26 research outputs found
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
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
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
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 and a momentum . Additional control variables
vary the energy. Dettmann's description includes a time-scaling
variable , as does Nos\'e's original work. Time scaling controls the rates
at which the variables change. The ergodic Hoover-Holian
oscillator provides the stationary Gibbsian probability density for the
time-scaling variable . Yang considered {\it qualitative} features of
Nos\'e-Hoover dynamics. He showed that longtime Nos\'e-Hoover trajectories
change energy, repeatedly crossing the 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