Ergodic hypothesis - Revision history

Carl McBride: Added a recent publication

2011-10-21T11:01:07Z

Added a recent publication

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 A restatement of the ergodic hypothesis is to say that all allowed states are equally probable. This holds true if
-the ''metrical transitivity'' of general [[Hamiltonian]] systems holds true.
+the ''metrical transitivity'' of general [[Hamiltonian]] systems holds true. Recent experiments have demonstrated the hypothesis <ref>[http://dx.doi.org/10.1002/anie.201105388   Florian Feil, Sergej Naumov, Jens Michaelis, Rustem Valiullin, Dirk Enke, Jörg Kärger, and Christoph Bräuchle "Single-Particle and Ensemble Diffusivities—Test of Ergodicity", Angewandte Chemie International Edition Early View (2011)]</ref>.
 ==See also==
 *[[Fermi-Pasta-Ulam experiment]]
 *[[Markov chain]]
 *[[Mixing systems]]
 [[category: Computer simulation techniques]]

Carl McBride: /* See also */ Added mixing systems to list.

2008-07-29T10:45:45Z

Carl McBride: /* References */

2007-11-19T14:29:39Z

References

← Older revision		Revision as of 16:29, 19 November 2007
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	#[http://dx.doi.org/10.1070/RM1970v025n02ABEH003794 Ya G Sinai "Dynamical systems with elastic reflections", Russian Mathematical Surveys '''25''' pp. 137-189 (1970)]		#[http://dx.doi.org/10.1070/RM1970v025n02ABEH003794 Ya G Sinai "Dynamical systems with elastic reflections", Russian Mathematical Surveys '''25''' pp. 137-189 (1970)]
	#[http://library.lanl.gov/cgi-bin/getfile?15-18.pdf Adrian Patrascioiu "The Ergodic-Hypothesis, A Complicated Problem in Mathematics and Physics", Los Alamos Science, '''15''' pp. 263- (1987)]		#[http://library.lanl.gov/cgi-bin/getfile?15-18.pdf Adrian Patrascioiu "The Ergodic-Hypothesis, A Complicated Problem in Mathematics and Physics", Los Alamos Science, '''15''' pp. 263- (1987)]
			#[http://dx.doi.org/10.1007/BF00384333 Jan von Plato "Boltzmann's ergodic hypothesis", Archive for History of Exact Sciences '''42''' pp. 71-89 (1991)]
	# Domokos Ssász "Boltzmann's Ergodic Hypothesis, a Conjecture for Centuries?", Studia Scientiarum Mathematicarum Hungarica '''31''' pp. 299-322 (1996) [http://iml.univ-mrs.fr/~lafont/rencontres/esi098.pdf (reprint)]		# Domokos Ssász "Boltzmann's Ergodic Hypothesis, a Conjecture for Centuries?", Studia Scientiarum Mathematicarum Hungarica '''31''' pp. 299-322 (1996) [http://iml.univ-mrs.fr/~lafont/rencontres/esi098.pdf (reprint)]
	[[category: Computer simulation techniques]]		[[category: Computer simulation techniques]]

Carl McBride at 14:19, 19 November 2007

2007-11-19T14:19:00Z

← Older revision		Revision as of 16:19, 19 November 2007
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	*[[Markov chain]]		*[[Markov chain]]
	*[[Fermi-Pasta-Ulam experiment]]		*[[Fermi-Pasta-Ulam experiment]]

	==References==		==References==
	#[http://www.pnas.org/cgi/reprint/17/12/656 George D. Birkhoff, "Proof of the Ergodic Theorem", PNAS '''17''' pp. 656-660 (1931)]		#[http://www.pnas.org/cgi/reprint/17/12/656 George D. Birkhoff, "Proof of the Ergodic Theorem", PNAS '''17''' pp. 656-660 (1931)]
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	#[http://www.pnas.org/cgi/reprint/18/3/263 J. V. Neumann "Physical Applications of the Ergodic Hypothesis", PNAS '''18''' pp. 263-266 (1932)]		#[http://www.pnas.org/cgi/reprint/18/3/263 J. V. Neumann "Physical Applications of the Ergodic Hypothesis", PNAS '''18''' pp. 263-266 (1932)]
	#[http://www.pnas.org/cgi/reprint/18/3/279 G. D. Birkhoff and B. O. Koopman "Recent Contributions to the Ergodic Theory", PNAS '''18''' pp. 279-282 (1932)]		#[http://www.pnas.org/cgi/reprint/18/3/279 G. D. Birkhoff and B. O. Koopman "Recent Contributions to the Ergodic Theory", PNAS '''18''' pp. 279-282 (1932)]
	#Ya. G. Sinai "On the Foundation of the Ergodic Hypothesis for a Dynamical System of Statistical Mechanics", Soviet Math. Doklady '''4''' pp. 1818-1822 (1963)		#Ya. G. Sinai "On the Foundation of the Ergodic Hypothesis for a Dynamical System of Statistical Mechanics", Doklady Akademii Nauk SSSR '''153''' pp. 1261–1264 (1963) (English version: Soviet Math. Doklady '''4''' pp. 1818-1822 (1963))
	#[http://dx.doi.org/10.1070/RM1970v025n02ABEH003794 Ya G Sinai "Dynamical systems with elastic reflections", Russian Mathematical Surveys '''25''' pp. 137-189 (1970)]		#[http://dx.doi.org/10.1070/RM1970v025n02ABEH003794 Ya G Sinai "Dynamical systems with elastic reflections", Russian Mathematical Surveys '''25''' pp. 137-189 (1970)]
	#[http://library.lanl.gov/cgi-bin/getfile?15-18.pdf Adrian Patrascioiu "The Ergodic-Hypothesis, A Complicated Problem in Mathematics and Physics", Los Alamos Science, '''15''' pp. 263- (1987)]		#[http://library.lanl.gov/cgi-bin/getfile?15-18.pdf Adrian Patrascioiu "The Ergodic-Hypothesis, A Complicated Problem in Mathematics and Physics", Los Alamos Science, '''15''' pp. 263- (1987)]
	# Domokos Ssász "Boltzmann's Ergodic Hypothesis, a Conjecture for Centuries?", Studia Scientiarum Mathematicarum Hungarica '''31''' pp. 299-322 (1996) [http://iml.univ-mrs.fr/~lafont/rencontres/esi098.pdf (reprint)]		# Domokos Ssász "Boltzmann's Ergodic Hypothesis, a Conjecture for Centuries?", Studia Scientiarum Mathematicarum Hungarica '''31''' pp. 299-322 (1996) [http://iml.univ-mrs.fr/~lafont/rencontres/esi098.pdf (reprint)]
	[[category: Computer simulation techniques]]		[[category: Computer simulation techniques]]

Carl McBride: /* References */

2007-11-19T13:56:47Z

References

← Older revision		Revision as of 15:56, 19 November 2007
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	#[http://www.pnas.org/cgi/reprint/18/3/263 J. V. Neumann "Physical Applications of the Ergodic Hypothesis", PNAS '''18''' pp. 263-266 (1932)]		#[http://www.pnas.org/cgi/reprint/18/3/263 J. V. Neumann "Physical Applications of the Ergodic Hypothesis", PNAS '''18''' pp. 263-266 (1932)]
	#[http://www.pnas.org/cgi/reprint/18/3/279 G. D. Birkhoff and B. O. Koopman "Recent Contributions to the Ergodic Theory", PNAS '''18''' pp. 279-282 (1932)]		#[http://www.pnas.org/cgi/reprint/18/3/279 G. D. Birkhoff and B. O. Koopman "Recent Contributions to the Ergodic Theory", PNAS '''18''' pp. 279-282 (1932)]
			#Ya. G. Sinai "On the Foundation of the Ergodic Hypothesis for a Dynamical System of Statistical Mechanics", Soviet Math. Doklady '''4''' pp. 1818-1822 (1963)
			#[http://dx.doi.org/10.1070/RM1970v025n02ABEH003794 Ya G Sinai "Dynamical systems with elastic reflections", Russian Mathematical Surveys '''25''' pp. 137-189 (1970)]
	#[http://library.lanl.gov/cgi-bin/getfile?15-18.pdf Adrian Patrascioiu "The Ergodic-Hypothesis, A Complicated Problem in Mathematics and Physics", Los Alamos Science, '''15''' pp. 263- (1987)]		#[http://library.lanl.gov/cgi-bin/getfile?15-18.pdf Adrian Patrascioiu "The Ergodic-Hypothesis, A Complicated Problem in Mathematics and Physics", Los Alamos Science, '''15''' pp. 263- (1987)]
			# Domokos Ssász "Boltzmann's Ergodic Hypothesis, a Conjecture for Centuries?", Studia Scientiarum Mathematicarum Hungarica '''31''' pp. 299-322 (1996) [http://iml.univ-mrs.fr/~lafont/rencontres/esi098.pdf (reprint)]
	[[category: Computer simulation techniques]]		[[category: Computer simulation techniques]]

Carl McBride: /* See also */

2007-09-04T14:10:38Z

Carl McBride at 11:14, 14 August 2007

2007-08-14T11:14:23Z

← Older revision		Revision as of 13:14, 14 August 2007
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	:<math>\lim_{T \rightarrow \infty} \overline{O}_T (\{q_0(t)\},\{p_0(t)\}) = \langle O \rangle_\mu.</math>		:<math>\lim_{T \rightarrow \infty} \overline{O}_T (\{q_0(t)\},\{p_0(t)\}) = \langle O \rangle_\mu.</math>

	A restatement of the ergodic hypothesis is to say that all allowed states are equally probable.		A restatement of the ergodic hypothesis is to say that all allowed states are equally probable. This holds true if
			the ''metrical transitivity'' of general [[Hamiltonian]] systems holds true.
	==See also==		==See also==
	*[[Markov chain]]		*[[Markov chain]]

Carl McBride at 11:05, 14 August 2007

2007-08-14T11:05:11Z

← Older revision		Revision as of 13:05, 14 August 2007
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	The Ergodic hypothesis ~~(Ref 1 and 2)~~ essentially states that an ensemble average (i.e. [[Monte Carlo]]) of an observable, <math> \langle O \rangle_\mu</math> is equivalent to the time average, <math>\overline{O}_T</math> of an observable (i.e. [[molecular dynamics]]). ''i.e.''		The '''Ergodic hypothesis''' essentially states that an ensemble average (i.e. an instance of a [[Monte Carlo]] simulation) of an observable, <math> \langle O \rangle_\mu</math> is equivalent to the time average, <math>\overline{O}_T</math> of an observable (i.e. [[molecular dynamics]]). ''i.e.''

	:<math>\lim_{T \rightarrow \infty} \overline{O}_T (\{q_0(t)\},\{p_0(t)\}) = \langle O \rangle_\mu.</math>		:<math>\lim_{T \rightarrow \infty} \overline{O}_T (\{q_0(t)\},\{p_0(t)\}) = \langle O \rangle_\mu.</math>

Carl McBride: /* References */

2007-08-14T11:03:41Z

References

← Older revision		Revision as of 13:03, 14 August 2007
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	#[http://www.pnas.org/cgi/reprint/18/3/263 J. V. Neumann "Physical Applications of the Ergodic Hypothesis", PNAS '''18''' pp. 263-266 (1932)]		#[http://www.pnas.org/cgi/reprint/18/3/263 J. V. Neumann "Physical Applications of the Ergodic Hypothesis", PNAS '''18''' pp. 263-266 (1932)]
	#[http://www.pnas.org/cgi/reprint/18/3/279 G. D. Birkhoff and B. O. Koopman "Recent Contributions to the Ergodic Theory", PNAS '''18''' pp. 279-282 (1932)]		#[http://www.pnas.org/cgi/reprint/18/3/279 G. D. Birkhoff and B. O. Koopman "Recent Contributions to the Ergodic Theory", PNAS '''18''' pp. 279-282 (1932)]
	#Adrian Patrascioiu "The Ergodic-Hypothesis, A Complicated Problem in Mathematics and Physics", Los Alamos Science, '''15''' pp. 263- (1987)		#[http://library.lanl.gov/cgi-bin/getfile?15-18.pdf Adrian Patrascioiu "The Ergodic-Hypothesis, A Complicated Problem in Mathematics and Physics", Los Alamos Science, '''15''' pp. 263- (1987)]
	[[category: Computer simulation techniques]]		[[category: Computer simulation techniques]]

Carl McBride at 11:01, 14 August 2007

2007-08-14T11:01:57Z

← Older revision		Revision as of 13:01, 14 August 2007
Line 4:		Line 4:

	A restatement of the ergodic hypothesis is to say that all allowed states are equally probable.		A restatement of the ergodic hypothesis is to say that all allowed states are equally probable.
			==See also==
			*[[Markov chain]]
	==References==		==References==
	#[http://www.pnas.org/cgi/reprint/17/12/656 George D. Birkhoff, "Proof of the Ergodic Theorem", PNAS '''17''' pp. 656-660 (1931) ]		#[http://www.pnas.org/cgi/reprint/17/12/656 George D. Birkhoff, "Proof of the Ergodic Theorem", PNAS '''17''' pp. 656-660 (1931)]
			#[http://www.pnas.org/cgi/reprint/18/1/70 J. V. Neumann "Proof of the Quasi-ergodic Hypothesis", PNAS '''18''' pp. 70-82 (1932)]
			#[http://www.pnas.org/cgi/reprint/18/3/263 J. V. Neumann "Physical Applications of the Ergodic Hypothesis", PNAS '''18''' pp. 263-266 (1932)]
			#[http://www.pnas.org/cgi/reprint/18/3/279 G. D. Birkhoff and B. O. Koopman "Recent Contributions to the Ergodic Theory", PNAS '''18''' pp. 279-282 (1932)]
	#Adrian Patrascioiu "The Ergodic-Hypothesis, A Complicated Problem in Mathematics and Physics", Los Alamos Science, '''15''' pp. 263- (1987)		#Adrian Patrascioiu "The Ergodic-Hypothesis, A Complicated Problem in Mathematics and Physics", Los Alamos Science, '''15''' pp. 263- (1987)
	[[category: Computer simulation techniques]]		[[category: Computer simulation techniques]]

← Older revision		Revision as of 12:45, 29 July 2008
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	the ''metrical transitivity'' of general [[Hamiltonian]] systems holds true.		the ''metrical transitivity'' of general [[Hamiltonian]] systems holds true.
	==See also==		==See also==
			*[[Fermi-Pasta-Ulam experiment]]
	*[[Markov chain]]		*[[Markov chain]]
	*[[~~Fermi-Pasta-Ulam experiment~~]]		*[[Mixing systems]]

	==References==		==References==
	#[http://www.pnas.org/cgi/reprint/17/12/656 George D. Birkhoff, "Proof of the Ergodic Theorem", PNAS '''17''' pp. 656-660 (1931)]		#[http://www.pnas.org/cgi/reprint/17/12/656 George D. Birkhoff, "Proof of the Ergodic Theorem", PNAS '''17''' pp. 656-660 (1931)]

← Older revision		Revision as of 16:10, 4 September 2007
Line 7:		Line 7:
	==See also==		==See also==
	*[[Markov chain]]		*[[Markov chain]]
			*[[Fermi-Pasta-Ulam experiment]]

	==References==		==References==
	#[http://www.pnas.org/cgi/reprint/17/12/656 George D. Birkhoff, "Proof of the Ergodic Theorem", PNAS '''17''' pp. 656-660 (1931)]		#[http://www.pnas.org/cgi/reprint/17/12/656 George D. Birkhoff, "Proof of the Ergodic Theorem", PNAS '''17''' pp. 656-660 (1931)]