ABSTRACT

In our discussion of the random phase approximation, we saw that the particles in a Coulomb system move so as to produce a decided shielding effect. They reduce the effect of slowly varying external forces applied to the system. In particular the applied field U ( R , T ) $ U({\mathbf{R}}, T) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math13_1.tif"/> produces the reduced total potential field a U eff ( R , T ) = U ( R , T ) + ∫ d R ′ e 2 | R - R ′ | ( ⟨ n ^ ( R ′ , T ) ⟩ - n ) = U ( R , T ) + ∫ d R ′ e 2 | R - R ′ | × [ ± i ( 2 S + 1 ) G < ( R ′ , T ; R ′ , T ; U ) - n ] $$ \begin{gathered} U_{{{\text{eff}}}} ({\mathbf{R}},~T) = U({\mathbf{R}},~T) + \mathop \smallint \limits_{{}}^{{}} d{\mathbf{R}}^{'} \frac{{e^{2} }}{{|{\mathbf{R}} - {\mathbf{R^{\prime}}}|}}(\langle \hat{n}({\mathbf{R}}^{'} ,~T)\rangle - n) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = U({\mathbf{R}},~T) + \mathop \smallint \limits_{{}}^{{}} d{\mathbf{R}}^{'} \frac{{e^{2} }}{{|{\mathbf{R}} - {\mathbf{R}}^{'} |}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \times [ \pm i(2S + 1)G^{ < } ({\mathbf{R}}^{'} ,~T;{\mathbf{R}}^{'} ,~T;U) - n] \hfill \\ \end{gathered} $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/math13_1.tif"/>