the stationary shock, as shownin the top view of Figure 2 ) cold crystal enters from the left, with u = u s = 2 u p , and exitsat the right with u = u s − u p = u p = ( u s /
2) = 0 .
97. A time-reversal of this nonequilibriumshock process is easily implemented in a Runge-Kutta simulation by changing the sign ofthe timestep, dt = 0 . → dt = − .
01, or changing the signs of all the velocities in theproblem.
Figures 4 and 5 illustrate the surprising result of this straightforward “reversal”. Itmotivated the present work. Rather than seeing the shock travel backward unchanged, atleast for a reasonable time, instead we found that a rarefaction wave soon appears. Sucha wave is typically generated by the nearly isentropic expansion of a compressed fluid andis discussed in standard fluid mechanics texts for simple fluid models. An accurateLeapfrog integrator, likewise conserving energy throughout the run to an accuracy of sevendigits, produces a similar, likewise surprising, rarefaction. The “reversed motion” generatedwith either Runge-Kutta integration or Leapfrog is actually anything but! Notice the holesdeveloping in the reversed solution. To investigate the mechanism for this convincing fail-ure of algorithmic reversibility we turned to an analysis of the Lyapunov instability of the6rocess. We expected to see an analog of the symmetry breaking found for two collidingcrystallites as shown in
Figure 1 . We will shortly discuss this investigation, in the next Sec-tion, IV. First we remind the reader how Lyapunov instability is characterized in numericalsimulations . A. Lyapunov Instability with a Satellite Simulation
The largest Lyapunov exponent identifies that part of a system in which the mechanicsis least stable, with the highest growth rate of perturbations. It is evaluated in practice byfollowing the progress of two neighboring trajectories, the “reference” and the “satellite”,rescaling their separation at the end of each timestep. The magnitude of this offset–herewe use 0.0001–can be measured in coordinate q , momentum p , or ( q, p ) phase space. Tocarry out a precisely-reversed simulation one could use either Levesque and Verlet’s bit-reversible algorithm or our more-nearly-accurate implementation of one of Milne’s fourth-order algorithms . Both these approaches express the particle coordinates as (large) inte-gers. Typical force contributions, ¨ xdt or ¨ ydt , become considerably smaller integers, butare still large relative to unity. Consistent floating-point computations of the force contribu-tions, truncated to integers, then provide integer coordinate increments which are identical,apart from sign, in a pair of precisely-reversed motions. B. A Simpler Time-Reversed Algorithm
For enhanced accuracy and simplicity we choose here a simpler time-reversible methodof simulation, first storing an accurate Runge-Kutta reference trajectory for thousands oftimesteps and then separately computing two nearby satellite trajectories, one forward andone reversed. The offset lengths of both satellite trajectories from the reference are returnedfrom | δ ( t ) | to a fixed length δ at the completion of each timestep, giving the instantaneousvalue of the largest Lyapunov exponent, λ ( t ) ≡ ln( | δ ( t ) | /δ ) /dt , for small dt , ± .
01 inour simulations. All three trajectories, the reference and two satellites, are generated withthe same Runge-Kutta integrator. A novel vital detail is that the positions of the satelliteand reference trajectories often straddle a periodic boundary (in the y direction when thewave propagation direction is parallel to the x axis). To avoid discontinuous jumps in the7ector separating the two solutions it is necessary to detect and correct satellite coordinateswhich straddle the boundary, adding or subtracting L y as the case may be, resulting in acontinuously varying offset vector δ ( t ).An interesting consequence of the Lyapunov analysis is that the (largest) Lyapunov ex-ponent is uniformly positive in both time directions. Its numerical value is mostly in therange from 1 to 2 throughout both shockwave and rarefaction wave simulations. Insight intothe Lyapunov instability of the motion comes from identifying which particles contributemost to the offset vector. In a pioneering effort Stoddard and Ford calculated the largestLyapunov exponent of a Lennard-Jones fluid in 1967, maintaining the offset in coordinatespace.In 1998, with Kevin Boercker and Harald Posch , Bill simulated a nonequilibrium field-driven manybody particle flow and followed the largest local Lyapunov exponent, separatelyand instantaneously, in coordinate space and momentum space. The two identificationsof the exponent’s “important particles” (those with above-average separations, δ x + δ y or δ p x + δ p y ), were very similar. Nearly all important particles in coordinate space were alsoimportant in momentum space, and vice versa . One could quantify a particle’s contributionsto Lyapunov instability in at least three ways, in terms of δ x + δ y or δ p x + δ p y or δ x + δ y + δ p x + δ p y . Though different in principle , all three measures are in practice very similar in the particlesthey emphasize . Figures 4 and 5 display the result of an important-particle Lyapunovanalysis in coordinate space using the straightforward Runge-Kutta integrator, forward for6000 timesteps and backward for another 6000, with dt = ± .
01. Here and in
Figures 6-9 we use 2048 particles rather than 8192 in order better to visualize details on an individualparticle scale.
Figures 4 and 5 make the point quite convincingly that shockwaves areirreversible, even with very accurate integrators. Let us clarify the meaning of this obser-vation by storing the (forward) evolution of the shockwave trajectory and then analyzing itfor Lyapunov instability in both time directions.8
V. PRECISELY-REVERSIBLE SHOCK WAVE ANALYSES
Here
Figures 6 and 7 compare 2048-particle Lyapunov analyses forward and backwardfor the precisely-reversible (as the coordinates and momenta are all stored) simulations ofthat “most irreversible” shock process, the process shown in
Figure 3 for 8192 particles.The configurationally important particles have been colored brown in
Figures 4-9 . Noticethat only in the reversed direction is the shockwave itself the maximally unstable portionof the system. Exactly the same configurations, when analyzed forward in time ratherthan backward, show that the shockwave is relatively stable (as opposed to unstable) at the shockfront. Maximal instabilities instead occur here and there throughout the hotfluid, in relatively small transient clumps when the propagation is analyzed forward in time.Similar clump formation was found in the field-driven motion analyzed in Reference 23. Thedifference in the location of “important particles” (backward in time, found at the shock, butforward in time, located in distant clumps) is a significant positive indication that Lyapunovanalyses of Newtonian mechanics can provide a detailed understanding of the Second Law ofThermodynamics through the measurement of local instabilities. By including informationlocal in space and time from past history the Lyapunov offset vectors, { λ ( t ± dt ) ←→ δ ( t ) } quantify the simultaneous relative instabilities of microscopic motions. The difference foundhere between the forward and backward stability analyses of shocks is qualitative, not justquantitative, in the shockwave problem. We will come back to this analysis in our Summarysection. V. PRECISELY-REVERSIBLE RAREFACTION WAVE ANALYSES
In an effort to learn more here, we next generated, analyzed, and studied the evolutionof instability in a rarefaction wave. Apparently the lower-density boundary condition inthe reversed version of
Figure 5 provides an unnecessary perturbation of such a wave. Toinitiate a simpler pure-rarefaction simulation we first carry out an equilibrium Nos´e-Hoover isothermal high-density simulation (2048 particles with ρ = 2 q (4 /
3) and T = 0 . Figures 4 and 6 . Rather than using periodic boundaries in both the9 and y directions, as is usual in equilibrium situations, here we impose quartic boundarypotentials, dx / dx beyond the limits x = ± ( L x / x boundaries. We choose to release therighthand boundary. Figures 8 and 9 compare the forward and backward instability analyses of the resultingrarefaction wave. To make the details clear we again use only 2048 particles. The result-ing wave was constructed with a three-step process, first simulating 20000 equilibrationtimesteps at the high-temperature high-pressure thermodynamic state reached earlier byshock compression. Next, the righthand boundary was released and the resulting expansionfollowed for 4000 Runge-Kutta timesteps, a time of 40. Finally, the velocities were reversedfor a time of 40, returning to a close approximation of the initial high-temperature high-pressure state. This preliminary investigation surprised us yet again. Expansion (forming ararefaction wave), followed by time reversal, showed no tendency toward shock formation.Instead the reversed flow closely approximated the rarefaction configurations. To analyzethe motion precisely after equilibration, we followed and stored the 4000 { x, y, p x , p y } rar-efaction states, analyzing them in both directions so as to see the local “important particles”. Figures 8 and 9 shows the important particles found in both time directions for the rar-efaction wave. Here the unstable portions of both the forward and the backward rarefactionflows are all distributed in the hotter denser part of the wave. It is interesting, and was sur-prising to us, to see that reversing a rarefaction wave showed no tendency toward shockwaveformation.
VI. CONTINUUM FIELD VARIABLES FROM ( q, p ) PARTICLE INFORMATION
Figure 10 displays thermodynamic data from the stored forward = backward trajec-tory of
Figures 8 and 9 . The velocities stored for the latter figure show no essentialdifference between the longitudinal and transverse temperatures, indicating that the rar-efaction wave is indeed nearly isentropic. Such a wave provides the chance to measure theisentropic equation of state over a range of density and temperature. Let us do so now.We calculate “smoothed” values of the density and the longitudinal and transverse tem-peratures, { ρ ( x ) , T xx ( x ) , T yy ( x ) } . To reduce fluctuations for Figure 10 we use data for10192 = 128 ×
64 rather than 2048 particles. These data are smoothed with a properlynormalized one-dimensional form of Lucy’s short-ranged smooth-particle weight function , w ( x, h ) = (5 / hL )(1 − z + 8 z − z ) ; z ≡ ( | x | /h ) → Z + ∞−∞ dx Z + L/ − L/ w ( x ) dy ≡ .L = L y is the height of the system. The weight function vanishes for | x | > h . In theinitial hot fluid the 8192-particle system length was L x = 128 q (3 / ρ initial =2 q (4 /
3) = 2 . x grid point ρ ( x g ) is given by theintegrated density (delta functions) of particles nearby in their x coordinate, ρ ( x ): ρ ( x g ) ≡ N X i w ( x i − x g ) ≃ Z + L/ − L/ dy Z x g + hx g − h w ( x − x g ) ρ ( x ) dx . The smoothing distributes the influence of each particle over a region of width 2 h in x . Thekinetic temperatures are given by similarly-averaged differences h p i − h p i . Figure 10 shows these local temperatures as functions of the local density for a smoothing length h= 3 at the conclusion of the rarefaction simulation. The plot approximates a straight linefrom the origin to the point ( ρ, T ) = (2 . , . v × T constant. VII. SUMMARY AND SUGGESTED RESEARCH DIRECTIONS
Lyapunov analyses provide atomistic demonstrations and explanations of the symmetry-breaking instabilities associated with nonequilibrium states obeying standard classical me-chanics. Developing robust algorithms for stationary shock and rarefaction waves is a worthyresearch goal. We encourage readers to consider these problems. A research goal stimulatedby the present work is to quantify an instability metric. Such a metric would necessarily de-pend upon offset-vector components distinguishing the past from the future. Such a metricshould also be related to entropy production and the Second Law of Thermodynamics.A Lyapunov analysis of stationary states, as opposed to the transients treated here, ishighly desirable. Steady-state shockwave simulations, with particles entering at the left andexiting at the right, just as in the stationary view of
Figure 2 , would make it possible tocarry out longtime averages of instability properties. Most likely such an approach wouldassign to each particle in a variable-number system private forward and backward vectors,11oth offset from the reference trajectory. These vectors would give pairs of private Lyapunovexponents, N forward and N backward at any time. Histories of these pairs could then beaveraged to minimize fluctuations.The continuum entropy production, depending as it does on gradients of thermodynamicproperties, cannot distinguish between the two time directions. On the other hand the dif-ference between the instability metrics forward and backward in time, because they dependonly on their “pasts”, offers the chance better to quantify the relative stability of motionsobeying and disobeying macroscopic thermodynamics. VIII. ACKNOWLEDGMENTS
We very much enjoyed the chance to help honor Berni at his 90th Birthday Symposiumat the Livermore Laboratory on 20 August 2015. In turn, Berni very kindly delivered thekeynote address at Bill’s 80th Birthday Celebration at Sheffield the following year, on 26 July2016. Our description of this pedagogical example of the irreversibility inherent in Newtoniandynamics was motivated in part by email correspondence with Marcus Bannerman and KrisWojciechowski. We are grateful for their interest. We are grateful to the anonymous refereewho pointed out some typographical errors in an earlier version of the manuscript andsuggested that the direction of increasing time be indicated by arrows in Figures 4, 6, and8. 12
B. J. Alder and T. E. Wainwright,“Molecular Motions”, Scientific American , 113-126(1959). B. J. Alder and T. E. Wainwright, “Molecular Dynamics by Electronic Computers”, pages 97-131 in the Proceedings of the 27-31 August 1956 Symposium in Brussels,
Transport Processesin Statistical Mechanics , edited by I. Prigogine (Interscience, New York, 1958). M. Ross and B. Alder, “Shock Compression of Argon II. Nonadditive Repulsive Potential”,Journal of Chemical Physics , 4203-4210 (1967). B. J. Alder, W. G. Hoover, and T. E. Wainwright,“Cooperative Motion of Hard Disks Leadingto Melting”, Physical Review Letters , 241-243 (1963). W. G. Hoover, B. J. Alder, and F.H. Ree,“Dependence of Lattice Gas Properties on Mesh Size”,Journal of Chemical Physics , 3528-3533 (1964). W. G. Hoover and B. J. Alder, “Cell Theories for Hard Particles”, Journal of Chemical Physics , 2361-2367 (1966). W. G. Hoover and B. J. Alder, “Studies in Molecular Dynamics. IV. The Pressure, CollisionRate, and Their Number-Dependence for Hard Disks”, Journal of Chemical Physics , 686-691(1967). B. J. Alder, W. G. Hoover, and D.A. Young, “Studies in Molecular Dynamics. V. High-DensityEquation of State and Entropy for Hard Disks and Spheres”, Journal of Chemical Physics ,3688-3696 (1968). B. J. Alder and W. G. Hoover, “Numerical Statistical Mechanics”, pages 79-113 in
Physics ofSimple Liquids , edited by H. N.V. Temperley, J. S. Rowlinson, and G. S. Rushbrooke, (North-Holland, Amsterdam, 1968). W. G. Hoover and C. G. Hoover, “Time-Symmetry Breaking in Hamiltonian Mechanics”, Com-putational Methods in Science and Technology , 77-87 (2013). B. L. Holian, W. G. Hoover, and H. A. Posch, “Resolution of Loschmidt’s Paradox: The Originof Irreversible Behavior in Reversible Atomistic Dynamics”, Physical Review Letters , 10-13(1987). S. Nos´e, “A Molecular Dynamics Method for Simulations in the Canonical Ensemble”, MolecularPhysics , 255-268 (1984). S. Nos´e, “A Unified Formulation of the Constant Temperature Molecular Dynamics Methods”,Journal of Chemical Physics , 511-519 (1984). W. G. Hoover, H. A. Posch, B. L. Holian, M. J. Gillan, M. Mareschal, and C. M. Massobrio,“Dissipative Irreversibility from Nos´e’s Reversible Mechanics”, Molecular Simulation , 79-86(I987). B. Moran, W. G. Hoover, and S. Bestiale, “Diffusion in a Periodic Lorentz Gas”, Journal ofStatistical Physics , 709-726 (1987). B. L. Holian, W. G. Hoover, B. Moran, and G. K. Straub, “Shockwave Structure via
Nonequi-librium Molecular Dynamics and Navier-Stokes Continuum Mechanics”, Physical Review A ,2798-2808 (980). R. Courant and K. O. Friedrichs,
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Fluid Mechanics (Elsevier, Amsterdam, 1959 and 1987). S. D. Stoddard and J. Ford, “Numerical Experiments on the Stochastic Behavior of a Lennard-Jones Gas System”, Physical Review A , 1504-1512 (1973). I. Shimada and T. Nagashima, “A Numerical Approach to Ergodic Problems of DissipativeDynamical Systems”, Progress of Theoretical Physics , 1605-1616 (1979). G. Benettin, L. Galgani, A. Giorgilli, and J. M. Strelcyn, “Lyapunov Characteristic Exponentsfor Smooth Dynamical Systems and for Hamiltonian Systems; a Method for Computing All ofThem. Part 1: Theory”, Meccanica , 9-20 (1980). D. Levesque and L. Verlet, “Molecular Dynamics and Time Reversibility”, Journal of StatisticalPhysics , 519-537 (1993). Wm. G. Hoover, Kevin Boercker, and H. A. Posch, “Large-System Hydrodynamic Limit forColor Conductivity in Two Dimensions”, Physical Review E , 3911-3916 (1998). Wm. G. Hoover and C. G. Hoover, “Why Instantaneous Values of the ‘Covariant’ LyapunovExponents Depend upon the Chosen State-Space Scale”, Computational Methods in Scienceand Technology , 5-8 (2014). W. G. Hoover, “Canonical Dynamics: Equilibrium Phase-Space Distributions”, Physical ReviewA , 1695-1697 (1985). W. G. Hoover and C. G. Hoover, “SPAM-Based Recipes for Continuum Simulations”, Comput-ing in Science and Engineering (2), 78-85 (2001). orward Backward FIG. 1: Two identical snapshots from a “bit-reversible” precisely-time-reversible Newtonian colli-sion of two solid 400-particle balls . The important particles forward and backward in time showthat local mechanical instability, not phase volume, is the mechanism for Second Law irreversibility. (u p /2) − u s u p − u s StationaryStagnationSymmetric u s u p u p /2 − u p /2u s − u p FIG. 2: Three mechanisms for generating one-dimensional shockwaves. We use stagnation geom-etry here. The symmetric mechanism leads to the Hugoniot Relation ∆ e = h P i ∆ v , where h P i isthe average of the cold and hot pressures and ∆ v is the difference of the cold and hot volumes.FIG. 3: A one-dimensional leftmoving shockwave. Initially cold solid at density p (4 /
3) movesrightward at u p = 0 .
97, stagnates at a fixed barrier at x = 128 p (3 /
4) = 110 .
85, launches a twofold-compressed shockwave leftward, at u p − u s = − .