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Entropy $S,$ is usually defined as an additive measure because mathematically and physically speaking it's usually easier to work with.

I'm wondering how to write entropy as a multiplicative measure...How would it look compared to say Boltzmann's formula for entropy?

Here is Boltzmann's formula:

$$ S=k_{B}\log(W)$$ where $W$ is the number of micro states that make up a macro state of a gas and $k_{B}$ is a constant.

So clearly we have an additive measure:

$S_1+S_2=k_{B}\log(W_1W_2)=k_{B}\log(W_1)+k_{B}\log(W_2)$

Then here's my guess for a multiplicative measure for entropy:

$$ e^S=W^{k_B}.$$

Then $$ e^{S_1}e^{S_2}=(W_1W_2)^{k_{B}}. $$

I'd like to be able to write down a probability distribution function that involves entropy as a parameter.

Could I do it like so?

$$ \varphi_S(x)=e^{\frac{S}{\log x}}$$

Is entropy ever used as a parameter in a (probability) distribution function in statistical mechanics?

What physical quantity or quantities would make sense for $x$?

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The multiplicative measure that you seek is perhaps most sensibly understood to simply be $W$ itself - the number of microstates that are compatible with the macrostate. You cannot do the exponentiation

$$W^{k_B}$$

because $k_B$ has units. Unitful quantities cannot appear as exponents - just ask yourself, "what is $2^\mathrm{m}$, i.e. '2 to the power of meter'?" In terms of $S$,

$$W = e^{\frac{S}{k_B}}$$

.

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  • $\begingroup$ thank you. $\varphi_S(x)=e^{\frac{S}{k_{B}\log x}}.$ so $x$ must be dimensionless because of $\log$? $\endgroup$
    – geocalc33
    Commented Sep 21, 2021 at 12:19
  • $\begingroup$ $k_B$ is just a conversion factor, it's set equal to $1$ i.e. ignored in statistical mechanics, so of course one can do the exponentiation - I would recommend checking this point carefully e.g. in this book. $\endgroup$
    – bolbteppa
    Commented Sep 21, 2021 at 23:12
  • $\begingroup$ @geocalc33 : I am not sure what the purpose of $x$ is supposed to be in your formula. There is no need to introduce an additional parameter besides the entropy itself. $\endgroup$
    – The_Sympathizer
    Commented Sep 22, 2021 at 0:07

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