11.03.02 · stat-mech-physics / ensembles

Grand canonical ensemble: chemical potential, fugacity, and variable particle number

shipped3 tiersLean: none

Anchor (Master): Landau & Lifshitz, Statistical Physics, Part 1, 3rd ed. (1980), §35–36; Huang, Statistical Mechanics, 2nd ed. (1987), Ch. 6–7

Intuition Beginner

The canonical ensemble describes a system at fixed temperature $T$ and fixed particle number $N$ . The system exchanges energy with a reservoir but not particles. Many physical systems do exchange particles with their environment: water in an open glass evaporates (molecules leave for the air), a semiconductor doped with impurities gains and loses charge carriers, a chemical reactor has reactants flowing in and products flowing out.

The grand canonical ensemble handles this case. The system exchanges both energy and particles with a large reservoir. The reservoir fixes the temperature $T$ and the chemical potential $μ$ — a quantity that measures how much the energy of the system changes when you add one particle. High chemical potential means particles "want" to enter the system; low chemical potential means they "want" to leave.

The probability of finding the system in a state with energy $E$ and $N$ particles is proportional to $e^{- (E - μ N) / k T}$ . The extra factor $e^{μ N / k T}$ rewards states with more particles when $μ > 0$ (adding particles lowers the effective energy) and penalises them when $μ < 0$ . The normaliser — the total of all these Boltzmann-weighted terms over every possible $N$ and every state at that $N$ — is the grand partition function $Ξ$ .

Visual Beginner

Picture a container connected to a large reservoir by a small opening. Molecules hop in and out. The reservoir is so big that its temperature $T$ and chemical potential $μ$ are fixed regardless of what the small system does. The small system fluctuates in both energy and particle number.

At any instant, the system might have $N = 100$ particles or $N = 102$ or $N = 98$ . The average $⟨ N ⟩$ is set by $μ$ : increase $μ$ and the system fills up; decrease $μ$ and it empties. The fluctuations $Δ N = ⟨ N^{2} ⟩ - ⟨ N ⟩^{2}$ are tiny for macroscopic systems ( $Δ N / ⟨ N ⟩ \sim 1/ N$ ) but can be significant for small systems.

Worked example Beginner

A system has just one energy level at energy $ϵ$ , and any number of distinguishable particles can occupy it (a simplification). The grand partition function adds up the weights for $N = 0, 1, 2, \dots$ particles:

Ξ = 1 + e^{- (ϵ - μ) / k T} + e^{- 2 (ϵ - μ) / k T} + e^{- 3 (ϵ - μ) / k T} + \dots .

This is a geometric series. If $e^{- (ϵ - μ) / k T} < 1$ (i.e., $μ < ϵ$ ), it sums to

Ξ = \frac{1}{1 - e ^{- (ϵ - μ) / k T}} .

The mean particle number is obtained from the rate of change of $ln Ξ$ with respect to $μ / k T$ , giving

⟨ N ⟩ = \frac{1}{e ^{(ϵ - μ) / k T} - 1} .

When $μ$ is far below $ϵ$ , the occupation is small ( $⟨ N ⟩ ≪ 1$ ). As $μ \to ϵ$ from below, the occupation diverges — this is the signal of Bose-Einstein condensation in a more realistic system.

Check your understanding Beginner

Formal definition Intermediate+

The grand canonical ensemble describes a system at fixed temperature $T$ , fixed chemical potential $μ$ , and fixed volume $V$ . The probability of finding the system in a microstate $∣ n ⟩$ with energy $E_{n}$ and particle number $N_{n}$ is

P_{n} = \frac{e ^{- β (E_{n} - μ N_{n})}}{Ξ},

where $β = 1/ (k_{B} T)$ and the grand partition function (or grand canonical sum over states) is

Ξ (β, μ, V) = N \sum {n : N_{n} = N} \sum e^{- β (E_{n} - μ N_{n})} = N \sum e^{β μ N} Z_{N} (β, V) .

Here $Z_{N}$ is the canonical partition function at fixed $N$ . The fugacity is $z := e^{β μ}$ , giving the alternative form

Ξ = N = 0 \sum \infty z^{N} Z_{N} .

The grand potential (Landau potential, or "big Phi") is

Φ_{G} := - k_{B} T ln Ξ = - P V,

where the second equality holds for a homogeneous system. Thermodynamic quantities follow from derivatives of $Φ_{G}$ :

⟨ N ⟩ = - (\frac{\partial Φ _{G}}{\partial μ})_{T, V}, S = - (\frac{\partial Φ _{G}}{\partial T})_{μ, V}, P = - (\frac{\partial Φ _{G}}{\partial V})_{T, μ} .

Fluctuation-response relations

The particle-number variance is

(Δ N)^{2} = ⟨ N^{2} ⟩ - ⟨ N ⟩^{2} = k_{B} T (\frac{\partial ⟨ N ⟩}{\partial μ})_{T, V} .

This connects a fluctuation (left side) to a thermodynamic derivative (right side). Using $(\partial N / \partial μ)_{T, V} = N^{2} κ_{T} / V$ where $κ_{T}$ is the isothermal compressibility, one finds

\frac{( Δ N ) ^{2}}{⟨ N ⟩ ^{2}} = \frac{k _{B} T κ _{T}}{V} .

For an ideal gas, $κ_{T} = 1/ P$ , giving $Δ N / N = 1/ N$ — the relative fluctuation vanishes in the thermodynamic limit.

Key derivation: Grand partition function for non-interacting particles Intermediate+

Proposition. For a system of non-interacting identical particles with single-particle energy levels $ϵ_{k}$ , the grand partition function factorises as

Ξ = k \prod Ξ_{k},

where the product runs over all single-particle states $k$ . For fermions, $Ξ_{k} = 1 + z e^{- β ϵ_{k}}$ . For bosons, $Ξ_{k} = 1/ (1 - z e^{- β ϵ_{k}})$ for $∣ z e^{- β ϵ_{k}} ∣ < 1$ .

Proof. The total energy of a configuration ${n_{k}}$ (occupation numbers) is $E = \sum_{k} n_{k} ϵ_{k}$ and the total particle number is $N = \sum_{k} n_{k}$ . The grand canonical weight is

e^{- β (E - μ N)} = e^{- β \sum_{k} n_{k} (ϵ_{k} - μ)} = k \prod e^{- β n_{k} (ϵ_{k} - μ)} .

The grand partition function is

Ξ = {n_{k}} \sum k \prod e^{- β n_{k} (ϵ_{k} - μ)} = k \prod (n_{k} \sum e^{- β n_{k} (ϵ_{k} - μ)}) = k \prod Ξ_{k} .

For fermions, $n_{k} \in {0, 1}$ : $Ξ_{k} = 1 + z e^{- β ϵ_{k}}$ .

For bosons, $n_{k} \in {0, 1, 2, \dots}$ : $Ξ_{k} = \sum_{n_{k} = 0}^{\infty} (z e^{- β ϵ_{k}})^{n_{k}} = 1/ (1 - z e^{- β ϵ_{k}})$ for $z e^{- β ϵ_{k}} < 1$ . $□$

The mean occupation number of state $k$ is

⟨ n_{k} ⟩ = - \frac{1}{β} \frac{\partial ln Ξ _{k}}{\partial ϵ _{k}} .

For fermions: $⟨ n_{k} ⟩ = 1/ (e^{β (ϵ_{k} - μ)} + 1)$ — the Fermi-Dirac distribution 11.05.02.

For bosons: $⟨ n_{k} ⟩ = 1/ (e^{β (ϵ_{k} - μ)} - 1)$ — the Bose-Einstein distribution 11.05.01.

Bridge. The factorisation of the grand partition function builds toward the quantum statistics of ideal Bose and Fermi gases [11.05.01, 11.05.02], where the single-state occupation numbers $n_{k}$ carry the entire physics. This is exactly the structure that produces the Bose-Einstein and Fermi-Dirac distributions from a single unified framework: the grand canonical ensemble treats particles as excitations of single-particle modes, and the mode occupations are independent because the particles are non-interacting. The foundational reason the grand canonical ensemble is the natural framework for quantum statistics is that particle number is not conserved in relativistic quantum field theory (particles can be created and destroyed), and the chemical potential is the Lagrange multiplier that controls the expected number. Putting these together, the grand partition function generalises from the canonical ensemble what the partition function generalises from the microcanonical: a generating function for the thermodynamics of an open system.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Show that the energy fluctuation in the grand canonical ensemble is $⟨ E^{2} ⟩ - ⟨ E ⟩^{2} = k_{B} T^{2} C_{V} + (⟨ E N ⟩ - ⟨ E ⟩ ⟨ N ⟩)^{2} / (⟨ N^{2} ⟩ - ⟨ N ⟩^{2})$ .

Hint

Take two derivatives of $ln Ξ$ with respect to $β$ at fixed $z$ .

Answer

From $ln Ξ = - β Φ_{G}$ , and $⟨ E ⟩ = - (\partial ln Ξ/ \partial β)_{z, V}$ :

$⟨ E ⟩ = - (\frac{\partial l n Ξ}{\partial β})_{z, V}$ .

$⟨ E^{2} ⟩ - ⟨ E ⟩^{2} = (\frac{\partial ^{2} l n Ξ}{\partial β ^{2}})_{z, V}$ .

At fixed $μ$ (not fixed $z$ ), $z = e^{β μ}$ varies with $β$ . The relation at fixed $μ$ is:

$(\frac{\partial}{\partial β})_{μ} = (\frac{\partial}{\partial β})_{z} + \frac{μ}{β} (\frac{\partial}{\partial μ})_{β}$ .

The energy fluctuation at fixed $μ$ has two contributions: the heat-capacity term (energy fluctuation at fixed $N$ ) and a cross-term involving energy-particle-number correlations. The full expression is $⟨(Δ E)^{2} ⟩ = k_{B} T^{2} C_{V} + (\partial ⟨ E ⟩ / \partial ⟨ N ⟩)_{T}^{2} ⟨(Δ N)^{2} ⟩$ , connecting energy fluctuations to both the heat capacity and the compressibility through the particle-number fluctuations.

Exercise 7 (hard, symbolic).

For a single bosonic state at energy $ϵ$ , compute the divergence of $⟨ n ⟩$ as $μ \to ϵ^{-}$ and interpret this as the onset of macroscopic occupation.

Hint

$⟨ n ⟩ = 1/ (e^{β (ϵ - μ)} - 1)$ . Let $δ = ϵ - μ > 0$ and take $δ \to 0$ .

Answer

$⟨ n ⟩ = 1/ (e^{β δ} - 1) \approx 1/ (β δ)$ for $β δ ≪ 1$ .

As $μ \to ϵ^{-}$ ( $δ \to 0$ ), $⟨ n ⟩ \to \infty$ . The bosonic state accumulates an arbitrarily large number of particles as the chemical potential approaches the state energy from below. In a real system with many states, the ground state ( $ϵ_{0} = 0$ ) saturates first: $μ$ approaches zero from below, and when $μ$ gets close enough the ground-state occupation becomes macroscopic — this is Bose-Einstein condensation 11.05.04. The critical condition is $μ = 0$ (or $μ = ϵ_{0}$ for a general ground-state energy).

Exercise 8 (hard, symbolic).

Show that for a single bosonic mode (harmonic oscillator), the grand partition function gives $⟨ E ⟩ = ϵ / (e^{β ϵ} - 1) + ϵ /2$ (Planck distribution plus zero-point energy) when the chemical potential is zero. Relate this to blackbody radiation.

Hint

A photon gas has $μ = 0$ because photon number is not conserved. The partition function for a single mode is $Ξ = 1/ (1 - e^{- β ϵ})$ .

Answer

For a single mode with $μ = 0$ : $Ξ = \sum_{n = 0}^{\infty} e^{- n β ϵ} = 1/ (1 - e^{- β ϵ})$ .

The energy is $⟨ E ⟩ = - \partial ln Ξ/ \partial β = ϵ e^{- β ϵ} / (1 - e^{- β ϵ}) = ϵ / (e^{β ϵ} - 1)$ .

This is the Planck distribution — the mean energy of a single photon mode at temperature $T$ . Adding the zero-point energy $ϵ /2$ (which comes from the quantum mechanical ground state of the harmonic oscillator) gives $⟨ E ⟩ = ϵ / (e^{β ϵ} - 1) + ϵ /2$ . The zero-point energy is temperature-independent and does not contribute to heat capacity.

For blackbody radiation, summing over all photon modes with $ϵ = ℏ ω$ and density of states $ρ (ω) \propto ω^{2}$ gives the Stefan-Boltzmann law 11.05.03.

Lean formalization Intermediate+

The grand canonical ensemble has no Mathlib formalization. The grand partition function $Ξ = \sum_{N} z^{N} Z_{N}$ is a generating function for the canonical partition functions, and the thermodynamic quantities follow from its logarithmic derivatives. Formalising this requires: (1) a probability distribution on the disjoint union $⨆_{N} Ω_{N}$ of state spaces at each particle number; (2) the factorisation $Ξ = \prod_{k} Ξ_{k}$ for non-interacting systems; (3) the identification $Φ_{G} = - P V$ . The first item is the load-bearing prerequisite.

Advanced results Master

Lee-Yang zeros and phase transitions

The grand partition function is a polynomial in $z$ (for a finite system) or an entire function (in the thermodynamic limit). The roots of $Ξ (z) = 0$ — the Lee-Yang zeros — lie on the unit circle $∣ z ∣ = 1$ for ferromagnetic systems (Lee-Yang circle theorem, 1952). In the thermodynamic limit, these roots can pinch the positive real $z$ -axis, creating a singularity in $ln Ξ$ that is the signature of a phase transition.

The grand canonical approach thus provides a mathematically precise characterisation of phase transitions: a phase transition occurs when $Ξ$ develops a non-analyticity as a function of $z$ in the thermodynamic limit. For first-order transitions, a zero of $Ξ$ crosses the positive real axis; for continuous transitions, the zeros accumulate and approach the axis.

Chemical potential and thermodynamic equilibrium

The condition for chemical equilibrium between two phases (or two species) is equality of chemical potentials: $μ_{1} = μ_{2}$ . In the grand canonical framework this is automatic: a system in contact with a reservoir at chemical potential $μ$ equilibrates to $μ_{system} = μ_{reservoir}$ . For a reacting mixture $a A + b B ⇌ c C + d D$ , the equilibrium condition is $a μ_{A} + b μ_{B} = c μ_{C} + d μ_{D}$ , which in the ideal-gas limit becomes the law of mass action $K = \prod_{i} (P_{i} / P^{0})^{ν_{i}}$ .

Functional derivatives and the density functional

In density-functional theory, the grand potential is written as a functional of the one-particle density $ρ (r)$ :

Φ_{G} [ρ] = F [ρ] - \int ρ (r) (μ - V_{ext} (r)) d^{3} r,

where $F [ρ]$ is the intrinsic Helmholtz free energy functional and $V_{ext}$ is the external potential. The equilibrium density minimises $Φ_{G} [ρ]$ , and the Euler-Lagrange equation for this minimisation reproduces the grand canonical equilibrium condition.

Synthesis. The grand canonical ensemble builds toward the quantum statistics of ideal gases [11.05.01, 11.05.02] and the theory of phase transitions 11.06.01, where the chemical potential controls the occupation of quantum states. The central insight is that the grand partition function factorises for non-interacting particles, reducing the many-body problem to a product of single-particle partition functions. This is dual to the canonical ensemble, where factorisation works for independent energy modes but not for particles. The foundational reason is that the chemical potential $μ$ is the Lagrange multiplier for the particle-number constraint, just as $β$ is the Lagrange multiplier for the energy constraint; putting these together, the grand canonical ensemble treats both energy and particle number on equal footing, which is exactly the structure needed for quantum field theory where particle number is not conserved. The generalisation from canonical to grand canonical mirrors the Legendre transform from Helmholtz to Gibbs free energy in thermodynamics.

Full proof set Master

Proposition. The grand canonical ensemble is equivalent to the canonical ensemble in the thermodynamic limit: the probability distribution $P (N) = z^{N} Z_{N} /Ξ$ is sharply peaked around $⟨ N ⟩$ with relative width $Δ N / ⟨ N ⟩ \sim 1/ ⟨ N ⟩$ .

Proof. Consider $P (N) = z^{N} Z_{N} /Ξ$ . Write $P (N) = e^{f (N)}$ where $f (N) = N ln z + ln Z_{N} - ln Ξ$ . Expand $f$ around its maximum at $N = N^{*}$ :

$f (N) \approx f (N^{*}) + \frac{1}{2} f^{''} (N^{*}) (N - N^{*})^{2}$

where $f^{'} (N^{*}) = 0$ gives $N^{*} = ⟨ N ⟩$ and $f^{''} (N^{*}) = - 1/ (Δ N)^{2}$ . The distribution is Gaussian with width $Δ N = 1/ ∣ f^{''} (N^{*}) ∣$ . From the fluctuation relation, $(Δ N)^{2} = k T (\partial ⟨ N ⟩ / \partial μ)_{T, V}$ . For an ideal gas, $(\partial ⟨ N ⟩ / \partial μ) = ⟨ N ⟩ / (k T)$ , giving $Δ N = ⟨ N ⟩$ and $Δ N / ⟨ N ⟩ = 1/ ⟨ N ⟩ \to 0$ as $⟨ N ⟩ \to \infty$ . $□$

Connections Master

11.05.01 The Bose-Einstein distribution $⟨ n_{k} ⟩ = 1/ (e^{β (ϵ_{k} - μ)} - 1)$ is the mean occupation number from the bosonic grand partition function $Ξ_{k} = 1/ (1 - z e^{- β ϵ_{k}})$ .
11.05.02 The Fermi-Dirac distribution $⟨ n_{k} ⟩ = 1/ (e^{β (ϵ_{k} - μ)} + 1)$ is the mean occupation number from the fermionic grand partition function $Ξ_{k} = 1 + z e^{- β ϵ_{k}}$ .
11.05.04 Bose-Einstein condensation occurs when the chemical potential $μ$ reaches the ground-state energy $ϵ_{0}$ and the ground-state occupation becomes macroscopic — the grand canonical signature of the transition.
11.04.01 The canonical ensemble is the special case $μ = 0$ (or more precisely, the $N$ -fixed sector of the grand canonical ensemble); the grand potential reduces to the Helmholtz free energy.
11.06.01 The Lee-Yang zeros of the grand partition function provide a mathematically precise characterisation of phase transitions in the Ising model.

Historical and philosophical context Master

Gibbs introduced the grand canonical ensemble in his 1902 Elementary Principles in Statistical Mechanics, recognising that systems that exchange particles with their environment require a distribution that accounts for variable particle number. The chemical potential, which had been introduced by Gibbs in his earlier thermodynamic work (1875–78), found its natural statistical-mechanical interpretation as the parameter conjugate to particle number in the grand canonical distribution.

The grand canonical ensemble was initially viewed as a mathematical convenience — equivalent to the canonical ensemble in the thermodynamic limit but sometimes easier to compute with. Its importance grew with the development of quantum statistics: the Bose-Einstein and Fermi-Dirac distributions are most naturally derived in the grand canonical framework, where particle number is a derived quantity controlled by $μ$ rather than a fixed parameter.

Bibliography Master

@book{gibbs1902,
  author = {Gibbs, J. Willard},
  title = {Elementary Principles in Statistical Mechanics},
  publisher = {Yale University Press},
  year = {1902}
}

@book{pathria-beale2011,
  author = {Pathria, R. K. and Beale, Paul D.},
  title = {Statistical Mechanics},
  edition = {3rd},
  publisher = {Academic Press},
  year = {2011}
}

@book{reif1965,
  author = {Reif, Frederick},
  title = {Fundamentals of Statistical and Thermal Physics},
  publisher = {McGraw-Hill},
  year = {1965}
}

@book{huang1987,
  author = {Huang, Kerson},
  title = {Statistical Mechanics},
  edition = {2nd},
  publisher = {Wiley},
  year = {1987}
}

Prerequisites

11.03.01
11.04.01

Used in

11.03.03
11.04.03
11.05.01

Tier anchors

beginner: Schroeder, An Introduction to Thermal Physics (2000), Ch. 7 (Quantum statistics); Susskind, Statistical Mechanics: The Theoretical Minimum (2014), Lecture 9
intermediate: Reif, Fundamentals of Statistical and Thermal Physics (1965), Ch. 6; Pathria & Beale, Statistical Mechanics, 3rd ed. (2011), Ch. 4
master: Landau & Lifshitz, Statistical Physics, Part 1, 3rd ed. (1980), §35–36; Huang, Statistical Mechanics, 2nd ed. (1987), Ch. 6–7

References

Gibbs — Elementary Principles in Statistical Mechanics (Yale, 1902) · Ch. IV–V The canonical and grand canonical distributions. Gibbs introduced the grand ensemble as the natural framework for systems that exchange particles with a reservoir.
Pathria & Beale — Statistical Mechanics, 3rd ed. (Academic Press, 2011) · Ch. 4 Other ensembles and equivalence in the thermodynamic limit. Grand partition function Xi = sum_N Z(N) z^N where z = e^{beta mu} is the fugacity. Grand potential Phi_G = -PV = -kT ln Xi.
Reif — Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965) · Ch. 6 The canonical and grand canonical distributions. Derivation of the grand canonical ensemble by coupling to a particle reservoir.
Landau & Lifshitz — Statistical Physics, Part 1, 3rd ed. (Pergamon, 1980) · Sec. 35 The Gibbs distribution for a variable number of particles; Sec. 36 The derivation of the thermodynamic relations from the Gibbs distribution.
Huang — Statistical Mechanics, 2nd ed. (Wiley, 1987) · Ch. 6 The grand canonical ensemble and the thermodynamic limit; Ch. 7 Simple gases (ideal Bose and Fermi gases in the grand canonical framework).

Estimated time

beginner: 15 min
intermediate: 40 min
master: 70 min