Groupoid algebra

The Zwanzig projection operator 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.

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 ρ_k(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 Hamiltonian H(q).

A projection operator requires a scalar product. A scalar product between two arbitrary phase space functions f₁(q) and f₂(q) is defined by the equilibrium correlation

$\left(f_{1},f_{2}\right)=\int dq\rho _{0}\left(q\right)f_{1}\left(q\right)f_{2}\left(q\right),$

where

$\rho _{0}\left(q\right)={\frac {\delta \left(H\left(q\right)-E\right)}{\int dq^{\prime }\delta \left(H\left(q^{\prime }\right)-E\right)}},$

denotes the 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 Φ_a(q) = δ(A(q) - a) = Π_nδ(A_n(q)-a_n) with a = {a_n} constant. Any phase space function G(A(q)) depending on q only through A(q) is a function of the Φ_a, namely

$G(A\left(q\right))=\int daG\left(a\right)\delta \left(A\left(q\right)-a\right).$

A generic phase space function f(q) decomposes according to

$f\left(q\right)=F\left(A\left(q\right)\right)+R\left(q\right),$

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),

$\int dq\rho _{0}\left(q\right)f\left(q\right)\delta \left(A\left(q\right)-a\right)=\int dq\rho _{0}\left(q\right)F\left(A\left(q\right)\right)\delta \left(A\left(q\right)-a\right)=F\left(a\right)\int dq\rho _{0}\left(q\right)\delta \left(A\left(q\right)-a\right).$

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),

$P\cdot f\left(q\right)=F\left(A\left(q\right)\right)={\frac {\int dq^{\prime }\rho _{0}\left(q^{\prime }\right)f\left(q^{\prime }\right)\delta \left(A\left(q^{\prime }\right)-A\left(q\right)\right)}{\int dq^{\prime }\rho _{0}\left(q^{\prime }\right)\delta \left(A\left(q^{\prime }\right)-A\left(q\right)\right)}}.$

This expression agrees with the expression given by Zwanzig, 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 $\rho (q,t)=\rho _{0}(q)\sigma (q,t)$ denote the time-dependent probability distribution in phase space. The phase space density $\sigma (q,t)$ (as well as $\rho (q,t)$) is a solution of the Liouville equation

$i{\frac {\partial }{\partial t}}\sigma (q,t)=L\sigma (q,t).$

The crucial step then is to write $\rho _{1}=P\sigma$, $\rho _{2}=(1-P)\sigma$ and to project the Liouville equation onto the slow and the fast subspace,

$i{\frac {\partial }{\partial t}}\rho _{1}=PL\rho _{1}+PL\rho _{2},$

$i{\frac {\partial }{\partial t}}\rho _{2}=\left(1-P\right)L\rho _{2}+\left(1-P\right)L\rho _{1}.$

Solving the second equation for $\rho _{2}$ and inserting $\rho _{2}(q,t)$ into the first equation gives a closed equation for $\rho _{1}$. The latter equation finally gives an equation for $p(A(q),t)=p_{0}(A(q))\rho _{1}(q,t)$, where $p_{0}(a)$ 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 Φ_a(q) = δ(A(q) - a) of functions one also might use some enumerable set of functions Φ_n(A(q)). If these functions constitute a complete orthonormal function set then the projection operator simply reads

$P\cdot f\left(q\right)=\sum _{n}\left(f,\Phi _{n}\right)\Phi _{n}\left(A\left(q\right)\right).$

A special choice for Φ_n(A(q)) are orthonormalized linear combinations of the slow variables A(q). This leads to the Mori projection operator. 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

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