The Microcanonical Ensemble

The Microcanonical Ensemble: Geometry & Equipartition

This guide details the statistical mechanics of isolated systems (fixed \(E\), \(V\), \(N\)). By analyzing the geometry of phase space surfaces, we derive the fundamental definition of temperature and provide a rigorous proof of the Equipartition Theorem.

1. Statement and Hypothesis

The Ergodic Hypothesis: If we follow the time evolution of a particular point in phase space, it will visit any region of the available phase space with equal probability over a long period.

For a closed system with fixed Energy \(E\) and Particle Number \(N\), the Hamiltonian is constrained such that \(H(q,p) = E\). Geometrically, this means the system will always exist on a hypersurface of dimension \(2f-1\) (where \(f\) is the number of degrees of freedom).

2. The Density of States

To describe the system mathematically, we define the probability density \( \rho(q,p) \). Since the system is confined strictly to the energy surface \( E \), we use the Dirac delta function.

The Probability Density

\[ \rho(q,p) = \frac{\delta(E - H(q,p))}{\Sigma} \]

Surface Area vs. Volume

We distinguish between two key geometric quantities in phase space:

• Surface Area (\(\Sigma\)): The area of the constant energy hypersurface. \[ \Sigma(E) = \int d\Gamma \, \delta(E - H(q,p)) \]
• Phase Volume (\(\Omega\)): The total volume of phase space where energy is less than \(E\). \[ \Omega(E) = \int d\Gamma \, \Theta(E - H(q,p)) \] (where \( \Theta \) is the step function).

These two are related by differentiation. The surface area is the rate of change of the volume with respect to energy:

\[ \Sigma = \frac{\partial \Omega}{\partial E} \]

3. Geometric Definition of Temperature

We connect these geometric concepts to thermodynamics using the Boltzmann Hypothesis:

\[ S = k \ln W \approx k \ln \Omega \]

From thermodynamics, we know the definition of temperature is \( \frac{1}{T} = \frac{\partial S}{\partial E} \). Let us derive \( T \) using our phase space variables:

\[ \frac{1}{T} = k \frac{\partial (\ln \Omega)}{\partial E} = \frac{k}{\Omega} \frac{\partial \Omega}{\partial E} \]

Substituting \( \Sigma = \frac{\partial \Omega}{\partial E} \), we get:

\[ \frac{1}{T} = k \frac{\Sigma}{\Omega} \]

Rearranging this gives a beautiful geometric definition of thermal energy:

\[ kT = \frac{\Omega}{\Sigma} = \frac{\text{Phase Volume}}{\text{Surface Area}} \]

4. Proof of the Equipartition Theorem

We wish to calculate the ensemble average of the quantity \( p_n \frac{\partial H}{\partial p_n} \). This calculation will rigorously prove why each degree of freedom contributes \( \frac{1}{2}kT \) to the energy.

Step 1: Setup the Integral

The average over the microcanonical ensemble is an integral over the energy surface area \( \Sigma \):

\[ \left\langle p_n \frac{\partial H}{\partial p_n} \right\rangle = \frac{1}{\Sigma} \int d\Gamma \, p_n \frac{\partial H}{\partial p_n} \delta(E - H) \]

Step 2: Relate Delta to Step Function

Recall that the derivative of the step function \( \Theta \) is the delta function. By chain rule:

\[ \frac{\partial}{\partial p_n} \Theta(E - H) = -\delta(E - H) \frac{\partial H}{\partial p_n} \]

Therefore, we can replace the \( \frac{\partial H}{\partial p_n} \delta(E-H) \) term in our integral with \( -\frac{\partial \Theta}{\partial p_n} \):

\[ \text{Integral} = -\frac{1}{\Sigma} \int d\Gamma \, p_n \frac{\partial}{\partial p_n} \Theta(E - H) \]

Step 3: Integration by Parts

We perform integration by parts with respect to \( p_n \).
Boundary term: \( [p_n \Theta(E-H)]_{-\infty}^{+\infty} \). This vanishes because the energy shell is bounded (density goes to zero at infinite momentum).
Remaining term: Since \( \frac{\partial p_n}{\partial p_n} = 1 \), we are left with:

\[ = \frac{1}{\Sigma} \int d\Gamma \, \Theta(E - H) \]

Step 4: Final Result

The integral of the step function \( \Theta \) over all phase space is simply the total volume \( \Omega \). Thus:

\[ \left\langle p_n \frac{\partial H}{\partial p_n} \right\rangle = \frac{\Omega}{\Sigma} \]

Using our previously derived definition of temperature (\( kT = \Omega/\Sigma \)), we arrive at the final theorem:

\[ \left\langle p_n \frac{\partial H}{\partial p_n} \right\rangle = kT \]

Application: The Ideal Gas

For an ideal gas, \( H = \sum \frac{p_n^2}{2m} \).
Then \( p_n \frac{\partial H}{\partial p_n} = p_n \frac{p_n}{m} = 2 \left( \frac{p_n^2}{2m} \right) \).
Therefore, \( 2 \langle \text{Kinetic Energy} \rangle = kT \), or:

\[ \langle K.E. \rangle = \frac{1}{2} kT \]