page_title: Ergodic hypothesis
hierarchical_section_title: Ergodic hypothesis
caption_attribution_description: English: Trap for fruit flies. Heavy orange paper is rolled into a cone, and taped. A tiny hole is cut at the point of the cone. The cone is inserted into a jar containing the bait. In this trap, the bait is several ounces of fermenting kombucha tea, which strongly attracts the target organism, fruit or vinegar flies. Tape is then applied around the outside of the cone and jar, so that flies that enter the jar through the bottom of the cone cannot escape. After all the flies have drowned, or the bait has naturally dried out and the flies have died, the trap can be discarded. Fermenting fruit of any kind can be used; add a little water or apple cider vinegar to keep the trap moist for several days.
context_page_description: In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a system in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., that all accessible microstates are equiprobable over a long period of time.
  Liouville's theorem states that, for Hamiltonian systems, the local density of microstates following a particle path through phase space is constant as viewed by an observer moving with the ensemble. Thus, if the microstates are uniformly distributed in phase space initially, they will remain so at all times. But Liouville's theorem does not imply that the ergodic hypothesis holds for all Hamiltonian systems.
  The ergodic hypothesis is often assumed in the statistical analysis of computational physics. The analyst would assume that the average of a process parameter over time and the average over the statistical ensemble are the same. This assumption—that it is as good to simulate a system over a long time as it is to make many independent realizations of the same system—is not always correct.
context_section_description: In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a system in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., that all accessible microstates are equiprobable over a long period of time.
  Liouville's theorem states that, for Hamiltonian systems, the local density of microstates following a particle path through phase space is constant as viewed by an observer moving with the ensemble (i.e., the convective time derivative is zero). Thus, if the microstates are uniformly distributed in phase space initially, they will remain so at all times. But Liouville's theorem does not imply that the ergodic hypothesis holds for all Hamiltonian systems.
  The ergodic hypothesis is often assumed in the statistical analysis of computational physics. The analyst would assume that the average of a process parameter over time and the average over the statistical ensemble are the same. This assumption—that it is as good to simulate a system over a long time as it is to make many independent realizations of the same system—is not always correct. (See, for example, the Fermi–Pasta–Ulam–Tsingou experiment of 1953.)
  Assumption of the ergodic hypothesis allows proof that certain types of perpetual motion machines of the second kind are impossible.
caption: This device can trap fruit flies, but if it trapped atoms when placed in gas that already uniformly fills the available phase space, then both Liouville's theorem and the second law of thermodynamics would be violated.
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