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In terms of a dielectric, it means there is a linear constitutive relation between the vectors. $${\boldsymbol D} = \epsilon_0 {\boldsymbol E} + {\boldsymbol P}$$ Or $${\boldsymbol D} = \epsilon_r \epsilon_0 {\boldsymbol E}$$ where $\epsilon_r \epsilon_0$ is a scalar relative and vacuum permitivitty. This way, there is now a linear relation between the ...

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Just to add, nonlocal dielectric response also leads to the permittivity being dependent on the wavevector. Nonlocality is thus tightly bound to the notion of spatial dispersion. This has profound implications on the light propagation is nonlocal media. The dispersion curves can be bent upwards or downwards with the wavenumber. Therefore, it may occur that ...

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This peak is typical for the resonance curve of any (weakly damped) harmonic oscillator. It naturally comes as a solution of its differential equation for different frequencies. Driving the oscillator at a frequency slightly below its resonance frequency leads to big amplitude of oscillations. Therefore, even a weak electric field causes a strong electric ...

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I found a paper that discusses this nonlocality in the context of intermolecular interactions. Not the same context as the article you linked, but it may help with understanding the concept. "The nonlocal dielectric function of a molecule determines the effective potential at a certain point due to an applied external potential at a different point, within ...

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It means polarization $\mathbf P$ at point $\mathbf x$ is an integral involving electric field $\mathbf E$ at points $\mathbf x'$ different from $\mathbf x$.

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First question: I do not have Griffiths book but I hope this is what you mean. You can use Gauss law on a surface that encloses the surface charge, you just can't use it on the surface itself, because the electric field is infinite there. Mathematically you can state: \iint \mathrm{d}S \, \, \vec{n} \cdot \vec{D}= \sum_i \iiint_{V_i} \mathrm{d}V \, \, ...

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