Statistical Physics PHYS-0461: Phase Space, Systems and Ensembles

House of Physics October 02, 2025
Statistical Physics: Complete Guide to Position Space, Momentum Space, and Phase Space | HRK Physics

Statistical Physics: Complete Guide to Position Space, Momentum Space, and Phase Space

Mastering Statistical Mechanics Fundamentals: Systems, Assemblies, Ensembles, and Phase Space Concepts
Statistical Physics Position Space Momentum Space Phase Space Statistical Mechanics Reading Time: 25 min

📋 Table of Contents

📜 Historical Background

The development of statistical physics revolutionized our understanding of thermodynamics and the behavior of systems with many particles:

  • James Clerk Maxwell (1860s): Developed the kinetic theory of gases
  • Ludwig Boltzmann (1870s): Formulated statistical mechanics and entropy relationship
  • J. Willard Gibbs (1902): Developed the concept of ensembles in statistical mechanics
  • Albert Einstein (1905): Applied statistical mechanics to Brownian motion
  • Paul Ehrenfest (1911): Refined the foundations of statistical mechanics

These developments established statistical physics as the bridge between microscopic particle behavior and macroscopic thermodynamic properties.

Introduction to Statistical Physics

🔬 What is Statistical Physics?

Statistical physics is the branch of physics that uses probability theory and statistics to study the behavior of systems composed of a large number of particles. It provides the microscopic foundation for thermodynamics and explains macroscopic properties in terms of the statistical behavior of microscopic constituents.

The fundamental idea is that while the behavior of individual particles may be unpredictable, the collective behavior of many particles follows statistical laws that can be precisely described.

💡 Key Insight

Statistical physics bridges the microscopic world of atoms and molecules with the macroscopic world of thermodynamics. It explains how the random motion of countless particles gives rise to predictable thermodynamic behavior through statistical averaging.

Position Space

📍 Position Space Definition

Consider a system consisting of \( N \) particles distributed in a given volume. If the system is STATIC, all particles will remain fixed at different points in space.

To completely specify the location of any particle in 3D space, we must know three Cartesian coordinates \( x, y, z \).

As there are \( N \) particles, knowledge of 3N coordinates gives complete information about such a static system.

📐 Position Space Characteristics

The three-dimensional space in which the location of a particle is completely described by three position coordinates is called position space. The volume element in this space is \( dx\,dy\,dz \).

Static Systems

⚓ Static System Properties

In a static system, particles remain fixed at their positions. The complete description requires:

  • 3 coordinates per particle (\( x, y, z \))
  • Total of 3N coordinates for N particles
  • No time evolution of positions

Examples of static systems include crystals at absolute zero temperature or fixed mechanical structures.

Volume Element in Position Space

🧮 Position Space Volume Element

Volume Element Definition

For a single particle in 3D space, the infinitesimal volume element is:

\[ dV = dx\,dy\,dz \]

N-Particle System

For a system of N particles, the volume element in position space becomes:

\[ dV_N = \prod_{i=1}^{N} dx_i\,dy_i\,dz_i \]
\[ = dx_1\,dy_1\,dz_1\,dx_2\,dy_2\,dz_2\,\ldots\,dx_N\,dy_N\,dz_N \]

Dimensionality

Position space for N particles has 3N dimensions, with each dimension corresponding to one coordinate of one particle.

Momentum Space

🚀 Momentum Space Definition

If a system is dynamic, its particles move with different velocities and hence momenta. A complete specification of such a system cannot be described in terms of position coordinates only.

For a dynamic system, we must specify three components of momentum. In the PDF, these are given as:

\[ R_x = mv_x^2, \quad R_y = mv_y^2, \quad R_z = mv_z^2 \]

Note: There appears to be an error in the PDF. The standard momentum components are \( p_x = mv_x \), \( p_y = mv_y \), \( p_z = mv_z \), not squared as shown.

📊 Momentum Space Characteristics

The space in which a system is completely specified by three momentum coordinates \( p_x, p_y, p_z \) is known as momentum space. The volume element in this space is \( dp_x\,dp_y\,dp_z \).

Dynamic Systems

🌀 Dynamic System Properties

In a dynamic system, particles are in motion. The complete description requires:

  • 3 momentum components per particle (\( p_x, p_y, p_z \))
  • Total of 3N momentum components for N particles
  • Time evolution of both positions and momenta

Most physical systems are dynamic, including gases, liquids, and solids at finite temperatures.

Volume Element in Momentum Space

🧮 Momentum Space Volume Element

Volume Element Definition

For a single particle, the infinitesimal volume element in momentum space is:

\[ dV_p = dp_x\,dp_y\,dp_z \]

N-Particle System

For a system of N particles, the volume element in momentum space becomes:

\[ dV_{p,N} = \prod_{i=1}^{N} dp_{x_i}\,dp_{y_i}\,dp_{z_i} \]
\[ = dp_{x_1}\,dp_{y_1}\,dp_{z_1}\,dp_{x_2}\,dp_{y_2}\,dp_{z_2}\,\ldots\,dp_{x_N}\,dp_{y_N}\,dp_{z_N} \]

Dimensionality

Momentum space for N particles also has 3N dimensions, with each dimension corresponding to one momentum component of one particle.

Phase Space

🌌 Phase Space Definition

A combination of position and momentum space is called phase space. Thus phase space has 6 dimensions for a single particle.

A point in phase space has coordinates \( (x, y, z, p_x, p_y, p_z) \).

If there are \( N \) particles, knowledge of 6N coordinates gives a complete picture of a dynamic system.

🎯 Phase Space Significance

Phase space provides the most complete description of a physical system. Each point in phase space represents a unique microscopic state of the system. The evolution of the system corresponds to a trajectory through phase space.

Complete System Description

📋 Complete Microscopic Description

A complete microscopic description of a system of N particles requires:

  • 3N position coordinates: \( x_1, y_1, z_1, x_2, y_2, z_2, \ldots, x_N, y_N, z_N \)
  • 3N momentum coordinates: \( p_{x_1}, p_{y_1}, p_{z_1}, p_{x_2}, p_{y_2}, p_{z_2}, \ldots, p_{x_N}, p_{y_N}, p_{z_N} \)
  • Total of 6N coordinates

This complete set of coordinates defines a point in the 6N-dimensional phase space.

Volume Element in Phase Space

🧮 Phase Space Volume Element

Single Particle

For a single particle, the volume element in phase space is:

\[ d\Gamma = dx\,dy\,dz\,dp_x\,dp_y\,dp_z \]

N-Particle System

For a system of N particles, the volume element in phase space becomes:

\[ d\Gamma_N = \prod_{i=1}^{N} dx_i\,dy_i\,dz_i\,dp_{x_i}\,dp_{y_i}\,dp_{z_i} \]
\[ = dx_1\,dy_1\,dz_1\,dp_{x_1}\,dp_{y_1}\,dp_{z_1}\,\ldots\,dx_N\,dy_N\,dz_N\,dp_{x_N}\,dp_{y_N}\,dp_{z_N} \]

Dimensionality

Phase space for N particles has 6N dimensions, making it a high-dimensional space even for modest numbers of particles.

6N
Dimensions in phase space for N particles

Systems in Statistical Physics

🔧 System Classification

In statistical physics, systems are classified based on their interactions with the surroundings:

Open Systems

An open system can exchange both energy and matter with its surroundings. Most real-world systems are open systems.

Examples: A cup of coffee, living organisms, chemical reactors

Closed Systems

A closed system can exchange energy but not matter with its surroundings. The total mass remains constant.

Examples: A sealed container of gas, a thermos flask

Isolated Systems

An isolated system cannot exchange either energy or matter with its surroundings. Total energy and mass remain constant.

Examples: The universe (approximately), perfectly insulated systems

Adiabatic Systems

An adiabatic system cannot exchange heat with its surroundings, but may exchange work. The system is thermally insulated.

Examples: Systems with perfect thermal insulation

Assemblies in Statistical Physics

🔢 Assembly Concept

An assembly is a collection of a large number of systems, all identical in composition and macroscopic properties, but possibly in different microscopic states.

Each system in the assembly is a replica of the actual system being studied, prepared under identical macroscopic conditions.

📊 Statistical Ensemble

The concept of an assembly leads to the idea of a statistical ensemble - a collection of all possible systems that have different microscopic states but identical macroscopic properties.

🎲 Ensemble Averaging

In statistical physics, we calculate macroscopic properties by taking averages over the ensemble:

  • Each system in the ensemble represents a possible microscopic state
  • Macroscopic observables are ensemble averages
  • For large systems, ensemble averages equal time averages (ergodic hypothesis)

Ensembles in Statistical Mechanics

📈 Types of Ensembles

Different physical situations correspond to different types of ensembles, depending on what quantities are fixed in the system.

Ensemble Type Fixed Quantities Fluctuating Quantities Physical Situation
Microcanonical N, V, E Temperature, Pressure Isolated systems
Canonical N, V, T Energy Systems in thermal equilibrium
Grand Canonical V, T, μ Energy, Particle Number Open systems

Microcanonical Ensemble

🔒 Microcanonical Ensemble

The microcanonical ensemble describes isolated systems with fixed:

  • Number of particles (N)
  • Volume (V)
  • Energy (E)

All accessible microstates are equally probable (fundamental postulate of statistical mechanics).

The entropy is given by:

\[ S = k_B \ln \Omega \]

where \( \Omega \) is the number of accessible microstates and \( k_B \) is Boltzmann's constant.

Canonical Ensemble

🌡️ Canonical Ensemble

The canonical ensemble describes systems in thermal equilibrium with a heat bath at temperature T, with fixed:

  • Number of particles (N)
  • Volume (V)
  • Temperature (T)

The probability of a microstate with energy \( E_i \) is given by the Boltzmann factor:

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

where \( \beta = 1/(k_B T) \) and Z is the canonical partition function.

Grand Canonical Ensemble

🔁 Grand Canonical Ensemble

The grand canonical ensemble describes open systems that can exchange both energy and particles with a reservoir, with fixed:

  • Volume (V)
  • Temperature (T)
  • Chemical potential (μ)

The probability of a microstate with energy \( E_i \) and particle number \( N_i \) is:

\[ P_i = \frac{e^{-\beta (E_i - \mu N_i)}}{\Xi} \]

where \( \Xi \) is the grand canonical partition function.

Mathematical Framework

📐 Mathematical Tools

Statistical physics employs various mathematical tools to describe the behavior of many-particle systems:

🧮 Partition Functions

Canonical Partition Function

For a system with discrete energy levels:

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

For a classical system:

\[ Z = \frac{1}{N! h^{3N}} \int e^{-\beta H(\mathbf{q},\mathbf{p})} d^{3N}q\,d^{3N}p \]

where H is the Hamiltonian and h is Planck's constant.

Grand Canonical Partition Function

\[ \Xi = \sum_{N=0}^{\infty} e^{\beta \mu N} Z_N \]

where \( Z_N \) is the canonical partition function for N particles.

🧮 Thermodynamic Relations

From Canonical Ensemble

\[ F = -k_B T \ln Z \]
\[ U = -\frac{\partial \ln Z}{\partial \beta} \]
\[ S = k_B \left( \ln Z - \beta \frac{\partial \ln Z}{\partial \beta} \right) \]
\[ P = k_B T \frac{\partial \ln Z}{\partial V} \]

From Grand Canonical Ensemble

\[ \Omega = -k_B T \ln \Xi \]
\[ \langle N \rangle = k_B T \frac{\partial \ln \Xi}{\partial \mu} \]
\[ U = -\left( \frac{\partial \ln \Xi}{\partial \beta} \right)_{\mu,V} \]

Practice Problems with Solutions

Problem 1: Phase Space Dimensions

Calculate the number of dimensions in phase space for:

(a) A single monatomic particle

(b) A system of N monatomic particles

(c) A system of N diatomic molecules (considering rotational degrees of freedom)

(a) For a single monatomic particle:
\[ \text{Position coordinates} = 3 \]
\[ \text{Momentum coordinates} = 3 \]
\[ \text{Total dimensions} = 3 + 3 = 6 \]
(b) For N monatomic particles:
\[ \text{Position coordinates} = 3N \]
\[ \text{Momentum coordinates} = 3N \]
\[ \text{Total dimensions} = 3N + 3N = 6N \]
(c) For N diatomic molecules:
\[ \text{Position coordinates per molecule} = 3 \times 2 = 6 \]
\[ \text{Momentum coordinates per molecule} = 3 \times 2 = 6 \]
\[ \text{Rotational coordinates per molecule} = 2 \text{ (angles)} \]
\[ \text{Rotational momenta per molecule} = 2 \]
\[ \text{Total per molecule} = 6 + 6 + 2 + 2 = 16 \]
\[ \text{Total for N molecules} = 16N \]
Problem 2: Volume Element in Phase Space

Write the expression for the volume element in phase space for a system of N identical particles in three dimensions.

For a single particle:
\[ d\Gamma_1 = dx\,dy\,dz\,dp_x\,dp_y\,dp_z \]
For N particles:
\[ d\Gamma_N = \prod_{i=1}^{N} dx_i\,dy_i\,dz_i\,dp_{x_i}\,dp_{y_i}\,dp_{z_i} \]
For identical particles, we must account for indistinguishability:
\[ d\Gamma_N = \frac{1}{N!} \prod_{i=1}^{N} dx_i\,dy_i\,dz_i\,dp_{x_i}\,dp_{y_i}\,dp_{z_i} \]
The factor of \( 1/N! \) corrects for overcounting of identical particle configurations.
Problem 3: Classical Ideal Gas Partition Function

Calculate the canonical partition function for a classical ideal gas of N monatomic particles in volume V at temperature T.

The Hamiltonian for an ideal gas is:
\[ H = \sum_{i=1}^{N} \frac{\mathbf{p}_i^2}{2m} \]
The canonical partition function is:
\[ Z = \frac{1}{N! h^{3N}} \int e^{-\beta H} d^{3N}q\,d^{3N}p \]
Separate position and momentum integrals:
\[ Z = \frac{1}{N! h^{3N}} \left( \int d^{3N}q \right) \left( \int e^{-\beta \sum \frac{\mathbf{p}_i^2}{2m}} d^{3N}p \right) \]
Position integral:
\[ \int d^{3N}q = V^N \]
Momentum integral (Gaussian):
\[ \int e^{-\beta \frac{p^2}{2m}} dp = \sqrt{\frac{2\pi m}{\beta}} \]
\[ \int e^{-\beta \sum \frac{\mathbf{p}_i^2}{2m}} d^{3N}p = \left( \frac{2\pi m}{\beta} \right)^{3N/2} \]
Combine results:
\[ Z = \frac{1}{N! h^{3N}} V^N \left( \frac{2\pi m}{\beta} \right)^{3N/2} \]
\[ = \frac{1}{N!} \left( \frac{V}{h^3} \right)^N \left( \frac{2\pi m}{\beta} \right)^{3N/2} \]
\[ = \frac{1}{N!} \left( \frac{V}{\lambda_T^3} \right)^N \]
where \( \lambda_T = \frac{h}{\sqrt{2\pi m k_B T}} \) is the thermal de Broglie wavelength.

Frequently Asked Questions

❓ What is the difference between position space, momentum space, and phase space?

Position space describes where particles are located, using position coordinates (x, y, z).

Momentum space describes how particles are moving, using momentum coordinates (p_x, p_y, p_z).

Phase space combines both position and momentum spaces, providing a complete description of both where particles are and how they're moving.

For N particles in 3D:

  • Position space has 3N dimensions
  • Momentum space has 3N dimensions
  • Phase space has 6N dimensions
❓ Why do we need statistical physics when we already have thermodynamics?

Thermodynamics provides relationships between macroscopic quantities (temperature, pressure, entropy) but doesn't explain why these relationships hold or how they emerge from microscopic behavior.

Statistical physics provides the microscopic foundation for thermodynamics by:

  • Deriving thermodynamic laws from statistical principles
  • Calculating thermodynamic quantities from microscopic properties
  • Explaining concepts like entropy in terms of probability
  • Predicting behavior of systems where thermodynamics alone is insufficient

Statistical physics connects the microscopic world of atoms and molecules to the macroscopic world of thermodynamics.

❓ What is the significance of the 1/N! factor in the partition function?

The 1/N! factor accounts for the indistinguishability of identical particles in quantum mechanics. This factor is necessary to avoid the Gibbs paradox and obtain extensive thermodynamic properties.

Without this factor:

  • Entropy would not be extensive (would not scale with system size)
  • Mixing of identical gases would appear to increase entropy
  • Thermodynamic relations would be inconsistent

The 1/N! factor is a quantum correction that appears in the classical limit. In fully quantum statistical mechanics, particle indistinguishability is built into the formalism from the beginning.

❓ How does the concept of ensembles help in statistical physics?

Ensembles provide a powerful framework for calculating thermodynamic properties:

  • Microcanonical ensemble: Used for isolated systems with fixed energy
  • Canonical ensemble: Used for systems in thermal equilibrium with a heat bath
  • Grand canonical ensemble: Used for open systems that exchange particles

Each ensemble is appropriate for different physical situations:

  • Choose microcanonical when energy is fixed (isolated systems)
  • Choose canonical when temperature is fixed (systems in contact with heat bath)
  • Choose grand canonical when both temperature and chemical potential are fixed (open systems)

In the thermodynamic limit (large N), all ensembles give equivalent results for intensive properties.

📚 Master Statistical Physics

Understanding statistical physics is fundamental to modern physics, chemistry, and engineering. It provides the foundation for thermodynamics, explains phase transitions, and is essential for understanding complex systems from gases to condensed matter to astrophysical objects.

Read More: Physics HRK Notes of Statistical Physics

© House of Physics | HRK Physics: Statistical Physics

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

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