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absorption of radiation from a 'cooler' source

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This post specifically queries the absorption, by matter, of (thermal) radiation which has been emitted from a source of a cooler temperature than the receiving matter.

As I understand it, when matter 'receives' radiation, that radiation is either a) absorbed, b) reflected (or absorbed and instantly re-emitted) or c) transmitted.

When radiation from a warmer source is absorbed, the molecules of the matter gain an increase in energy which is manifested as an increase in temperature. In this sense, the radiation is absorbed 'for energy gain'.

My questions are:

What happens when radiation from a cooler body meets the receiving matter?
And,
Can receiving matter gain (thermal) energy from any 'absorption' from a cooler source? (Or is it 'transparent' due to the wavelength of the 'cooler' radiation?

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