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

The general properties of matter in the neighborhood of T = 0 are summarized by two independent observations: (i) the vanishing of the isothermal change of entropy (Nernst heat theorem) and (ii) the vanishing of the specific heats. 1,2 The two observations traditionally form the third law of thermodynamics, as they could not be deduced from the first and second laws of thermodynamics. [3][4][5][6] The relationship between these two observations and the unattainability of the absolute zero has sparked discussion in the past. [7][8][9][10][11][12][13][14] Recently, it was assumed that, within the framework of the second law, T = 0 must be determined by a Carnot thermometer. Therefore, Einstein's objection that a Carnot engine at T = 0 cannot operate "in practice" 15,16 was rebutted, and the Nernst theorem probed. 17 This leaves the vanishing of the specific heats as the only general property of matter in the neighborhood of T = 0 that cannot be predicted by the first and second law.

Traditionally, the vanishing of the specific heats as T → 0 + has been associated with the definiteness of Clausius’ entropy at T = 0. 18 Alternatively, the vanishing of the specific heats is strong evidence of the validity of Boltzmann’s entropy, since the number of accessible microstates Ω at T = 0 cannot vanish in a real system and, therefore, S ∼ log Ω must remain finite. Finally, classical models such as the ideal gas or the crystalline solid fail to predict the vanishing of the specific heats. 19 Only a) Electronic mail: olalla@us.es when quantum mechanics is incorporated do these models satisfy the requirement. 20 Therefore, the origin of this requirement is tipically linked to quantum physics. 21,22 The goal of this manuscript is to present a thermodynamic argument that explains why real systems exhibit vanishing specific heats as T → 0 + at least as faster as T . The argument relies solely on the thermal stability of the states in the phase space boundary, which is the condition T (S, X) = 0, 23 where S is the entropy and X is a suitable mechanical parameter such as volume or magnetization. If successful, every aspect related to the third law of thermodynamics would then have been derived from the framework of the first and second law of thermodynamics.

Thermodynamic equilibrium is a spatially homogeneous state. Its stability means that every spatial inhomogeneity arising from an equilibrium state will initiate a process that mitigates the inhomogeneity, restoring the original state.

With equilibrium states determined by minimizing of the energy U (or maximizing the entropy) for a given entropy (or energy), stability conditions are associated with:

U (S e +δS, X e +δX)+U (S e -δS, X e -δX) > 2U (S e , X e ),

(1) where (S e , X e ) is the state under study and (δS, δX) drives the inhomogeneity. Condition (1) means U (S, X) must be a convex function of (S, X). Physically, this im-plies that U (S, X) increases with the inhomogeneity and, therefore, it can be followed by a regular process that, by decreasing U (S, X), restores back U (S e , X e ).

Sufficient conditions for (1) are: 6

where U ii stands for (∂ 2 U/∂i 2 ) and det H u is the determinant of the Hessian matrix. These conditions are obtained by expanding U (S, X) and analysing the leading term, which is quadratic in (δS, δX).

The above analysis becomes problematic at the phase space boundary (T (S, X) = 0). After the minimum entropy S 0 is reached at the boundary -the Nernst theorem is assumed to be valid 17 -, an inhomogeneity such as S 0 -δS seems not possible. However, the principle of continuity suggests this analysis remains valid in the neighborhood of T → 0 + . This study focuses solely in the thermal stability, associated with entropy inhomogeneities that develops at constant mechanical parameter X (δX = 0) or its conjugate Y (δY = 0). They can be expressed from (2) as:

where C i is the heat capacity at i constant and H(S, Y ) is the enthalpy. It must be noted that det

Condition (3b) directly conveys the information associated with the requirement det H u > 0. As an example, ordinary first-order phase transitions in hydrostatic systems -a paradigm of stability loss-occur when H ss vanishes due to C y → ∞.

Conditions (3) can be analyzed in the limit of T → 0 + for a leading analytical term C i ∼ T b , where b ≥ 0 is a constant. a The following results arise. First, for b ≥ 1, the heat capacities vanish at a rate at least as fast as T . In this case, the equilibrium states at the phase space boundary (T = 0) are thermally stable, as conditions (3) upheld as T → 0 + . The entropy at T = 0 would remain finite.

Second, for b ∈ (0, 1), the heat capacities vanish at a slower rate than T . Consequently, U ss → 0 + and H ss → 0 + as T → 0 + . This indicates that the equilibrium states at the phase space boundary (T = 0) would be thermally unstable even though S(T, X) would still remain finite at T = 0.

a This choice excludes specific heats that vanish no

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