Remark on the (non)convergence of ensemble densities in dynamical systems.

@article{Goldstein1998RemarkOT,
  title={Remark on the (non)convergence of ensemble densities in dynamical systems.},
  author={Sheldon Goldstein and Joel L Lebowitz and Yakov G. Sinai},
  journal={Chaos},
  year={1998},
  volume={8 2},
  pages={
          393-395
        }
}
We consider a dynamical system with state space M, a smooth, compact subset of some R(n), and evolution given by T(t), x(t)=T(t)x, x in M; T(t) is invertible and the time t may be discrete, t in Z, T(t)=T(t), or continuous, t in R. Here we show that starting with a continuous positive initial probability density rho(x,0)>0, with respect to dx, the smooth volume measure induced on M by Lebesgue measure on R(n), the expectation value of logrho(x,t), with respect to any stationary (i.e., time… 
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References

SHOWING 1-10 OF 12 REFERENCES
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
Steady-state electrical conduction in the periodic Lorentz gas
We study nonequilibrium steady states in the Lorentz gas of periodic scatterers when an electric external field is applied and the particle kinetic energy is held fixed by a “thermostat” constructed
Topics on chaotic dynamics
Various kinematical quantities associated with the statistical properties of dynamical systems are examined: statistics of the motion, dynamical bases and Lyapunov exponents. Markov partitions for
Microscopic reversibility and macroscopic behavior: Physical explanatoins and mathematical derivations
The observed general tune-asymmetric behavior of macroscopic systems—embodied in the second law of thermodynainics—arises naturally from time-symmetric microscopic laws due to the great disparity
Boltzmann's Entropy and Time's Arrow
Given the success of Ludwig Boltzmann's statistical approach in explaining the observed irreversible behavior of macroscopic systems in a manner consistent with their reversible microscopic dynamics,
Entropy evolution for the Baker map.
TLDR
This entropy conundrum is resolved by considering the difference between weak and strong convergence, and a binary representation is used to make these points transparent.
Topics in Ergodic Theory
Preface Introduction 1. The principal ergodic theorems 2. Martingales and the ergodic theorem of information theory 3. Mixing 4. Entropy 5. Some examples Appendix References Further literature Index.
Dynamical Systems Approach to Nonequilibrium Statistical Mechanics: An Introduction, IHES/Rutgers, Lecture Notes
  • Dynamical Systems Approach to Nonequilibrium Statistical Mechanics: An Introduction, IHES/Rutgers, Lecture Notes
  • 1997
Lebowitz's article and his reply
  • Physics Today
  • 1994
...
1
2
...