Ergodicity - Revision history

Comp.arch at 23:28, 17 September 2024

2024-09-17T23:28:26Z

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	The mathematical definition of ergodicity aims to capture ordinary every-day ideas about [[randomness]]. This includes ideas about systems that move in such a way as to (eventually) fill up all of space, such as [[diffusion]] and [[Brownian motion]], as well as common-sense notions of mixing, such as mixing paints, drinks, cooking ingredients, [[Mixing (process engineering)\|industrial process mixing]], smoke in a smoke-filled room, the dust in [[Saturn's rings]] and so on. To provide a solid mathematical footing, descriptions of ergodic systems begin with the definition of a [[measure-preserving dynamical system]]. This is written as <math>(X, \mathcal{A}, \mu, T).</math>		The mathematical definition of ergodicity aims to capture ordinary every-day ideas about [[randomness]]. This includes ideas about systems that move in such a way as to (eventually) fill up all of space, such as [[diffusion]] and [[Brownian motion]], as well as common-sense notions of mixing, such as mixing paints, drinks, cooking ingredients, [[Mixing (process engineering)\|industrial process mixing]], smoke in a smoke-filled room, the dust in [[Saturn's rings]] and so on. To provide a solid mathematical footing, descriptions of ergodic systems begin with the definition of a [[measure-preserving dynamical system]]. This is written as <math>(X, \mathcal{A}, \mu, T).</math>

	The set <math>X</math> is understood to be the total space to be filled: the mixing bowl, the smoke-filled room, ''etc.'' The [[measure (mathematics)\|measure]] <math>\mu</math> is understood to define the natural [[volume]] of the space <math>X</math> and of its subspaces. The collection of subspaces is denoted by <math>\mathcal{A}</math>, and the size of any given [[subset]] <math>A\subset X</math> is <math>\mu(A)</math>; the size is its volume. Naively, one could imagine <math>\mathcal{A}</math> to be the [[power set]] of <math>X</math>; this doesn't quite work, as not all subsets of a space have a volume (famously, the [[~~Banach-Tarski~~ paradox]]). Thus, conventionally, <math>\mathcal{A}</math> consists of the measurable subsets—the subsets that do have a volume. It is always taken to be a [[Borel set]]—the collection of subsets that can be constructed by taking [[set intersection\|intersections]], [[set union\|unions]] and [[set complement]]s of open sets; these can always be taken to be measurable.		The set <math>X</math> is understood to be the total space to be filled: the mixing bowl, the smoke-filled room, ''etc.'' The [[measure (mathematics)\|measure]] <math>\mu</math> is understood to define the natural [[volume]] of the space <math>X</math> and of its subspaces. The collection of subspaces is denoted by <math>\mathcal{A}</math>, and the size of any given [[subset]] <math>A\subset X</math> is <math>\mu(A)</math>; the size is its volume. Naively, one could imagine <math>\mathcal{A}</math> to be the [[power set]] of <math>X</math>; this doesn't quite work, as not all subsets of a space have a volume (famously, the [[Banach–Tarski paradox]]). Thus, conventionally, <math>\mathcal{A}</math> consists of the measurable subsets—the subsets that do have a volume. It is always taken to be a [[Borel set]]—the collection of subsets that can be constructed by taking [[set intersection\|intersections]], [[set union\|unions]] and [[set complement]]s of open sets; these can always be taken to be measurable.

	The time evolution of the system is described by a [[map (mathematics)\|map]] <math>T:X\to X</math>. Given some subset <math>A\subset X</math>, its map <math>T(A)</math> will in general be a deformed version of <math>A</math> – it is squashed or stretched, folded or cut into pieces. Mathematical examples include the [[baker's map]] and the [[horseshoe map]], both inspired by [[bread]]-making. The set <math>T(A)</math> must have the same volume as <math>A</math>; the squashing/stretching does not alter the volume of the space, only its distribution. Such a system is "measure-preserving" (area-preserving, volume-preserving).		The time evolution of the system is described by a [[map (mathematics)\|map]] <math>T:X\to X</math>. Given some subset <math>A\subset X</math>, its map <math>T(A)</math> will in general be a deformed version of <math>A</math> – it is squashed or stretched, folded or cut into pieces. Mathematical examples include the [[baker's map]] and the [[horseshoe map]], both inspired by [[bread]]-making. The set <math>T(A)</math> must have the same volume as <math>A</math>; the squashing/stretching does not alter the volume of the space, only its distribution. Such a system is "measure-preserving" (area-preserving, volume-preserving).

Ira Leviton: Fixed a reference. Please see Category:CS1 errors: dates.

2024-09-01T19:31:18Z

Fixed a reference. Please see Category:CS1 errors: dates.

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	==Informal explanation==		==Informal explanation==
	{{unreferenced section\|date=November 2021}}		{{unreferenced section\|date=November 2021}}
	Ergodicity occurs in broad settings in [[physics]] and [[mathematics]].<ref>{{Cite journal \|last=Schöpf \|first=H.‐G. \|date=1970~~-01~~ \|title=<scp>V. I. Arnold</scp> and<scp> A. Avez</scp>, Ergodic Problems of Classical Mechanics. (The Mathematical Physics Monograph Series) IX + 286 S. m. Fig. New York/Amsterdam 1968. W. A. Benjamin, Inc. Preis geb. $ 14.75, brosch. $ 6.95 . \|url=http://dx.doi.org/10.1002/zamm.19700500721 \|journal=ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik \|volume=50 \|issue=7-9 \|pages=506–506 \|doi=10.1002/zamm.19700500721 \|issn=0044-2267}}</ref> All of these settings are unified by a common mathematical description, that of the [[measure-preserving dynamical system]]. Equivalently, ergodicity can be understood in terms of [[stochastic process]]es. They are one and the same, despite using dramatically different notation and language.		Ergodicity occurs in broad settings in [[physics]] and [[mathematics]].<ref>{{Cite journal \|last=Schöpf \|first=H.‐G. \|date=January 1970 \|title=<scp>V. I. Arnold</scp> and<scp> A. Avez</scp>, Ergodic Problems of Classical Mechanics. (The Mathematical Physics Monograph Series) IX + 286 S. m. Fig. New York/Amsterdam 1968. W. A. Benjamin, Inc. Preis geb. $ 14.75, brosch. $ 6.95 . \|url=http://dx.doi.org/10.1002/zamm.19700500721 \|journal=ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik \|volume=50 \|issue=7-9 \|pages=506–506 \|doi=10.1002/zamm.19700500721 \|issn=0044-2267}}</ref> All of these settings are unified by a common mathematical description, that of the [[measure-preserving dynamical system]]. Equivalently, ergodicity can be understood in terms of [[stochastic process]]es. They are one and the same, despite using dramatically different notation and language.

	===Measure-preserving dynamical systems===		===Measure-preserving dynamical systems===

Costron Systems: added reference of a paper from 1969 study of ergodic as a mathematical tool

2024-08-31T09:52:32Z

added reference of a paper from 1969 study of ergodic as a mathematical tool

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	==Informal explanation==		==Informal explanation==
	{{unreferenced section\|date=November 2021}}		{{unreferenced section\|date=November 2021}}
	Ergodicity occurs in broad settings in [[physics]] and [[mathematics]]. All of these settings are unified by a common mathematical description, that of the [[measure-preserving dynamical system]]. Equivalently, ergodicity can be understood in terms of [[stochastic process]]es. They are one and the same, despite using dramatically different notation and language.		Ergodicity occurs in broad settings in [[physics]] and [[mathematics]].<ref>{{Cite journal \|last=Schöpf \|first=H.‐G. \|date=1970-01 \|title=<scp>V. I. Arnold</scp> and<scp> A. Avez</scp>, Ergodic Problems of Classical Mechanics. (The Mathematical Physics Monograph Series) IX + 286 S. m. Fig. New York/Amsterdam 1968. W. A. Benjamin, Inc. Preis geb. $ 14.75, brosch. $ 6.95 . \|url=http://dx.doi.org/10.1002/zamm.19700500721 \|journal=ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik \|volume=50 \|issue=7-9 \|pages=506–506 \|doi=10.1002/zamm.19700500721 \|issn=0044-2267}}</ref> All of these settings are unified by a common mathematical description, that of the [[measure-preserving dynamical system]]. Equivalently, ergodicity can be understood in terms of [[stochastic process]]es. They are one and the same, despite using dramatically different notation and language.

	===Measure-preserving dynamical systems===		===Measure-preserving dynamical systems===

Magriteappleface: /* Ergodic measure */ corrected a definition

2024-08-17T03:02:44Z

Ergodic measure: corrected a definition

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	\| title = An introduction to infinite ergodic theory		\| title = An introduction to infinite ergodic theory
	\| volume = 50		\| volume = 50
	\| year = 1997}}</ref> relax the requirement that <math>T</math> preserves <math>\mu</math> to the requirement that <math>T</math> is a non-singular transformation with respect to <math>\mu</math>, meaning that if <math>N</math> is a subset of zero measure then so is <math>T(N)</math>.		\| year = 1997}}</ref> relax the requirement that <math>T</math> preserves <math>\mu</math> to the requirement that <math>T</math> is a non-singular transformation with respect to <math>\mu</math>, meaning that if <math>N</math> is a subset so that <math>T^{-1}(N)</math> has zero measure, then so does <math>T(N)</math>.

	===Examples===		===Examples===

76.71.13.148: updated broken links

2024-07-15T05:09:43Z

updated broken links

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	===Criterion for ergodicity===		===Criterion for ergodicity===

	The measure <math>\mu_\nu</math> is always ergodic for the shift map if the associated Markov chain is [[Markov chain#~~Reducibility~~\|irreducible]]~~{{Broken anchor\|date=2024-06-13\|bot=User:Cewbot/log/20201008/configuration\|target_link=Markov chain#Reducibility\|reason= The anchor (Reducibility) [[Special:Diff/970694186\|has been deleted]].}}~~ (any state can be reached with positive probability from any other state in a finite number of steps).{{sfn\|Walters\|1982\|p=42}}		The measure <math>\mu_\nu</math> is always ergodic for the shift map if the associated Markov chain is [[Markov chain#Irreducibility\|irreducible]] (any state can be reached with positive probability from any other state in a finite number of steps).{{sfn\|Walters\|1982\|p=42}}

	The hypotheses above imply that there is a unique stationary measure for the Markov chain. In terms of the matrix <math>P</math> a sufficient condition for this is that 1 be a simple eigenvalue of the matrix <math>P</math> and all other eigenvalues of <math>P</math> (in <math>\mathbb C</math>) are of modulus <1.		The hypotheses above imply that there is a unique stationary measure for the Markov chain. In terms of the matrix <math>P</math> a sufficient condition for this is that 1 be a simple eigenvalue of the matrix <math>P</math> and all other eigenvalues of <math>P</math> (in <math>\mathbb C</math>) are of modulus <1.

	Note that in probability theory the Markov chain is called [[Markov chain#Ergodicity\|ergodic]] if in addition each state is [[Markov chain#~~Periodicity~~\|aperiodic]]~~{{Broken anchor\|date=2024-06-13\|bot=User:Cewbot/log/20201008/configuration\|target_link=Markov chain#Periodicity\|reason= The anchor (Periodicity) [[Special:Diff/970694186\|has been deleted]].}}~~ (the times where the return probability is positive are not multiples of a single integer >1). This is not necessary for the invariant measure to be ergodic; hence the notions of "ergodicity" for a Markov chain and the associated shift-invariant measure are different (the one for the chain is strictly stronger).<ref>{{cite web \|url=https://mathoverflow.net/questions/74503/different-uses-of-the-word-ergodic/74503 \|title=Different uses of the word "ergodic" \|date=September 4, 2011 \|website=MathOverflow \|access-date=May 16, 2020}}</ref>		Note that in probability theory the Markov chain is called [[Markov chain#Ergodicity\|ergodic]] if in addition each state is [[Markov chain#Properties\|aperiodic]] (the times where the return probability is positive are not multiples of a single integer >1). This is not necessary for the invariant measure to be ergodic; hence the notions of "ergodicity" for a Markov chain and the associated shift-invariant measure are different (the one for the chain is strictly stronger).<ref>{{cite web \|url=https://mathoverflow.net/questions/74503/different-uses-of-the-word-ergodic/74503 \|title=Different uses of the word "ergodic" \|date=September 4, 2011 \|website=MathOverflow \|access-date=May 16, 2020}}</ref>

	Moreover the criterion is an "if and only if" if all communicating classes in the chain are [[Markov chain#~~Transience and recurrence~~\|recurrent]]{{Broken anchor\|date=2024-06-13\|bot=User:Cewbot/log/20201008/configuration\|target_link=Markov chain#Transience and recurrence\|reason= The anchor (Transience and recurrence) [[Special:Diff/970694186\|has been deleted]].}} and we consider all stationary measures.		Moreover the criterion is an "if and only if" if all communicating classes in the chain are [[Markov chain#Properties\|recurrent]] and we consider all stationary measures.

	===Examples===		===Examples===

Cewbot: Fixing broken anchor: Reminder of an inactive anchor: irreducible, Reminder of an inactive anchor: aperiodic, Reminder of an inactive anchor: recurrent

2024-06-13T20:03:42Z

Fixing broken anchor: Reminder of an inactive anchor: irreducible, Reminder of an inactive anchor: aperiodic, Reminder of an inactive anchor: recurrent

← Previous revision		Revision as of 20:03, 13 June 2024
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	===Criterion for ergodicity===		===Criterion for ergodicity===

	The measure <math>\mu_\nu</math> is always ergodic for the shift map if the associated Markov chain is [[Markov chain#Reducibility\|irreducible]] (any state can be reached with positive probability from any other state in a finite number of steps).{{sfn\|Walters\|1982\|p=42}}		The measure <math>\mu_\nu</math> is always ergodic for the shift map if the associated Markov chain is [[Markov chain#Reducibility\|irreducible]]{{Broken anchor\|date=2024-06-13\|bot=User:Cewbot/log/20201008/configuration\|target_link=Markov chain#Reducibility\|reason= The anchor (Reducibility) [[Special:Diff/970694186\|has been deleted]].}} (any state can be reached with positive probability from any other state in a finite number of steps).{{sfn\|Walters\|1982\|p=42}}

	The hypotheses above imply that there is a unique stationary measure for the Markov chain. In terms of the matrix <math>P</math> a sufficient condition for this is that 1 be a simple eigenvalue of the matrix <math>P</math> and all other eigenvalues of <math>P</math> (in <math>\mathbb C</math>) are of modulus <1.		The hypotheses above imply that there is a unique stationary measure for the Markov chain. In terms of the matrix <math>P</math> a sufficient condition for this is that 1 be a simple eigenvalue of the matrix <math>P</math> and all other eigenvalues of <math>P</math> (in <math>\mathbb C</math>) are of modulus <1.

	Note that in probability theory the Markov chain is called [[Markov chain#Ergodicity\|ergodic]] if in addition each state is [[Markov chain#Periodicity\|aperiodic]] (the times where the return probability is positive are not multiples of a single integer >1). This is not necessary for the invariant measure to be ergodic; hence the notions of "ergodicity" for a Markov chain and the associated shift-invariant measure are different (the one for the chain is strictly stronger).<ref>{{cite web \|url=https://mathoverflow.net/questions/74503/different-uses-of-the-word-ergodic/74503 \|title=Different uses of the word "ergodic" \|date=September 4, 2011 \|website=MathOverflow \|access-date=May 16, 2020}}</ref>		Note that in probability theory the Markov chain is called [[Markov chain#Ergodicity\|ergodic]] if in addition each state is [[Markov chain#Periodicity\|aperiodic]]{{Broken anchor\|date=2024-06-13\|bot=User:Cewbot/log/20201008/configuration\|target_link=Markov chain#Periodicity\|reason= The anchor (Periodicity) [[Special:Diff/970694186\|has been deleted]].}} (the times where the return probability is positive are not multiples of a single integer >1). This is not necessary for the invariant measure to be ergodic; hence the notions of "ergodicity" for a Markov chain and the associated shift-invariant measure are different (the one for the chain is strictly stronger).<ref>{{cite web \|url=https://mathoverflow.net/questions/74503/different-uses-of-the-word-ergodic/74503 \|title=Different uses of the word "ergodic" \|date=September 4, 2011 \|website=MathOverflow \|access-date=May 16, 2020}}</ref>

	Moreover the criterion is an "if and only if" if all communicating classes in the chain are [[Markov chain#Transience and recurrence\|recurrent]] and we consider all stationary measures.		Moreover the criterion is an "if and only if" if all communicating classes in the chain are [[Markov chain#Transience and recurrence\|recurrent]]{{Broken anchor\|date=2024-06-13\|bot=User:Cewbot/log/20201008/configuration\|target_link=Markov chain#Transience and recurrence\|reason= The anchor (Transience and recurrence) [[Special:Diff/970694186\|has been deleted]].}} and we consider all stationary measures.

	===Examples===		===Examples===

Jean Raimbault: /* Non-ergodic Markov chains */

2024-06-06T07:38:28Z

Non-ergodic Markov chains

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	====Non-ergodic Markov chains====		====Non-ergodic Markov chains====
	Markov chains with recurring communicating classes are not irreducible are not ergodic~~{{Incomprehensible inline}}~~, and this can be seen immediately as follows. If <math>S_1 \subsetneq S</math> are two distinct recurrent communicating classes there are nonzero stationary measures <math>\nu_1, \nu_2</math> supported on <math>S_1, S_2</math> respectively and the subsets <math>S_1^\mathbb{Z}</math> and <math>S_2^\mathbb{Z}</math> are both shift-invariant and of measure 1.2 for the invariant probability measure <math display="inline">\frac{1}{2}(\nu_1 + \nu_2)</math>. A very simple example of that is the chain on <math>S = \{1, 2\}</math> given by the matrix <math display="inline">\left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right)</math> (both states are stationary).		Markov chains with recurring communicating classes which are not irreducible are not ergodic, and this can be seen immediately as follows. If <math>S_1, S_2 \subsetneq S</math> are two distinct recurrent communicating classes there are nonzero stationary measures <math>\nu_1, \nu_2</math> supported on <math>S_1, S_2</math> respectively and the subsets <math>S_1^\mathbb{Z}</math> and <math>S_2^\mathbb{Z}</math> are both shift-invariant and of measure 1/2 for the invariant probability measure <math display="inline">\frac{1}{2}(\nu_1 + \nu_2)</math>. A very simple example of that is the chain on <math>S = \{1, 2\}</math> given by the matrix <math display="inline">\left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right)</math> (both states are stationary).

	====A periodic chain====		====A periodic chain====

Pot: Marked incomprehensible phrase (probably a typo where "are" should be "and"

2024-06-06T06:39:27Z

Marked incomprehensible phrase (probably a typo where "are" should be "and"

← Previous revision		Revision as of 06:39, 6 June 2024
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	====Non-ergodic Markov chains====		====Non-ergodic Markov chains====
	Markov chains with recurring communicating classes are not irreducible are not ergodic, and this can be seen immediately as follows. If <math>S_1 \subsetneq S</math> are two distinct recurrent communicating classes there are nonzero stationary measures <math>\nu_1, \nu_2</math> supported on <math>S_1, S_2</math> respectively and the subsets <math>S_1^\mathbb{Z}</math> and <math>S_2^\mathbb{Z}</math> are both shift-invariant and of measure 1.2 for the invariant probability measure <math display="inline">\frac{1}{2}(\nu_1 + \nu_2)</math>. A very simple example of that is the chain on <math>S = \{1, 2\}</math> given by the matrix <math display="inline">\left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right)</math> (both states are stationary).		Markov chains with recurring communicating classes are not irreducible are not ergodic{{Incomprehensible inline}}, and this can be seen immediately as follows. If <math>S_1 \subsetneq S</math> are two distinct recurrent communicating classes there are nonzero stationary measures <math>\nu_1, \nu_2</math> supported on <math>S_1, S_2</math> respectively and the subsets <math>S_1^\mathbb{Z}</math> and <math>S_2^\mathbb{Z}</math> are both shift-invariant and of measure 1.2 for the invariant probability measure <math display="inline">\frac{1}{2}(\nu_1 + \nu_2)</math>. A very simple example of that is the chain on <math>S = \{1, 2\}</math> given by the matrix <math display="inline">\left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right)</math> (both states are stationary).

	====A periodic chain====		====A periodic chain====

David Eppstein: it's a book, not a journal paper

2024-06-03T07:03:00Z

it's a book, not a journal paper

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	In other words, there are no [[invariant sigma-algebra\|<math>T</math>-invariant subsets]] up to measure 0 (with respect to <math>\mu</math>).		In other words, there are no [[invariant sigma-algebra\|<math>T</math>-invariant subsets]] up to measure 0 (with respect to <math>\mu</math>).

			Some authors<ref>{{cite book
⚫	~~Some authors<ref>{{cite journal~~ \| ~~last = Aaronson \| first = Jon \| date~~ = 1997 ~~\| title = An introduction to infinite ergodic theory \|page = 21\|publisher = American Mathematical Soc.\|isbn = 9780821804940~~}}</ref> relax the requirement that <math>T</math> preserves <math>\mu</math> to the requirement that <math>T</math> is a non-singular transformation with respect to <math>\mu</math>, meaning that if <math>N</math> is a subset of zero measure then so is <math>T(N)</math>.
			\| last = Aaronson \| first = Jon
			\| doi = 10.1090/surv/050
			\| isbn = 0-8218-0494-4
			\| mr = 1450400
			\| page = 21
			\| publisher = American Mathematical Society \| location = Providence, Rhode Island
			\| series = Mathematical Surveys and Monographs
			\| title = An introduction to infinite ergodic theory
			\| volume = 50
		⚫	\| year = 1997}}</ref> relax the requirement that <math>T</math> preserves <math>\mu</math> to the requirement that <math>T</math> is a non-singular transformation with respect to <math>\mu</math>, meaning that if <math>N</math> is a subset of zero measure then so is <math>T(N)</math>.

	===Examples===		===Examples===

Twotwice: /* Ergodic processes */ subsection capitalization

2024-06-03T01:13:19Z

Ergodic processes: subsection capitalization

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	[[Mixing (mathematics)\|Mixing]] is a stronger statement than ergodicity. Mixing asks for this ergodic property to hold between any two sets <math>A, B</math>, and not just between some set <math>A</math> and <math>X</math>. That is, given any two sets <math>A, B\in\mathcal{A}</math>, a system is said to be (topologically) mixing if there is an integer <math>N</math> such that, for all <math>A, B</math> and <math>n>N</math>, one has that <math>T^n(A) \cap B \ne \varnothing</math>. Here, <math>\cap</math> denotes [[set intersection]] and <math>\varnothing</math> is the [[empty set]]. Other notions of mixing include strong and weak mixing, which describe the notion that the mixed substances intermingle everywhere, in equal proportion. This can be non-trivial, as practical experience of trying to mix sticky, gooey substances shows.		[[Mixing (mathematics)\|Mixing]] is a stronger statement than ergodicity. Mixing asks for this ergodic property to hold between any two sets <math>A, B</math>, and not just between some set <math>A</math> and <math>X</math>. That is, given any two sets <math>A, B\in\mathcal{A}</math>, a system is said to be (topologically) mixing if there is an integer <math>N</math> such that, for all <math>A, B</math> and <math>n>N</math>, one has that <math>T^n(A) \cap B \ne \varnothing</math>. Here, <math>\cap</math> denotes [[set intersection]] and <math>\varnothing</math> is the [[empty set]]. Other notions of mixing include strong and weak mixing, which describe the notion that the mixed substances intermingle everywhere, in equal proportion. This can be non-trivial, as practical experience of trying to mix sticky, gooey substances shows.

	===Ergodic ~~Processes~~===		===Ergodic processes===
	{{main\|Ergodic process}}		{{main\|Ergodic process}}

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	The mathematical definition of ergodicity aims to capture ordinary every-day ideas about [[randomness]]. This includes ideas about systems that move in such a way as to (eventually) fill up all of space, such as [[diffusion]] and [[Brownian motion]], as well as common-sense notions of mixing, such as mixing paints, drinks, cooking ingredients, [[Mixing (process engineering)\|industrial process mixing]], smoke in a smoke-filled room, the dust in [[Saturn's rings]] and so on. To provide a solid mathematical footing, descriptions of ergodic systems begin with the definition of a [[measure-preserving dynamical system]]. This is written as <math>(X, \mathcal{A}, \mu, T).</math>		The mathematical definition of ergodicity aims to capture ordinary every-day ideas about [[randomness]]. This includes ideas about systems that move in such a way as to (eventually) fill up all of space, such as [[diffusion]] and [[Brownian motion]], as well as common-sense notions of mixing, such as mixing paints, drinks, cooking ingredients, [[Mixing (process engineering)\|industrial process mixing]], smoke in a smoke-filled room, the dust in [[Saturn's rings]] and so on. To provide a solid mathematical footing, descriptions of ergodic systems begin with the definition of a [[measure-preserving dynamical system]]. This is written as <math>(X, \mathcal{A}, \mu, T).</math>

	The set <math>X</math> is understood to be the total space to be filled: the mixing bowl, the smoke-filled room, ''etc.'' The [[measure (mathematics)\|measure]] <math>\mu</math> is understood to define the natural [[volume]] of the space <math>X</math> and of its subspaces. The collection of subspaces is denoted by <math>\mathcal{A}</math>, and the size of any given [[subset]] <math>A\subset X</math> is <math>\mu(A)</math>; the size is its volume. Naively, one could imagine <math>\mathcal{A}</math> to be the [[power set]] of <math>X</math>; this doesn't quite work, as not all subsets of a space have a volume (famously, the [[~~Banach-Tarski~~ paradox]]). Thus, conventionally, <math>\mathcal{A}</math> consists of the measurable subsets—the subsets that do have a volume. It is always taken to be a [[Borel set]]—the collection of subsets that can be constructed by taking [[set intersection\|intersections]], [[set union\|unions]] and [[set complement]]s of open sets; these can always be taken to be measurable.		The set <math>X</math> is understood to be the total space to be filled: the mixing bowl, the smoke-filled room, ''etc.'' The [[measure (mathematics)\|measure]] <math>\mu</math> is understood to define the natural [[volume]] of the space <math>X</math> and of its subspaces. The collection of subspaces is denoted by <math>\mathcal{A}</math>, and the size of any given [[subset]] <math>A\subset X</math> is <math>\mu(A)</math>; the size is its volume. Naively, one could imagine <math>\mathcal{A}</math> to be the [[power set]] of <math>X</math>; this doesn't quite work, as not all subsets of a space have a volume (famously, the [[Banach–Tarski paradox]]). Thus, conventionally, <math>\mathcal{A}</math> consists of the measurable subsets—the subsets that do have a volume. It is always taken to be a [[Borel set]]—the collection of subsets that can be constructed by taking [[set intersection\|intersections]], [[set union\|unions]] and [[set complement]]s of open sets; these can always be taken to be measurable.

	The time evolution of the system is described by a [[map (mathematics)\|map]] <math>T:X\to X</math>. Given some subset <math>A\subset X</math>, its map <math>T(A)</math> will in general be a deformed version of <math>A</math> – it is squashed or stretched, folded or cut into pieces. Mathematical examples include the [[baker's map]] and the [[horseshoe map]], both inspired by [[bread]]-making. The set <math>T(A)</math> must have the same volume as <math>A</math>; the squashing/stretching does not alter the volume of the space, only its distribution. Such a system is "measure-preserving" (area-preserving, volume-preserving).		The time evolution of the system is described by a [[map (mathematics)\|map]] <math>T:X\to X</math>. Given some subset <math>A\subset X</math>, its map <math>T(A)</math> will in general be a deformed version of <math>A</math> – it is squashed or stretched, folded or cut into pieces. Mathematical examples include the [[baker's map]] and the [[horseshoe map]], both inspired by [[bread]]-making. The set <math>T(A)</math> must have the same volume as <math>A</math>; the squashing/stretching does not alter the volume of the space, only its distribution. Such a system is "measure-preserving" (area-preserving, volume-preserving).