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Ensemble Theory

# Ensemble Theory: Complete Step-by-Step Derivation

This document provides the foundational derivation of statistical mechanics, starting from the classical phase space description to the formulation of the Microcanonical, Canonical, and Grand Canonical ensembles.

STEP 1: Phase Space Description

Consider a classical system consisting of \(N\) particles. The state of each particle \(i\) is defined by:

  • Position: \(\mathbf{q}_i = (q_{ix}, q_{iy}, q_{iz})\)
  • Momentum: \(\mathbf{p}_i = (p_{ix}, p_{iy}, p_{iz})\)

The state of the entire system at any instant is represented by a single point in a \(6N\)-dimensional space called Phase Space.

\[ \Gamma = (\mathbf{q}_1, \dots, \mathbf{q}_N; \mathbf{p}_1, \dots, \mathbf{p}_N) \]

Dimension of Phase Space: \(\dim(\Gamma) = 6N\).

STEP 2: Density Function

Since we cannot track the exact trajectory of \(10^{23}\) particles, we use a statistical approach. We define an Ensemble as a collection of a very large number of mental copies of the system, identical in macroscopic properties but different in microscopic states.

We define the Phase Space Density \(\rho(\mathbf{q}, \mathbf{p}, t)\) such that:

\[ \rho(\mathbf{q}, \mathbf{p}, t)\, d\Gamma \]

represents the fraction of ensemble members (systems) found within the volume element \(d\Gamma\) around the point \((\mathbf{q}, \mathbf{p})\) at time \(t\).

Normalization Condition: Since the system must exist somewhere in phase space, the total probability is 1:

\[ \int \rho(\mathbf{q}, \mathbf{p}, t)\, d\Gamma = 1 \]

STEP 3: The Continuity Equation

The number of ensemble members is conserved (systems are neither created nor destroyed). Therefore, the flow of probability in phase space behaves like an incompressible fluid. This leads to a continuity equation:

\[ \frac{\partial \rho}{\partial t} + \nabla_\Gamma \cdot (\rho \dot{\Gamma}) = 0 \]

where:

  • \(\nabla_\Gamma\) is the gradient operator in \(6N\) dimensions.
  • \(\dot{\Gamma} = (\dot{\mathbf{q}}, \dot{\mathbf{p}})\) is the velocity vector of the system point in phase space.

STEP 4: Hamilton’s Equations and Divergence

The time evolution of the system is governed by Hamilton's Equations of Motion:

\[ \dot{q}_i = \frac{\partial H}{\partial p_i}, \quad \dot{p}_i = -\frac{\partial H}{\partial q_i} \]

We calculate the divergence term \(\nabla_\Gamma \cdot \dot{\Gamma}\):

\[ \nabla_\Gamma \cdot \dot{\Gamma} = \sum_{i=1}^{N} \left[ \frac{\partial \dot{q}_i}{\partial q_i} + \frac{\partial \dot{p}_i}{\partial p_i} \right] \]

Substituting Hamilton's equations:

\[ \sum_{i=1}^{N} \left[ \frac{\partial}{\partial q_i}\left(\frac{\partial H}{\partial p_i}\right) + \frac{\partial}{\partial p_i}\left(-\frac{\partial H}{\partial q_i}\right) \right] \] \[ = \sum_{i=1}^{N} \left[ \frac{\partial^2 H}{\partial q_i \partial p_i} - \frac{\partial^2 H}{\partial p_i \partial q_i} \right] = 0 \]

The divergence of the phase velocity is zero because the order of partial differentiation does not matter.

STEP 5: Liouville’s Theorem

Since \(\nabla_\Gamma \cdot \dot{\Gamma} = 0\), the continuity equation simplifies:

\[ \frac{\partial \rho}{\partial t} + \dot{\Gamma} \cdot \nabla_\Gamma \rho + \rho (\nabla_\Gamma \cdot \dot{\Gamma}) = 0 \] \[ \frac{\partial \rho}{\partial t} + \dot{\Gamma} \cdot \nabla_\Gamma \rho = 0 \]

The left side is exactly the convective derivative (total time derivative) of \(\rho\):

\[ \frac{d\rho}{dt} = 0 \]

Physical Meaning: The local density of representative points in phase space remains constant as we move along with the dynamical trajectory of the system. This is Liouville's Theorem.

Equilibrium Condition: For a system in statistical equilibrium, the ensemble distribution must not depend explicitly on time (\(\partial \rho / \partial t = 0\)). By Liouville's theorem, this implies:

\[ \{\rho, H\} = 0 \]

Thus, \(\rho\) must be a function of the constants of motion, primarily the Hamiltonian \(H\) (Energy).

STEP 6: Ensemble Averages

In statistical mechanics, the macroscopic observable value of a physical quantity \(A\) is the ensemble average:

\[ \langle A \rangle = \int A(\Gamma) \rho(\Gamma, t) \, d\Gamma \]

This replaces the time-average measurement used in experiments, based on the Ergodic Hypothesis (time average = ensemble average).

STEP 7: The Microcanonical Ensemble

This ensemble describes an Isolated System with fixed:

  • Number of particles (\(N\))
  • Volume (\(V\))
  • Energy (\(E\)) within a narrow range \((E, E+\delta E)\)

Postulate of Equal a Priori Probability

For an isolated system in equilibrium, all accessible microstates (those with energy \(E\)) are equally likely to be occupied.

\[ \rho(\Gamma) = \begin{cases} \text{const}, & E \le H(\Gamma) \le E+\delta E \\ 0, & \text{otherwise} \end{cases} \]

The constant is \(1/\Omega(E)\), where \(\Omega(E)\) is the total volume of phase space accessible to the system:

\[ \Omega(E) = \int \delta(E - H(\Gamma)) \, d\Gamma \]

Boltzmann Entropy

The connection to thermodynamics is made via entropy \(S\):

\[ S = k_B \ln \Omega \]

Temperature is then defined as:

\[ \frac{1}{T} = \left(\frac{\partial S}{\partial E}\right)_{N,V} \]

STEP 8: The Canonical Ensemble

This ensemble describes a system in thermal contact with a large Heat Reservoir (Heat Bath) at temperature \(T\).

  • Total Energy \(E_{tot} = E_S + E_R = \text{const}\)
  • \(E_S \ll E_{tot}\)

The probability \(P(E_S)\) of finding the system in a state with energy \(E_S\) is proportional to the number of microstates available to the reservoir, \(\Omega_R(E_{tot} - E_S)\).

Derivation via Taylor Expansion

\[ \ln \Omega_R(E_{tot} - E_S) \approx \ln \Omega_R(E_{tot}) - E_S \left(\frac{\partial \ln \Omega_R}{\partial E}\right)_{E_{tot}} \]

Recognizing \(\partial \ln \Omega / \partial E = \beta = 1/k_B T\):

\[ \ln \Omega_R(E_{tot} - E_S) \approx \text{const} - \beta E_S \]

Taking the exponential gives the Canonical Distribution:

\[ \rho(\Gamma) = \frac{1}{Z} e^{-\beta H(\Gamma)} \]

Partition Function

The normalization factor \(Z\) is the Canonical Partition Function:

\[ Z = \int e^{-\beta H(\Gamma)} \, d\Gamma \]

Thermodynamic Potentials

  • Helmholtz Free Energy: \(F = -k_B T \ln Z\)
  • Average Energy: \(\langle E \rangle = -\frac{\partial \ln Z}{\partial \beta}\)
  • Entropy: \(S = -\left(\frac{\partial F}{\partial T}\right)_V\)

STEP 9: Energy Fluctuations

In the canonical ensemble, energy is not fixed. The fluctuation is given by:

\[ \langle (\Delta E)^2 \rangle = \frac{\partial^2 \ln Z}{\partial \beta^2} = k_B T^2 C_V \]

The relative fluctuation scales as:

\[ \frac{\sqrt{\langle (\Delta E)^2 \rangle}}{\langle E \rangle} \propto \frac{1}{\sqrt{N}} \]

As \(N \to \infty\) (Thermodynamic Limit), fluctuations vanish, and the Canonical and Microcanonical ensembles become equivalent.

STEP 10: The Grand Canonical Ensemble

This ensemble describes a system exchanging both Energy and Particles with a reservoir.

  • Fixed Variables: \(T\) (Temperature), \(V\) (Volume), \(\mu\) (Chemical Potential).

Following a similar derivation (Taylor expanding entropy with respect to \(E\) and \(N\)), we get the Grand Canonical Distribution:

\[ \rho(\Gamma, N) = \frac{1}{\mathcal{Z}} e^{-\beta(H(\Gamma) - \mu N)} \]

Grand Partition Function

\[ \mathcal{Z} = \sum_{N=0}^{\infty} \frac{1}{h^{3N}N!} \int e^{-\beta(H - \mu N)} \, d\Gamma \]

(Note: The factor \(1/h^{3N}N!\) corrects for the Gibbs paradox and phase space unit cell quantization).

Thermodynamic Connection:

\[ \Phi_G = -k_B T \ln \mathcal{Z} = PV \]

where \(\Phi_G\) is the Grand Potential.