en>Frietjes: cleanup (wikitables, html markup, layout, etc.)

2014-11-28T19:17:48Z

cleanup (wikitables, html markup, layout, etc.)

en>Helpful Pixie Bot: ISBNs (Build KE)

2012-05-10T22:00:00Z

ISBNs (Build KE)

New page

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]].
It operates in the linear space of [[phase space]] functions and projects onto the linear subspace of "slow"
phase space functions. It was introduced by R. Zwanzig to derive a generic [[master equation]]. It is
mostly used in this or similar context in a formal way to derive equations of motion for some "slow"
[[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>

==Slow variables and scalar product==
The Zwanzig projection operator operates on functions in the 6-''N''-dimensional phase space ''q''={'''''x'''''i, '''''p'''''i} of ''N'' point particles with coordinates '''''x'''''i and momenta '''''p'''''i.
A special subset of these functions is an enumerable set of "slow variables" ''A''(''q'')={(''A''n(''q'')}. Candidates for some of these variables might be the long-wavelength Fourier components ρk(''q'') of the mass density and the long-wavelength Fourier components '''π'''k(''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)''.

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''1(''q'') and ''f''2(''q'') is defined by the equilibrium correlation
:<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>
where
:<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>

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.

==The projection operator==
Consider the continuous set of functions Φa(''q'') = δ(''A(q) - a'') = Πnδ(''An(q)-an'') with ''a = {an}'' constant. Any phase space function ''G(A(q))'' depending on ''q'' only through ''A(q)'' is a function of the Φa, namely
:<math>G(A\left( q\right) )=\int daG\left( a\right) \delta \left( A\left( q\right)-a\right).</math>
A generic phase space function ''f(q)'' decomposes according to
:<math>f\left( q\right) =F\left( A\left( q\right) \right) +R\left( q\right),</math>
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''),
:<math>
\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).
</math>
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)'',
:<math>
P\cdot f\left( q\right) =F\left( A\left( q\right) \right) =\frac{\int dq^{\prime }\rho
_{0}\left( q^{\prime }\right) f\left( q^{\prime }\right) \delta \left(
A\left( q^{\prime }\right) -A\left( q\right) \right) }{\int dq^{\prime }\rho
_{0}\left( q^{\prime }\right) \delta \left( A\left( q^{\prime }\right)
-A\left( q\right) \right) }.
</math>
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 ''P2 = P''. The fast part of ''f(q)'' is ''(1-P)f(q)''.

==Connection with Liouville and Master equation ==
The ultimate justification for the definition of ''P'' as given above is that
it allows to derive a master equation for the time dependent probability
distribution ''p(a,t)'' of the slow variables (or Langevin equations for the slow variables themselves).

To sketch the typical steps, let <math>\rho(q,t)=\rho_{0}(q)\sigma(q,t)</math>
denote the time-dependent probability distribution in phase space.
The phase space density <math>\sigma(q,t)</math> (as well as <math>\rho(q,t)</math>) is a
solution of the [[Liouville's theorem (Hamiltonian)|Liouville equation]]
:<math>i\frac{\partial}{\partial t}\sigma (q,t)=L\sigma (q,t).</math>
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"/>
:<math>i\frac{\partial}{\partial t}\rho_{1} =PL\rho_{1}+PL\rho_{2},</math>
:<math>i\frac{\partial}{\partial t}\rho_{2} =\left(1-P\right) L\rho_{2}+\left(1-P\right)L\rho_{1}.</math>
Solving the second equation for <math>\rho_{2}</math> and inserting <math>\rho_{2}(q,t)</math> into the first
equation gives a closed equation for <math>\rho _{1}</math>.
The latter equation finally gives an equation for <math>p(A(q),t)=p_{0}(A(q))\rho_{1}(q,t)</math>,
where <math>p_{0}(a)</math> denotes the equilibrium distribution of the slow variables.

==Discrete set of functions, relation to the Mori projection operator==
Instead of expanding the slow part of ''f(q)'' in the continuous set Φa(''q'') = δ(''A(q) - a'') of functions one also might use some enumerable set of functions Φn(''A(q)''). If these functions constitute a complete orthonormal function set then the projection operator simply reads
:<math>P\cdot f\left( q\right) =\sum_{n}\left( f,\Phi _{n}\right) \Phi _{n}\left(A\left( q\right) \right).</math>

A special choice for Φn(''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.

== References ==

{{Reflist}}

[[Category:Statistical mechanics]]
[[Category:Articles created via the Article Wizard]]

Groupoid algebra - Revision history

en>Frietjes: cleanup (wikitables, html markup, layout, etc.)

en>Helpful Pixie Bot: ISBNs (Build KE)