[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/nakajima-zwanzig-equation-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/nakajima-zwanzig-equation-wikipedia\/","headline":"Nakajima\u2013Zwanzig equation – Wikipedia","name":"Nakajima\u2013Zwanzig equation – Wikipedia","description":"From Wikipedia, the free encyclopedia Integral equation in quantum simulations The Nakajima\u2013Zwanzig equation (named after the physicists who developed it,","datePublished":"2019-09-17","dateModified":"2019-09-17","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d1ce8d3dacefcd6793ddace1f72a96b96fc112e0","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d1ce8d3dacefcd6793ddace1f72a96b96fc112e0","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/nakajima-zwanzig-equation-wikipedia\/","wordCount":4712,"articleBody":"From Wikipedia, the free encyclopediaIntegral equation in quantum simulationsThe Nakajima\u2013Zwanzig equation (named after the physicists who developed it, Sadao Nakajima[1] and Robert Zwanzig[2]) is an integral equation describing the time evolution of the “relevant” part of a quantum-mechanical system. It is formulated in the density matrix formalism and can be regarded a generalization of the master equation.The equation belongs to the Mori-Zwanzig formalism within the statistical mechanics of irreversible processes (named after Hazime Mori). By means of a projection operator the dynamics is split into a slow, collective part (relevant part) and a rapidly fluctuating irrelevant part. The goal is to develop dynamical equations for the collective part.Table of ContentsDerivation[edit]See also[edit]References[edit]External links[edit]Derivation[edit]The starting point[note 1] is the quantum mechanical version of the von Neumann equation, also known as the Liouville equation:\u2202t\u03c1=i\u210f[\u03c1,H]=L\u03c1,{displaystyle partial _{t}rho ={frac {i}{hbar }}[rho ,H]=Lrho ,}where the Liouville operator L{displaystyle L} is defined as LA=i\u210f[A,H]{displaystyle LA={frac {i}{hbar }}[A,H]}.The density operator (density matrix) \u03c1{displaystyle rho } is split by means of a projection operatorP{displaystyle {mathcal {P}}}into two parts\u03c1=(P+Q)\u03c1{displaystyle rho =left({mathcal {P}}+{mathcal {Q}}right)rho },where Q\u22611\u2212P{displaystyle {mathcal {Q}}equiv 1-{mathcal {P}}}. The projection operator P{displaystyle {mathcal {P}}} selects the aforementioned relevant part from the density operator,[note 2] for which an equation of motion is to be derived.The Liouville \u2013 von Neumann equation can thus be represented as\u2202t(PQ)\u03c1=(PQ)L(PQ)\u03c1+(PQ)L(QP)\u03c1.{displaystyle {partial _{t}}left({begin{matrix}{mathcal {P}}\\{mathcal {Q}}\\end{matrix}}right)rho =left({begin{matrix}{mathcal {P}}\\{mathcal {Q}}\\end{matrix}}right)Lleft({begin{matrix}{mathcal {P}}\\{mathcal {Q}}\\end{matrix}}right)rho +left({begin{matrix}{mathcal {P}}\\{mathcal {Q}}\\end{matrix}}right)Lleft({begin{matrix}{mathcal {Q}}\\{mathcal {P}}\\end{matrix}}right)rho .}The second line is formally solved as[note 3]Q\u03c1=eQLtQ\u03c1(t=0)+\u222b0tdt\u2032eQLt\u2032QLP\u03c1(t\u2212t\u2032).{displaystyle {mathcal {Q}}rho ={{e}^{{mathcal {Q}}Lt}}Qrho (t=0)+int _{0}^{t}dt'{e}^{{mathcal {Q}}Lt’}{mathcal {Q}}L{mathcal {P}}rho (t-{t}’).}By plugging the solution into the first equation, we obtain the Nakajima\u2013Zwanzig equation:\u2202tP\u03c1=PLP\u03c1+PLeQLtQ\u03c1(t=0)\u23df=0+PL\u222b0tdt\u2032eQLt\u2032QLP\u03c1(t\u2212t\u2032).{displaystyle partial _{t}{mathcal {P}}rho ={mathcal {P}}L{mathcal {P}}rho +underbrace {{mathcal {P}}L{{e}^{{mathcal {Q}}Lt}}{mathcal {Q}}rho (t=0)} _{=0}+{mathcal {P}}Lint _{0}^{t}{dt'{{e}^{{mathcal {Q}}Lt’}}{mathcal {Q}}L{mathcal {P}}rho (t-{t}’)}.}Under the assumption that the inhomogeneous term vanishes[note 4] and usingK(t)\u2261PLeQLtQLP,{displaystyle {mathcal {K}}left(tright)equiv {mathcal {P}}L{{e}^{{mathcal {Q}}Lt}}{mathcal {Q}}L{mathcal {P}},}P\u03c1\u2261\u03c1rel,{displaystyle {mathcal {P}}rho equiv {{rho }_{mathrm {rel} }},} as well asP2=P,{displaystyle {mathcal {P}}^{2}={mathcal {P}},}we obtain the final form\u2202t\u03c1rel=PL\u03c1rel+\u222b0tdt\u2032K(t\u2032)\u03c1rel(t\u2212t\u2032).{displaystyle partial _{t}{rho }_{mathrm {rel} }={mathcal {P}}L{{rho }_{mathrm {rel} }}+int _{0}^{t}{dt'{mathcal {K}}({t}’){{rho }_{mathrm {rel} }}(t-{t}’)}.}See also[edit]^ A derivation analogous to that presented here is found, for instance, in Breuer, Petruccione The theory of open quantum systems, Oxford University Press 2002, S.443ff^ P\u03c1={displaystyle {mathcal {P}}rho =} (relevant part)\u00a0\u00b7  (constant). The relevant part is called the reduced density operator of the system, the constant part is the density matrix of the thermal bath at equilibrium.^ To verify the equation it suffices to write the function under the integral as a derivative, deQLt\u2032QeL(t\u2212t\u2032)=\u2212eQLt\u2032QLPeL(t\u2212t\u2032)dt\u2032.{displaystyle de^{{mathcal {Q}}Lt’}{mathcal {Q}}e^{L(t-t’)}=-e^{{mathcal {Q}}Lt’}{mathcal {Q}}L{mathcal {P}}e^{L(t-t’)}dt’.} ^ Such an assumption can be made if we assume that the irrelevant part of the density matrix is 0 at the initial time, so that the projector for t=0 is the identity. This is true if the correlation of fluctuations on different sites caused by the thermal bath is zero.References[edit]E. Fick, G. Sauermann: The Quantum Statistics of Dynamic Processes Springer-Verlag, 1983, ISBN\u00a03-540-50824-4.Heinz-Peter Breuer, Francesco Petruccione: Theory of Open Quantum Systems. Oxford, 2002  ISBN\u00a09780198520634Hermann Grabert Projection operator techniques in nonequilibrium statistical mechanics, Springer Tracts in Modern Physics, Band 95, 1982R. K\u00fchne, P. Reineker: Nakajima-Zwanzig’s generalized master equation: Evaluation of the kernel of the integro-differential equation, Zeitschrift f\u00fcr Physik B (Condensed Matter), Band 31, 1978, S. 105\u2013110, doi:10.1007\/BF01320131External links[edit]  "},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/nakajima-zwanzig-equation-wikipedia\/#breadcrumbitem","name":"Nakajima\u2013Zwanzig equation – Wikipedia"}}]}]