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Is there a physical (intuitive) explanation for why most systems are not ergodic? As my book states, it is a natural assumption that a system is at least quasi-ergodic; it then proceeds to state that this hypothesis is, in fact, false, and that we need a different basis for statistical mechanics. I don't understand why most systems aren't, and how we can prove this.

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I'm not sure how to quantify "most" systems, but off the top of my head, there are many processes that we encounter in our daily lives that break ergodicity within the timeframe under which we consider them.

  1. Symmetry breaking in general breaks ergodicity. Take, for example, a magnet. Ergodicity would imply that a magnet's magnetization would point in all directions (as every direction of magnetization occupies the same volume of phase space) with equal probability if sampled over a sufficiently long time. However, human timeframes is manifestly not anywhere near "sufficiently long".
  2. Disordered materials (like wood and glass) also break ergodicity. A block of wood has equal energy in many different configurations, but it only explores a tiny portion of that phase space volume. (i.e. if your block is a cube, it doesn't spontaneously move into a position rotated 90 degrees about one of its axes, again within human timeframes.)

Since statistical mechanics also aims to (and does) explain the behavior of these types of materials as well, it shouldn't need to depend on ergodicity to do so.

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