Relation between Polarization and electric field for instantaneous response

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In materials that respond instantaneously to an external electric field, the polarization can be expressed as P(r,t) = χE(t), suggesting that susceptibility may be time-independent. However, it is noted that susceptibility does not have to be time-independent, as established by Noll's general restrictions on constitutive relations. The Kramers-Kronig relation indicates that the susceptibility must be causal, meaning P(t) relies on E(t') for t' < t. To derive equations in the frequency domain, the standard approach involves applying a Fourier or Laplace transform to the original equation. This discussion highlights the complexities of susceptibility in relation to electric fields and polarization.
Algarion
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Dear all

In case of a material that instantaneous responds to an external applied field is

P(r,t)=χE(t)

Is the suseptibility then time independent? And if so, how it is possible to derive from the above equation an equation in the frequency domain?

Yours
 
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The most simple relation (the Kramers-Kronig relation) is found by only demanding the susceptibility be causal- P(t) can only depend on E(t'), where t' < t (extend the results to a lightcone for spatially-varying E).

The general restrictions on the susceptibility (or any constitutive relation) were laid down by Noll in the '60s, and are fairly general. AFAIK, there is no requirement that the susceptibility *may not* be time-dependent. This is covered in several books, including the Encyclopedia of Physics (Vol III, Non-linear field theories).

For the frequency domain, the usual procedure is to Fourier transform (or Laplace transform) the equation.
 
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