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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 ''6N''-dimensional phase space ''Γ''= of ''N'' point particles with coordinates ''q''i and momenta ''p''i. A special subset of these functions is an enumerable set of "slow variables" ''A''(''Γ'')=. Candidates for some of these variables might be the long-wavelength Fourier components ''ρk(''Γ'')'' of the mass density and the long-wavelength Fourier components πk(''Γ'') 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(Γ)''. A scalar product between two arbitrary phase space functions ''f''1(''Γ'') and ''f''2(''Γ'') is defined by the equilibrium correlation : where : denotes the microcanonical equilibrium distribution. "Fast" variables, by definition, are orthogonal to all functions ''G(A(Γ))'' of ''A(Γ)'' under this scalar product. This definition states that fluctuations of fast and slow variables are uncorrelated, and according to the ergodic hypothesis this also is true for time averages. If a generic function ''f(Γ)'' is correlated with some slow variables, then one may subtract functions of slow variables until there remains the uncorrelated fast part of ''f(Γ)''. The product of a slow and a fast variable is a fast variable. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Zwanzig projection operator」の詳細全文を読む スポンサード リンク
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