The Grand Canonical Ensemble

Hello! In our previous lessons, we developed the canonical ensemble to describe systems with a fixed number of particles ( $N$ ) in thermal contact with a heat reservoir at a constant temperature ( $T$ ). This led us to the Boltzmann distribution and the powerful relation $F = - k_{B} T ln Z$ , which connects the microscopic partition function $Z$ to the macroscopic Helmholtz free energy $F$ .

This framework is excellent for closed systems, but many physical and chemical processes occur in "open" systems that can exchange not only energy but also particles with their surroundings. Consider a gas adsorbing onto a surface, a chemical reaction where molecules are created or consumed, or the electron gas in a semiconductor connected to leads. To handle these situations, we need to generalize our statistical framework.

This lesson introduces the grand canonical ensemble, which describes systems in equilibrium with a reservoir that fixes both temperature and chemical potential. Our primary goal is to derive the probability distribution for a system in this ensemble. We will find that the probability of a microstate depends not only on its energy but also on its particle number, leading to a new "Boltzmann-like" factor and its corresponding normalization constant, the grand canonical partition function.

1. The Grand Canonical Setup: Open Systems

Imagine a system $S$ that can exchange both energy and particles with a very large reservoir $R$ . The combined entity $S + R$ forms an isolated system, so its total energy $E_{tot} = E_{S} + E_{R}$ and total particle number $N_{tot} = N_{S} + N_{R}$ are conserved.

Because the reservoir is vast, any exchange leaves its intensive properties unchanged. It acts as a heat bath, fixing the temperature $T$ of our system, and as a particle bath, fixing a new quantity called the chemical potential, denoted by $μ$ . The macroscopic state of our system $S$ is therefore specified by the variables $(μ, V, T)$ . In this ensemble, the system's energy $E_{S}$ and particle number $N_{S}$ are no longer fixed but are fluctuating quantities whose average values are determined by $T$ and $μ$ .

Our task is to find the probability $p_{k}$ of finding the system $S$ in a specific microstate $k$ , which is now characterized by both its energy $E_{k}$ and its particle number $N_{k}$ .

2. Derivation via the Principle of Maximum Entropy

Just as we did for the canonical ensemble, we can derive the equilibrium distribution from a fundamental information-theoretic standpoint using the Principle of Maximum Entropy (MaxEnt), which you first encountered in Module 1. This principle states that the most unbiased probability distribution is the one that maximizes the Gibbs entropy, subject only to the constraints imposed by our knowledge of the system.

For the grand canonical ensemble, we have three constraints:

Normalization: The probabilities must sum to one. $\sum_{k} p_{k} = 1$ .
Mean Energy: The system is in thermal equilibrium with a reservoir at temperature $T$ , which fixes its average energy. $\sum_{k} p_{k} E_{k} = ⟨ E ⟩$ .
Mean Particle Number: The system is in diffusive equilibrium with a reservoir at chemical potential $μ$ , which fixes its average particle number. $\sum_{k} p_{k} N_{k} = ⟨ N ⟩$ .

Our goal is to maximize the entropy $S = - k_{B} \sum_{k} p_{k} ln p_{k}$ subject to these three constraints. This is a classic optimization problem solved using the method of Lagrange multipliers.

Statistical Physics—Section 2: Assignment of Probabilities ...

This document from the University of Edinburgh provides a clear, step-by-step derivation using the MaxEnt formalism. It first reviews the method of Lagrange multipliers and then applies it directly to the grand canonical case.

First, briefly review the general method of Lagrange multipliers in the section "Assignment of Probability" on page 5, particularly Example 2, which shows how to maximize entropy with multiple constraints. Next, read the section titled "Grand Canonical Ensemble" on page 7. This section applies the same method to derive the probability distribution for a system that can exchange both energy and particles. Focus on how the constraints on \langle E \rangle and \langle N \rangle lead to the final form of the probability, pi,N = (1/Zgc)exp(...). Note that the Lagrange multipliers (\lambda_E, \lambda_N) are not yet identified with physical quantities.

As you saw in the reading, maximizing the entropy subject to constraints on $⟨ E ⟩$ and $⟨ N ⟩$ yields a probability distribution of the form:

p_{k} = \frac{1}{Ξ} exp (- \frac{λ _{E}}{k _{B}} E_{k} - \frac{λ _{N}}{k _{B}} N_{k})

Here, $λ_{E}$ and $λ_{N}$ are the Lagrange multipliers associated with the energy and particle number constraints, respectively. The normalization constant, which we denote by $Ξ$ (Greek letter "Ksi"), is the grand canonical partition function.

3. Identifying the Lagrange Multipliers

The expression above is general but not yet physical. To make the connection to thermodynamics, we must identify $λ_{E}$ and $λ_{N}$ . This requires comparing the statistical mechanical form of the first law of thermodynamics with its established thermodynamic form for an open system.

The fundamental thermodynamic relation, which you've seen in your prior studies, generalizes to open systems as:

d E = T d S - P d V + μ d N

This equation defines temperature $T$ and chemical potential $μ$ as partial derivatives of the energy:

T = (\frac{\partial E}{\partial S})_{V, N} and μ = (\frac{\partial E}{\partial N})_{S, V}

The chemical potential $μ$ is the energy cost of adding one particle to the system while keeping entropy and volume constant.

By a careful argument that relates the statistical differential $d ⟨ E ⟩$ to the statistical entropy differential $d S$ , one can show that the Lagrange multipliers must be:

λ_{E} = \frac{1}{T} and λ_{N} = - \frac{μ}{T}

Statistical Physics—Section 2: Assignment of Probabilities ...

The same notes you just read provide the elegant argument for identifying the Lagrange multipliers. This is a key step that connects the abstract mathematical framework to measurable thermodynamic quantities.

Please read the subsection "Grand Canonical distribution" on page 13. Follow the derivation that starts with the differential of energy, dE = ..., and compares it to the thermodynamic first law. This comparison directly leads to the identification of the multipliers, λE and λN, and the final, physically meaningful form of the grand canonical distribution.

Substituting these multipliers back into our probability expression and setting $β = 1/ (k_{B} T)$ , we arrive at the grand canonical distribution:

p_{k} = \frac{1}{Ξ} e^{- β (E_{k} - μ N_{k})}

This is the central result of our lesson. It gives the probability of finding the system in a microstate $k$ with energy $E_{k}$ and particle number $N_{k}$ when it is in contact with a reservoir at temperature $T$ and chemical potential $μ$ . The term $e^{- β (E_{k} - μ N_{k})}$ is the grand canonical Boltzmann factor.

4. An Alternative View: Analogy to the Canonical Ensemble

An alternative and intuitive way to derive this distribution is to directly follow the logic we used for the canonical ensemble. The probability $p_{k}$ must be proportional to the number of states available to the reservoir, $Ω_{R}$ , when our system is in state $k$ .

p_{k} \propto Ω_{R} (E_{tot} - E_{k}, N_{tot} - N_{k})

Taking the logarithm, we get $ln p_{k} \propto ln Ω_{R} = S_{R} / k_{B}$ . We can then perform a Taylor expansion of the reservoir's entropy $S_{R}$ around the total energy and particle number, since $E_{k} ≪ E_{tot}$ and $N_{k} ≪ N_{tot}$ :

S_{R} (E_{tot} - E_{k}, N_{tot} - N_{k}) \approx S_{R} (E_{tot}, N_{tot}) - E_{k} (\frac{\partial S _{R}}{\partial E _{R}}) - N_{k} (\frac{\partial S _{R}}{\partial N _{R}})

From thermodynamics, we identify the derivatives as $(\partial S_{R} / \partial E_{R}) = 1/ T$ and $(\partial S_{R} / \partial N_{R}) = - μ / T$ . Plugging this into the expansion and exponentiating gives the probability:

p_{k} \propto exp (- \frac{E _{k}}{k _{B} T} + \frac{μ N _{k}}{k _{B} T}) = e^{- β (E_{k} - μ N_{k})}

This confirms our result from the MaxEnt approach. The following video provides a concise walkthrough of this line of reasoning.

Teach Yourself Statistical Mechanics In One Video | New & Improved

This video from the "Physics Daemon" channel offers a clear, visual derivation of the grand canonical distribution using the method of expanding the reservoir's entropy, which is a direct generalization of the canonical case.

Please watch the segment from the beginning of the derivation. This part covers the Taylor expansion of the logarithm of \Omega_R, the identification of the chemical potential \mu, and the final form of the distribution. The video also includes a nice proof that particles flow from high to low chemical potential, which you can watch from here to solidify your intuition about \mu.

5. The Grand Canonical Partition Function

The normalization constant $Ξ$ , called the grand canonical partition function, ensures that $\sum_{k} p_{k} = 1$ . It is defined by summing the grand canonical Boltzmann factors over all possible microstates for the system:

Ξ (μ, V, T) = k \sum e^{- β (E_{k} - μ N_{k})}

It is crucial to understand the structure of this sum. The index $k$ runs over all microstates for all possible particle numbers. We can make this more explicit by writing it as a double summation: first, we sum over all possible particle numbers $N$ , from 0 to $\infty$ . Then, for each fixed $N$ , we sum over all microstates $i (N)$ available to that $N$ -particle system.

Ξ = N = 0 \sum \infty i (N) \sum e^{- β (E_{i (N)} - μ N)}

We can factor out the term depending on $N$ from the inner sum:

Ξ = N = 0 \sum \infty e^{β μ N} i (N) \sum e^{- β E_{i (N)}}

The term in the parenthesis is precisely the canonical partition function $Z (N, V, T)$ for a system with a fixed number of particles $N$ . This gives us a fundamental and extremely useful relationship:

Ξ (μ, V, T) = N = 0 \sum \infty e^{β μ N} Z (N, V, T)

The quantity $z \equiv e^{β μ}$ is called the fugacity. It acts as a weighting factor that controls the average number of particles in the system. High fugacity (high $μ$ ) favors states with larger $N$ . Using the fugacity, the grand partition function takes the compact form:

Ξ (μ, V, T) = N = 0 \sum \infty z^{N} Z (N, V, T)

This shows that the grand partition function is a "generating function" for the canonical partition functions.

Lecture 24-Grand Canonical Ensemble

This NPTEL lecture provides a very clear explanation of how the sum over all microstates in the grand canonical ensemble can be broken down into a sum over particle number N, where each term involves the canonical partition function for that N.

Watch the segment from deriving the partition function. The lecturer carefully transforms the unrestricted sum over all microstates into a restricted sum over states for a fixed particle number, explicitly showing how the canonical partition function Z_N emerges inside the sum for \Xi.

Conclusion

In this lesson, we have extended our statistical framework to accommodate open systems, which can exchange both energy and particles with their environment. This led us to the grand canonical ensemble.

Key Takeaways:

The grand canonical ensemble describes systems at constant volume $V$ , temperature $T$ , and chemical potential $μ$ . Energy and particle number are fluctuating quantities.
The probability of finding the system in a microstate $k$ with energy $E_{k}$ and particle number $N_{k}$ is given by the grand canonical distribution:
$p_{k} = \frac{1}{Ξ} e^{- β (E_{k} - μ N_{k})}$
The normalization constant $Ξ$ is the grand canonical partition function. It is found by summing the grand canonical Boltzmann factors over all microstates for all possible particle numbers.
$Ξ$ is related to the canonical partition functions $Z (N, V, T)$ via the fugacity $z = e^{β μ}$ :
$Ξ (μ, V, T) = \sum_{N = 0}^{\infty} z^{N} Z (N, V, T)$

Just as the canonical partition function $Z$ gave us a bridge to the Helmholtz free energy $F$ , the grand partition function $Ξ$ provides a bridge to another thermodynamic potential. In our next lesson, we will define this potential—the grand potential $Φ$ —and establish the final master equation for this ensemble, $Φ = - k_{B} T ln Ξ$ , completing our set of tools for equilibrium statistical mechanics.

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