Ergodic hypothesis

Liouville's theorem states that, for a Hamiltonian system, 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).

Assumption of the ergodic hypothesis allows proof that certain types of perpetual motion machines of the second kind are impossible.

The fact that macroscopic systems often violate the literal form of the ergodic hypothesis is an example of spontaneous symmetry breaking.

However, complex disordered systems such as a spin glass show an even more complicated form of ergodicity breaking where the properties of the thermodynamic equilibrium state seen in practice are much more difficult to predict purely by symmetry arguments.

Nassim Nicholas Taleb has argued that a very important part of empirical reality in finance and investment is non-ergodic.

A path where an individual, firm or country hits a "stop"—an absorbing barrier, "anything that prevents people with skin in the game from emerging from it, and to which the system will invariably tend.

The emerging field of ergodicity economics is beginning to show how including non-ergodic dynamics addresses some of the criticisms of neoclassical and pluralist economics;[4][5] and, practically, what investors and entrepreneurs can do[6] to correct for the typical outcome of a business or investment fund (under non-ergodic capital dynamics) being less than the expectation value.

The question of ergodicity in a perfectly collisionless ideal gas with specular reflections .
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.