# Some thermodynamics derivations

This note is a tiny bit of thermodynamics foundation.

In this case energy can be any conserved quantity, a definition of temperature follows considering thermal equilibrium.

*NOTE*, it is obviously not entirely rigorously defined, and by extension derived..

Assume:

- An ensemble has a set of states to be in with equal probability.
- Ensembles have some value
**E**, which can be exchanged, but the sum is constant. - Connecting two ensembles, we call thermal equilibrium when the two sets of states maximizing the number of states.

Connecting two independent ensembles, there is some total **E=E1+E2** and the two ensembles have
numbers of states to be in **N(E1)**, **N(E2)**. They’re independent so the combination has

**N = N(E1)⋅N(E2)**

And now we can just optimize that with the constraint. (simply filling in **E2=E-E1**)

**dN/dE1 = N'(E1)⋅N(E-E1) - N(E1)⋅N'(E-E1) = 0**

So that **N'(E1)/N(E1) = N'(E2)/N(E2)** lets consider **exp(S)=N** it simplies;
**N'=exp(S)S'** or the equation:

**exp(S1(E1))S'(E1)/exp(S1(E1)) = dS1(E1)/dE2 = dS2/E2 ≡ T**

So as is a known equation, the derivative of entropy by energy is constant among
ensembles in thermal equilibrium, and we call this constant the **T**.

We could call it temperature, but maybe we shouldn’t! After all, we have specified
nothing other than constancy about the energy. Each extensive sum-conserved value
about an ensemble, like **E**, has a intensive constant like **T**.

For instance, chemical potential has “**T**” here as **μ/T** and
“**E**” the number of particles. The division by temperature is probably largely
incidental. Possibly because it occurs equations like
this one on internal energy.

### Some notes on the assumptions.

Really, the optimization maximizing the amount of possibilities is about maximum
likelyhood. There is thermal noise around it, and there might be awkward **S(E)**
functions that have a lot of probability away from the maximum.

Also, things are discrete, limiting posibilities of redistributing **E**. Awkward
cases might frustrate the derivative.

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