Thermal stability originates the vanishing of the specific heats at the absolute zero

TL;DR

This work addresses why specific heats vanish as temperature approaches zero by deriving a thermodynamic stability argument grounded in the convexity of the internal energy $U(S,X)$ and the phase-space boundary condition $T(S,X)=0$. By analyzing stability through $U_{ss}>0$, $U_{xx}>0$, and $\det\mathcal{H}_u>0$ with $U_{ss}=(\partial T/\partial S)_x=T/C_x$ and $H_{ss}= (\partial T/\partial S)_y=T/C_y$, the authors show that if the heat capacities scale as $C_i\sim T^b$, then $b\ge1$ yields stable boundaries (finite $S$ at $T\to0^+$), while $0<b<1$ or $b=0$ lead to instability. They relate the scaling to a critical condition $\alpha=1+1/b\le 2$ (with $\alpha=2$ at $b=1$) and interpret the vanishing $C$ as a confirmatory result of the first two laws rather than a new law. The analysis aligns with known models and emphasizes that only sufficiently fast vanishing maintains stability at $T=0$, integrating the third law into the framework of thermodynamics. This provides a coherent thermodynamic basis for the observed behavior of specific heats near absolute zero and clarifies the role of stability in the third law.

Abstract

The relationship between the vanishing of the heat capacities as $T\to0^+$ and the thermal stability is examined. The heat capacities vanish as fast as or faster than $T$ as $T\to0^+$ for states at the phase space boundary ($T=0$) to sustain the standard thermal stability criterion $U_{ss}>0$. Conversely, weakly vanishing heat capacities, which signify a loss of curvature in $U(S)$ at $T=0$, are the signature of a critical condition precisely at $T=0$, as exemplified in marginal Fermi liquids. Therefore, the vanishing of the specific heat should be viewed not as a new law but as a confirmatory result of the existing framework of thermodynamics.

Figures (1)

• Figure 1: Panel a, the $S-T$ chart for $C_x\sim T^b$ for $b=1/2$ (gray), $b=1$ (dark, thick) and $b=2$ (dark, thin). All three cases yield a finite $S_0$ but $b=1/2$ is unstable due to the diverging slope at $T=0$. Panel b, as panel a but for the case $b=0$ which translates in a $\log T$ dependence. This is unstable and unbounded. Panel c, the $U-S$ chart for the cases displayed in panel a. Stability requires a stiff relationship between $U$ and $S$, with a parabola (dark thick line) as the critical condition. The case $b=1/2$ (gray line) yields a very shallow flat arrival at $U_0(X)$ and the equilibrium state is unstable. Panel d, the same as panel c but for the case $b=0$. Here $U(S,X)$ is too flat to sustain stability, and the domain of $S$ is unbounded.