- #1
Tio Barnabe
Is it ok to assume that the entropy ##S## of an arbritary system can be written as a power series as a function of the system's internal energy ##U##? Like
$$S(U) = \sum_{i=1}^{\infty}a_iU^i = a_1 U + a_2 U^2 + \ ...$$ with ##a_i \in \mathbb{R}##.
What results could be obtained from such expression? For instance, if the temperature could be defined as ##T = (dS / dU)^{-1}##, then we have an interesting result if ##|U| <<1## and we neglect terms higher than power one in the expansion above. In other words, in such a case we would have ##T = (dS / dU)^{-1} = 1 / a_1## i.e. constant temperature.
On the other hand, if ##U## takes on considerable values, then the entropy would increase exponentially, like ##S(U) \propto e^U##, and of course, the temperature would be ##T \propto 1 / e^U##.
The ##U## above perhaps could be the normalized original ##U##, i.e. the original internal energy.
$$S(U) = \sum_{i=1}^{\infty}a_iU^i = a_1 U + a_2 U^2 + \ ...$$ with ##a_i \in \mathbb{R}##.
What results could be obtained from such expression? For instance, if the temperature could be defined as ##T = (dS / dU)^{-1}##, then we have an interesting result if ##|U| <<1## and we neglect terms higher than power one in the expansion above. In other words, in such a case we would have ##T = (dS / dU)^{-1} = 1 / a_1## i.e. constant temperature.
On the other hand, if ##U## takes on considerable values, then the entropy would increase exponentially, like ##S(U) \propto e^U##, and of course, the temperature would be ##T \propto 1 / e^U##.
The ##U## above perhaps could be the normalized original ##U##, i.e. the original internal energy.
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