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An extension of the principle of thermal equilibrium is known as the zeroth law of thermodynamics, which states that

two systems in thermal equilibrium with a third system are in equilibrium with each other.

Are there alternative descriptions for the zeroth law of thermodynamics?

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    $\begingroup$ Are you looking for something more profound than "'thermal equilibrium' is a relationship between two systems, and it is transitive?" There's a lot of ways to rephrase the transitive property, but the one provided is by far the simplest. $\endgroup$
    – Cort Ammon
    Commented Dec 4, 2020 at 7:35

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Notwithstanding a very long history of careful scrutiny and alternative restatements of thermodynamics laws, the zeroth law has been challenged as an independent postulate, but, to the best of my knowledge, no real alternative formulation has been used. This is probably due to its logical simplicity. It is just the statement that the property of being in mutual thermal equilibrium can be used to establish an equivalence relation between equilibrium thermodynamic systems.

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The zeroth law of classical thermodynamics is a direct consequence of the 2 postulates of statistical thermodynamics, simply stated as follows: postulate 1: If the microscopic accessible states of an isolated system are equally probable, the system is in thermodynamic equilibrium. postulate 2: if the microscopic accessible states of an isolated system are not equally probable, the system evolves in time towards equilibrium.

The 2 postulates of statistical thermodynamics may be considered an alternative rendering of the zeroth law.

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