In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent 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 equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same. Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), nature) is the science of the natural world dealing with the fundamental constituents of the universe, the forces they exert on one another, and the results produced by these forces. ... Thermodynamics (from the Greek thermos meaning heat and dynamis meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... Phase space of a dynamical system with focal stability. ... In statistical mechanics, a microstate describes a specific detailed microscopic configuration of a system, that the system visits in the course of its thermal fluctuations. ... In physics, a statistical ensemble is a very large set of similar systems, considered all at once. ...

Liouville's Theorem shows that, for conserved classical systems, the local density of microstates following a particle path through phase space is constant (i.e., the total or convective time derivative is zero). Thus, if the microstates are uniformly distributed in phase space initially, this will remain true at later time. In mathematical physics, Liouvilles theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. ... Classical mechanics is a model of the physics of forces acting upon bodies. ... In mathematics, the uniform distributions are simple probability distributions. ...

Ergodic theory is a branch of mathematics which deals with dynamical systems which satisfy a version of this hypothesis, phrased in the language of measure theory. In mathematics, a measure-preserving transformation T on a probability space is said to be ergodic if the only measurable sets invariant under T have measure 0 or 1. ... Euclid, detail from The School of Athens by Raphael. ... A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. ... In mathematics, a measure is a function that assigns a number, e. ...

In macroscopic systems, the timescales over which a system can truly explore the entirety of its own phase space can be sufficiently large that the thermodynamic equilibrium state exhibits some form of ergodicity breaking. A common example is that of spontaneous magnetisation in ferromagnetic systems, whereby below the Curie temperature the system preferentially adopts a non-zero magnetisation even though the ergodic hypothesis would imply that no net magnetisation should exist by virtue of the system exploring all states whose time-averaged magnetisation should be zero. 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. Phase space of a dynamical system with focal stability. ... Ferromagnetism is a phenomenon by which a material can exhibit a spontaneous magnetization, and is one of the strongest forms of magnetism. ... In physics, the Curie point, or Curie temperature, is the temperature above which a ferromagnet loses its ferromagnetic ability to possess a net (spontaneous) magnetization in the absence of an external magnetic field. ... Spontaneous symmetry breaking in physics takes place when a system that is symmetric with respect to some symmetry group goes into a vacuum state that is not symmetric. ... A spin glass is a disordered material exhibiting high magnetic frustration. ...

See also

• Poincaré recurrence theorem
• Loschmidt's paradox