Nonexistence of equilibrium states at absolute negative temperatures

We show that states of macroscopic systems with purported absolute negative temperatures are not stable under small, yet arbitrary, perturbations. We prove the previous statement using the fact that, in equilibrium, the entropy takes its maximum value. We discuss that, while Ramsey theoretical reformulation of the second law for systems with negative temperatures is logically correct, it must be a priori assumed that those states are in thermodynamic equilibrium. Since we argue that those states cannot occur, reversible processes are impossible, and, thus, Ramsey identification of absolute negative temperatures is untenable.

Figure 1

Entropies $s=S/N_0{k}_b$ versus energy $e=E/N_0$, for a paramagnet $n$ of ten moles $N_n=10N_0$, thick-dashed blue line, and for one mole $N_p=N_0$ of an ideal Fermi gas $p$, solid red line. The energy units are arbitrary with ${\mu }_{0}B=15$. The volume of the gas was chosen such that Fermi energy is ${\epsilon }_F=5$. The dashed vertical lines indicate the initial energies $e_n=50$ and $e_p=10$. The equilibrium state is indicated with vertical dotted lines and correspond to $e_n^{\mathrm{eq}}=-11.7$ and $e_p^{\mathrm{eq}}=71.7$. At those states, we have plotted the slope of the curves $s$ vs $e$, respectively, to show that they correspond to the same temperature $k_{B}T=47.6$.

Figure 2

Total entropy $s_T=S_T/N_0{k}_B$ versus energy $e_p=E_p/N_0$ of the system $p$ (ideal gas). The maximum occurs at $e_p^{\mathrm{eq}}=71.7$; see Fig. 1.

Figure 3

Equation of state $M=M(B,T)$ of an ideal paramagnet; see Eq. (7). Positive temperatures correspond to the magnetization $M$ and magnetic field $B$ being parallel (solid blue curves), while negative ones to the antiparallel situation (dashed red curves).

Figure 4

Change of magnetization $M$ for a fixed value of the magnetic field $B$. Initially, the system has magnetization $M_0$ at positive temperature $T_0$. The system is reversibly heated up to $T\to \infty$, indicated by the upper arrow. Then, irreversibly, the magnetization is inverted reaching a temperature $T\to -\infty$. Afterwards, it continues to be reversibly heated up to $-T_0$ and $-M_0$.

Figure 5

Change of magnetic field $B$ for a fixed value of the magnetization $M$. Initially, the system has magnetic field $B_0$ at positive temperature $T_0$. The system is reversibly cooled down to $T\to +0$ at constant magnetization, as indicated by the right arrow. Then, the field is irreversibly inverted reaching a temperature $T\to -0$. After that, it continues to be reversibly cooled down to $-T_0$ and $-B_0$.