Stability and concavity of the entropy function

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Est120
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I am struggling to understand Callen's explanation for stability, I understand that the concavity of S(U) must be negative because otherwise we can show that this means that the temperature increases as the internal energy decreases (dT/dU<0) but I cannot understand equation (8.1) which basically says that the entropy must decrease and if the system is isolated that is absurd, in addition to what it refers to with "internal inhomogeneities"

the worst thing is that the text says that it is evident, it is geometrically clear, but what physical meaning does a final entropy resulting less than the initial one have? imagining 2 bodies isolated from the outside

I honestly believe that only the author understood that book
Graph of entropy S against energy U for a convex-then-concave curve, with points marked at U - ΔU, U and U + ΔU and a dashed chord connecting two points on the curve.
 

Discussion

@Est120

Right, not an explanation that is simply followed, since it is a bit brief.
Anyways, the graph is an example of how the system does not work.
Note that for the graph, if the two isolated systems start out at U-ΔU and at U+ΔU, they never come to equilibrium, as they are at a ( maximal ) entropy greater than that at state U.

This is more of a mathematical problem in convexity rather than that of just for entropy

You should review convexity of functions for more understanding.
And how some definitions with what one is familiar with may not be that which another interprets.

Equation 8.1 does not represent the graph, which can be said to be convex or concave depending upon whether one wants to add the words upwards or downwards.

See the example which has :
Two inequality definitions of convexity: f(x1) + f(x2) ≥ 2f(x0), and f((x1 + x2)/2) ≤ (f(x1) + f(x2))/2, with the line 'hat is according to the definition, the function f(x) is convex downward.'

( or cancave upward )

in
https://www.math24.net/convex-functions/

or
https://www.ge.infn.it/~zanghi/FS/ConvexThermoTEXT.pdf

You could search also for concavity of entropy, and you might come up with something better.
 
correct let me see if I understand, as they are identical systems in theory they should reach thermal equilibrium in U but that would mean a lower total entropy which cannot happen (considering the 2 subsystems isolated from the outside) so they stay as they are? also it is an easy proof to show that a positive concavity in S implies dt/dU <0 which is nonsense but i can't understand what callen's tries to say with "internal inhomogeneities" anyway your explanation is pretty good
 
Est120 said:
correct let me see if I understand, as they are identical systems in theory they should reach thermal equilibrium in U but that would mean a lower total entropy which cannot happen (considering the 2 subsystems isolated from the outside) so they stay as they are? also it is an easy proof to show that a positive concavity in S implies dt/dU <0 which is nonsense but i can't understand what callen's tries to say with "internal inhomogeneities" anyway your explanation is pretty good
@Est120
The two isolated systems not reaching equilibrium is the counter proof, resulting in the concept that S(U ) is concave. Inhomogeneties ( pockets of increased density, condensation, .. ) would be a consequence of the counterproof. The energy in the counterproof can result in being not spread evenly across the system.

a positive concavity in S implies dt/dU <0
dS/dU = 1/T --> the slope of the S(U) function is 1/T

We have two isolated systems prepared to be identical.
Graph of a concave curve S(U,V,N) plotted as entropy S against internal energy U, with points U1, U, U2 and S(U1), S(U), S(U2) marked and arrows showing energy changes.

The S curve S(U, V , N ) for both isolated systems is as given in the graph, each with an internal energy U and entropy S. Removing, or adding an amount of E from either system, will affect the internal energy of either system giving U1 and U2. The entropy will move along the curve to S1 and S2 respectively.( This act can be either reversible - moving along the curve-, or irreversible - not along the curve for intermediate states-, but in either case the final state 1 and 2 will be the same reversible of irreversible ).

For a stable system, as Callen states,
Inequality S(U + ΔU, V, N) + S(U − ΔU, V, N) ≤ 2S(U, V, N), for all Δ, showing concavity of entropy as a function of internal energy.


You could put some numbers in there such as S1 = 1, S2 = 5 and S = 4
S1 +S2 ? 2S
1 +5 < 8
Surely this would not happen spontaneously for a stable system, as the entropy has decreased.
For isolated systems, the entropy is constant or can increase.

Going the other way, with a hotter and colder systems in contact, reaching equilibrium, the entropy would increase. The lower temperature S1 dominates over the higher temperature S2.

( Hopefully I have my ups and downs correct, as I think this is a difficult subject to keep straight, even though it should be straight forward, it really isn't )