Microcanonical Ensemble: Complete Statistical Mechanics Guide with Sackur-Tetrode Formula & Gibbs Paradox Resolution

House of Physics October 02, 2025
Microcanonical Ensemble: Complete Statistical Mechanics Guide | HRK Physics

Microcanonical Ensemble: Complete Statistical Mechanics Guide

Mastering Statistical Mechanics Foundations: Equal A Priori Probability, Sackur-Tetrode Formula, Gibbs Paradox, and Entropy Calculations
Microcanonical Ensemble Statistical Mechanics Sackur-Tetrode Formula Gibbs Paradox Chemical Potential Reading Time: 25 min

📋 Table of Contents

📜 Historical Background

The development of statistical mechanics and the microcanonical ensemble concept revolutionized thermodynamics in the late 19th and early 20th centuries:

  • Ludwig Boltzmann (1870s): Developed statistical interpretation of entropy
  • J. Willard Gibbs (1902): Formalized ensemble theory in his book "Elementary Principles in Statistical Mechanics"
  • Max Planck (1900-1901): Applied statistical methods to blackbody radiation
  • Albert Einstein (1905): Used statistical mechanics to explain Brownian motion
  • Otto Sackur & Hugo Tetrode (1912): Derived the entropy formula for ideal gases

These developments established statistical mechanics as the bridge between microscopic physics and macroscopic thermodynamics.

1. Microcanonical Ensemble Definition

🔬 What is a Microcanonical Ensemble?

A microcanonical ensemble represents a collection of completely isolated thermodynamic systems characterized by:

  • Fixed total energy (E)
  • Fixed number of particles (N)
  • Fixed volume (V)
  • No exchange of energy or matter with surroundings
  • Insulated boundaries

Mathematically, we describe such systems as having energies confined to a narrow range between E and E + ΔE, where ΔE is an infinitesimally small energy width.

💡 Key Insight

The microcanonical ensemble provides the most fundamental description of equilibrium statistical mechanics. All other ensembles (canonical, grand canonical) can be derived from it. It represents systems that are completely isolated from their environment.

Microcanonical Ensemble Canonical Ensemble Grand Canonical Ensemble
Fixed E, N, V Fixed T, N, V Fixed T, μ, V
Isolated system Heat bath Heat and particle bath
All states equally probable Boltzmann distribution Gibbs distribution

2. Probable Distribution of Energy

2.1 Fundamental Postulate

🎯 Equal A Priori Probability Postulate

The cornerstone of microcanonical ensemble is the equal a priori probability postulate:

"An isolated system in equilibrium is equally likely to be found in any of its accessible microstates."

This postulate forms the foundation of statistical mechanics and allows us to calculate thermodynamic properties from microscopic descriptions.

2.2 Probability Distribution

📊 Probability in Microcanonical Ensemble

For a system with discrete energy states Eᵢ, the probability βᵢ of finding the system in state i is:

\[ \beta_i = \frac{1}{\Omega} \quad \text{if } E \leq E_i \leq E + \Delta E \]
\[ \beta_i = 0 \quad \text{otherwise} \]

Where Ω is the total number of accessible microstates.

📈 Continuous Representation

For continuous systems, we use the Dirac delta function:

\[ \beta = \text{constant} \times \delta(E - E_0) \]

This ensures that only states with energy exactly equal to E₀ are accessible in the idealized case.

3. Analytical Nature of Probability

3.1 Phase Space Analysis

🌌 Phase Space Description

Consider N particles in a volume V with total energy E. The phase space has 6N dimensions (3 position + 3 momentum coordinates per particle).

The probability density in phase space is:

\[ P(q,p) = \text{constant} \quad \text{within accessible region} \]
\[ P(q,p) = 0 \quad \text{outside accessible region} \]

3.2 Two Particles in 1D Example

🧮 Example: Two Particles in One Dimension

Position Space

Coordinates: \( q_1, q_2 \in [0, L] \)
Forms a square of area \( L^2 \) in configuration space

Momentum Space

Energy constraint: \( \frac{p_1^2}{2M} + \frac{p_2^2}{2M} = E_0 \)
This describes a circle: \( p_1^2 + p_2^2 = 2ME_0 \)
Radius: \( R = \sqrt{2ME_0} \)

Accessible Phase Space Volume

Position space volume: \( L^2 \)
Momentum space volume: \( \pi R^2 = \pi (2ME_0) \)
Total accessible volume: \( L^2 \cdot \pi (2ME_0) \)

4. Derivation of Sackur-Tetrode Formula

4.1 Phase Space Volume

🧮 Phase Space Calculation for N Particles

Step 1: Position Space Volume

For N identical particles in volume V:
\[ \text{Position space volume} = V^N \]

Step 2: Momentum Space Volume

Spherical shell in 3N dimensions:
\[ \Omega_p = \int \delta(E - H(p)) \, d^{3N}p \]
\[ = \frac{(2\pi mE)^{3N/2}}{(3N/2)!} \]

Step 3: Total Microstates

\[ \Omega = \frac{1}{h^{3N} N!} \times V^N \times \Omega_p \]
\[ = \frac{1}{h^{3N} N!} V^N \frac{(2\pi mE)^{3N/2}}{(3N/2)!} \]

4.2 Entropy Calculation

🧮 Sackur-Tetrode Formula Derivation

Step 1: Boltzmann Entropy

\[ S = k_B \ln \Omega \]
\[ = k_B \ln \left[ \frac{1}{h^{3N} N!} V^N \frac{(2\pi mE)^{3N/2}}{(3N/2)!} \right] \]

Step 2: Apply Stirling's Approximation

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

Step 3: Simplify Expression

\[ S = k_B N \left[ \ln \left( \frac{V}{N} \left( \frac{4\pi mE}{3h^2 N} \right)^{3/2} \right) + \frac{5}{2} \right] \]

Step 4: Final Sackur-Tetrode Formula

\[ S = N k_B \left[ \ln \left( \frac{V}{N} \left( \frac{4\pi m E}{3h^2 N} \right)^{3/2} \right) + \frac{5}{2} \right] \]

📏 Important Note

The Sackur-Tetrode formula gives the absolute entropy of an ideal gas, not just entropy differences. This was a major achievement of statistical mechanics, as classical thermodynamics could only determine entropy differences.

5. Gibbs Paradox and Its Resolution

5.1 The Paradox

⚖️ Gibbs Paradox Statement

Consider two identical ideal gases separated by a partition. When the partition is removed:

  • Classical thermodynamics predicts an entropy increase of \( 2Nk_B \ln 2 \)
  • But experimentally, no entropy change is observed for identical gases

This contradiction is known as the Gibbs paradox.

Gibbs Paradox Calculation

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

Initial entropy:
\[ S_i = 2 \times Nk_B \left[ \ln \left( \frac{V}{N} \lambda^{-3} \right) + \frac{5}{2} \right] \]
Final entropy (2N particles in 2V):
\[ S_f = 2Nk_B \left[ \ln \left( \frac{2V}{2N} \lambda^{-3} \right) + \frac{5}{2} \right] \]
\[ = 2Nk_B \left[ \ln \left( \frac{V}{N} \lambda^{-3} \right) + \frac{5}{2} \right] \]
Entropy change:
\[ \Delta S = S_f - S_i = 0 \]

5.2 Resolution

🔍 Resolution of Gibbs Paradox

The resolution comes from the indistinguishability of identical particles in quantum mechanics:

  • Classical counting overcounts states by N!
  • Quantum mechanically, permutations of identical particles don't create new states
  • The correct counting includes the 1/N! factor in the phase space volume

💡 Key Insight

The Gibbs paradox highlights the quantum nature of identical particles, even in what appears to be a classical system. The resolution requires the introduction of the 1/N! factor, which has no classical justification but is essential for consistency with thermodynamics.

6. Chemical Potential in Microcanonical Ensemble

⚗️ Chemical Potential Definition

The chemical potential μ is defined as:

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

For an ideal gas using the Sackur-Tetrode formula:

\[ \mu = -k_B T \ln \left[ \frac{V}{N} \left( \frac{2\pi m k_B T}{h^2} \right)^{3/2} \right] \]

🧮 Chemical Potential Derivation

Step 1: Start with Sackur-Tetrode Formula

\[ S = N k_B \left[ \ln \left( \frac{V}{N} \left( \frac{4\pi m E}{3h^2 N} \right)^{3/2} \right) + \frac{5}{2} \right] \]

Step 2: Differentiate with Respect to N

\[ \left( \frac{\partial S}{\partial N} \right)_{E,V} = k_B \left[ \ln \left( \frac{V}{N} \left( \frac{4\pi m E}{3h^2 N} \right)^{3/2} \right) + \frac{5}{2} \right] + N k_B \left( -\frac{1}{N} - \frac{3}{2N} \right) \]
\[ = k_B \left[ \ln \left( \frac{V}{N} \left( \frac{4\pi m E}{3h^2 N} \right)^{3/2} \right) \right] \]

Step 3: Use E = (3/2)Nk_BT

\[ \left( \frac{\partial S}{\partial N} \right)_{E,V} = k_B \ln \left[ \frac{V}{N} \left( \frac{2\pi m k_B T}{h^2} \right)^{3/2} \right] \]

Step 4: Chemical Potential

\[ \mu = -T \left( \frac{\partial S}{\partial N} \right)_{E,V} \]
\[ = -k_B T \ln \left[ \frac{V}{N} \left( \frac{2\pi m k_B T}{h^2} \right)^{3/2} \right] \]

7. Thermodynamic vs Statistical Entropy

Thermodynamic Entropy Statistical Entropy
Defined for equilibrium states Defined for any microscopic state
Based on heat and temperature Based on number of microstates
Only differences are measurable Absolute values can be calculated
Macroscopic description Microscopic foundation

🔗 Connection Between Approaches

The statistical definition of entropy \( S = k_B \ln \Omega \) provides the microscopic foundation for the thermodynamic concept. This connection, established by Boltzmann, explains why entropy increases in irreversible processes - systems evolve toward macrostates with more microstates.

8. Dimensionless Phase Space

📏 Making Phase Space Dimensionless

To count states properly, we need to make phase space dimensionless. This is achieved by:

\[ \text{Dimensionless volume} = \frac{d^{3N}q \, d^{3N}p}{h^{3N}} \]

Where h is Planck's constant. This division:

  • Makes phase space volume dimensionless
  • Accounts for quantum uncertainty principle
  • Gives correct absolute entropy

9. Distributive Properties

📊 Probability Distributions in Microcanonical Ensemble

For a system in microcanonical ensemble, the probability distribution for any subsystem is:

\[ P(\text{subsystem in state i}) = \frac{\Omega(\text{reservoir with energy } E - E_i)}{\Omega(\text{total system})} \]

This leads to the canonical distribution when the subsystem is much smaller than the total system.

Practice Problems with Solutions

Problem 1: Entropy of Monatomic Ideal Gas

Calculate the entropy of 1 mole of monatomic ideal gas at STP using the Sackur-Tetrode formula.

Given:
\[ N = 6.022 \times 10^{23} \]
\[ V = 22.4 \, \text{L} = 0.0224 \, \text{m}^3 \]
\[ T = 273.15 \, \text{K} \]
\[ m = 6.64 \times 10^{-27} \, \text{kg (for helium)} \]
\[ k_B = 1.38 \times 10^{-23} \, \text{J/K} \]
\[ h = 6.626 \times 10^{-34} \, \text{J·s} \]
Thermal wavelength:
\[ \lambda = \frac{h}{\sqrt{2\pi m k_B T}} \]
\[ = \frac{6.626 \times 10^{-34}}{\sqrt{2\pi \times 6.64 \times 10^{-27} \times 1.38 \times 10^{-23} \times 273.15}} \]
\[ = 7.32 \times 10^{-11} \, \text{m} \]
Sackur-Tetrode formula:
\[ S = N k_B \left[ \ln \left( \frac{V}{N \lambda^3} \right) + \frac{5}{2} \right] \]
\[ = 6.022 \times 10^{23} \times 1.38 \times 10^{-23} \left[ \ln \left( \frac{0.0224}{6.022 \times 10^{23} \times (7.32 \times 10^{-11})^3} \right) + \frac{5}{2} \right] \]
\[ = 8.314 \left[ \ln(1.38 \times 10^5) + 2.5 \right] \]
\[ = 8.314 \times (11.83 + 2.5) \]
\[ = 119.2 \, \text{J/K} \]
Problem 2: Chemical Potential Calculation

Calculate the chemical potential of nitrogen gas at room temperature (300 K) and atmospheric pressure.

Given:
\[ T = 300 \, \text{K} \]
\[ P = 1 \, \text{atm} = 1.013 \times 10^5 \, \text{Pa} \]
\[ m = 4.65 \times 10^{-26} \, \text{kg (N₂ molecule)} \]
\[ k_B = 1.38 \times 10^{-23} \, \text{J/K} \]
\[ h = 6.626 \times 10^{-34} \, \text{J·s} \]
Volume per particle:
\[ \frac{V}{N} = \frac{k_B T}{P} \]
\[ = \frac{1.38 \times 10^{-23} \times 300}{1.013 \times 10^5} \]
\[ = 4.09 \times 10^{-26} \, \text{m}^3 \]
Thermal wavelength:
\[ \lambda = \frac{h}{\sqrt{2\pi m k_B T}} \]
\[ = \frac{6.626 \times 10^{-34}}{\sqrt{2\pi \times 4.65 \times 10^{-26} \times 1.38 \times 10^{-23} \times 300}} \]
\[ = 1.88 \times 10^{-11} \, \text{m} \]
Chemical potential:
\[ \mu = -k_B T \ln \left( \frac{V}{N \lambda^3} \right) \]
\[ = -1.38 \times 10^{-23} \times 300 \times \ln \left( \frac{4.09 \times 10^{-26}}{(1.88 \times 10^{-11})^3} \right) \]
\[ = -4.14 \times 10^{-21} \times \ln(6.15 \times 10^6) \]
\[ = -4.14 \times 10^{-21} \times 15.63 \]
\[ = -6.47 \times 10^{-20} \, \text{J} \]
\[ = -0.404 \, \text{eV} \]

Frequently Asked Questions

❓ Why do we need the 1/N! factor in the microcanonical ensemble?

The 1/N! factor is essential for resolving the Gibbs paradox and obtaining extensive thermodynamic quantities. Without it:

  • Entropy would not be extensive (would not scale with system size)
  • Mixing of identical gases would appear to increase entropy
  • The chemical potential would not be intensive

This factor accounts for the quantum mechanical indistinguishability of identical particles, even in classical statistical mechanics.

❓ What is the physical meaning of the thermal wavelength?

The thermal wavelength \( \lambda = h/\sqrt{2\pi m k_B T} \) represents the quantum wavelength associated with a particle at temperature T. It has several important interpretations:

  • De Broglie wavelength of a particle with thermal kinetic energy
  • Quantum resolution limit - particles closer than λ cannot be distinguished
  • Quantum regime indicator - when \( n\lambda^3 \approx 1 \), quantum effects become important

In the Sackur-Tetrode formula, \( V/(N\lambda^3) \) represents the number of quantum cells available per particle.

❓ How does the microcanonical ensemble relate to other ensembles?

The microcanonical ensemble is the most fundamental ensemble in statistical mechanics:

  • Canonical ensemble can be derived by considering a small system in thermal contact with a large microcanonical reservoir
  • Grand canonical ensemble can be derived by considering a small system exchanging both energy and particles with a large reservoir
  • All ensembles become equivalent in the thermodynamic limit for large systems

The choice of ensemble depends on what quantities are controlled in the physical situation being modeled.

📚 Master Statistical Mechanics

Understanding the microcanonical ensemble is fundamental to statistical mechanics, thermodynamics, and many applications in physics, chemistry, and engineering. Continue your journey into the fascinating world of statistical physics and its applications to real systems.

Read More: Physics HRK Notes of Statistical Mechanics

© House of Physics | HRK Physics: Microcanonical Ensemble

Based on Halliday, Resnick, and Krane's "Physics" with additional insights from university physics curriculum

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