Micro-Canonical Ensemble: Complete Statistical Physics Guide with Entropy Derivation

House of Physics October 02, 2025
Micro-Canonical Ensemble: Complete Statistical Physics Guide | Thermal Physics

Micro-Canonical Ensemble: Complete Statistical Physics Guide

Mastering Isolated Systems, Equal Probability Postulate, Phase Space, Entropy Derivation, and the Sackur-Tetrode Equation
Micro-Canonical Ensemble Statistical Mechanics Entropy Phase Space Sackur-Tetrode Equation Gibbs Paradox Reading Time: 25 min

📋 Table of Contents

📜 Historical Background

The development of statistical mechanics and the micro-canonical ensemble concept revolutionized our understanding of thermodynamics:

  • Ludwig Boltzmann (1870s): Developed statistical interpretation of entropy
  • J. Willard Gibbs (1902): Formulated the ensemble theory in statistical mechanics
  • Max Planck (1900s): Applied statistical mechanics to blackbody radiation
  • Otto Sackur & Hugo Tetrode (1912): Derived the entropy formula for monatomic ideal gas
  • Albert Einstein (1905): Applied statistical concepts to Brownian motion

These developments established the foundation of statistical mechanics, connecting microscopic physics with macroscopic thermodynamics.

Introduction to Micro-Canonical Ensemble

🔬 What is a Micro-Canonical Ensemble?

A micro-canonical ensemble corresponds to a collection of isolated systems. In this ensemble, the systems neither exchange mass nor energy with their surroundings, meaning they have:

  • Fixed volume (V)
  • Fixed energy (E)
  • Fixed number of particles (N)
  • Insulated boundaries

This ensemble is fundamental in statistical mechanics as it provides the simplest starting point for deriving thermodynamic properties from microscopic physics.

💡 Key Insight

The micro-canonical ensemble describes systems in thermodynamic equilibrium that are completely isolated from their environment. All accessible microstates (states with energy between E and E+dE) are equally probable, which is the fundamental postulate of statistical mechanics.

Fundamental Postulate of Statistical Mechanics

⚖️ Equal Probability Postulate

For an isolated system in equilibrium, all accessible microstates are equally probable. This is the fundamental postulate of statistical mechanics.

A system is in equilibrium when its macroscopic parameters remain unchanged over time. If changes occur in one or more parameters, the system is not in equilibrium.

Equilibrium Condition Non-Equilibrium Condition
Macroscopic parameters constant Macroscopic parameters changing
All accessible microstates equally probable Probability distribution not uniform
Maximum entropy Entropy increasing toward maximum

Micro-Canonical Distribution

📊 Probability Distribution

Let the energies of systems lie between \( E \) and \( E + dE \). According to the fundamental postulate, the probability \( \beta_i \) of finding the system in a particular microstate i is:

\[ \beta_i = \begin{cases} \text{constant} & \text{if } E < E_i < E + dE \\ 0 & \text{if } E_i \text{ doesn't lie in range } E - E + dE. \end{cases} \]

This probability distribution is known as the micro-canonical distribution.

🧮 Alternative Formulation

Using Dirac Delta Function

The micro-canonical distribution can also be expressed using the Dirac delta function:

\[ \beta_i = P(E) = \text{constant} \cdot \delta (E - E_0) \]

where \( \delta \) is the Dirac delta function, which ensures that only states with energy exactly equal to \( E_0 \) have non-zero probability.

Phase Space and Volume Element

🌌 Phase Space Concept

In statistical mechanics, the state of a system is described by its position in phase space. For a system with N particles, the phase space has 6N dimensions (3 position and 3 momentum coordinates for each particle).

The fundamental volume element in phase space is:

\[ d\Gamma = \frac{d^3q_1 d^3p_1 d^3q_2 d^3p_2 \cdots d^3q_N d^3p_N}{h^{3N}} \]

where \( h \) is Planck's constant, which provides the natural scale for quantum systems.

📏 Quantum Considerations

The factor \( h^{3N} \) in the phase space volume element arises from quantum mechanical considerations. It represents the smallest distinguishable volume in phase space according to the uncertainty principle, which states that \( \Delta q \Delta p \geq h \).

Entropy in Micro-Canonical Ensemble

🔥 Boltzmann's Entropy Formula

The thermodynamic entropy S is related to the number of accessible microstates Ω by Boltzmann's famous formula:

\[ S = k_B \ln \Omega \]

where \( k_B \) is Boltzmann's constant and Ω is the number of microstates accessible to the system with energy between E and E+dE.

This formula provides the fundamental connection between thermodynamics (macroscopic) and statistical mechanics (microscopic).

💡 Statistical Interpretation

Entropy is a measure of the number of ways a system can be arranged at the microscopic level while maintaining the same macroscopic properties. Higher entropy means more possible microstates and greater disorder.

Derivation of Sackur-Tetrode Equation

📐 Entropy of Ideal Monatomic Gas

The Sackur-Tetrode equation gives the entropy of an ideal monatomic gas in the micro-canonical ensemble. Consider a system with N particles of mass m confined in volume V.

Phase Space Volume Calculation

🧮 Step 1: Phase Space Volume

Total Phase Space Volume

The phase space volume for a system with energy between E and E+ΔE is:

\[ \Delta \Gamma = \frac{V^N}{h^{3N}} \Delta F_p \]

Momentum Space Integral

The momentum space integral with constraint \( E - \Delta E \leq \sum_{k=1}^{3N} \frac{P_k^2}{2M} \leq E \) is:

\[ \Delta F_p = \frac{(\pi)^{3N/2}}{(\frac{3N}{2})!} (2mE)^{3N/2} \]
\[ = \frac{(2\pi mE)^{3N/2}}{(\frac{3N}{2})!} \]

Complete Expression

Combining both parts:

\[ \Delta \Gamma = \frac{V^N}{h^{3N}} \left( 2\pi mE \right)^{3N/2} \cdot \frac{1}{(\frac{3N}{2})!} \]

Applying Stirling's Approximation

🧮 Step 2: Entropy Calculation

Boltzmann Entropy Formula

\[ S = k_B \ln \Delta \Gamma \]
\[ = k_B \ln \left[ \frac{V^N \left( \frac{2\pi mE}{h^2} \right)^{3N/2}}{\left( \frac{3N}{2} \right)!} \right] \]

Stirling's Approximation

Using Stirling's approximation \( \ln n! \approx n \ln n - n \):

\[ \ln \left( \frac{3N}{2} \right)! \approx \frac{3N}{2} \ln \left( \frac{3N}{2} \right) - \frac{3N}{2} \]

Substituting and Simplifying

\[ S = k_B \left[ N \ln V + \frac{3N}{2} \ln \left( \frac{2\pi mE}{h^2} \right) - \frac{3N}{2} \ln \left( \frac{3N}{2} \right) + \frac{3N}{2} \right] \]

Final Entropy Expression

🧮 Step 3: Sackur-Tetrode Equation

Final Simplification

\[ S = k_B N \left[ \ln V + \frac{3}{2} \ln \left( \frac{2\pi mE}{h^2} \right) - \frac{3}{2} \ln \left( \frac{3N}{2} \right) + \frac{3}{2} \right] \]

Alternative Form

Using \( E = \frac{3}{2} N k_B T \) for monatomic ideal gas:

\[ S = N k_B \left[ \ln \left( \frac{V}{N} \right) + \frac{3}{2} \ln \left( \frac{2\pi m k_B T}{h^2} \right) + \frac{5}{2} \right] \]

Standard Sackur-Tetrode Equation

\[ S = N k_B \left[ \ln \left( \frac{V}{N \lambda_T^3} \right) + \frac{5}{2} \right] \]

where \( \lambda_T = \frac{h}{\sqrt{2\pi m k_B T}} \) is the thermal de Broglie wavelength.

Sample Problem: Entropy Calculation

Calculate the entropy of 1 mole of monatomic ideal gas at standard temperature and pressure (STP) using the Sackur-Tetrode equation.

Given:
\[ N = 6.022 \times 10^{23} \]
\[ V = 22.4 \, \text{L} = 0.0224 \, \text{m}^3 \]
\[ T = 273.15 \, \text{K} \]
\[ m = 6.69 \times 10^{-27} \, \text{kg} \quad \text{(for helium)} \]
\[ k_B = 1.38 \times 10^{-23} \, \text{J/K} \]
\[ h = 6.626 \times 10^{-34} \, \text{J·s} \]
Thermal de Broglie wavelength:
\[ \lambda_T = \frac{h}{\sqrt{2\pi m k_B T}} \]
\[ = \frac{6.626 \times 10^{-34}}{\sqrt{2\pi \times 6.69 \times 10^{-27} \times 1.38 \times 10^{-23} \times 273.15}} \]
\[ = 7.84 \times 10^{-11} \, \text{m} \]
Entropy calculation:
\[ S = N k_B \left[ \ln \left( \frac{V}{N \lambda_T^3} \right) + \frac{5}{2} \right] \]
\[ = 6.022 \times 10^{23} \times 1.38 \times 10^{-23} \left[ \ln \left( \frac{0.0224}{6.022 \times 10^{23} \times (7.84 \times 10^{-11})^3} \right) + \frac{5}{2} \right] \]
\[ = 8.31 \left[ \ln \left( \frac{0.0224}{2.91 \times 10^{-8}} \right) + 2.5 \right] \]
\[ = 8.31 \left[ \ln (7.70 \times 10^5) + 2.5 \right] \]
\[ = 8.31 \left[ 13.55 + 2.5 \right] \]
\[ = 8.31 \times 16.05 \]
\[ = 133.4 \, \text{J/K} \]

Gibbs Paradox and Its Resolution

⚖️ The Gibbs Paradox

The Gibbs paradox concerns the entropy change when two identical gases are mixed. According to classical thermodynamics, mixing two identical gases at the same temperature and pressure should result in no entropy change. However, early statistical mechanics calculations predicted an entropy increase.

🧮 Resolution of Gibbs Paradox

Classical Calculation

For two identical gases with N particles each in volume V:

\[ S_{\text{initial}} = 2 \times k_B N \ln \left( \frac{V}{N \lambda_T^3} \right) \]
\[ S_{\text{final}} = k_B (2N) \ln \left( \frac{2V}{2N \lambda_T^3} \right) \]
\[ \Delta S = S_{\text{final}} - S_{\text{initial}} = 2k_B N \ln 2 \]

This gives a positive entropy change, which contradicts thermodynamics.

Quantum Resolution

The resolution comes from quantum mechanics and the indistinguishability of identical particles. The correct counting of microstates requires dividing by N!:

\[ \Omega_{\text{correct}} = \frac{\Omega_{\text{classical}}}{N!} \]
\[ S = k_B \ln \Omega_{\text{correct}} = k_B \ln \left( \frac{\Omega_{\text{classical}}}{N!} \right) \]

Corrected Entropy

\[ S = N k_B \left[ \ln \left( \frac{V}{N \lambda_T^3} \right) + \frac{5}{2} \right] \]

With this correction, mixing identical gases gives \( \Delta S = 0 \), resolving the paradox.

🔍 Indistinguishability of Particles

The Gibbs paradox resolution highlights a fundamental quantum mechanical principle: identical particles are indistinguishable. This means that permutations of identical particles do not create new microstates, which is why we divide by N! in the microstate counting.

Applications and Examples

🎯 Ideal Gas Thermodynamics

The micro-canonical ensemble provides the foundation for deriving all thermodynamic properties of ideal gases, including pressure, temperature, and heat capacities, from first principles.

🔬 Paramagnetic Systems

Micro-canonical ensemble analysis of paramagnetic systems explains temperature dependence of magnetization and the behavior of spins in magnetic fields.

⚛️ Quantum Statistical Mechanics

The concepts developed for the micro-canonical ensemble form the basis for understanding more complex quantum statistical ensembles like Fermi-Dirac and Bose-Einstein statistics.

🌡️ Thermal Equilibrium

The micro-canonical ensemble provides insights into how systems reach thermal equilibrium through the equal probability postulate and entropy maximization.

Practice Problems with Solutions

Problem 1: Entropy of Two-Level System

Consider a system of N distinguishable particles, each of which can be in one of two energy states: 0 or ε. The total energy of the system is E = Mε, where M is an integer between 0 and N. Calculate the entropy of this system.

Number of microstates:
\[ \Omega = \frac{N!}{M!(N-M)!} \]
Entropy:
\[ S = k_B \ln \Omega \]
\[ = k_B \ln \left( \frac{N!}{M!(N-M)!} \right) \]
Using Stirling's approximation:
\[ S \approx k_B \left[ N \ln N - M \ln M - (N-M) \ln (N-M) \right] \]
Problem 2: Phase Space Volume

Calculate the phase space volume for a single particle of mass m confined in a volume V with energy between E and E+ΔE.

Position space volume:
\[ \int d^3q = V \]
Momentum space volume (spherical shell):
\[ \int_{E}^{E+\Delta E} d^3p = 4\pi p^2 dp \]
\[ = 4\pi (2mE) \frac{m}{p} dE \]
\[ = 4\pi m \sqrt{2mE} \, \Delta E \]
Total phase space volume:
\[ \Delta \Gamma = \frac{V}{h^3} \cdot 4\pi m \sqrt{2mE} \, \Delta E \]
Problem 3: Entropy of Einstein Solid

An Einstein solid consists of N quantum harmonic oscillators. Each oscillator has energy levels \( E_n = (n + \frac{1}{2})\hbar\omega \), where n = 0, 1, 2, ... If the total energy is E = q\hbar\omega (where q >> N), show that the entropy is approximately \( S \approx k_B N [\ln(\frac{q}{N}) + 1] \).

Number of microstates:
\[ \Omega = \frac{(q + N - 1)!}{q!(N-1)!} \]
Using Stirling's approximation:
\[ \ln \Omega \approx (q + N) \ln(q + N) - q \ln q - N \ln N \]
For q >> N:
\[ \ln \Omega \approx N \ln \left( \frac{q}{N} \right) + N \]
Entropy:
\[ S = k_B \ln \Omega \approx k_B N \left[ \ln \left( \frac{q}{N} \right) + 1 \right] \]

Frequently Asked Questions

❓ Why is the micro-canonical ensemble important in statistical mechanics?

The micro-canonical ensemble is fundamental because:

  • It describes isolated systems, which are the simplest to analyze theoretically
  • It provides the foundation for deriving other ensembles (canonical, grand canonical)
  • It directly connects to the fundamental postulate of statistical mechanics
  • It allows derivation of thermodynamic quantities from microscopic physics
  • It provides the most direct interpretation of entropy through Boltzmann's formula

While real systems are rarely perfectly isolated, understanding the micro-canonical ensemble is essential for building the theoretical framework of statistical mechanics.

❓ What is the physical significance of the N! factor in the entropy expression?

The N! factor accounts for the indistinguishability of identical particles, which is a quantum mechanical concept. Classically, particles were considered distinguishable, leading to the Gibbs paradox.

The physical significance includes:

  • It ensures that entropy is extensive (scales with system size)
  • It resolves the Gibbs paradox (mixing identical gases gives zero entropy change)
  • It reflects the quantum mechanical principle that permutations of identical particles don't create new states
  • It makes the entropy expression consistent with the third law of thermodynamics

Without the N! factor, entropy would not be properly extensive, and we would get unphysical results like the entropy of mixing for identical gases.

❓ How does the micro-canonical ensemble relate to the second law of thermodynamics?

The micro-canonical ensemble provides a statistical interpretation of the second law of thermodynamics:

  • The second law states that entropy tends to increase in isolated systems
  • In the micro-canonical ensemble, systems evolve toward states with higher multiplicity (more microstates)
  • Since S = k_B ln Ω, higher multiplicity means higher entropy
  • The equilibrium state corresponds to maximum entropy, which is the state with the largest number of accessible microstates

This connection shows that the second law is fundamentally statistical in nature - systems tend toward states that are more probable at the microscopic level.

❓ What are the limitations of the micro-canonical ensemble?

While fundamental, the micro-canonical ensemble has several limitations:

  • Real systems are rarely perfectly isolated - they typically exchange energy with their environment
  • It can be mathematically challenging to work with for complex systems
  • For systems with continuous energy spectra, we need to introduce an energy width ΔE
  • It's not always the most convenient ensemble for calculations, especially for systems in thermal contact with reservoirs

For these reasons, other ensembles like the canonical ensemble (fixed temperature) and grand canonical ensemble (fixed chemical potential) are often more practical for real-world applications.

📚 Master Statistical Mechanics

Understanding the micro-canonical ensemble is fundamental to statistical mechanics, thermodynamics, and modern physics. It provides the foundation for connecting microscopic physics with macroscopic thermodynamics. Continue your journey into the fascinating world of statistical physics.

Read More: Statistical Physics Notes

© House of Physics | Thermal and Statistical Physics: Micro-Canonical Ensemble

Based on university physics curriculum with additional insights from statistical mechanics textbooks

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