Instances of negative mobility, where a system responds to a perturbation in a way opposite to naive expectation, have been studied theoretically and experimentally in numerous nonequilibrium systems. In this work we show that Absolute Negative Mobility (ANM), whereby current is produced in a direction opposite to the drive, can occur around equilibrium states. This is demonstrated with a simple one-dimensional lattice model with a driven tracer. We derive analytical predictions in the linear response regime and elucidate the mechanism leading to ANM by studying the high-density limit. We also study numerically a model of hard Brownian disks in a narrow planar channel, for which the lattice model can be viewed as a toy model. We find that the model exhibits Negative Differential Mobility (NDM), but no ANM.

Steady-state properties of a driven tracer moving in a narrow two-dimensional (2D) channel of quiescent medium are studied. The tracer drives the system out of equilibrium, perturbs the density and pressure fields, and gives the bath particles a nonzero average velocity, creating a current in the channel. Three models in which the confining effect of the channel is probed are analyzed and compared in this study: the first is the simple symmetric exclusion process (SSEP), for which the stationary density profile and the pressure on the walls in the frame of the tracer are computed. We show that the tracer acts like a dipolar source in an average velocity field. The spatial structure of this 2D strip is then simplified to a one-dimensional (1D) SSEP, in which exchanges of position between the tracer and the bath particles are allowed. Using a combination of mean-field theory and exact solution in the limit where no exchange is allowed gives good predictions of the velocity of the tracer and the density field. Finally, we show that results obtained for the 1D SSEP with exchanges also apply to a gas of overdamped hard disks in a narrow channel. The correspondence between the parameters of the SSEP and of the gas of hard disks is systematic and follows from simple intuitive arguments. Our analytical results are checked numerically.

The Oseledec splitting of the tangent space into covariant subspaces for a hyperbolic dynamical system is numerically accessible by computing the full set of covariant Lyapunov vectors. In this paper, the covariant Lyapunov vectors, the orthogonal Gram-Schmidt vectors, and the corresponding local (time-dependent) Lyapunov exponents, are analyzed for a planar system of rough hard disks (RHDS). These results are compared to respective results for a smooth-hard-disk system (SHDS). We find that the rotation of the disks deeply affects the Oseledec splitting and the structure of the tangent space. For both the smooth and rough hard disks, the stable, unstable and central manifolds are transverse to each other, although the minimal angle between the unstable and stable manifolds of the RHDS typically is very small. Both systems are hyperbolic. However, the central manifold is precisely orthogonal to the rest of the tangent space only for the smooth-particle case and not for the rough disks. We also demonstrate that the rotations destroy the Hamiltonian character for the rough-hard-disk system.

The pressure components of 'soft' disks in a two-dimensional narrow channel are analyzed in the dilute gas regime using the Mayer cluster expansion and molecular dynamics. Channels with either periodic or reflecting boundaries are considered. It is found that when the two-body potential, u(r), is singular at some distance r(0), the dependence of the pressure components on the channel width exhibits a singularity at one or more channel widths which are simply related to r(0). In channels with periodic boundary conditions and for potentials which are discontinuous at r(0), the transverse and longitudinal pressure components exhibit a 1/2 and a 3/2 singularity, respectively. Continuous potentials with a power-law singularity result in weaker singularities of the pressure components. For channels with reflecting boundary conditions the singularities are found to be weaker than those corresponding to periodic boundaries.

We explore and compare numerical methods for the determination of multifractal dimensions for a doubly-thermostatted harmonic oscillator. The equations of motion are continuous and time-reversible. At equilibrium the distribution is a four-dimensional Gaussian, so that all the dimension calculations can be carried out analytically. Away from equilibrium the distribution is a surprisingly isotropic multifractal strange attractor, with the various fractal dimensionalities in the range 1<D<4. The attractor is relatively homogeneous, with projected two-dimensional information and correlation dimensions which are nearly independent of direction. Our data indicate that the Kaplan–Yorke conjecture (for the information dimension) fails in the full four-dimensional phase space. We also find no plausible extension of this conjecture to the projected fractal dimensions of the oscillator. The projected growth rate associated with the largest Lyapunov exponent is negative in the one-dimensional coordinate space.

The ergodic properties of many-body systems with repulsive-core interactions are the basis of classical statistical mechanics and are well established. This is not the case for systems of purely-attractive or gravitational particles. Here we consider two examples, (i) a family of one-dimensional systems with attractive power-law interactions, \(|x_i -x_j|^{\nu} \;, \; \nu > 0\), and (ii) a system of N gravitating particles confined to a finite compact domain. For (i) we deduce from the numerically-computed Lyapunov spectra that chaos, measured by the maximum Lyapunov exponent or by the Kolmogorov–Sinai entropy, increases linearly for positive and negative deviations of ν from the case of a non-chaotic harmonic chain (ν = 2). For \(2 < \nu \le 3\) there is numerical evidence for two additional hitherto unknown phase-space constraints. For the theoretical interpretation of model (ii) we assume ergodicity and show that for a small-enough system the reduction of the allowed phase space due to any other conserved quantity, in addition to the total energy, renders the system asymptotically stable. Without this additional dynamical constraint the particle collapse would continue forever. These predictions are supported by computer simulations.

has eigenvalues {±i} corresponding to oscillation about the fixed-point origin {q,p} = {0, 0}.For more complicated systems -a pendulum is the usual example -it is usual to define stable and unstable "manifolds", unions of those phase-space directions which correspond to exponential decay and to exponential growth, respectively [1].In even more complicated situations, including ones indicate decay, toward the fixed point.Complex eigenvalues can also occur.For example, the undamped harmonic oscillator, with its phase-space flow velocity Positive eigenvalues of this matrix correspond to exponential growth.Negative matrix" oscillator.We find that it breaks down at a dense set of singular points, where the four eigenvectors span only a three-dimensional subspace.We believe that the concepts of stable and unstable global manifolds are problematic for this simple nonequilibrium system.

We simulate the far-from-equilibrium irreversible expansion of a compressed ideal gas in two space dimensions. For this problem the particle trajectories from conventional smooth particle applied mechanics are isomorphic to those from a corresponding molecular dynamics simulation. The smooth-particle “weight function” used to describe the expanding gas is identical to the pair potential governing the molecular dynamics simulation. These many-body particle simulations are compared with those using a modified smooth-particle algorithm invented by Monaghan, as well as with those based on conventional grid-based Eulerian and Lagrangian methods.

Recent computer simulations have contributed significantly to our understanding of the Lyapunov instability of hard particle systems in equilibrium and in nonequilibrium steady states. We discuss a very general method for the computation of the full Lyapunov spectra and apply it to billiards and to many-body hard disk and hard sphere systems. The velocity correlation function of billiard flows is also discussed. For hard disk and hard sphere systems the perturbed states associated with the smallest Lyapunov exponents (in absolute magnitude) are shown to reveal collective dynamic modes. We study the properties of these modes and provide examples for hard disk systems in two dimensions. It is suggested that there is a connection with the dynamic modes familiar from fluctuating hydrodynamics. The largest Lyapunov exponent, however, is associated with localized perturbations in the fluid.

Hard Ball Systems and the Lorentz Gas are fundamental models arising in the theory of Hamiltonian dynamical systems. Moreover, in these models, some key laws of statistical physics can also be tested

Mathematical Physics 2000, pp. 289-305 (2000) No AccessTHE CLASSICAL THREE-BODY PROBLEM – WHERE IS ABSTRACT MATHEMATICS, PHYSICAL INTUITION, COMPUTATIONAL PHYSICS MOST POWERFUL?H. A. POSCH and W. THIRRINGH. A. POSCHInstitut für Experimentalphysik, Universität Wien, Boltzmanngasse 5, A-1090 Wien, Austria and W. THIRRINGInstitut für Theoretische Physik, Universität Wien, Boltzmanngasse 5, A-1090 Wien, Austriahttps://doi.org/10.1142/9781848160224_0015Cited by:0 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: We show how different aspects of the restricted three-body problem can be understood with physical intuition, rigorous mathematics and computer simulations. The first explains the short time stability, the second tells us when it is stable for all times, and the third shows when and why chaos takes over. FiguresReferencesRelatedDetails Mathematical Physics 2000Metrics History PDF download

We report on the computation of full Lyapunov spectra of the boundary-driven Chernov–Lebowitz model for stationary planar shear flow. The Lyapunov exponents are calculated with a recently developed formalism for systems with elastic hard collisions. Although the Chernov–Lebowitz model is strictly energy conserving, any phase-space volume is subjected to a contraction due to the reflection rules of the hard disks colliding with the walls. Consequently, the sum of Lyapunov exponents is negative. As expected for an inhomogeneously driven system, the Lyapunov spectra do not obey the conjugate pairing rule. The external driving makes the system less chaotic, which is reflected in a decrease of the Kolmogorov–Sinai entropy if the driving is increased.

Systems composed of particles with purely attractive potentials are thermodynamically unstable. Lowering their total energy below a certain threshold results in a phase transition associated with a significant increase in temperature. A single cluster is formed which floats in the rest atmosphere of the remaining particles. The microcanonical and canonical equilibrium properties and transient behaviour of such classical systems have been studied by computer simulation. We find that clustered equilibrium states are well described by stationary solutions of the Vlasov equation, but the Vlasov dynamics is unable to account for the collapse. Reduced phase-space distributions have been used to compute entropy changes for systems subjected to periodic expansions and compressions. From model considerations we deduce that in the thermodynamic limit active states violating the second law are fractally distributed in phase space.

We discuss computer solutions of Newton's equations of motion for unstable systems in a container with time-dependent walls. An expansion leads to the formation of a cluster and a significant increase of the temperature. The question of entropy increase for expansion and compression of the system and the related problem of the feasibility of a perpetum mobile of the second kind are investigated.

We derive a generalized Langevin equation for the simultaneous motion of two structureless particles (solute molecules) in a fluid. The resulting equations contain, in addition to the usual self memory integral term, a cross memory (or solvent-mediated drag) term which is a time-dependent generalization of the so-called hydrodynamic interaction. The cross memory function is expressed explicitly in terms of velocity cross correlation functions and may thus be evaluated numerically in molecular dynamics experiments.

Nose's Hamiltonian mechanics makes possible the efficient simulation of irreversible flows of mass, momentum and energy. Such flows illustrate the paradox that reversible microscopic equations of motion underlie the irreversible behavior described by the second law of thermodynamics. This generic behavior of molecular many-body systems is illustrated here for the simplest possible system, with only one degree of freedom: a one-body Frenkel-Kontorova model for isothermal electronic conduction. This model system, described by Nose-Hoover Hamiltonian dynamics, exhibits several interesting features: (1) deterministic and reversible equations of motion; (2) Lyapunov instability, with phase-space offsets increasing exponentially with time; (3) limit cycles; (4) dissipative conversion of work (potential energy) into heat (kinetic energy); and (5) phase-space contraction, a characteristic feature of steady irreversible flows. The model is particularly instructive in illustrating and explaining a paradox associated with steady-state statistical mechanics: the Gibbs entropy of a nonequilibrium steady state decreases continuously to minus infinity.

After a brief introduction with a discussion of the concept of the correlation function needed for the formulation of the problem, a progress report is given concerning our knowledge of microdynamic processes in liquids covering the fields of neutron scattering, IR-absorption and light-scattering, dielectric and nuclear magnetic relaxation as well as studies of diffusion and electric conductance.

Depolarized Rayleigh spectra for four different thermodynamic states of krypton are reported. At large liquidlike densities and for low frequency shifts (<20 cm−1) the existence of a Lorentzian component previously discovered in liquid argon is confirmed. For large ν the line shapes are mainly exponential with the exception of a weak shoulder located at around 50 cm−1. These spectral features are interpreted in the light of a recent theory by Madden. In comparing the results for krypton and argon the law of corresponding states is found to hold. The dependence of the Lorentzian half-width on the diffusion coefficient is not the same as inferred from the Madden theory.