One thing I was blown away by from Thermodynamics is that you can derive a perfectly reasonable and intuitive definition of Temperature that makes no reference whatsoever to kinetic energy, colliding particles, jiggling -- any microscopic dynamics whatsoever.

So... let's do it!

Thermo works with macroscopic systems: forgetting all knowledge of the fine-scale workings of the system & only focusing on characteristics we can observe macroscopically.

We call these systems "simple." (homogenous, isotropic, uncharged, unaffected by electromag./grav. fields)

In these "simple" systems, we can characterize the state of the system *completely* by simply describing:

- the energy U
- the volume V
- the amounts of each of it's chemical components (usually mole numbers) N₁, N₂, etc.

If a system can be completely described/characterized in this way, we say the system is in an "equilibrium state."

This definition is circular: Thermodynamics deals with equilibrium states, but equilibrium states are states whose properties are consistent with thermodynamics!

These state variables are so special we give them a name: "extensive" -- they depend on the "extent" of the system:

If you take two identical systems with the same state variables and combine them, the state variables describing the combined system are *twice* the original ones.

The basic problem of Thermo is this:

- Given 2 or more simple systems, form a single composite system.
- Keep this composite system closed (no change in total U, V, N)
- Determine the equilibrium state if you remove internal constraints (allow them to exchange U, V, N)

If you're me, in an intro Thermo class, you might think... okay... that's obviously impossible.

We'll have to come up with a new theory for every composite system. Think of all the different systems with energy, volume, & different numbers/types of particles!

Cool class, thx!

But, as it turns out, all you need is a nice little function called the "entropy".

Now, don't run, I know.

This isn't the scary, confusing entropy you're used to from Stat. Mech.*

This is Thermo entropy, and it's so much gentler and more subtle.

*(it's the same one, but shh)

In Thermo the entropy is defined like this:

The entropy is a function S of the extensive parameters: S(U, V, N₁, N₂, ...) with the following property ⟶ the values obtained by the extensive parameters in the absence of internal constraints are those that maximize the entropy.

That's it. That's entropy. It's a function.

Entropy is more powerful w/ a few additional (reasonable) properties we force the function to have:

(1) The entropy of a composite system is the sum of the entropies of the constituents
(2) It's continuous and differentiable
(3) It's monotonically increasing as a function of U

(There's one other property, which we won't actually need here, but will be important when we define the temperature: the entropy vanishes in the state for which (dU/dS)_(V, N) = 0. This is often called the Third Law of Thermodynamics, very important in the history of the field)

Now, I'll point something out, which may make this whole enterprise feel more familiar.

If we can define the entropy S in terms of U, V, and N, with S monotonically increasing in U, can't we define U in terms of S, V, and N with U monotonically *decreasing* in S?

Yup!

In that case, the definitions above would give us a different extremum principle:

- The values obtained by the extensive variables in the absence of internal constraints are those which *minimize* the energy U over all the different possible S, V, N values!

Sound familiar?

Now, we're going to pull a trick.

When I first saw this trick I thought it was annoying. A fancy, symbol based Three-card Monte.

But just trust me.

Let's stick with the energy representation for now -- the system is described by:

U(S, V, N₁, N₂, ...)

Let's say we change the energy by a little (dU).

How can it change?

Well, the entropy can change, as can the volume, and the number of particles of each type -- all independently.

In other words:

dU = (dU/dS) dS + (dU/dV) dV + (dU/N₁) dN₁ + (dU/dN₂) dN₂ + ...

Let's give each of these (partial) derivatives a name:

- (dU/dS) will be called T or "the temperature"
- (dU/dV) will be called -P or "the negative of the pressure"
- (dU/dN) will be called μ or "the chemical potential"

So:

dU = T dS - P dV + μ₁ dN₁ + μ₂ dN₂ + ...

These could be any symbols or any names, we just *happen* to be giving them these names for no particular reason at all

All this theorizing is nice, but let's get practical, eh?

Let's construct a closed system:

- Two chambers separated by a wall that is fixed, and impermeable to each subsystems particles (no exchange of V, or N between them), but which *does* allow for exchange of energy.

Because the whole composite system is closed (no exchange of U, V, or N with the outside), we know that the total energy is constant!

So we can define a conservation law for the energies U₁ and U₂ of chambers 1 and 2:

U₁ + U₂ = 0 ⟶ dU₁ + dU₂ = 0 ⟶ dU₁ = - dU₂

Our definitions above told us that the values our extensive variables take on in the absence of internal constraints are those that maximize the entropy, so we're looking for the state where dS = 0 (the maximum happens when the function stops increasing).

So, since entropy is additive over the subsystems, let's look at the composite entropy:

S = S₁(U₁, V₁, N₁) + S₂(U₂, V₂, N₂).

Because V and N can't change, the change in entropy will only be due to U!

dS = (dS₁/dU₁)dU₁ + (dS₂/dU₂) dU₂

But hey! Those partial derivatives look familiar...

Since (dU/dS) was defined to be equal to T:

(dS/dU) = 1/T !

So, our equation above becomes:

dS = (1/T₁) dU₁ + (1/T₂) dU₂

From our conservation law from up above, we can substitute dU₂ for -dU₁ to get:

dS = (1/T₁) dU₁ - (1/T₂) dU₁ = (1/T₁ - 1/T₂) dU₁

But remember, we're looking for where dS = 0!

If dS = (1/T₁ - 1/T₂) dU₁, the only way dS can be zero is if:

- dU₁ = 0 (no energy transfer between subsystems -- BORING)

or

- (1/T₁ - 1/T₂) ⟶ T₁ = T₂

The systems will exchange energy until the temperatures are equal!

Not only that, but let's look at what happens *before* the temperatures are equal: let's say T₁ > T₂.

Well, we know the composite systems' extensive variables change to maximize the entropy, so, if we're not there yet 𝚫S > 0

Well, if

𝚫S ≃ (1/T₁ - 1/T₂) 𝚫U₁

then if T₁ > T₂, the only way to have 𝚫S > 0 will be if 𝚫U₁ < 0!

If the temperature in chamber 1 starts out larger than chamber 2, the way the entropy will be maximized will be for chamber 1 to *lose energy* to chamber 2!

With only a few simple, reasonable definitions for this made up function, the entropy, and made up names for partial derivatives, we get a definition of temperature that, when applied, behaves *exactly* like the temperature we're used to!

Systems at higher temperature put in contact with systems at lower temperature lose energy to the latter until the temperatures are equal!