physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...

and

thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...

, the ergodic hypothesis says that, over long periods of time, the time spent by a system in some region of the

phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...

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

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 Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers propertie ...

of computational physics. The analyst would assume that the

average In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7 ...

of a process parameter over

time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

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. Systems that are ergodic are said to have the property of ergodicity; a broad range of systems in

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

and stochastic probability theory are ergodic. Ergodic systems are studied in ergodic theory.

Phenomenology

In macroscopic systems, the timescales over which a system can truly explore the entirety of its own

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 Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...

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. Also conventional glasses (e.g. window glasses) violate ergodicity in a complicated manner. In practice this means that on sufficiently short time scales (e.g. those of parts of seconds, minutes, or a few hours) the systems may behave as ''solids'', i.e. with a positive shear modulus, but on extremely long scales, e.g. over millennia or eons, as ''liquids'', or with two or more time scales and ''plateaux'' in between.The introduction of the practical aspect of ergodicity breaking by introducing a "non-ergodicity time scale" is due to . Also related to these time-scale phenomena are the properties of

ageing Ageing ( BE) or aging ( AE) is the process of becoming older. The term refers mainly to humans, many other animals, and fungi, whereas for example, bacteria, perennial plants and some simple animals are potentially biologically immortal. ...

and the Mode-Coupling theory of

References

{{DEFAULTSORT:Ergodic Hypothesis Ergodic theory Hypotheses Statistical mechanics Philosophy of thermal and statistical physics Concepts in physics

Phenomenology

See also

References