Canonical Ensemble: Complete Guide to Statistical Mechanics & Thermodynamics

House of Physics October 02, 2025
Canonical Ensemble: Complete Guide to Statistical Mechanics & Thermodynamics

Canonical Ensemble: Complete Guide to Statistical Mechanics

Master the Canonical Ensemble: Boltzmann Distribution, Partition Function, Thermodynamic Relations, and Applications with Solved Examples
Statistical Mechanics Canonical Ensemble Partition Function Boltzmann Distribution Reading Time: 25 min

📋 Table of Contents

Introduction to Canonical Ensemble

⚛️ What is the Canonical Ensemble?

A statistical ensemble where the energy is not known exactly but the number of particles is fixed is called the canonical ensemble. In place of energy, the temperature is specified. The canonical ensemble is appropriate for describing a closed system which is in weak thermal contact with a heat bath.

🔬 Key Characteristics of Canonical Ensemble

  • Fixed parameters: Number of particles (N), volume (V), and temperature (T)
  • Energy exchange: System can exchange energy with a heat reservoir
  • No particle exchange: System remains closed to particle exchange
  • Thermal equilibrium: System is in equilibrium with a heat bath at temperature T

Probability and Partition Function

📊 Probability in Canonical Ensemble

Consider a system at certain temperature which is in equilibrium with a heat reservoir. The probability of finding the system in state \( i \) with energy \( E_i \) is given by:

\[ P_i = Ce^{-\beta E_i} \]
\[ \text{where } \beta = \frac{1}{k_B T} \]

Here, \( k_B \) is Boltzmann's constant and T is the absolute temperature.

Normalization Condition

The normalization condition requires that the sum of all probabilities equals 1:

\[ \sum_i P_i = 1 \]
\[ \Rightarrow \sum_i Ce^{-\beta E_i} = 1 \]
\[ \Rightarrow C \sum_i e^{-\beta E_i} = 1 \]
\[ \Rightarrow C = \frac{1}{\sum_i e^{-\beta E_i}} \]

Boltzmann Distribution

📈 The Boltzmann Distribution

Substituting the value of C from the normalization condition, we obtain the Boltzmann distribution:

\[ P_i = \frac{e^{-\beta E_i}}{\sum_i e^{-\beta E_i}} \]

This is called the Boltzmann distribution and tells us the probability that a particular quantum state of the system is occupied.

📐 Partition Function Definition

The denominator in the Boltzmann distribution is called the partition function, denoted by Z:

\[ Z = \sum_{i} e^{-\beta E_i} = \sum_{i} e^{-E_i/k_B T} \]

If there are \( g_j \) quantum states all with energy \( E_j \), then the partition function becomes:

\[ Z = \sum_{i} g_i e^{-E_i/k_B T} \]

Importance of Partition Function

🔑 Central Role of Partition Function

We have found the partition function by the normalization condition needed to get the sum of probabilities equal to one. Its importance lies in the fact that it enables us to make a direct connection between the quantum states of the system and its thermodynamic properties such as free energy, pressure, and entropy.

📊 Partition Function as a Bridge

The partition function serves as a bridge between:

  • Microscopic description: Quantum states and their energies
  • Macroscopic description: Thermodynamic properties

We can find all thermodynamic properties of a system from knowledge of the partition function.

Thermodynamic Relations

⚖️ Connecting Statistics to Thermodynamics

In the canonical ensemble, we can derive all thermodynamic quantities from the partition function. These relations form the foundation of statistical thermodynamics.

Internal Energy

\[ U = k_B T^2 \frac{\partial (\ln Z)}{\partial T} \]

Helmholtz Free Energy

\[ F = -k_B T \ln Z \]

Entropy

\[ S = -k_B \sum_i P_i \ln P_i \]

Energy in Canonical Ensemble

⚡ Mean Energy Calculation

In the canonical ensemble, the probability of being in quantum state \( \psi_i \) is:

\[ P_i = \frac{e^{-E_i/k_B T}}{\sum_{i} e^{-E_i/k_B T}} \]

The mean energy is then:

\[ E = \sum_{i} P_i E_i = \frac{\sum_{i} E_i e^{-E_i/k_B T}}{\sum_{i} e^{-E_i/k_B T}} \]

Derivation of Internal Energy Formula

Starting with the partition function:

\[ Z = \sum_{i} e^{-E_i/k_B T} \]

Differentiate with Respect to Temperature

\[ \frac{\partial Z}{\partial T} = \sum_{i} e^{-E_i/k_B T} \times \frac{E_i}{k_B T^2} \]
\[ \Rightarrow k_B T^2 \frac{\partial Z}{\partial T} = \sum_{i} e^{-E_i/k_B T} E_i \]

Express Mean Energy

\[ ZE = \sum_{i} e^{-E_i/k_B T} E_i \]
\[ \Rightarrow k_B T^2 \frac{\partial Z}{\partial T} = ZE \]
\[ \Rightarrow E = k_B T^2 \frac{1}{Z} \frac{\partial Z}{\partial T} \]
\[ \Rightarrow E = k_B T^2 \frac{\partial (\ln Z)}{\partial T} \]

📐 Internal Energy Formula

The internal energy U in the canonical ensemble is:

\[ U = k_B T^2 \frac{\partial (\ln Z)}{\partial T} \]

Helmholtz Free Energy

🔮 Definition of Helmholtz Free Energy

Helmholtz free energy is defined thermodynamically as:

\[ F = U - TS \]

In statistical mechanics, it is related to the partition function by:

\[ F = -k_B T \ln Z \]

Derivation from Thermodynamic Relation

Starting from the definition:

\[ F = U - TS \]
\[ \Rightarrow U = F + TS \]
\[ \Rightarrow \frac{U - F}{T} = S \]

Differentiate at Constant Volume

\[ \frac{U}{T^2} = \frac{\partial}{\partial T} \left( \frac{F}{T} \right) \]

Substitute Internal Energy Expression

\[ -\frac{1}{T^2} k_B T^2 \frac{\partial (\ln Z)}{\partial T} = \frac{\partial}{\partial T} \left( \frac{F}{T} \right) \]
\[ \Rightarrow \frac{\partial}{\partial T} \left( \frac{F}{T} \right) = -k_B \frac{\partial (\ln Z)}{\partial T} \]

Integrate Both Sides

\[ \frac{F}{T} = -k_B \ln Z \]
\[ \Rightarrow F = -k_B T \ln Z \]

Entropy in Canonical Ensemble

📊 Statistical Definition of Entropy

From the relation \( F = U - TS \), we have:

\[ TS = U - F \]
\[ \Rightarrow S = \frac{U - F}{T} \]

Express in Terms of Probabilities

\[ S = \frac{\sum_i P_i E_i - \sum_i P_i F}{T} \]
\[ \Rightarrow S = \frac{\sum_i P_i (E_i - F)}{T} \]
\[ \Rightarrow S = \frac{k_B \sum_i P_i (E_i - F)}{k_B T} \]

Use Probability Expression

\[ P_i = e^{-(E_i - F)/k_B T} \]
\[ \Rightarrow \ln P_i = \frac{-(E_i - F)}{k_B T} \]
\[ \Rightarrow E_i - F = -k_B T \ln P_i \]

Substitute into Entropy Expression

\[ S = k_B \sum_i P_i (-\ln P_i) \]
\[ \Rightarrow S = -k_B \sum_i P_i \ln P_i \]

📐 Statistical Entropy Formula

The entropy in the canonical ensemble is given by:

\[ S = -k_B \sum_i P_i \ln P_i \]

This is the famous Gibbs entropy formula, which connects the microscopic probabilities to macroscopic entropy.

Useful Features of Canonical Ensemble

🌟 Advantages of Canonical Ensemble

The canonical ensemble offers several important advantages in statistical mechanics:

Temperature Dependence

The probability density depends upon energy E and temperature T, making it suitable for systems in thermal contact with a reservoir.

Additive Property

The additive property of \( \ln P_i \) allows us to couple two canonical ensembles with the same temperature so that the resulting ensemble is again a canonical ensemble.

Universal Applicability

Canonical ensemble applies equally to macroscopic systems or microscopic systems, providing a unified framework.

Classical Connection

There is a strong resemblance between probability density of a canonical ensemble and distribution function of classical statistics.

Energy Fluctuations

As the system is in thermal equilibrium with a heat reservoir, fluctuations do not occur in temperature but appear in energy.

Entropy of Ideal Gas (Sackur-Tetrode)

🌡️ Entropy of Ideal Boltzmann Gas

For an ideal Boltzmann gas consisting of N atoms of mass M, the energy is:

\[ E = \sum_{i=1}^{3N} \frac{p_i^2}{2M} = \frac{1}{2M} \sum_{i=1}^{3N} \left( p_{xi}^2 + p_{yi}^2 + p_{zi}^2 \right) \]

📐 Sackur-Tetrode Equation

The Sackur-Tetrode equation gives the entropy of an ideal monatomic gas in the canonical ensemble:

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

This equation correctly accounts for the indistinguishability of particles and provides the correct extensive behavior of entropy.

Heat Capacity and Energy Fluctuations

📈 Energy Fluctuations in Canonical Ensemble

In the canonical ensemble, the system is in thermal contact with a heat reservoir, so energy can fluctuate. The mean square fluctuation in energy is given by:

\[ \langle (\Delta E)^2 \rangle = \langle E^2 \rangle - \langle E \rangle^2 \]

Relation to Heat Capacity

The mean square fluctuation in energy is related to the heat capacity at constant volume:

\[ \langle (\Delta E)^2 \rangle = k_B T^2 C_V \]

Derivation

\[ \langle E \rangle = \frac{1}{Z} \sum_i E_i e^{-\beta E_i} \]
\[ \langle E^2 \rangle = \frac{1}{Z} \sum_i E_i^2 e^{-\beta E_i} \]
\[ \frac{\partial \langle E \rangle}{\partial T} = \frac{1}{k_B T^2} \left( \langle E^2 \rangle - \langle E \rangle^2 \right) \]
\[ \Rightarrow \langle (\Delta E)^2 \rangle = k_B T^2 \frac{\partial \langle E \rangle}{\partial T} \]
\[ \Rightarrow \langle (\Delta E)^2 \rangle = k_B T^2 C_V \]

📐 Relative Energy Fluctuation

The relative fluctuation in energy is:

\[ \frac{\sqrt{\langle (\Delta E)^2 \rangle}}{\langle E \rangle} = \frac{\sqrt{k_B T^2 C_V}}{\langle E \rangle} \]

For macroscopic systems, this ratio is extremely small, which explains why energy fluctuations are negligible in everyday thermodynamics.

Solved Examples

Example 1: Two-Level System

Consider a system with two energy levels: \( E_1 = 0 \) and \( E_2 = \varepsilon \). Find the partition function, mean energy, and entropy.

Solution:

\[ Z = e^{-\beta E_1} + e^{-\beta E_2} \]
\[ = e^0 + e^{-\beta \varepsilon} \]
\[ = 1 + e^{-\beta \varepsilon} \]
\[ \langle E \rangle = \frac{0 \cdot e^0 + \varepsilon \cdot e^{-\beta \varepsilon}}{1 + e^{-\beta \varepsilon}} \]
\[ = \frac{\varepsilon e^{-\beta \varepsilon}}{1 + e^{-\beta \varepsilon}} \]
\[ P_1 = \frac{1}{1 + e^{-\beta \varepsilon}}, \quad P_2 = \frac{e^{-\beta \varepsilon}}{1 + e^{-\beta \varepsilon}} \]
\[ S = -k_B (P_1 \ln P_1 + P_2 \ln P_2) \]
Example 2: Classical Harmonic Oscillator

Find the partition function for a one-dimensional classical harmonic oscillator with Hamiltonian \( H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2 \).

Solution:

\[ Z = \frac{1}{h} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-\beta H} dx dp \]
\[ = \frac{1}{h} \int_{-\infty}^{\infty} e^{-\beta p^2/2m} dp \int_{-\infty}^{\infty} e^{-\beta m \omega^2 x^2/2} dx \]
\[ = \frac{1}{h} \sqrt{\frac{2\pi m}{\beta}} \sqrt{\frac{2\pi}{\beta m \omega^2}} \]
\[ = \frac{1}{h} \frac{2\pi}{\beta \omega} \]
\[ = \frac{k_B T}{\hbar \omega} \]

Practice Problems

Problem 1: Three-Level System

A system has three energy levels: \( E_1 = 0 \), \( E_2 = \varepsilon \), \( E_3 = 2\varepsilon \). Calculate:

  1. The partition function
  2. The probabilities of each level
  3. The mean energy
  4. The entropy at temperature T
Problem 2: Ideal Monatomic Gas

For an ideal monatomic gas of N particles in volume V at temperature T:

  1. Show that the partition function is \( Z = \frac{1}{N!} \left( \frac{V}{\lambda^3} \right)^N \) where \( \lambda = \frac{h}{\sqrt{2\pi m k_B T}} \)
  2. Calculate the Helmholtz free energy
  3. Derive the entropy using \( S = -\left( \frac{\partial F}{\partial T} \right)_V \)

Multiple Choice Questions

1. In the canonical ensemble, which of the following is fixed?

A) Energy

B) Temperature

C) Entropy

D) Pressure

Answer: B) Temperature

2. The partition function Z is defined as:

A) \( \sum_i E_i e^{-\beta E_i} \)

B) \( \sum_i e^{-\beta E_i} \)

C) \( \sum_i E_i \)

D) \( \sum_i \ln E_i \)

Answer: B) \( \sum_i e^{-\beta E_i} \)

3. The Helmholtz free energy is related to the partition function by:

A) \( F = k_B T \ln Z \)

B) \( F = -k_B T \ln Z \)

C) \( F = \frac{1}{k_B T} \ln Z \)

D) \( F = -\frac{1}{k_B T} \ln Z \)

Answer: B) \( F = -k_B T \ln Z \)

Frequently Asked Questions

❓ What is the difference between canonical and microcanonical ensembles?

In the microcanonical ensemble, the energy is fixed and the system is isolated. In the canonical ensemble, the temperature is fixed and the system can exchange energy with a heat reservoir. The canonical ensemble is more practical for most experimental situations where systems are in thermal contact with their surroundings.

❓ Why is the partition function so important in statistical mechanics?

The partition function contains all the information about the thermodynamic properties of a system. Once we know the partition function, we can calculate all thermodynamic quantities like internal energy, free energy, entropy, pressure, and heat capacity. It serves as a bridge between the microscopic description (quantum states) and macroscopic thermodynamics.

❓ How do energy fluctuations relate to heat capacity?

The mean square fluctuation in energy is directly proportional to the heat capacity at constant volume: \( \langle (\Delta E)^2 \rangle = k_B T^2 C_V \). This means that systems with larger heat capacities have larger energy fluctuations. For macroscopic systems, these fluctuations are typically very small compared to the total energy.

❓ What is the physical significance of the Helmholtz free energy?

The Helmholtz free energy F = U - TS represents the maximum amount of work that can be extracted from a system at constant temperature and volume. In statistical mechanics, it is directly related to the partition function through F = -k_B T ln Z, making it a fundamental quantity for connecting microscopic and macroscopic descriptions.

❓ How does the canonical ensemble handle indistinguishable particles?

For indistinguishable particles, we must account for the fact that permutations of particles don't create new states. This is typically handled by including a factor of 1/N! in the partition function for ideal gases. For quantum systems, we use the appropriate statistics (Bose-Einstein or Fermi-Dirac) depending on the particle type.

📚 Continue Your Statistical Mechanics Journey

Mastering the Canonical Ensemble is fundamental to understanding statistical mechanics and its applications to thermodynamics, quantum systems, and condensed matter physics. These comprehensive notes provide a solid foundation for further studies in statistical physics and its applications.

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© 2025 Physics Education Initiative | Canonical Ensemble: Statistical Mechanics

These comprehensive notes are designed to help B.Sc. and M.Sc. Physics students understand fundamental concepts of Statistical Mechanics based on Halliday, Resnick and Krane

Author: Physics Education | Contact: physics.education@example.com

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