Ergodic hypothesis
Encyclopedia
In physics
and thermodynamics
, the ergodic hypothesis says that, over long periods of time, the time spent by a particle 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. A long period of time will depend on the time constants of the processes involved and the number of different components, both initial and possible. Use caution in assuming ergodic performance when the number of components, N, is large; it is likely that the time to completion will vary as N2. For complex biological systems, ergodic performance is seldom possible.
Ludwig Boltzmann
accomplished the feat of showing that the second law of thermodynamics is only a statistical fact. The gradual disordering of energy is analogous to the disordering of an initially ordered pack of cards under repeated shuffling, and just as the cards will finally return to their original order if shuffled a gigantic number of times, so the entire universe must some day regain, by pure chance, the state from which it first set out. (This optimistic coda
to the idea of the dying universe becomes somewhat muted when one attempts to estimate the timeline which will probably elapse before it spontaneously occurs.) The tendency for entropy increase seems to cause difficulty to beginners in thermodynamics, but is easy to understand from the standpoint of the theory of probability. Consider two ordinary dice
, with both sixes face up. After the dice are shaken, the chance of finding these two sixes face up is small (1 in 36); thus one can say that the random motion (the agitation) of the dice, like the chaotic collisions of molecules because of thermal energy, causes the less probable state to change to one that is more probable. With millions of dice, like the millions of atoms involved in thermodynamic calculations, the probability of their all being sixes becomes so vanishingly small that the system must move to one of the more probable states. However, mathematically the odds of all the dice results not being a pair sixes is also as hard as the ones of all of them being sixes, and since statistically the data tend to balance, one in every 36 pairs of dice will tend to be a pair of sixes. And the cards, when shuffled, will sometimes present a certain temporary sequence order even if in its whole they are disordered.
The ergodic hypothesis is often assumed in statistical analysis. The analyst would assume that the average
of a process parameter over time
and the average over the statistical ensemble are the same. Right or not, the analyst assumes that it is as good to observe a process for a long time as sampling many independent realisations of the same process.
Liouville's Theorem
shows that, for conserved classical 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 total or convective time derivative is zero). Thus, if the microstates are uniformly distributed
in phase space initially, they will remain so at all times. Liouville's theorem ensures that the notion of time average makes sense, but ergodicity does not follow from Liouville's theorem.
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. Also conventional glasses (e.g. window glasses) violate ergodicity in a complicated manner. In praxis 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. in millennia or eons, as liquids, or with two or more time scales and plateaux in between.
is a branch of mathematics
which deals with dynamical system
s that satisfy a version of this hypothesis, phrased in the language of measure theory.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
, the ergodic hypothesis says that, over long periods of time, the time spent by a particle in some region of the phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
of microstates
Microstate (statistical mechanics)
In statistical mechanics, a microstate is a specific microscopic configuration of a thermodynamic system that the system may occupy with a certain probability in the course of its thermal fluctuations...
with the same energy is proportional to the volume of this region, i.e., that all accessible microstates are equiprobable
Equiprobable
Equiprobability is a philosophical concept in probability theory that allows one to assign equal probabilities to outcomes when they are judged to be equipossible or to be "equally likely" in some sense...
over a long period of time. A long period of time will depend on the time constants of the processes involved and the number of different components, both initial and possible. Use caution in assuming ergodic performance when the number of components, N, is large; it is likely that the time to completion will vary as N2. For complex biological systems, ergodic performance is seldom possible.
Ludwig Boltzmann
Ludwig Boltzmann
Ludwig Eduard Boltzmann was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics...
accomplished the feat of showing that the second law of thermodynamics is only a statistical fact. The gradual disordering of energy is analogous to the disordering of an initially ordered pack of cards under repeated shuffling, and just as the cards will finally return to their original order if shuffled a gigantic number of times, so the entire universe must some day regain, by pure chance, the state from which it first set out. (This optimistic coda
Coda
Coda can denote any concluding event, summation, or section.Coda may also refer to:-Acronyms:* Calgary Olympic Development Association, the former name of the Canadian Winter Sport Institute, a non profit organization...
to the idea of the dying universe becomes somewhat muted when one attempts to estimate the timeline which will probably elapse before it spontaneously occurs.) The tendency for entropy increase seems to cause difficulty to beginners in thermodynamics, but is easy to understand from the standpoint of the theory of probability. Consider two ordinary dice
Dice
A die is a small throwable object with multiple resting positions, used for generating random numbers...
, with both sixes face up. After the dice are shaken, the chance of finding these two sixes face up is small (1 in 36); thus one can say that the random motion (the agitation) of the dice, like the chaotic collisions of molecules because of thermal energy, causes the less probable state to change to one that is more probable. With millions of dice, like the millions of atoms involved in thermodynamic calculations, the probability of their all being sixes becomes so vanishingly small that the system must move to one of the more probable states. However, mathematically the odds of all the dice results not being a pair sixes is also as hard as the ones of all of them being sixes, and since statistically the data tend to balance, one in every 36 pairs of dice will tend to be a pair of sixes. And the cards, when shuffled, will sometimes present a certain temporary sequence order even if in its whole they are disordered.
The ergodic hypothesis is often assumed in statistical analysis. The analyst would assume that the average
Average
In mathematics, an average, or central tendency of a data set is a measure of the "middle" value of the data set. Average is one form of central tendency. Not all central tendencies should be considered definitions of average....
of a process parameter over time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
and the average over the statistical ensemble are the same. Right or not, the analyst assumes that it is as good to observe a process for a long time as sampling many independent realisations of the same process.
Liouville's Theorem
Liouville's theorem (Hamiltonian)
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics...
shows that, for conserved classical 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 total or convective time derivative is zero). Thus, if the microstates are uniformly distributed
Uniform distribution (continuous)
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by...
in phase space initially, they will remain so at all times. Liouville's theorem ensures that the notion of time average makes sense, but ergodicity does not follow from Liouville's theorem.
Phenomenology
In macroscopic systems, the timescales over which a system can truly explore the entirety of its own phase spacePhase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the 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
Spontaneous symmetry breaking
Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
.
However, complex disordered systems such as a spin glass
Spin glass
A spin glass is a magnet with frustrated interactions, augmented by stochastic disorder, where usually ferromagnetic and antiferromagnetic bonds are randomly distributed...
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 praxis 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. in millennia or eons, as liquids, or with two or more time scales and plateaux in between.
Mathematics
Ergodic theoryErgodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
is a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
which deals with dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s that satisfy a version of this hypothesis, phrased in the language of measure theory.
See also
- Poincaré recurrence theoremPoincaré recurrence theoremIn mathematics, the Poincaré recurrence theorem states that certain systems will, after a sufficiently long time, return to a state very close to the initial state. The Poincaré recurrence time is the length of time elapsed until the recurrence. The result applies to physical systems in which...
- Loschmidt's paradoxLoschmidt's paradoxLoschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics...
- Ergodic theoryErgodic theoryErgodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
, a branch of mathematics concerned with a more general formulation of ergodicity - Ergodic processErgodic processIn signal processing, a stochastic process is said to be ergodic if its statistical properties can be deduced from a single, sufficiently long sample of the process.- Specific definitions :...
- ErgodicityErgodicityIn mathematics, the term ergodic is used to describe a dynamical system which, broadly speaking, has the same behavior averaged over time as averaged over space. In physics the term is used to imply that a system satisfies the ergodic hypothesis of thermodynamics.-Etymology:The word ergodic is...