Remarks on non-Hamiltonian statistical mechanics: Lyapunov exponents and phase-space dimensionality loss

  title={Remarks on non-Hamiltonian statistical mechanics: Lyapunov exponents and phase-space dimensionality loss},
  author={Wm. G. Hoover and Harald A. Posch and Kenichiro Aoki and Dimitri Kusnezov},
The dissipation associated with nonequilibrium flow processes is reflected by the formation of strange attractor distributions in phase space. The information dimension of these attractors is less than that of the equilibrium phase space, corresponding to the extreme rarity of nonequilibrium states. Here we take advantage of a simple model for heat conduction to demonstrate that the nonequilibrium dimensionality loss can definitely exceed the number of phase-space dimensions required to… 

Figures from this paper

On the Statistical Mechanics of Non-Hamiltonian Systems: The Generalized Liouville Equation, Entropy, and Time-Dependent Metrics

Several questions in the statistical mechanics of non-Hamiltonian systems are discussed. The theory of differential forms on the phase space manifold is applied to provide a fully covariant

Nosé–Hoover nonequilibrium dynamics and statistical mechanics

At equilibrium Nosé's 1984 revolutionary thermostat idea linked Newton's mechanics with Gibbs' statistical mechanics. His work expanded the scope of isothermal and isobaric simulations. Nosé–Hoover

Phase-space metric for non-Hamiltonian systems

We consider an invariant skew-symmetric phase-space metric for non-Hamiltonian systems. We say that the metric is an invariant if the metric tensor field is an integral of motion. We derive the

Evolution equation for thermodynamic systems with correlated states

We have derived time evolution equations for thermodynamic systems with correlated states, considering the change in a probability density, entropy, and phase space metric at fixed phase-space

Lyapunov spectra and conjugate-pairing rule for confined atomic fluids.

This work presents nonequilibrium molecular dynamics simulation results for the Lyapunov spectra of atomic fluids confined in narrow channels of the order of a few atomic diameters and demonstrates how the spectrum reflects the presence of two different dynamics in the system: one for the unthermostatted fluid atoms and the other for the thermostatted and tethered wall atoms.

Non-equilibrium Statistical Mechanics of Classical and Quantum Systems

We study the statistical mechanics of classical and quantum systems in non-equilibrium steady states. Emphasis is placed on systems in strong thermal gradients. Various measures and functional forms

The Liouville Equation : A rapid review

This short essay has two purposes:(I) A rapid review of classical and stochastic quantum Liouville equation and proof of that and some use of Liouville equation in statistical mechanics and (II) to



Lyapunov Exponents, Transport and the Extensivity of Dimensional Loss

An explicit relation between the dimensional loss ($\Delta D$), entropy production and transport is established under thermal gradients, relating the microscopic and macroscopic behaviors of the

Fractality of the hydrodynamic modes of diffusion.

This work relates the Hausdorff dimension to the diffusion coefficient and the Lyapunov exponent for long-wavelength modes, and tests the relationship numerically on two Lorentz gases with hard repulsive forces.

Chaos, Scattering and Statistical Mechanics

1. Dynamical systems and their linear stability 2. Topological chaos 3. Liouvillian dynamics 4. Probabalistic chaos 5. Chaotic scattering 6. Scattering theory of transport 7. Hydrodynamic modes of

Derivation of Ohm's law in a deterministic mechanical model.

We study the Lorentz gas in small external electric and magnetic fields, with the particle kinetic energy held fixed by a Gaussian ``thermostat.'' Starting from any smooth initial density, a unique


Nonequilibrium systems in thermodynamic steady states can be studied by computer simulation, and the calculated transport coefficients are in agreement with results obtained by equilibrium methods.

Physica 7D

  • 153
  • 1983


  • Lett. 59, 319
  • 2002