Wikipedia describes thermal energy as quoted:
Thermal energy (of some state of matter - gas, plasma, solid, etc.) is the energy associated with the microscopical random motion of particles constituting the media. For example, in case of monoatomic gas it is just a kinetic energy of motion of atoms of gas as measured in the reference frame of the center of mass of gas. In case of molecules in the gas rotational and vibrational energy is involved. In the case of liquids and solids there is also potential energy (of interaction of atoms) involved, and so on.
With regard to the last sentence, what exactly does potential energy (of interaction of atoms) mean?
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Thermal energy is the net amount of energy that a substance has associated with the motion of it's particles.
The particles of solids and liquids are close enough to have mutual force interactions (that enables the vibrations to occur) .. any vibrating system must be able to interchange energy between at least two different forms .. in the case of masses joined by springs (analogous to vibrating molecules), the interchanging forms are KE and elastic-strain energy (e-s being a PE form)
The bonding between molecules in solids or liquids involves forces that enables them to vibrate .. these forces of attraction or repulsion (when the particles are within or beyond their equilibrium separations) are electrostatic in origin, operating on each other via the electric fields they create.
As charges move their position (vibrate) in an electric field changes in electrical PE occur (like mass changing position in a grav. field has changes to it's GPE) .. an EPE loss corresponds to the gain in KE during the vibration and vice versa.
So .. PE of interaction refers to the amounts of energy stored in electrical PE form as the atoms interact .. ie vibrate.
btw .. the comment made in the previous answer .. " .. potential energy is never connected to any form of motion or interaction", is entirely incorrect.
In the last sentence, " the potential energy of the interaction of atoms" means :--
When the solid or liquid substance is heated, the atoms become unstable and hence have a tendency to move or relocate. When they move their individual potential energies get converted to Kinetic energies which is generated due to Interaction of the constituent atoms. This very kinetic energy is expressed as Thermal Energy. To be more specific, Thermal energy is nothing else but the heat effect caused due to any kind of internal or external motion of the constituent atoms or molecules.
In the above example, in the last sentence though, the word Kinetic energy is misrepresented as Potential energy. It should be Kinetic energy written over there, since potential energy is never connected to any form of motion or interaction . You are right in your argument.