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| {{Use dmy dates|date=July 2013}}
| | If you present photography effectively, it helps you look much more properly at the globe around you. Offshore expert Word - Press developers high level of interactivity, accessibility, functionality and usability of our website can add custom online to using. SEO Ultimate - I think this plugin deserves more recognition than it's gotten up till now. Keep reading for some great Word - Press ideas you can start using today. If you beloved this article and also you would like to be given more info pertaining to [http://l.zennethe.net/wordpressbackupplugin67642 wordpress backup plugin] kindly visit the internet site. The number of options offered here is overwhelming, but once I took the time to begin to review the video training, I was amazed at how easy it was to create a squeeze page and a membership site. <br><br> |
| {{technical|date=November 2010}}
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| The '''Lieb–Liniger model''' describes a gas of particles moving in one dimension and satisfying [[Bose–Einstein statistics]].
| | Right starting from social media support to search engine optimization, such plugins are easily available within the Word - Press open source platform. You may either choose to link only to the top-level category pages or the ones that contain information about your products and services. There are number of web services that offer Word press development across the world. It primarily lays emphasis on improving the search engine results of your website whenever a related query is typed in the search box. W3C compliant HTML and a good open source powered by Word - Press CMS site is regarded as the prime minister. <br><br>It is very easy to install Word - Press blog or website. It was also the very first year that the category of Martial Arts was included in the Parents - Connect nationwide online poll, allowing parents to vote for their favorite San Antonio Martial Arts Academy. I hope this short Plugin Dynamo Review will assist you to differentiate whether Plugin Dynamo is Scam or a Genuine. To turn the Word - Press Plugin on, click Activate on the far right side of the list. So, if you are looking online to hire dedicated Wordpress developers, India PHP Expert can give a hand you in each and every best possible way. <br><br>Whether your Word - Press themes is premium or not, but nowadays every theme is designed with widget-ready. High Quality Services: These companies help you in creating high quality Word - Press websites. Enterprise, when they plan to hire Word - Press developer resources still PHP, My - SQL and watch with great expertise in codebase. It's now become a great place to sell it thanks to Woo - Commerce. OSDI, a Wordpress Development Company based on ahmedabad, India. <br><br>Every single module contains published data and guidelines, usually a lot more than 1 video, and when pertinent, incentive links and PDF files to assist you out. Being a Plugin Developer, it is important for you to know that development of Word - Press driven website should be done only when you enable debugging. By the time you get the Gallery Word - Press Themes, the first thing that you should know is on how to install it. Thus, Word - Press is a good alternative if you are looking for free blogging software. I have never seen a plugin with such a massive array of features, this does everything that platinum SEO and All In One SEO, also throws in the functionality found within SEO Smart Links and a number of other plugins it is essentially the swiss army knife of Word - Press plugins. |
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| ==Introduction==
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| A model of a gas of particles moving in one dimension and satisfying
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| [[Bose–Einstein statistics]] was introduced in 1963 <ref name="ll63">Elliott H. Lieb and Werner Liniger, ''Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State'', Physical
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| Review 130: 1605–1616, 1963</ref><ref name="l63">Elliott H. Lieb, ''Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum'', Physical Review 130:1616–1624,1963</ref> in order to study whether the available approximate theories
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| of such gases, specifically Bogolubov's theory, would conform to the
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| actual properties of the model gas. The model is based on a well
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| defined
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| Schrödinger Hamiltonian for particles interacting with each other
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| via a two-body potential, and all the eigenfunctions and eigenvalues of
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| this Hamiltonian can, in principle, be calculated exactly. Sometimes it is called one dimensional [[Bose gas]] with delta interaction. It also can be considered as quantum [[Non-linear Schroedinger equation]].
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| The ground state as well as the low-lying excited states were computed
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| and found to be in agreement with Bogolubov's theory when the potential
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| is small, except for the fact that there are actually two types of
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| elementary excitations instead of one, as predicted by Bogolubov's and
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| other theories.
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| The model seemed to be only of academic interest until, with the
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| sophisticated experimental techniques developed in the first decade of
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| the 21<math>^{\rm st}</math> century, it became possible to produce this
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| kind of gas using real atoms as particles.
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| ==Definition and solution of the Model==
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| There are <math>N</math> particles with coordinates <math> x </math> on the line <math>[0,L]</math>, with periodic boundary conditions. Thus, an allowed wave function <math>\psi(x_1, x_2, \dots, x_j, \dots,x_N)</math> is symmetric, i.e., <math>\psi(\dots, x_i,\dots, x_j, \dots) = \psi(\dots, x_j,\dots, x_i, \dots) </math> for all <math>i \neq j</math> and <math>\psi</math> satisfies <math>\psi( \dots, x_j=0, \dots ) =\psi(\dots, x_j=L,\dots )</math> for all <math>j</math>. The Hamiltonian, in appropriate units, is
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| : <math> H = -\sum\nolimits_{j=1}^N \partial^2/\partial x_j^2 +2c
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| \sum\nolimits_{1\leq i< j\leq N} \delta(x_i-x_j)\ , </math>
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| where <math>\delta </math> is the [[Dirac delta function]], i.e., the interaction is a contact interaction. The constant <math>c\geq 0</math> denotes its strength. The delta function gives rise to a boundary condition when two coordinates, say <math>x_1 </math> and <math>x_2</math> are equal; this condition is that as <math>x_2 \searrow x_1</math>, the derivative satisfies <math>(\frac{\partial}{\partial x_2} - \frac{\partial}{\partial x_1} ) \psi (x_1, x_2)|_{x_2=x_1+}= c \psi (x_1=x_2)</math>. The hard core limit <math>c=\infty</math> is known as the [[Tonks–Girardeau gas]].<ref>{{cite journal |first=Marvin |last=Girardeau |title=Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension |journal=Journal of Mathematical Physics |volume=1 |issue=6 |pages=516–523 |year=1960 |doi=10.1063/1.1703687 |bibcode = 1960JMP.....1..516G }}</ref>
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| Schrödinger's time independent equation, <math>H\psi = E\psi</math> is solved by explicit construction of <math>\psi</math>. Since <math>\psi </math> is symmetric it is completely determined by its values in the simplex <math>\mathcal{R} </math>, defined by the condition that <math>0\leq x_1\leq x_2 \leq \dots, \leq x_N \leq L</math>. In this region one looks for a <math>\psi </math> of the form considered by H.A. Bethe in 1931 in the context of magnetic spin systems—the [[Bethe ansatz]]. That is, for certain real numbers <math>k_1< k_2 < \cdots <k_N</math>, to be determined,
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| : <math> \psi(x_1, \dots, x_N) = \sum_P a(P)\exp \left( i \sum_{j=1}^N k_{P
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| j} x_j\right) </math>
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| where the sum is over all <math>N !</math> permutations, <math>P</math>, of the integers <math>1,2, \dots, N</math>, and <math>P</math> maps <math>1,2,\dots,N</math> to <math> P1,P2,\dots,PN</math>. The coefficients <math>a(P)</math>, as well as the <math>k</math>'s are determined by the condition <math>H\psi =E\psi</math>, and this leads to
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| : <math> E= \sum\nolimits_{j=1}^N\, k_j^2 </math>
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| : <math> a(P) = \prod\nolimits_{1\leq i<j \leq N}
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| \left(1+\frac{ic}{k_{Pi} -k_{Pj}}\right) \ . </math>
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| [[Teunis C. Dorlas|Dorlas]] (1993) proved that all eigenfunctions of <math>H</math> are of this form.<ref>{{cite journal |first=Teunis C. |last=Dorlas |title=Orthogonality and Completeness of the Bethe Ansatz Eigenstates of the nonlinear Schrödinger model |journal=Communications in Mathematical Physics |volume=154 |issue=2 |pages=347–376 |year=1993 |doi=10.1007/BF02097001 |bibcode = 1993CMaPh.154..347D }}</ref>
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| These equations determine <math>\psi</math> in terms of the <math>k</math>'s, which, in turn, are determined by the periodic boundary conditions. These lead to <math>N</math> equations:
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| : <math> L\, k_j= 2\pi I_j\ -2 \sum\nolimits_{i=1}^N \arctan
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| \left(\frac{k_j-k_i}{c} \right) \qquad \qquad \text{for } j=1, \,
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| \dots,\, N \ , </math>
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| where <math>I_1 < I_2<\cdots < I_N</math> are integers when <math>N</math> is odd and, when <math>N</math> is even, they take values <math>\pm \frac12, \pm \frac32, \dots</math> . For the ground state the <math>I</math>'s satisfy
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| : <math> I_{j+1} -I_j = 1, \quad {\rm for} \ 1\leq j <N \qquad
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| \text{and } I_1=-I_N. \, </math>
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| The first kind of elementary excitation consists in choosing <math>I_1,\dots, I_{N-1} </math> as before, but increasing <math>I_N</math> by an amount <math>n>0</math> (or decreasing <math>I_1</math> by <math>n</math>). The momentum of this state is <math>p= 2\pi n /L</math> (or <math>-2\pi n /L</math>).
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| For the second kind, choose some <math>0< n \leq N/2</math> and increase <math>I_i\to I_i+1</math> for all <math>i\geq n</math>. The momentum of this state is <math>p= \pi - 2\pi n/L</math>. Similarly, there is a state with <math>p= -\pi +2\pi n/L</math>. The momentum of this type of excitation is limited to <math>|p| \leq \pi.</math>
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| These excitations can be combined and repeated many times. Thus, they are bosonic-like. If we denote the ground state (= lowest) energy by <math>E_0</math> and the energies of the states mentioned above by <math>E_{1,2}(p)</math> then <math>\epsilon_{1}(p) = E_{1}(p)-E_0</math> and <math>\epsilon_{2}(p) = E_{2}(p)-E_0</math> are the excitation energies of the two modes. Later the model was also solved by [[Quantum inverse scattering method]]. This help to calculate correlation functions and to find conservation laws. The Hamiltonian of the model has a family of commuting operators: conservation laws. An explicit expression for higher conservation laws can be found in Davies and Korepin (1989).<ref>http://insti.physics.sunysb.edu/~korepin/davis.pdf</ref> | |
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| ==Thermodynamic limit==
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| [[File:lieb-liniger-1.gif|thumb|450px|right|F1| Fig. 1: The ground state
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| energy, from.<ref name="ll63"/> See text.]]
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| To discuss a gas we take a limit <math>N</math> and <math>L</math> to
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| infinity with the density <math>\rho =N/L</math> fixed. The ground state
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| energy per particle <math>e = E_0/N</math>, and the
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| <math>\epsilon_{1,2}(p)</math> all have limits as <math>N\to
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| \infty</math>. While there are two parameters, <math>\rho</math> and
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| <math>c</math>, simple length scaling <math>x\to \rho x</math> shows
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| that there is really only one, namely <math>\gamma =c/\rho</math>.
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| To evaluate <math>E_0</math> we assume that the ''N'' <math>k</math>'s lie between numbers <math>K</math> and −''K'', to be determined, and with a density <math>L\, f(k)</math>. This <math>f</math> is found to satisfy the equation (in the interval <math>-K \leq k \leq K</math>)
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| : <math> 2c\int\nolimits_{-K}^K \frac{f(p)}{c^2 +(p-k)^2} dp = 2\pi f(k)
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| -1 \quad {\rm and} \quad \int\nolimits_{-K}^K f(p) dp = \rho \ , </math>
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| which has a unique positive solution. An excitation distorts this density <math>f</math> and similar integral equations determine these distortions. The ground state energy per particle is given by
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| : <math> e = \frac{1}{\rho}\int\nolimits_{-K}^K k^2 f(k) dk .</math>
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| Figure 1 shows how <math>e</math> depends on <math>\gamma</math> and also shows Bogolubov's approximation to
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| <math>e</math>. The latter is asymptotically exact to second order in <math>\gamma</math>, namely, <math>e\approx \gamma
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| -4\gamma^{3/2}/\pi</math>. At <math>\gamma =\infty</math>, <math>e = | |
| \pi^2/3</math>.
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| [[File:lieb-liniger-2.gif|thumb|200px|right|F2| Fig. 2: The energies of the two types of excitations, from.<ref name="l63"/> See text.]]
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| Figure 2 shows the two excitation energies
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| <math>\epsilon_1(p)</math> and <math>\epsilon_2 (p)</math> for a small value of <math>\gamma = 0.787</math>. The two curves are similar to these for all values of <math>\gamma >0</math>, but the Bogolubov approximation (dashed) becomes worse as <math>\gamma </math> increases.
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| [[Correlation function (statistical mechanics)]] of the model was evaluated in the book entitled of ''Quantum Inverse Scattering Method and Correlation Functions'' by
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| V.E. Korepin, N.M. Bogoliubov and A.G. Izergin (Cambridge University Press, 1993).
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| <ref>V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, ''Quantum Inverse Scattering Method and Correlation Functions'', Cambridge University Press, 1993</ref>
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| [[Quantum entanglement]] of the model was calculated in Physical Review Letters, vol 92, issue 9, electronic identifier 096402, 5 March 2004 .
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| ==From three to one dimension.==
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| This one-dimensional gas can be made using real, three-dimensional atoms as particles. One can prove, mathematically, from the Schrödinger equation for three-dimensional particles in a long cylindrical container, that the low energy states are described by the one-dimensional Lieb–Liniger model. This was done for the ground state<ref>{{cite journal |first=Elliott H. |last=Lieb |first2=Robert |last2=Seiringer |first3=Jakob |last3=Yngvason |title=One-dimensional Bosons in Three-dimensional Traps |journal=[[Physical Review Letters]] |volume=91 |issue=15 |pages=150401 |year=2003 |doi=10.1103/PhysRevLett.91.150401 |arxiv=cond-mat/0304071 |bibcode = 2003PhRvL..91o0401L }}</ref> and for excited states.<ref>{{cite journal |first=Robert |last=Seiringer |first2=Jun |last2=Yin |title=The Lieb–Liniger Model as a Limit of Dilute Bosons in Three Dimensions |journal=Communications in Mathematical Physics |volume=284 |issue=2 |pages=459–479 |year=2008 |doi=10.1007/s00220-008-0521-6 |arxiv=0709.4022 |bibcode = 2008CMaPh.284..459S }}</ref> The cylinder does ''not'' have to be as narrow as the atomic diameter; it can be much wider if the excitation energy in the direction perpendicular to the axis is large compared to the energy per particle <math>e</math>.
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| ==References==
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| <!--- See [[Wikipedia:Footnotes]] on how to create references using <ref></ref> tags which will then appear here automatically -->
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| {{Reflist}}
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| ==External links==
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| * See also Elliott H. Lieb (2008), Scholarpedia, 3(12):8712.[http://www.scholarpedia.org/article/Lieb-Liniger_model_of_a_Bose_Gas]
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| * See also V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, ''Quantum Inverse Scattering Method and Correlation Functions'', Cambridge University Press, 1993.[http://books.google.com/books?id=kaZ0pKIHhxAC&dq=quantum+inverse+scattering+method&printsec=frontcover&source=bl&ots=4AaoICh4Q3&sig=8RoPelCcAOLTG1vY21DDLCulNGY&hl=en&ei=I6zuScSBE5a-M9q1pOwP&sa=X&oi=book_result&ct=result&resnum=8#v=onepage&q&f=false]
| |
| | |
| {{DEFAULTSORT:Lieb-Liniger model}}
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| [[Category:Statistical mechanics]]
| |
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