- The global condition for the concavity of entropy (6.9) as a function of internal energy and volume is written,
S (U − ΔU, V − ΔV) + S (U+ΔU, V+ΔV) ≤ 2S (U, V)
The series expansion of entropy S (U+ΔU, V+ΔV) to second-order in ΔU and ΔV is given by,
S (U+ΔU, V+ΔV) \simeq S (U, V) +\frac{\partial S(U,V)}{\partial U}\Delta U+\frac{\partial S(U,V)}{\partial V}\Delta V+\frac{1}{2!}\bigl(\frac{\partial^2 S(U,V)}{\partial U^2}\Delta U^2+2\frac{\partial^2 S(U,V)}{\partial U \partial V}\Delta U\Delta V+\frac{\partial^2 S(U,V)}{\partial V^2}\Delta V^2 \bigr).
and the series expansion of entropy S (U − ΔU, V − ΔV) to second-order in ΔU and
ΔV by,
S (U-ΔU, V-ΔV) \simeq S (U, V) -\frac{\partial S(U,V)}{\partial U}\Delta U-\frac{\partial S(U,V)}{\partial V}\Delta V+\frac{1}{2!}\bigl(\frac{\partial^2 S(U,V)}{\partial U^2}\Delta U^2+2\frac{\partial^2 S(U,V)}{\partial U \partial V}\Delta U\Delta V+\frac{\partial^2 S(U,V)}{\partial V^2}\Delta V^2 \bigr).
Substituting these two series expansions into the global condition for the entropy concavity, the zeroth-order terms and the first-order terms cancel each other out and the inequality, to second-order, becomes,
\frac{\partial^2 S(U,V)}{\partial U^2}\Delta U^2+2\frac{\partial^2 S(U,V)}{\partial U \partial V}\Delta U\Delta V+\frac{\partial^2 S(U,V)}{\partial V^2}\Delta V^2≤0.
In order to simplify this expression, the dependence of the partial derivatives on the state variables U and V will no longer be indicated explicitly. Thus, the previous result is written,
\frac{\partial^2 S}{\partial U^2}\Delta U^2+2\frac{\partial^2 S}{\partial U \partial V}\Delta U\Delta V+\frac{\partial^2 S}{\partial V^2}\Delta V^2≤0.
- Multiplying this inequality by the local condition for entropy concavity (6.12) as a function of internal energy,
\frac{\partial^2 S}{\partial U^2}≤0. and \frac{\partial^2 S}{\partial V^2}≤0.
\frac{\partial^2 S}{\partial U^2}≤0.
yields the following inequality,
\bigl(\frac{\partial^2 S}{\partial U^2} \bigr)^2 ΔU^2+2\frac{\partial^2 S}{\partial U^2}\frac{\partial^2 S}{\partial U \partial V}ΔUΔV+\frac{\partial^2 S}{\partial U^2}\frac{\partial^2 S}{ \partial V^2}ΔV^2≥0.
It can be recast as,
\bigl(\frac{\partial^2 S}{\partial U^2}\Delta U+\frac{\partial^2 S}{\partial U\partial V}\Delta V \bigr)^2+\Bigl(\frac{\partial^2 S}{\partial U^2} \frac{\partial^2 S}{\partial V^2}-\bigl(\frac{\partial^2 S}{\partial U\partial V} \bigr)^2\Bigr)\Delta V^2\geq 0.
- The internal energy variation ΔU and volume variation ΔV can always be chosen such that the terms in the first brackets on the left-hand side vanish. Thus, in order for this inequality to be alway satisfied, the difference of terms in the brackets above has to be
non-negative, i.e.
\frac{\partial^2 S}{\partial U^2}\frac{\partial^2 S}{\partial V^2}-\bigl(\frac{\partial^2 S}{\partial U\partial V} \bigr)^2≥ 0.
This corresponds to the local condition for entropy concavity (6.13) as a function of internal energy and volume.
\frac{\partial^2 S}{\partial U^2}\frac{\partial^2 S}{\partial V^2}-\Bigl(\frac{\partial^2 S}{\partial U \partial V}\Bigr)^2≥0.
- Since the inverse of the entropy function S (U) is monotonous and increasing, the inverses on both sides of the global condition for the entropy concavity (Fig. 6.13) have to satisfy the inequality,
S^{−1} (λ S_1 + (1 − λ) S_2) ≤ λU_1 + (1 − λ) U_2 and λ ∈ [ 0, 1 ]
where S_1 = S (U_1), S_2 = S (U_2), U_1 = U(S_1). and U_2 = U(S_2). Moreover,
U(λ S_1 + (1 − λ) S_2) = S_1 (λ S_1 + (1 − λ) S_2). where λ ∈ [ 0, 1 ]
These two equations imply the global condition for the internal energy convexity,
λU(S_1) + (1 − λ) U(S_2) ≥ U(λ S_1 + (1 − λ) S_2) where λ ∈ [ 0, 1 ]
- In the particular case where λ = 1/2, U_1 = U− ΔU, U_2 = U+ΔU, S_1 = S− ΔS and S_2 = S+ΔS, the global conditions for entropy concavity and internal energy convexity become,
\frac{1}{2}S (U − ΔU)+\frac{1}{2}S (U+ΔU) ≤ S (U)
\frac{1}{2}U(S − ΔS)+\frac{1}{2}U(S+ΔS) ≥ U(S).
and correspond to the inequalities stated in the problem, up to a factor of 2.
