Chapter 1 Predator-Prey Dynamics for Rabbits, Trees, and Romance

The Lotka-Volterra equations represent a simple nonlinear model for the dynamic interaction between two biological species in which one species (the predator) benefits at the expense of the other (the prey). With a change in signs, the same model can apply to two species that compete for resources or that symbiotically interact

Deterministic Time-Reversible Thermostats : Chaos, Ergodicity, and the Zeroth Law of Thermodynamics

The relative stability and ergodicity of deterministic time-reversible thermostats, both singly and in coupled pairs, are assessed through their Lyapunov spectra. Five types of thermostat are coupled to one another through a single Hooke's-Law harmonic spring. The resulting dynamics shows that three specific thermostat types, Hoover-Holian, Ju-Bulgac, and Martyna-Klein-Tuckerman, have very similar Lyapunov spectra in their equilibrium four-dimensional phase spaces and when coupled in equilibrium or nonequilibrium pairs. All three of these oscillator-based thermostats are shown to be ergodic, with smooth analytic Gaussian distributions in their extended phase spaces ( coordinate, momentum, and two control variables ). Evidently these three ergodic and time-reversible thermostat types are particularly useful as statistical-mechanical thermometers and thermostats. Each of them generates Gibbs' universal canonical distribution internally as well as for systems to which they are coupled. Thus they obey the Zeroth Law of Thermodynamics, as a good heat bath should. They also provide dissipative heat flow with relatively small nonlinearity when two or more such bath temperatures interact and provide useful deterministic replacements for the stochastic Langevin equation.Comment: Eight figures and 22 pages, prepared for a Molecular Physics issue honoring Jean Pierre Hanse

Nonequilibrium Systems : Hard Disks and Harmonic Oscillators Near and Far From Equilibrium

We relate progress in statistical mechanics, both at and far from equilibrium, to advances in the theory of dynamical systems. We consider computer simulations of time-reversible deterministic chaos in small systems with three- and four-dimensional phase spaces. These models provide us with a basis for understanding equilibration and thermodynamic irreversibility in terms of Lyapunov instability, fractal distributions, and thermal constraintsComment: Thirteen Figures and Forty Pages, Invited article for Molecular Simulation issue on Nonequilibrium Systems, accepted 23 August 2015. The article was published a year later, but with the editing thoroughly botched. The Figures in the Molecular Simulation version are more detailed. Unfortunately the labels and captions suffered from the journal's editin

A Tutorial: Adaptive Runge-Kutta Integration for Stiff Systems : Comparing the Nos\'e and Nos\'e-Hoover Oscillator Dynamics

"Stiff" differential equations are commonplace in engineering and dynamical systems. To solve them we need flexible integrators that can deal with rapidly-changing righthand sides. This tutorial describes the application of "adaptive" [ variable timestep ] integrators to "stiff" mechanical problems encountered in modern applications of Gibbs' 1902 statistical mechanics. Linear harmonic oscillators subject to nonlinear thermal constraints can exhibit either stiff or smooth dynamics. Two closely-related examples, Nos\'e's 1984 dynamics and Nos\'e-Hoover 1985 dynamics, are both based on Hamiltonian mechanics, as was ultimately clarified by Dettmann and Morriss in 1996. Both these dynamics are consistent with Gibbs' canonical ensemble. Nos\'e's dynamics is "stiff" and can present severe numerical difficulties. Nos\'e-Hoover dynamics, though it follows exactly the same trajectory, is "smooth" and relatively trouble-free. Our tutorial emphasises the power of adaptive integrators to resolve stiff problems like the Nos\'e oscillator. The solutions obtained illustrate the power of computer graphics to enrich numerical solutions. Adaptive integration with computer graphics are basic to an understanding of dynamical systems and statistical mechanics. These tools lead naturally into the visualization of intricate fractal structures formed by chaos as well as elaborate knots tied by regular nonchaotic dynamics. This work was invited by the American Journal of Physics.Comment: 24 pages with ten figures written for the American Journal of Physics --- Both here and in the published AJP version of October 2016 the time ranges shown in Figure 3 were permuted. 0 to 200 corresponds to the upper portion and 750 to 790 to the lower portion. We regret the erro

Ergodicity of a Singly-Thermostated Harmonic Oscillator

Although Nose's thermostated mechanics is formally consistent with Gibbs' canonical ensemble, the thermostated Nose-Hoover ( harmonic ) oscillator, with its mean kinetic temperature controlled, is far from ergodic. Much of its phase space is occupied by regular conservative tori. Oscillator ergodicity has previously been achieved by controlling two oscillator moments with two thermostat variables. Here we use computerized searches in conjunction with visualization to find singly-thermostated motion equations for the oscillator which are consistent with Gibbs' canonical distribution. These models are the simplest able to bridge the gap between Gibbs' statistical ensembles and Newtonian single-particle dynamics.Comment: Fifteen pages and six figures, accepted 23 August 2015 by Communications in Nonlinear Science and Numerical Simulatio

Ergodic Time-Reversible Chaos for Gibbs' Canonical Oscillator

Nos\'e's pioneering 1984 work inspired a variety of time-reversible deterministic thermostats. Though several groups have developed successful doubly-thermostated models, single-thermostat models have failed to generate Gibbs' canonical distribution for the one-dimensional harmonic oscillator. Sergi and Ferrario's 2001 doubly-thermostated model, claimed to be ergodic, has a singly-thermostated version. Though neither of these models is ergodic this work has suggested a successful route toward singly-thermostated ergodicity. We illustrate both ergodicity and its lack for these models using phase-space cross sections and Lyapunov instability as diagnostic tools.Comment: Sixteen pages and six figures, originally intended for Physical Review E, version accepted by Physics Letters

Synchronization of two Rossler systems with switching coupling

In this paper, we study a system of two Rossler oscillators coupled through a time-varying link, periodically switching between two values. We analyze the system behavior with respect to the frequency of the switching. By applying an averaging technique under the hypothesis of a high switching frequency, we find that although each value of the coupling does not produce synchronization, switching between the two at a high frequency stabilizes the synchronization manifold. However, we also find windows of synchronization below the value predicted by this technique, and we develop a master stability function to explain the appearance of these windows. Spectral properties of the system are a useful tool for understanding the dynamical properties and the synchronization failure in some intervals of the switching frequency. Numerical and experimental results in agreement with the analysis are presented.Comment: 11 Pages, 11 figure

The Equivalence of Dissipation from Gibbs' Entropy Production with Phase-Volume Loss in Ergodic Heat-Conducting Oscillators

Gibbs' thermodynamic entropy is given by the logarithm of the phase volume, which itself responds to heat transfer to and from thermal reservoirs. We compare the thermodynamic dissipation described by phase-volume loss with heat-transfer entropy production. Their equivalence is documented for computer simulations of the response of an ergodic harmonic oscillator to thermostated temperature gradients. In the simulations one or two thermostat variables control the kinetic energy or the kinetic energy and its fluctuation. All of the motion equations are time-reversible. We consider both strong and weak control variables. In every case the time-averaged dissipative loss of phase-space volume coincides with the entropy produced by heat transfer. Linear response theory nicely reproduces the small-gradient results obtained by computer simulation. The thermostats considered here are ergodic and provide simple dynamical models, some of them with as few as three ordinary differential equations, while remaining capable of reproducing Gibbs' canonical phase-space distribution and precisely consistent with irreversible thermodynamics.Comment: 21 pages with five figures, accepted by the International Journal of Bifurcation and Chaos on 6 December 2015. This version begins with a historical glossary of statistical mechanical background as an aid to the mathematical readershi

The Nos\'e-Hoover, Dettmann, and Hoover-Holian Oscillators

To follow up recent work of Xiao-Song Yang on the Nos\'e-Hoover oscillator we consider Dettmann's harmonic oscillator, which relates Yang's ideas directly to Hamiltonian mechanics. We also use the Hoover-Holian oscillator to relate our mechanical studies to Gibbs' statistical mechanics. All three oscillators are described by a coordinate $q$ and a momentum $p$. Additional control variables $(\zeta, \xi)$ vary the energy. Dettmann's description includes a time-scaling variable $s$, as does Nos\'e's original work. Time scaling controls the rates at which the $(q,p,\zeta)$ variables change. The ergodic Hoover-Holian oscillator provides the stationary Gibbsian probability density for the time-scaling variable $s$. Yang considered {\it qualitative} features of Nos\'e-Hoover dynamics. He showed that longtime Nos\'e-Hoover trajectories change energy, repeatedly crossing the $\zeta = 0$ plane. We use moments of the motion equations to give two new, different, and brief proofs of Yang's long-time limiting result.Comment: Seven pages with two figures, as accepted by Computational Methods in Science and Technology on 27 July 201

Heat Conduction, and the Lack Thereof, in Time-Reversible Dynamical Systems: Generalized Nos\'e-Hoover Oscillators with a Temperature Gradient

We use nonequilibrium molecular dynamics to analyze and illustrate the qualitative differences between the one-thermostat and two-thermostat versions of equilibrium and nonequilibrium (heat-conducting) harmonic oscillators. Conservative nonconducting regions can coexist with dissipative heat conducting regions in phase space with exactly the same imposed temperature field.Comment: Fifteen pages and six figures, incorporating two rounds of referees' suggestions for Physical Review