Zeroth Law of Thermodynamics

The zeroth law of thermodynamics describes the fundamental behavior of the temperature of systems in thermal equilibrium [72]. If two given systems $\mathcal{S}_1$ and $\mathcal{S}_2$ are in thermal equilibrium with a third system $\mathcal{S}_3$ , then, the system $\mathcal{S}_1$ has also to be in thermal equilibrium with the system $\mathcal{S}_2$ :

$$\displaystyle{\left(\frac{\partial{{\sigma}}_1}{\partial{{U}}_1}\... ...\sigma}}_2}{\partial{{U}}_2}\right)}_{\stackrel{\scriptstyle{{N}}_2}{{{V}}_2}}.$$ (2.26)

Thus, the zeroth law of thermodynamics describes the transitivity and the symmetry [60] of the equilibrium relationship [73]. According to the definition of the temperature $\tau$ from (2.22), this law can be also expressed by their fundamental temperatures $\tau_i$

$$\tau_1 = \tau_3 \quad \wedge \quad \tau_2 = \tau_3 \quad\Rightarrow\quad \tau_1 = \tau_2,$$ (2.27)

where $\tau_i$ is proportional to $T_i$ according to (2.23).