Monatomic ideal gas derivation

Generalizing from discrete to continuous variables in statistical mechanics requires moving from counting microstates to measuring volumes in phase space.

The process is:

Find Ω_p(E' < E) = volume of momentum space with energy less than E
Take derivative: Ω_p'(E) = dΩ_p/dE = density at energy E
This gives you states per unit energy
Configuration space V^N just multiplies this result

Calculating Density of States

Step 1: Find cumulative volume $$\Omega(s_{AB} < s_0) = \begin{cases} \frac{1}{2}s_0^2 & s_0 \leq 1 \ 1 - \frac{1}{2}(1-s_0)^2 & s_0 > 1 \end{cases}$$
Step 2: Take derivative to get density $$\Omega'(s_0) = \frac{d\Omega(s_{AB} < s_0)}{ds_0}$$$$\Omega'(s_0) = \begin{cases} s_0 & s_0 \leq 1 \ (1-s_0) & s_0 > 1 \end{cases}$$

Probability Density

ρ (s_{A B}) = \frac{Ω^{'} (s_{A B})}{Ω}

Result: $$\rho(s_{AB}) = \begin{cases} s_{AB} & s_{AB} \leq 1 \ (1-s_{AB}) & s_{AB} \geq 1 \end{cases}$$

Phase Space of Billiards Model

Definition

Phase space = set of all possible positions $\vec{q}$ and momenta $\vec{p}$ of all particles
Dimensions:

$N$ particles in $d$ dimensions → $2 d N$ -dimensional phase space
2D billiards with $N$ particles → $4 N$ dimensions
3D gas with $N$ particles → $6 N$ dimensions

Density of States Formula

Ω (E, V, N) = ω_{d, N} \int_{H (\vec{p}, \vec{q}) = E} d^{d N} p, d^{d N} q

$H (\vec{p}, \vec{q})$ = Hamiltonian (total energy)
$ω_{d, N}$ = normalization constant
Constraint: $H (\vec{p}, \vec{q}) = E$ (energy shell)

Factorization

For ideal gas (energy independent of position): $$\Omega(E,V,N) = \omega_{d,N} \Omega_p(E,N) \times \Omega_q(V,N)$$
Configuration space $Ω_{q}$ : $$\Omega_q(V,N) = V^N$$

Each particle can be anywhere in volume $V$
Probability all particles on left half: $(1 / 2)^{N}$
Momentum space $Ω_{p}$ :
Energy constraint: $\sum_{i} \frac{| {\vec{p}}_{i} |^{2}}{2 m} = E$
This defines a hypersphere in momentum space

Key Results

2D Billiards Model

Ω (E, V = A, N) \approx ω_{2, N} m \sqrt{2 e} {(\frac{2 π e \times 2 m E}{2 N})}^{\frac{2 N - 2}{2}} A^{N}

3D Monatomic Ideal Gas

Ω (E, V, N) \approx ω_{3, N} m \sqrt{2 e} {(\frac{2 π e \times 2 m E}{3 N})}^{\frac{3 N - 2}{2}} V^{N}

Physical Interpretation

Energy spreads roughly evenly over all momentum components:

Average energy per momentum component: $\sim E / 2 N$ (2D) or $E / 3 N$ (3D)
$Ω$ grows extremely rapidly with $E$ (scales as $E^{N}$ )

Important Concepts

Volume vs Counting

Discrete systems: Count individual microstates
Continuous systems: Measure phase space volume
Volume is proportional to number of states

Energy Shell

Constraint $H (\vec{p}, \vec{q}) = E$ defines a surface in phase space
Actually consider thin shell: $E < H < E + d E$
Density: $Ω^{'} (E) = d Ω / d E$

Why This Matters

Foundation for thermodynamics: $S = k \ln Ω$
Explains ideal gas law
Shows how microscopic mechanics → macroscopic behavior