Distributed lag - Revision history

208.50.124.65: /* Infinite distributed lags */ ce

2014-11-13T18:03:42Z

Infinite distributed lags: ce

← Older revision		Revision as of 20:03, 13 November 2014
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	~~First introduced by M~~. ~~Pollak,<ref>M~~. ~~Pollak~~. ~~Discuss~~. ~~Faraday Soc~~. ~~50 (1970), p~~. 13		The Internet has become an extremely crowded place. Couple of years back the number of websites about the Web wasn"t as big because it is today. Today the number of sites online is in millions. In this example it is very hard for almost any website to stand-out of the group. No website today can endure without proper promotional efforts to create your website a well visited one. To make sure a regular movement of traffic towards your site you have to get it search-engine enhanced. Search Engine Optimization is vital nowadays. Learn further on our favorite partner portfolio - Click here: [http://pom.chem.uw.edu.pl/dokuwiki/doku.php?id=Enhance_Search_Engine_Rankings_Using_Online_Monopoly_Techniques_41652 enhance_search_engine_rankings_using_online_monopoly_techniques_41652 [POM Do]. No site may survive today without seo. Seo makes it possible to in getting the internet site discovered. Your site rating can be enhanced and this can be your key to online success. My co-worker found out about [http://about.me/recommendedreadingboard look at this] by searching the San Francisco Post. <br><br>
	~~</ref> the '''Coulomb gap'''~~ is ~~a soft gap in the Single~~-~~Particle~~ [[~~Density of states\|Density of States~~]~~] (DOS) of a system of interacting localized electrons~~.
	~~Due~~ to the ~~long~~-~~range Coulomb interactions, the single-particle DOS vanishes~~ at the ~~chemical potential, at low enough temperatures, such that thermal excitations do not wash out the gap~~.

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	~~At zero temperature, a classical treatment~~ of a system gives an upper bound for the DOS near the [[Fermi]]-energy, first suggested by Efros and Shklovskii.<ref name="EfrosShklovskii1">A L Efros and B I Shklovskii , J. Phys. C8, L49 (1975)</ref> The argument is ~~as follows:~~
	~~Let us look at~~ the ~~[[ground state]] configuration of the system~~. ~~Defining <math> E_i </math> as the energy of an~~ [~~[electron]] at site <math> i <~~/math>, due to the disorder and the Coulomb interaction with all other electrons (we define this both for occupied and unoccupied sites), it is easy to see that the energy needed to move an electron from an occupied site <math> i </~~math> to an unoccupied site <math> j <~~/~~math> is given by the expression:~~

	~~:<math>\Delta E=E_j~~-~~E_i~~-e^2/~~r_{ij} </math>.~~
	~~The subtraction of the last term accounts for the fact that <math> E_j </math> contains a term due~~ to ~~the interaction~~ with ~~the electron present at site <math> i </math>, but after moving the electron~~ this ~~term should not be considered~~. ~~It is easy to see from this that there exists an energy <math> E_f </math> such that all sites with energies above it are empty, and below it are full (this is~~ the ~~Fermi energy~~, ~~but since we are dealing with a system with interactions it is not obvious a-priori that it is still well-defined).~~
	~~Assume we have~~ a finite single-particle DOS at the Fermi energy, <math> g(E_f) </math>. For every possible transfer of an electron from an occupied site i to an unoccupied site j, the energy invested should be positive, since we are assuming we are in the ~~ground state~~ of ~~the system, i.e., <math>\Delta E>=0 </math>.~~
	~~Assuming we have~~ a ~~large system, let us consider all the sites with energies in the interval <math> [E_f-\epsilon, E_f+\epsilon]~~. ~~</math>~~ The ~~number~~ of ~~these, by assumption, is <math> N= 2 \epsilon g(E_f). </math> As explained, <math> N/2</math>~~ of ~~these would be occupied, and the others unoccupied~~. ~~Of all pairs~~ of ~~occupied and unoccupied sites, let us choose the one where the two~~ are ~~closest to each other~~. ~~If we assume the~~ sites ~~are randomly distributed in space~~, ~~we find that the distance between these two sites is of order:~~
	~~<math> R \sim (N/V)^{-1/d} </math>~~, ~~where d is the dimension~~ of ~~space.~~
	~~Plugging the expression for <math> N </math> into the previous equation, we obtain the inequality:~~
	~~<math> E_j-E_i-C e^2 (\epsilon g(E_f)/V)^{1/d} >0 </math> where <math> C</math>~~ is a ~~coefficient of order unity~~. ~~Since <math> E_j-E_i <2\epsilon </math>, this inequality will necessarily be violated for small enough <math> \epsilon </math>. Hence, assuming~~ a finite DOS at <math> E_f </math> led to a contradiction. Repeating the above calculation under the assumption that the DOS near <math> E_f </math> is proportional to <math> (E-E_f)^\alpha </math> shows that <math> \alpha>=d-1 </math>. This is an upper bound for the Coulomb gap. Efros <ref>A. L. Efros, ~~J. Phys. C: Solid State Phys 9, 2021 (1976)</ref> considered single electron excitations, and obtained an integro-differential equation for~~ the ~~DOS, showing the Coulomb gap~~ in ~~fact follows~~ the ~~above equation (i~~.~~e., the upper bound~~ is ~~a tight bound).~~

	~~Other treatments of~~ the ~~problem include a mean-field numerical approach~~,~~<ref>M. Grunewald, B. Pohlmann, L~~. ~~Schweitzer, and D.Wurtz,J. Phys. C: Solid State Phys., 15, L1153 (1982)</ref> as well as~~ more ~~recent treatments such as~~,~~<ref>M. Muller and S. Pankov, Phys. Rev. B. 75, 144201 (2007)</ref>~~ also ~~verifying the upper bound suggested above is a tight bound. Many Monte-Carlo simulations were also performed,<ref>J. H. Davies, P. A. Lee, and T. M. Rice, Phys. Rev. Lett. 49, 758 - 761 (1982)~~
	~~</ref><ref>A. Mobius, M. Richter, and B. Drittler, Phys. Rev. B 45, 11568 (1992)</ref> some of them~~ in ~~disagreement with the result quoted above~~. ~~Few works deal with~~ the ~~quantum aspect~~ of ~~the problem.<ref>G. Vignale~~, ~~Phys~~. ~~Rev. B 36, 8192(1987)~~
	~~</ref>~~

	~~== Experimental observations ==~~

	~~Direct experimental confirmation of the gap has been done via tunneling experiments, which probed the single-particle DOS~~ in ~~two and three dimensions~~,.<~~ref~~>~~J. G. Massey and M. Lee, Phys. Rev. Lett. 75, 4266 (1995)~~<~~/ref~~>~~<ref>V~~. Y. ~~Butko, J. F. Ditusa, and P. W. Adams, Phys. Rev. Lett. 84, 1543 (2000).</ref> The experiments clearly showed a linear gap in two-dimensions, and a parabolic gap in three-dimensions.~~
	~~Another experimental consequence of the Coulomb gap is found in~~ the ~~conductivity~~ of ~~samples in the localized regime~~.
	~~The existence of a gap in~~ the ~~spectrum~~ of ~~excitations would result in a lowered conductivity than that predicted by [[Variable range hopping\|Mott Variable range hopping]]. If~~ one ~~uses~~ the ~~analytical expression of the Single-Particle DOS~~ in ~~the Mott derivation, a universal <math> e^{-1/T^{1/2}} </math> is obtained, for any dimension~~.~~<ref>B. Shklovskii and A. Efros, Electronic properties of doped~~
	~~semiconductors (Springer-Verlag, Berlin, 1984).</ref> The observation of~~ this ~~is expected to occur below a certain temperature, such that the optimal energy of hopping would be smaller than the width of the Coulomb gap~~. ~~The transition from Mott to so-called Efros-Shklovskii Variable Range Hopping has been observed experimentally for various systems~~.<~~ref~~>~~A. Y. Rogatchev and U. Mizutani, Phys. Rev. B. 61, 15550~~
	~~(2000).~~<~~/ref~~> ~~Nevertheless, no rigorous derivation of the Efros-Shklovskii conductivity formula has been put forth,~~ and ~~in some experiments <math> e^{-1~~/~~T^{\alpha}} <~~/~~math> behavior is observed~~, ~~for alpha which fits neither~~ the ~~Mott nor the Efros-Shklovskii theories~~.

	~~==See also==~~
	* [[Coulomb's Law]]

	~~== References ==~~
	~~{{reflist}}~~

	~~==External links==~~

	~~{{DEFAULTSORT:Coulomb Gap}}~~
	~~[[Category:Condensed matter physics]]~~
	~~[[Category:Statistical mechanics]]~~
	~~[[Category:Concepts in physics]]~~

en>CmdrObot: sp: et. al.→et al.

2013-06-22T22:01:41Z

sp: et. al.→et al.

← Older revision		Revision as of 00:01, 23 June 2013
Line 1:		Line 1:
	~~Some much less like~~ to ~~play basketball as you~~,~~how~~ do ~~you choice your basketball boots~~ and ~~shoes?~~I ~~think many more choise~~ the ~~air jordan,~~as ~~people?Yeah~~,~~maybe you may not~~,~~but I know more people like~~ it.<br><br>~~Rack~~-~~and~~-~~field spike development contributes~~ to the ~~creation of~~ the ~~Elite~~, ~~one of air jordan shoes jordan best~~-~~fitting and best~~-~~selling racing and training shoes~~. ~~Continued innovative research and development yields~~ a ~~rapidly growing line~~ of ~~athletic shoes designed and built~~ to ~~air jordan specifications~~. The ~~intense future~~ is ~~dimmed when Steve Prefontaine dies before his era~~.<br><br>~~But what phrase will likely~~ be ~~considered essentially~~ the ~~most relevant associated with?~~ Of ~~course~~, ~~"Air Jordan shoes"~~. ~~Those webpages containing~~ the ~~idea~~ of ~~"Air Jordan shoes" would surely looked~~ into ~~very suitable.~~<br><br>~~Nike air jordan~~ is a ~~good shoe~~. ~~It is additionally expensive as all models are regarding excess of $100~~. ~~However~~, ~~now~~ that ~~there~~ is ~~a range picked online~~, ~~you can buy your shoes from online shoe stores at bargains~~. ~~Nike air jordan is offered nearly $100 off over these shoes~~. ~~The cherry on~~ the ~~cake proven~~ fact ~~that many stores offer free shipping~~. So, it is ~~discounts right up for you have to~~.<br><br>~~He won~~ the ~~scoring title ten times~~, ~~understanding that~~ in ~~only eleven seasons~~. ~~He and~~ the ~~legendary Wilt Chamberland are given that they ones to find scored three~~,~~000 points from a season~~. ~~Jordan did it~~ in ~~'87 with~~ three,~~041~~. ~~On average~~ and ~~over his full eleven year career his per game point score was 31+~~.<br><br>The ~~last highlight~~ is the ~~wonderful~~ a ~~sense passing and shooting crafted from~~ the ~~SF III dance shoes~~. ~~And you furthermore feel like touching~~ the ~~ball tremendously. This should be an objective~~ of the ~~Superlfy III Soccer cleats~~.<br><br>~~If you cherished~~ this ~~article and you simply~~ would ~~like~~ to ~~acquire more info relating to [http://www~~.~~plongeeo~~.~~com~~/~~chaussure~~-~~air~~-~~jordan-pas-cher~~/ ~~chaussure jordan pas cher] nicely visit our own web~~-~~page~~.		First introduced by M. Pollak,<ref>M. Pollak. Discuss. Faraday Soc. 50 (1970), p. 13
			</ref> the '''Coulomb gap''' is a soft gap in the Single-Particle [[Density of states\|Density of States]] (DOS) of a system of interacting localized electrons.
			Due to the long-range Coulomb interactions, the single-particle DOS vanishes at the chemical potential, at low enough temperatures, such that thermal excitations do not wash out the gap.

			== Theory ==
			At zero temperature, a classical treatment of a system gives an upper bound for the DOS near the [[Fermi]]-energy, first suggested by Efros and Shklovskii.<ref name="EfrosShklovskii1">A L Efros and B I Shklovskii , J. Phys. C8, L49 (1975)</ref> The argument is as follows:
			Let us look at the [[ground state]] configuration of the system. Defining <math> E_i </math> as the energy of an [[electron]] at site <math> i </math>, due to the disorder and the Coulomb interaction with all other electrons (we define this both for occupied and unoccupied sites), it is easy to see that the energy needed to move an electron from an occupied site <math> i </math> to an unoccupied site <math> j </math> is given by the expression:

			:<math>\Delta E=E_j-E_i-e^2/r_{ij} </math>.
			The subtraction of the last term accounts for the fact that <math> E_j </math> contains a term due to the interaction with the electron present at site <math> i </math>, but after moving the electron this term should not be considered. It is easy to see from this that there exists an energy <math> E_f </math> such that all sites with energies above it are empty, and below it are full (this is the Fermi energy, but since we are dealing with a system with interactions it is not obvious a-priori that it is still well-defined).
			Assume we have a finite single-particle DOS at the Fermi energy, <math> g(E_f) </math>. For every possible transfer of an electron from an occupied site i to an unoccupied site j, the energy invested should be positive, since we are assuming we are in the ground state of the system, i.e., <math>\Delta E>=0 </math>.
			Assuming we have a large system, let us consider all the sites with energies in the interval <math> [E_f-\epsilon, E_f+\epsilon]. </math> The number of these, by assumption, is <math> N= 2 \epsilon g(E_f). </math> As explained, <math> N/2</math> of these would be occupied, and the others unoccupied. Of all pairs of occupied and unoccupied sites, let us choose the one where the two are closest to each other. If we assume the sites are randomly distributed in space, we find that the distance between these two sites is of order:
			<math> R \sim (N/V)^{-1/d} </math>, where d is the dimension of space.
			Plugging the expression for <math> N </math> into the previous equation, we obtain the inequality:
			<math> E_j-E_i-C e^2 (\epsilon g(E_f)/V)^{1/d} >0 </math> where <math> C</math> is a coefficient of order unity. Since <math> E_j-E_i <2\epsilon </math>, this inequality will necessarily be violated for small enough <math> \epsilon </math>. Hence, assuming a finite DOS at <math> E_f </math> led to a contradiction. Repeating the above calculation under the assumption that the DOS near <math> E_f </math> is proportional to <math> (E-E_f)^\alpha </math> shows that <math> \alpha>=d-1 </math>. This is an upper bound for the Coulomb gap. Efros <ref>A. L. Efros, J. Phys. C: Solid State Phys 9, 2021 (1976)</ref> considered single electron excitations, and obtained an integro-differential equation for the DOS, showing the Coulomb gap in fact follows the above equation (i.e., the upper bound is a tight bound).

			Other treatments of the problem include a mean-field numerical approach,<ref>M. Grunewald, B. Pohlmann, L. Schweitzer, and D.Wurtz,J. Phys. C: Solid State Phys., 15, L1153 (1982)</ref> as well as more recent treatments such as,<ref>M. Muller and S. Pankov, Phys. Rev. B. 75, 144201 (2007)</ref> also verifying the upper bound suggested above is a tight bound. Many Monte-Carlo simulations were also performed,<ref>J. H. Davies, P. A. Lee, and T. M. Rice, Phys. Rev. Lett. 49, 758 - 761 (1982)
			</ref><ref>A. Mobius, M. Richter, and B. Drittler, Phys. Rev. B 45, 11568 (1992)</ref> some of them in disagreement with the result quoted above. Few works deal with the quantum aspect of the problem.<ref>G. Vignale, Phys. Rev. B 36, 8192(1987)
			</ref>

			== Experimental observations ==

			Direct experimental confirmation of the gap has been done via tunneling experiments, which probed the single-particle DOS in two and three dimensions,.<ref>J. G. Massey and M. Lee, Phys. Rev. Lett. 75, 4266 (1995)</ref><ref>V. Y. Butko, J. F. Ditusa, and P. W. Adams, Phys. Rev. Lett. 84, 1543 (2000).</ref> The experiments clearly showed a linear gap in two-dimensions, and a parabolic gap in three-dimensions.
			Another experimental consequence of the Coulomb gap is found in the conductivity of samples in the localized regime.
			The existence of a gap in the spectrum of excitations would result in a lowered conductivity than that predicted by [[Variable range hopping\|Mott Variable range hopping]]. If one uses the analytical expression of the Single-Particle DOS in the Mott derivation, a universal <math> e^{-1/T^{1/2}} </math> is obtained, for any dimension.<ref>B. Shklovskii and A. Efros, Electronic properties of doped
			semiconductors (Springer-Verlag, Berlin, 1984).</ref> The observation of this is expected to occur below a certain temperature, such that the optimal energy of hopping would be smaller than the width of the Coulomb gap. The transition from Mott to so-called Efros-Shklovskii Variable Range Hopping has been observed experimentally for various systems.<ref>A. Y. Rogatchev and U. Mizutani, Phys. Rev. B. 61, 15550
			(2000).</ref> Nevertheless, no rigorous derivation of the Efros-Shklovskii conductivity formula has been put forth, and in some experiments <math> e^{-1/T^{\alpha}} </math> behavior is observed, for alpha which fits neither the Mott nor the Efros-Shklovskii theories.

			==See also==
			* [[Coulomb's Law]]

			== References ==
			{{reflist}}

			==External links==

			{{DEFAULTSORT:Coulomb Gap}}
			[[Category:Condensed matter physics]]
			[[Category:Statistical mechanics]]
			[[Category:Concepts in physics]]

en>EmausBot: r2.7.2+) (Robot: Adding tr:Almon modeli

2012-08-15T17:50:28Z

r2.7.2+) (Robot: Adding tr:Almon modeli

New page

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