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The '''Jarzynski equality''' (JE) is an [[equation]] in [[statistical mechanics]] that relates [[Thermodynamic free energy|free energy]] differences between two ''equilibrium'' states and ''non-equilibrium'' processes. It is named after the physicist Christopher Jarzynski (then at [[Los Alamos National Laboratory]]) who derived it in 1997.
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In [[thermodynamics]], the free energy difference <math>\Delta F = F_B - F_A</math> between two states ''A'' and ''B'' is connected to the work ''W'' done on the system through the ''inequality'':
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: <math> \Delta F \leq W </math>,
 
which equality holding only in the case of a [[quasistatic process]], i.e. when one takes the system from ''A'' to ''B'' infinitely slowly.
 
In contrast to the thermodynamic statement above, the JE remains valid no matter how fast the process happens. The equality itself can be straightforwardly derived from the [[Crooks fluctuation theorem]]. The JE states:
 
: <math> e^ { -\Delta F / k T} = \overline{ e^{ -W/kT } }. </math>
 
Here ''k'' is the [[Boltzmann constant]] and ''T'' is the temperature of the system in the equilibrium state ''A'' or, equivalently, the temperature of the [[heat reservoir]] with which the system was thermalized before the process took place.
 
The over-line indicates an average over all possible realizations of an external process that takes the system from the equilibrium state ''A'' to a new, generally nonequilibrium state under the same external conditions as that of the equilibrium state ''B''. (For example, in the textbook case of a gas compressed by a piston, the gas is equilibrated at piston position ''A'' and compressed to piston position ''B''; in the Jarzynski equality, the final state of the gas does not need to be equilibrated at this new piston position). In the limit of an infinitely slow process, the work ''W'' performed on the system in each realization is numerically the same, so the average becomes irrelevant and the Jarzynski equality reduces to the thermodynamic equality <math>\Delta F = W</math> (see above). In general, however, ''W'' depends upon the specific initial [[microstate (statistical mechanics)|microstate]] of the system, though its average can still be related to <math>\Delta F</math> through an application of [[Jensen's inequality]] in the JE, viz.
 
: <math>\Delta F \leq \overline{W}, </math>
 
in accordance with the second law of thermodynamics.
 
Since its original derivation, the Jarzynski equality has been verified in a variety of contexts, ranging from experiments with biomolecules to numerical simulations. Many other theoretical derivations have also appeared, lending further confidence to its generality.
 
==History==
 
A question has been raised about who gave the earliest statement of the Jarzynski equality.  For example in 1977 the Russian physicists G.N. Bochkov and Yu. E. Kuzovlev (see Bibliography) proposed a generalized version of the Fluctuation-Dissipation relations which holds in the presence of arbitrary external time-dependent forces. Despite its close similarity to the JE, the Bochkov-Kuzovlev result does not relate free energy differences to work measurements, as discussed by Jarzynski himself in 2007 (see references below).
 
Another similar statement to the Jarzynski equality is the [[nonequilibrium partition identity]], which can be traced back to Yamada and Kawasaki. (The Nonequilibrium Partition Identity is the Jarzynski equality applied to two systems whose free energy difference is zero - like straining a fluid.) However, these early statements are very limited in their application. Both Bochkov and Kuzovlev as well as Yamada and Kawasaki consider a deterministic time reversible [[Hamiltonian system]]. As Kawasaki himself noted this precludes any treatment of nonequilibrium steady states. The fact that these nonequilibrium systems heat up forever because of the lack of any thermostatting mechanism leads to divergent integrals etc. No purely Hamiltonian description is capable of treating the experiments carried out to verify the [[Crooks fluctuation theorem]], Jarzynski equality and the [[Fluctuation theorem]]. These experiments involve thermostated systems in contact with heat baths.
 
==Bibliography==
* {{citation|first1=G. E.|last1=Crooks|title=Nonequilibrium measurements of free energy differences for microscopically reversible Markovian systems|journal= J. Stat. Phys.|volume=90|page=1481|year=1998|doi=10.1023/A:1023208217925}}
* {{citation|first1=C.|last1=Jarzynski|title=Nonequilibrium equality for free energy differences|journal=Phys. Rev. Lett.|volume=78| page=2690|year=1997|doi=10.1103/PhysRevLett.78.2690}} [http://www.papercore.org/Jarzynski1997 Papercore Summary http://www.papercore.org/Jarzynski1997]
* {{citation|first1=C.|last1=Jarzynski|title=Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach|journal=Phys. Rev. E|volume=56|page=5018|year=1997|doi=10.1103/PhysRevE.56.5018}}
 
For earlier results dealing with the statistics of work in adiabatic (i.e. Hamiltonian) nonequilibrium processes, see:
* {{citation|first1=G. N.| last1= Bochkov| first2=Yu. E. | last2=Kuzovlev|journal= Zh. Eksp. Teor. Fiz.| volume=72| page= 238| year= 1977}}; ''op. cit.'' '''76''', 1071 (1979)
* {{citation|first1=G. N.|last1= Bochkov| first2=Yu. E.|last2=Kuzovlev|journal=Physica A| title= Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics: I. Generalized fluctuation-dissipation theorem| volume=106| page=443| year=1981| doi=10.1016/0378-4371(81)90122-9}}; ''op. cit.'' '''106A''', 480 (1981)
* {{citation|first1=K.|last1=Kawasaki|first2=J.D.|last2=Gunton|title=Theory of Nonlinear Transport Processes: Nonlinear Shear Viscosity and Normal Stress Effects| journal=Phys. Rev. A|volume=8| page=2048| year=1973| doi=10.1103/PhysRevA.8.2048}}
* {{citation| first1=T.| last1=Yamada| first2=K.| last2=Kawasaki| journal=Prog. Theo. Phys.| title=Nonlinear Effects in the Shear Viscosity of Critical Mixtures| volume=38| page= 1031| year=1967 |doi=10.1143/PTP.38.1031}}
 
For a comparison of such results, see:
* {{citation|first1=C.|last1=Jarzynski|title=Comparison of far-from-equilibrium work relations|journal=Comptes Rendus Physique|volume=8|page=495|year=2007|doi=10.1016/j.crhy.2007.04.010}}
 
==See also==
* [[Fluctuation theorem]] - Provides an equality that quantifies fluctuations in time averaged entropy production in a wide variety of nonequilibrium systems.
* [[Crooks fluctuation theorem]] - Provides a fluctuation theorem between two equilibrium states.  Implies Jarzynski equality.
* [[Nonequilibrium partition identity]]
*[http://xstructure.inr.ac.ru/x-bin/theme2.py?arxiv=cond-mat&level=1&index1=6091 Jarzynski Equality on arxiv.org]
 
* "Fluctuation-Dissipation: Response Theory in Statistical Physics" by Umberto Marini Bettolo Marconi, Andrea Puglisi, Lamberto Rondoni, Angelo Vulpiani, [http://arxiv.org/abs/0803.0719]
 
[[Category:Statistical mechanics]]
[[Category:Non-equilibrium thermodynamics]]
[[Category:Equations]]

Revision as of 15:56, 4 March 2014

Normally, an aquarium like this can be about 48 inches long. The length and breadth usually are 13? and 21? 50 gallon aquarium , correspondingly. The size of tanks and aquariums still adjust as people today grow to be a lot more inventive using their patterns.

wikipedia.orgIf it is more desirable, your 55 gallon fish tank can be made vertically. One more convenient choice for you, is always to have it especially intended to squeeze into a wall surface. If you need to incorporate uniqueness, take into consideration making a crafted aquarium, Never stop there.

Can There Simply Be An individual Condition?

After you individualize your aquarium around, the chances are unlimited. For anyone who is happy with the usual unexciting rectangle aquarium, so whether it is. Nonetheless, if you prefer fish tank additional umph, then say the phrase and you may have what you may drive.

Benefits of Owning A 55 Gallon Fish Tank

1. Incredible Issue

The best advantage to having an aquarium is satisfaction - and bragging proper rights. Instantaneously it will become the focal point of each and every area that it must be viewable from. Your aquarium will take the gaze of the people who view it.

2. Feel Happy Impact

Different case reports have established that people whose lifestyles include things like aquariums have recognizable health improvements. They ordinarily imagine sharper, snooze sounder and communicate joy and happiness with greater frequency. Take advantage of the attractive hues, delicate tones and unlimited tranquility of your marine craft to cheer you up promptly.

3. Stress and anxiety, Stress and Anxiety Get Rid Of

Sick and tired of driving pills lower your palate to obtain by nerve-racking days or weeks? Avoid! You are able to aim your interest at a organic and beautiful treatment. Time will transfer even more without restraint that could also lessen any buildup of panic and discomfort.

4. Kids Come In Useful

Your young ones can have a break coming from the Television programs and relish the fish tank. They can have plenty of opportunities to discover something new. One more excelllent idea should be to work with the aquariums’ cleanup and eating plan to motivate young children to discover responsibilities.

5. Chill Out

The simple truth is that aquariums are seen to relax people. When looking at fish, even for youngsters that are hyperactive, mankind easily settle down. Aquariums have got a strange ability to have an impact on you on mental health, physical and emotional amounts, that is healthful.

6. Gaze To The Fish

Compared with hypnotherapy, your aquarium does far better at impacting you on emotional and physical levels all at once and right. Our company is products and solutions of character. There is certainly almost nothing that could effortlessly replace the effects which the tranquility of characteristics have for the our psyche. Provide underwater natural beauty in your office or home while not annoying the natural sequence of things.

Rid Yourself Of The Traditional Strategy

In the event you will, and this will develop very artistic layouts, imagine away from box, or the rectangle. Which I am sure one does, then join the revolution, in case you totally agree. Construct your individual aquatic work of art - 1 that one could be proud of.

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