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I wonder if there is any difference between these two terms. Is transverse conductivity always equal to Hall's conductivity?

I am asking in the context of the thermal and electrical conductivities of a material.

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If referring to electrical (not thermal) conductivities, then yes they are the same thing.

Though the term Hall usually pops up when dealing with the Hall effect (classical or quantum) meaning the transverse conductivity is dominated by the effect of an external magnetic field.

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