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| [[Image:Feynman ratchet.png|right|300px|thumb|Schematic figure of a Brownian Ratchet]]
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| In the [[philosophy of thermal and statistical physics]], the '''Brownian ratchet''' or '''Feynman-Smoluchowski ratchet''' is a [[thought experiment]] about an apparent [[perpetual motion]] machine first analysed in 1912 by Polish physicist [[Marian Smoluchowski]]<ref name="Smoluchowski">M. von Smoluchowski (1912) Experimentell nachweisbare, der Ublichen Thermodynamik widersprechende Molekularphenomene, ''Phys. Zeitshur.'' '''13''', p.1069 cited in [http://books.google.com/books?id=KfG_CKbmy1cC&pg=PA59 Freund, Jan (2000) Stochastic Processes in Physics, Chemistry, and Biology, Springer, p.59]</ref> and popularised by American [[Nobel laureate]] physicist [[Richard Feynman]] in a [[physics]] lecture at the [[California Institute of Technology]] on May 11, 1962, and in his text ''[[The Feynman Lectures on Physics]]''<ref name="Feynman" >{{cite book
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| | last = Feynman | first = Richard P.
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| | title = The Feynman Lectures on Physics, Vol. 1
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| | publisher = Addison-Wesley
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| | year = 1963
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| | location = Massachusetts, USA
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| | pages = Chapter 46
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| | isbn =0-201-02116-1
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| | nopp = true }}</ref> as an illustration of the laws of [[thermodynamics]]. The simple machine, consisting of a tiny [[paddle wheel]] and a [[ratchet (device)|ratchet]], appears to be an example of a [[Maxwell's demon]], able to extract useful work from [[thermal fluctuations|random fluctuations]] (heat) in a system at [[thermal equilibrium]] in violation of the [[second law of thermodynamics]]. Detailed analysis by Feynman and others showed why it cannot actually do this.
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| ==The machine==
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| The device consists of a gear known as a [[ratchet (device)|ratchet]] that rotates freely in one direction but is prevented from rotating in the opposite direction by a [[Ratchet_(device)#Ratchet_and_pawl|pawl]]. The ratchet is connected by an axle to a [[paddle wheel]] that is immersed in a fluid of molecules at temperature <math>T_1</math>. The molecules constitute a [[heat bath]] in that they undergo random [[Brownian motion]] with a mean [[kinetic energy]] that is determined by the [[temperature]]. The device is imagined as being small enough that the impulse from a single molecular collision can turn the paddle. Although such collisions would tend to turn the rod in either direction with equal probability, the pawl allows the ratchet to rotate in one direction only. The net effect of many such random collisions should be sufficient for the ratchet to rotate continuously in that direction. The ratchet's motion then can be used to do [[work (physics)|work]] on other systems, for example lifting a weight (''m'') against gravity. The energy necessary to do this work apparently would come from the heat bath, without any heat gradient. Were such a machine to work successfully, its operation would violate the [[second law of thermodynamics]], one form of which states: "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work."
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| ==Why it fails==
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| Although at first sight the Brownian ratchet seems to extract useful work from Brownian motion, Feynman demonstrated that if the entire device is at the same temperature, the ratchet will not rotate continuously in one direction but will move randomly back and forth, and therefore will not produce any useful work. The reason is that the pawl, since it is at the same temperature as the paddle, will also undergo Brownian motion, "bouncing" up and down. It therefore will intermittently fail by allowing a ratchet tooth to slip backward under the pawl while it is up. Another issue is that when the pawl rests on the sloping face of the tooth, the spring which returns the pawl exerts a sideways force on the tooth which tends to rotate the ratchet in a backwards direction. Feynman demonstrated that if the temperature <math>T_2</math> of the ratchet and pawl is the same as the temperature <math>T_1</math> of the paddle, then the failure rate must equal the rate at which the ratchet ratchets forward, so that no net motion results over long enough periods or in an ensemble averaged sense.<ref name="Feynman" /> A simple but rigorous proof that no net motion occurs no matter what shape the teeth are was given by Magnasco.<ref name="forced">{{cite journal
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| | last = Magnasco | first = Marcelo O.
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| | title = Forced Thermal Ratchets
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| | journal = Physical Review Letters
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| | volume = 71 | issue = 10 | pages = 1477–1481
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| | year = 1993
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| | pmid = 10054418
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| | doi = 10.1103/PhysRevLett.71.1477
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| | bibcode=1993PhRvL..71.1477M
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| }}</ref>
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| If, on the other hand, <math>T_2</math> is smaller than <math>T_1</math>, the ratchet will indeed move forward, and produce useful work. In this case, though, the energy is extracted from the temperature gradient between the two thermal reservoirs, and some waste heat is exhausted into the lower temperature reservoir by the pawl. In other words, the device functions as a miniature [[heat engine]], in compliance with the second law of thermodynamics. Conversely, if <math>T_2</math> is greater than <math>T_1</math>, the device will rotate in the opposite direction.
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| The Feynman ratchet model led to the similar concept of [[Brownian motor]]s, [[nanomachine]]s which can extract useful work not from thermal noise but from [[chemical potential]]s and other microscopic [[Thermodynamic equilibrium|nonequilibrium]] sources, in compliance with the laws of thermodynamics.<ref name="forced" /><ref>
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| {{cite journal
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| | last = Magnasco | first = Marcelo O.
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| | title = Molecular Combustion Motors
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| | journal = Physical Review Letters
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| | volume = 72 | issue = 16 | pages = 2656–2659
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| | year = 1994
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| | pmid = 10055939
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| | doi = 10.1103/PhysRevLett.72.2656
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| | bibcode=1994PhRvL..72.2656M
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| }}</ref> [[Diode]]s are an electrical analog of the ratchet and pawl, and for the same reason cannot produce useful work by rectifying [[Johnson noise]] in a circuit at uniform temperature.
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| ==History==
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| The [[ratchet and pawl]] was first discussed as a Second Law-violating device by [[Gabriel Lippmann]] in 1900.<ref name="Harmer">
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| {{cite web
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| | last = Harmer | first = Greg
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| | coauthors = [[Derek Abbott]]
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| | title = The Feynman-Smoluchowski ratchet
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| | work = Parrondo's Paradox Research Group
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| | publisher = [http://www1.eleceng.adelaide.edu.au/ School of Electrical & Electronic Engineering, Univ. of Adelaide]
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| | year = 2005
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| | url = http://www.eleceng.adelaide.edu.au/Groups/parrondo/ratchet.html
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| | accessdate = 2010-01-15}}</ref> In 1912, Polish physicist [[Marian Smoluchowski]]<ref name="Smoluchowski" /> gave the first correct qualitative explanation of why the device fails; thermal motion of the pawl allows the ratchet's teeth to slip backwards. Feynman did the first quantitative analysis of the device in 1962 using the [[Maxwell-Boltzmann distribution]], showing that if the temperature of the paddle ''T<sub>1</sub>'' was greater than the temperature of the ratchet ''T<sub>2</sub>'', it would function as a [[heat engine]], but if ''T<sub>1</sub>'' = ''T<sub>2</sub>'' there would be no net motion of the paddle. In 1996, [[Juan Parrondo]] and Pep Español used a variation of the above device in which no ratchet is present, only two paddles, to show that the axle connecting the paddles and ratchet conducts heat between reservoirs; they argued that although Feynman's conclusion was correct, his analysis was flawed because of his erroneous use of the [[Quasistatic process|quasistatic]] approximation, resulting in incorrect equations for efficiency.<ref>{{cite journal
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| | last = Parrondo | first = Juan M. R.
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| | coauthors = Pep Español
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| | title = Criticism of Feynman's analysis of the ratchet as an engine
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| | journal = American Journal of Physics
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| | volume = 64 | issue = 9 | pages = 1125
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| | date = March 8, 1996
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| | doi = 10.1119/1.18393
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| |bibcode = 1996AmJPh..64.1125P }}</ref> [[Marcelo Osvaldo Magnasco|Magnasco]] and Stolovitzky (1998) extended this analysis to consider the full ratchet device, and showed that the power output of the device is far smaller than the [[Carnot efficiency]] claimed by Feynman.<ref>{{cite journal
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| | last = Magnasco | first = Marcelo O.
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| | coauthors = Gustavo Stolovitzky
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| | title = Feynman's Ratchet and Pawl
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| | journal = Journal of Statistical Physics
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| | volume = 93 | issue = 3 | pages = 615
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| | year = 1998
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| | doi = 10.1023/B:JOSS.0000033245.43421.14
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| |bibcode = 1998JSP....93..615M }}</ref> A paper in 2000 by [[Derek Abbott]], [[Bruce R. Davis]] and Juan Parrondo, reanalyzed the problem and extended it to the case of multiple ratchets, showing a link with [[Parrondo's paradox]].<ref>
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| {{cite conference
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| | first = Derek | last = Abbott
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| | coauthors = Bruce R. Davis, Juan M. R. Parrondo
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| | title = The problem of detailed balance for the Feynman-Smoluchowski Engine and the multiple pawl paradox
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| | booktitle = Unsolved Problems of Noise and Fluctuations
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| | pages = 213–218
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| | publisher = American Institute of Physics
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| | year = 2000
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| | url = http://www.eleceng.adelaide.edu.au/Personal/dabbott/publications/UPN_abbott1999.pdf
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| | accessdate = 2010-01-15
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| }}</ref>
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| [[File:Brillouin-Paradoxon.svg|thumb|Brillouin paradox: an electrical analogue of the Brownian ratchet.]]
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| [[Léon Brillouin]] in 1950 discussed an electrical analogy that uses a [[rectifier]] (such as a diode) instead of a ratchet.<ref>{{cite doi|10.1103/PhysRev.78.627.2}}</ref> The idea was that the [[Johnson noise|thermal current fluctuations]] impinging on the diode should be rectified, and therefore spontaneously produce a nonzero constant voltage offset that can be used to perform work. In the detailed analysis it was shown that the thermal fluctuations within the diode generate an [[electromotive force]] that cancels the voltage from rectified current fluctuations. For this reason, the diode will only produce a nonzero voltage when the impinging current fluctuations have a different temperature than the diode itself.<ref>{{cite doi|10.1063/1.1652709}}</ref>
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| ==Granular gas==
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| Researchers from the University of Twente, the University of Patras in Greece, and the Foundation for Fundamental Research on Matter have constructed a Feynman-Smoluchowski engine which, when not in thermal equilibrium, converts pseudo-[[Brownian motion]] into [[work (physics)|work]] by means of a [[Granular_material#Granular_gases|granular gas]],<ref>[http://www.fom.nl/live/english/news/artikel.pag?objectnumber=120223 "Classical thought experiment brought to life in granular gas"], ''[[Foundation for Fundamental Research on Matter]]'', Utrecht, 18 June 2010. Retrieved on 2010-06-24.</ref> which is a conglomeration of solid particles vibrated with such vigour that the system assumes a gas-like state. The constructed engine consisted of four vanes which were allowed to rotate freely in a vibrofluidized granular gas.<ref name="eshuis">
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| {{Cite journal
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| | author = Peter Eshuis, Ko van der Weele, Detlef Lohse, and Devaraj van der Meer
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| |date=June 2010
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| | title = Experimental Realization of a Rotational Ratchet in a Granular Gas
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| | journal = Physical Review Letters
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| | volume = 104 | issue = 24 | pages = 4
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| | doi = 10.1103/PhysRevLett.104.248001
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| | pmid=20867337
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| | bibcode=2010PhRvL.104x8001E
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| }}</ref> Because the ratchet's gear and pawl mechanism, as described above, permitted the axle to rotate only in one direction, random collisions with the moving beads caused the vane to rotate. This seems to contradict Feynman's hypothesis. However, this system is not in perfect thermal equilibrium: energy is constantly being supplied to maintain the fluid motion of the beads. Vigorous vibrations on top of a shaking device mimic the nature of a molecular gas. Unlike an [[ideal gas]], though, in which tiny particles move constantly, stopping the shaking would simply cause the beads to drop. In the experiment, this necessary out-of-equilibrium environment was thus maintained. Work was not immediately being done, though; the ratchet effect only commenced beyond a critical shaking strength. For very strong shaking, the vanes of the paddle wheel interacted with the gas, forming a convection roll, sustaining their rotation.<ref name="eshuis" /> The experiment was [http://stilton.tnw.utwente.nl/dryquicksand/ratchet/ratchet.html filmed].
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| == See also == | |
| *[[Brownian motion]]
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| *[[Quantum stirring, ratchets, and pumping]]
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| *[[Geometric phase]] (section Stochastic Pump Effect)
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| *[[Parrondo's paradox]]
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| == Notes ==
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| {{Reflist}}
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| == External links ==
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| {{Commons category|Brownian ratchets}}
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| *[http://monet.physik.unibas.ch/~elmer/bm/#why Why is a Brownian motor not a ''perpetuum mobile'' of the second kind?]
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| *[http://www.kawai.phy.uab.edu/research/motor/ Coupled Brownian Motors - Can we get work out of unbiased fluctuation?]
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| *[http://www.physorg.com/news195819314.html Experiment finally proves 100-year-old thought experiment is possible (w/ Video)]
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| ;Articles
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| *{{cite journal |author=Astumian RD |title=Thermodynamics and kinetics of a Brownian motor |journal=Science |volume=276 |issue= 5314|pages=917–22 |year=1997 |doi=10.1126/science.276.5314.917 |pmid=9139648}}
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| *{{cite journal |author=Astumian RD, Hänggi P |title=Brownian Motors |journal=Physics Today |volume=55 |issue=11 |pages=33–9 |year=2002 |url=http://www.physik.uni-augsburg.de/theo1/hanggi/Papers/309.pdf |doi=10.1063/1.1535005|bibcode = 2002PhT....55k..33A }}
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| *{{cite journal |author=Hänggi P, Marchesoni F, Nori F |title=Brownian Motors |journal=A. Physik (Leipzig) |volume=14 |issue= 1-3|pages=51–70 |year=2005 |url=http://www.physik.uni-augsburg.de/theo1/hanggi/History/BrownianmotorsAnnPhys.pdf |doi=10.1002/andp.200410121|arxiv = cond-mat/0410033 |bibcode = 2005AnP...517...51H }}
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| *Lukasz Machura: ''Performance of Brownian Motors''. University of Augsburg, 2006 ([http://www.opus-bayern.de/uni-augsburg/volltexte/2006/222/ PDF])
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| *{{cite journal |author=Peskin CS, Odell GM, Oster GF |title=Cellular motions and thermal fluctuations: the Brownian ratchet |journal=Biophys. J. |volume=65 |issue=1 |pages=316–24 |date=July 1993 |pmid=8369439 |pmc=1225726 |doi=10.1016/S0006-3495(93)81035-X |bibcode=1993BpJ....65..316P}}
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| *{{cite journal |author=Hänggi P, Marchesoni F |title=Artificial Brownian motors: Controlling transport on the nanoscale: Review |url=http://www.physik.uni-augsburg.de/theo1/hanggi/Papers/515.pdf|arxiv=0807.1283 |year=2008 |bibcode = 2009RvMP...81..387H |doi = 10.1103/RevModPhys.81.387 }}
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| *{{cite journal | last1 = van Oudensaarden | first1 = A. | first2= S. G. | last2=Boxer | author-separator =, | author-name-separator= | year = 1999 | title = Brownian Ratchets: Molecular Separations in Lipid Bilayers Supported on Patterned Arrays | url = http://web.mit.edu/biophysics/papers/SCIENCE1999.pdf | journal = Science | volume = 285 | pages = 1046–1048 | }}
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| {{DEFAULTSORT:Brownian Ratchet}}
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| [[Category:Thought experiments in physics]]
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| [[Category:Richard Feynman]]
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| [[Category:Philosophy of thermal and statistical physics]]
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| [[Category:Concepts in physics]]
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| [[Category:Nanotechnology]]
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| [[Category:Perpetual motion]]
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