The thermal vibration phenomenon occurring in the dual-phase-lagging heat conduction violates the second law of thermodynamics under the local equilibrium assumption. In order to resolve this paradox, two types of the extended irreversible thermodynamics are developed in the present work, which make the dual-phase-lagging heat conduction model compatible with the second law of thermodynamics. It is also shown that these extended irreversible thermodynamics can give rise to the Maxwell model for the viscoelastic fluid flow.
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