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I'm reading the Wikipedia article on thermodynamic definition of entropy.

Because the energy of a particle in a classical thermodynamic system is a continuous variable, the number possible states is actually uncountably infinite. So, to be able to define entropy using the definition $S=k\ln(W)$, we need to somehow "coarse grain" the continuous distribution so that we end up with a finite number. We can do this by grouping together states, so that states with energies close to each other (within a, say, $dE$ from each other) are taken to be the same state.

According to Wikipedia, this means that entropy is actually "defined up to an additive constant". I read from somewhere else that this is analogous to potential energy, which is also defined up to an additive constant (by the choice of the zero point). It is also mentioned in Wikipedia that in the thermodynamic limit, the definition of entropy becomes independent of the choice of $dE$.

These points are where I am confused. Could somebody elaborate a bit more on what does it mean the entropy is defined up to a constant, and why the choice of $dE$ does not affect the definition of entropy in the "thermodynamic limit"?

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I'm assuming this is just due to the fact that derivatives are unique up to addition of a constant. For example, if we have the simple relationship $\text d y=\text dx$, then we know that $y=x+C$.

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