Editing Adiabatic invariant (section)

== Thermodynamics ==

In thermodynamics, adiabatic changes are those that do not increase the entropy. They occur slowly in comparison to the other characteristic timescales of the system of interest<ref>{{cite encyclopedia |first1=D. V. |last1=Anosov |first2=A. P. |last2=Favorskii |title=Adiabatic invariant |encyclopedia=Encyclopedia of Mathematics |editor-first=Michiel |editor-last=Hazewinkel |year=1988 |pages=43–44 |url=https://books.google.com/books?id=IEnqCAAAQBAJ&dq=adiabatic+invariant+slowly+in+comparison&pg=PA43 |publisher=Reidel, Dordrecht |volume=1 (A-B)|isbn=9789401512398 }}</ref> and allow heat flow only between objects at the same temperature. For isolated systems, an adiabatic change allows no heat to flow in or out.

=== Adiabatic expansion of an ideal gas ===

If a container with an [[ideal gas]] is expanded instantaneously, the temperature of the gas doesn't change at all, because none of the molecules slow down. The molecules keep their kinetic energy, but now the gas occupies a bigger volume. If the container expands slowly, however, so that the ideal gas pressure law holds at any time, gas molecules lose energy at the rate that they do work on the expanding wall. The amount of work they do is the pressure times the area of the wall times the outward displacement, which is the pressure times the change in the volume of the gas:
<math display="block">
dW = P \, dV = \frac{N k_\text{B} T}{V} \, dV.
</math>

If no heat enters the gas, the energy in the gas molecules is decreasing by the same amount. By definition, a gas is ideal when its temperature is only a function of the internal energy per particle, not the volume. So
<math display="block">
dT = \frac{1}{N C_v} \, dE,
</math>
where <math>C_v</math> is the specific heat at constant volume. When the change in energy is entirely due to work done on the wall, the change in temperature is given by
<math display="block">
N C_v \, dT = -dW = -\frac{N k_\text{B}T}{V} \, dV.
</math>

This gives a differential relationship between the changes in temperature and volume, which can be integrated to find the invariant. The constant <math>k_\text{B}</math> is just a [[natural units|unit conversion factor]], which can be set equal to one:
<math display="block">
d(C_v N \log T) = -d(N \log V).
</math>

So 
<math display="block">
C_v N \log T + N \log V
</math>
is an adiabatic invariant, which is related to the entropy 
<math display="block">
S = C_v N \log T + N \log V - N \log N = N \log \left(\frac{T^{C_v} V}{N}\right).
</math>

Thus entropy is an adiabatic invariant. The ''N''&nbsp;log(''N'') term makes the entropy additive, so the entropy of two volumes of gas is the sum of the entropies of each one.

In a molecular interpretation, ''S'' is the logarithm of the phase-space volume of all gas states with energy ''E''(''T'') and volume ''V''.

For a monatomic ideal gas, this can easily be seen by writing down the energy:
<math display="block">
E = \frac{1}{2m} \sum_k \left(p_{k1}^2 + p_{k2}^2 + p_{k3}^2 \right).
</math>

The different internal motions of the gas with total energy ''E'' define a sphere, the surface of a 3''N''-dimensional ball with radius <math>\sqrt{2mE}</math>. The volume of the sphere is
<math display="block">
\frac{2\pi^{3N/2}(2mE)^{(3N-1)/2}}{\Gamma(3N/2)},</math>
where <math>\Gamma</math> is the [[gamma function]].

Since each gas molecule can be anywhere within the volume ''V'', the volume in phase space occupied by the gas states with energy ''E'' is  
<math display="block">
\frac{2\pi^{3N/2}(2mE)^{(3N-1)/2} V^N}{\Gamma(3N/2)}.
</math>

Since the ''N'' gas molecules are indistinguishable, the phase-space volume is divided by <math>N! = \Gamma(N + 1)</math>, the number of permutations of ''N'' molecules.

Using [[Stirling's approximation]] for the gamma function, and ignoring factors that disappear in the logarithm after taking ''N'' large, 
<math display="block">
\begin{align}
S &= N \left( \tfrac{3}{2} \log(E) - \tfrac{3}{2} \log(\tfrac{3}{2}N) + \log(V) - \log(N) \right) \\
  &= N \left( \tfrac{3}{2} \log\left(\tfrac{2}{3} E/N\right) + \log\left(\frac{V}{N}\right)\right).
\end{align}
</math>

Since the specific heat of a monatomic gas is 3/2, this is the same as the thermodynamic formula for the entropy.

=== Wien's law – adiabatic expansion of a box of light ===

For a box of radiation, ignoring quantum mechanics, the energy of a classical field in thermal equilibrium is [[ultraviolet catastrophe|infinite]], since [[equipartition]] demands that each field mode has an equal energy on average, and there are infinitely many modes. This is physically ridiculous, since it means that all energy leaks into high-frequency electromagnetic waves over time.

Still, without quantum mechanics, there are some things that can be said about the equilibrium distribution from thermodynamics alone, because there is still a notion of adiabatic invariance that relates boxes of different size.

When a box is slowly expanded, the frequency of the light recoiling from the wall can be computed from the [[Doppler shift]]. If the wall is not moving, the light recoils at the same frequency. If the wall is moving slowly, the recoil frequency is only equal in the frame where the wall is stationary. In the frame where the wall is moving away from the light, the light coming in is bluer than the light coming out by twice the Doppler shift factor ''v''/''c'':
<math display="block">
\Delta f = \frac{2v}{c} f.
</math>

On the other hand, the energy in the light is also decreased when the wall is moving away, because the light is doing work on the wall by radiation pressure. Because the light is reflected, the pressure is equal to twice the momentum carried by light, which is ''E''/''c''. The rate at which the pressure does work on the wall is found by multiplying by the velocity:
<math display="block">
\Delta E = v \frac{2E}{c}.
</math>

This means that the change in frequency of the light is equal to the work done on the wall by the radiation pressure. The light that is reflected is changed both in frequency and in energy by the same amount:
<math display="block">
\frac{\Delta f}{f} = \frac{\Delta E}{E}.
</math>

Since moving the wall slowly should keep a thermal distribution fixed, the probability that the light has energy ''E'' at frequency ''f'' must only be a function of ''E''/''f''.

This function cannot be determined from thermodynamic reasoning alone, and Wien guessed at the form that was valid at high frequency. He supposed that the average energy in high-frequency modes was suppressed by a Boltzmann-like factor:
<math display="block">
\langle E_f \rangle = e^{-\beta h f}.
</math>
This is not the expected classical energy in the mode, which is <math>1/2\beta</math> by equipartition, but a new and unjustified assumption that fit the high-frequency data.

When the expectation value is added over all modes in a cavity, this is [[Wien approximation|Wien's distribution]], and it describes the thermodynamic distribution of energy in a classical gas of photons. Wien's law implicitly assumes that light is statistically composed of packets that change energy and frequency in the same way. The entropy of a Wien gas scales as the volume to the power ''N'', where ''N'' is the number of packets. This led Einstein to suggest that light is composed of localizable particles with energy proportional to the frequency. Then the entropy of the Wien gas can be given a statistical interpretation as the number of possible positions that the photons can be in.