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| The Zwanzig [[projection operator]]<ref name="Zwanzig1961">{{cite journal | title = Memory Effects in Irreversible Thermodynamics | journal = Phys. Rev. | year = 1961 | first = Robert | last = Zwanzig | volume = 124 | pages = 983| id = | accessdate = 2011-02-12}}</ref> is a mathematical device used in [[statistical mechanics]]. | | Yard Cards and Greetings - The New Way to Say Hello<br><br>Fans in the little Calico Critters will probably be thrilled to hear how the Calico Critters Luxury Townhome is really one of many hot Christmas toys for 2010 authored by Toys R Us, as there are a clear reason why. It isn't merely another dollhouse. The truth is, it gives your kids countless circumstances to discover and do, that they're going to be kept occupied and amused for hours on end.<br><br>This neat gift idea comes filled with four separate rooms for your child's Calico Critters, and there is a good divider that will enable your child to produce a fifth room. The Calico Critters Luxury Townhome will fulfill your son or daughter's should be creative as well as imaginative with its movable staircase and further floor room. The possibilities are limitless with this particular fantastic toy.<br><br>The Leaspter Explorer is a new handheld console from Leapster that combines an educational learning tool having a games machine. The third in Leapfrog's distinct Leapster toys this latest version comes with a touchscreen display capability. Games that come as downloads or cartridges bought separately, feature favourites like Tinker Bell, Dora, Disney Princesses and Toy Story.<br><br>So if you are concerned with deer, rabbits, critters, raccoons, gophers, opossums, elk, or chickens, you then need this book. It's solved the problem immensely over the years. You see, you may think than innocent squirrels, rabbits, and birds cannot hurt a garden or plants, nonetheless they can. And you might like them scampering or hovering, with out one can possibly blame you to the - I do too. But I let you know which you have some choices here.<br><br>The best part is the Calico Critters Townhouse is manufactured out of absolutely non-toxic materials and since a dad or mom you can be at ease knowing that your daughter is enough time of her life in a really safe manner. The vivid beauty and versatility from the Calico Critters Townhouse offer your daughter a chance at play-acting and pretend-playing. Remember, that merely while you dream of investing in you dream-house, your daughter also wants having fun with hers.<br><br>When you have almost any queries concerning wherever as well as the way to employ Inkd.us, you'll be able to call us with our own webpage. |
| It operates in the linear space of [[phase space]] functions and projects onto the linear subspace of "slow"
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| phase space functions. It was introduced by R. Zwanzig to derive a generic [[master equation]]. It is
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| mostly used in this or similar context in a formal way to derive equations of motion for some "slow"
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| [[collective variables]].<ref>{{cite book | last1 = Grabert | first1 = H. | title = Projection Operator Techniques in Nonequilibrium Statistical Mechanics | publisher = Springer Tracts in Modern Physics, 95 | year = 1982 | accessdate = 2011-02-12}}</ref>
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| ==Slow variables and scalar product==
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| The Zwanzig projection operator operates on functions in the 6-''N''-dimensional phase space ''q''={'''''x'''''<sub>i</sub>, '''''p'''''<sub>i</sub>} of ''N'' point particles with coordinates '''''x'''''<sub>i</sub> and momenta '''''p'''''<sub>i</sub>.
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| A special subset of these functions is an enumerable set of "slow variables" ''A''(''q'')={(''A''<sub>n</sub>(''q'')}. Candidates for some of these variables might be the long-wavelength Fourier components ρ<sub>k</sub>(''q'') of the mass density and the long-wavelength Fourier components '''π'''<sub>k</sub>(''q'') of the momentum density with the wave vector '''''k''''' identified with n. The Zwanzig projection operator relies on these functions but doesn't tell how to find the slow variables of a given [[Hamilton function|Hamiltonian]] ''H(q)''.
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| A projection operator requires a scalar product. A scalar product<ref name="Mori1965">{{cite journal | title = Transport, Collective Motion, and Brownian Motion | journal = Prog. Theor. Phys. | year = 1965 | first = H. | last = Mori | volume = 33 | pages = 423| id = | accessdate = 2011-02-12}}</ref> between two arbitrary phase space functions ''f''<sub>1</sub>(''q'') and ''f''<sub>2</sub>(''q'') is defined by the equilibrium correlation
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| :<math>\left( f_{1},f_{2}\right) =\int dq\rho _{0}\left( q\right) f_{1}\left(q\right) f_{2}\left( q\right),</math>
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| where
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| :<math>\rho _{0}\left( q\right) =\frac{\delta \left( H\left( q\right) -E\right) }{\int dq^{\prime }\delta \left( H\left( q^{\prime }\right) -E\right) },</math>
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| denotes the [[microcanonical ensemble|microcanonical]] equilibrium distribution. "Fast" variables, by definition, are orthogonal to all functions ''G(A(q))'' of ''A(q)'' under this scalar product. This definition states that fluctuations of fast and slow variables are uncorrelated. If a generic function ''f(q)'' is correlated with some slow variables, then one may subtract functions of slow variables until there remains the uncorrelated fast part of ''f(q)''. The product of a slow and a fast variable is a fast variable.
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| ==The projection operator==
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| Consider the continuous set of functions Φ<sub>a</sub>(''q'') = δ(''A(q) - a'') = Π<sub>n</sub>δ(''A<sub>n</sub>(q)-a<sub>n</sub>'') with ''a = {a<sub>n</sub>}'' constant. Any phase space function ''G(A(q))'' depending on ''q'' only through ''A(q)'' is a function of the Φ<sub>a</sub>, namely
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| :<math>G(A\left( q\right) )=\int daG\left( a\right) \delta \left( A\left( q\right)-a\right).</math>
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| A generic phase space function ''f(q)'' decomposes according to
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| :<math>f\left( q\right) =F\left( A\left( q\right) \right) +R\left( q\right),</math>
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| where ''R(q)'' is the fast part of ''f(q)''. To get an expression for the slow part ''F(A(q))'' of ''f'' take the scalar product with the slow function δ(''A(q) - a''),
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| :<math>
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| \int dq\rho _{0}\left( q\right) f\left( q\right) \delta \left( A\left(q\right) -a\right) =\int dq\rho _{0}\left( q\right) F\left( A\left(q\right) \right) \delta \left( A\left( q\right) -a\right) =F\left( a\right)\int dq\rho _{0}\left( q\right) \delta \left(A\left( q\right)-a\right).
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| </math>
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| This gives an expression for ''F(a)'', and thus for the operator ''P'' projecting an arbitrary function ''f(q)'' to its "slow" part depending on ''q'' only through ''A(q)'',
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| :<math>
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| P\cdot f\left( q\right) =F\left( A\left( q\right) \right) =\frac{\int dq^{\prime }\rho
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| _{0}\left( q^{\prime }\right) f\left( q^{\prime }\right) \delta \left(
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| A\left( q^{\prime }\right) -A\left( q\right) \right) }{\int dq^{\prime }\rho
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| _{0}\left( q^{\prime }\right) \delta \left( A\left( q^{\prime }\right)
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| -A\left( q\right) \right) }.
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| </math>
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| This expression agrees with the expression given by Zwanzig,<ref name="Zwanzig1961"/> except that Zwanzig subsumes ''H(q)'' in the slow variables. The Zwanzig projection operator fulfills ''PG(A(q)) = G(A(q)'' and ''P<sup>2</sup> = P''. The fast part of ''f(q)'' is ''(1-P)f(q)''.
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| ==Connection with Liouville and Master equation ==
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| The ultimate justification for the definition of ''P'' as given above is that
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| it allows to derive a master equation for the time dependent probability
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| distribution ''p(a,t)'' of the slow variables (or Langevin equations for the slow variables themselves).
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| To sketch the typical steps, let <math>\rho(q,t)=\rho_{0}(q)\sigma(q,t)</math>
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| denote the time-dependent probability distribution in phase space.
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| The phase space density <math>\sigma(q,t)</math> (as well as <math>\rho(q,t)</math>) is a
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| solution of the [[Liouville's theorem (Hamiltonian)|Liouville equation]]
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| :<math>i\frac{\partial}{\partial t}\sigma (q,t)=L\sigma (q,t).</math>
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| The crucial step then is to write <math>\rho_{1}=P\sigma</math>, <math>\rho_{2}=(1-P)\sigma</math> | |
| and to project the Liouville equation onto the slow and | |
| the fast subspace,<ref name="Zwanzig1961"/>
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| :<math>i\frac{\partial}{\partial t}\rho_{1} =PL\rho_{1}+PL\rho_{2},</math>
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| :<math>i\frac{\partial}{\partial t}\rho_{2} =\left(1-P\right) L\rho_{2}+\left(1-P\right)L\rho_{1}.</math>
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| Solving the second equation for <math>\rho_{2}</math> and inserting <math>\rho_{2}(q,t)</math> into the first
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| equation gives a closed equation for <math>\rho _{1}</math>.
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| The latter equation finally gives an equation for <math>p(A(q),t)=p_{0}(A(q))\rho_{1}(q,t)</math>,
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| where <math>p_{0}(a)</math> denotes the equilibrium distribution of the slow variables.
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| ==Discrete set of functions, relation to the Mori projection operator==
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| Instead of expanding the slow part of ''f(q)'' in the continuous set Φ<sub>a</sub>(''q'') = δ(''A(q) - a'') of functions one also might use some enumerable set of functions Φ<sub>n</sub>(''A(q)''). If these functions constitute a complete orthonormal function set then the projection operator simply reads
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| :<math>P\cdot f\left( q\right) =\sum_{n}\left( f,\Phi _{n}\right) \Phi _{n}\left(A\left( q\right) \right).</math>
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| A special choice for Φ<sub>n</sub>(''A(q)'') are orthonormalized linear combinations of the slow variables ''A(q)''. This leads to the Mori projection operator.<ref name="Mori1965"/> However, the set of linear functions isn't complete, and the orthogonal variables aren't fast or random if nonlinearity in ''A'' comes into play.
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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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| [[Category:Statistical mechanics]]
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| [[Category:Articles created via the Article Wizard]]
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Yard Cards and Greetings - The New Way to Say Hello
Fans in the little Calico Critters will probably be thrilled to hear how the Calico Critters Luxury Townhome is really one of many hot Christmas toys for 2010 authored by Toys R Us, as there are a clear reason why. It isn't merely another dollhouse. The truth is, it gives your kids countless circumstances to discover and do, that they're going to be kept occupied and amused for hours on end.
This neat gift idea comes filled with four separate rooms for your child's Calico Critters, and there is a good divider that will enable your child to produce a fifth room. The Calico Critters Luxury Townhome will fulfill your son or daughter's should be creative as well as imaginative with its movable staircase and further floor room. The possibilities are limitless with this particular fantastic toy.
The Leaspter Explorer is a new handheld console from Leapster that combines an educational learning tool having a games machine. The third in Leapfrog's distinct Leapster toys this latest version comes with a touchscreen display capability. Games that come as downloads or cartridges bought separately, feature favourites like Tinker Bell, Dora, Disney Princesses and Toy Story.
So if you are concerned with deer, rabbits, critters, raccoons, gophers, opossums, elk, or chickens, you then need this book. It's solved the problem immensely over the years. You see, you may think than innocent squirrels, rabbits, and birds cannot hurt a garden or plants, nonetheless they can. And you might like them scampering or hovering, with out one can possibly blame you to the - I do too. But I let you know which you have some choices here.
The best part is the Calico Critters Townhouse is manufactured out of absolutely non-toxic materials and since a dad or mom you can be at ease knowing that your daughter is enough time of her life in a really safe manner. The vivid beauty and versatility from the Calico Critters Townhouse offer your daughter a chance at play-acting and pretend-playing. Remember, that merely while you dream of investing in you dream-house, your daughter also wants having fun with hers.
When you have almost any queries concerning wherever as well as the way to employ Inkd.us, you'll be able to call us with our own webpage.