Open Systems: The Grand Canonical Ensemble
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You should be comfortable with the canonical ensemble, the definition of the Helmholtz free energy \( F \), and the concept of chemical potential as a partial derivative of free energy with respect to particle number.
Why this matters
The grand canonical ensemble is the essential tool for describing systems where particle number fluctuates, such as electrons in a metal or molecules in a gas phase.
In the canonical ensemble, we fixed the number of particles \( N \). However, in many physical situations, a small subsystem is in contact with a much larger reservoir that can exchange both energy and particles. This is the case for a droplet of liquid in a vapour, or gas molecules adsorbing on a surface. To describe such "open" systems, we must generalize the statistical weight to account for fluctuations in \( N \). The grand canonical ensemble provides the rigorous framework for this, introducing the grand partition function \( \Xi \) as the central object from which all thermodynamic properties of open systems are derived.
The Idea of Open Systems
Imagine a small box containing a few gas molecules. If the box is sealed, the number of molecules inside is fixed; this is a closed system. But if the box has a porous wall and is placed in a large room filled with the same gas, molecules can drift in and out. The number of molecules inside the box is no longer a fixed number but a fluctuating quantity. Sometimes there are more, sometimes fewer. To predict the average behaviour of this small box, we cannot simply pick one number \( N \) and calculate the probability for that specific case. Instead, we must consider all possible numbers of particles simultaneously. The grand canonical ensemble does exactly this: it sums over all possible microstates, where each microstate is defined not just by its energy \( E_s \), but also by the number of particles \( N_s \) it contains. The probability of finding the system in a particular state depends on both how much energy it has and how many particles it holds, relative to the reservoir's temperature and chemical potential.
The Grand Partition Function
The central quantity in the grand canonical ensemble is the grand partition function, denoted by \( \Xi \). It is defined as the sum of the Boltzmann factors over all possible microstates \( s \). Each microstate \( s \) has an energy \( E_s \) and a particle number \( N_s \). The weight of each state is determined by the factor \( e^{-\beta(E_s - \mu N_s)} \), where \( \beta = 1/kT \) and \( \mu \) is the chemical potential.
We can reorganise this sum by grouping states according to their particle number \( N \). For a fixed \( N \), the sum over states is just the canonical partition function \( Z(N, V, T) \). The term \( e^{\beta \mu N} \) can be factored out of the sum over states for that specific \( N \). Defining the fugacity \( z = e^{\beta \mu} \), we can write the grand partition function as a sum over \( N \):
This expression shows that \( \Xi \) is a generating function for the canonical partition functions. It encodes information about all possible particle numbers in a single mathematical object.
Thermodynamic Potentials
Just as the canonical partition function \( Z \) is related to the Helmholtz free energy \( F \) via \( F = -kT \ln Z \), the grand partition function \( \Xi \) is related to a new thermodynamic potential called the grand potential, often denoted by \( \Phi \) or \( \Omega \). The relationship is:
The grand potential \( \Phi \) is a function of temperature \( T \), volume \( V \), and chemical potential \( \mu \). It is particularly useful because it naturally describes systems in contact with a reservoir. The differential form of the grand potential is given by:
From this, we can derive several important thermodynamic relations. For example, the pressure \( P \) is given by \( P = -(\partial \Phi / \partial V)_{T, \mu} \), and the average particle number \( \langle N \rangle \) is given by \( \langle N \rangle = -(\partial \Phi / \partial \mu)_{T, V} \). These relations allow us to calculate macroscopic observables directly from the grand partition function.
Average Particle Number and Fluctuations
One of the key results of the grand canonical ensemble is the expression for the average particle number. Using the definition of \( \Phi \), we find that the average number of particles is:
Notice that \( \langle N \rangle \) is not a fixed integer but a continuous variable that depends on \( T \), \( V \), and \( \mu \). The fluctuations in the particle number are also of great interest. The variance of the particle number, \( \langle (\Delta N)^2 \rangle \), is given by:
This equation shows that the fluctuations in particle number are directly related to the isothermal compressibility of the system. In the thermodynamic limit, where \( N \) is very large, the relative fluctuations \( \sqrt{\langle (\Delta N)^2 \rangle} / \langle N \rangle \) become very small, justifying the use of the average value \( \langle N \rangle \) in place of the instantaneous value \( N \).
The derivation
The Grand Canonical Ensemble and Chemical Potential
A small system in thermal and diffusive contact with a large reservoir at temperature T and chemical potential μ occupies microstate s (energy Es, particle number Ns) with probability Ps = e−β(Es − μNs) / Ξ, where the grand partition function Ξ(T,V,μ) = Σs e−β(Es − μNs) and β = 1/kT. The chemical potential μ = (∂F/∂N)T,V is the free-energy cost of adding one particle at fixed temperature and volume.
A common misconception is that the chemical potential \( \mu \) is a constant property of a substance, like its molar mass. In reality, \( \mu \) is an intensive variable that depends on the state of the system (temperature, pressure, and composition). For an ideal gas, \( \mu \) depends logarithmically on the density. When comparing two systems in equilibrium, their chemical potentials must be equal, but this does not mean they have the same density or pressure if their temperatures differ. C
Worked example
Problem. A surface has 1000 independent adsorption sites. Each site is either empty (energy 0) or holds one molecule bound with energy \(\varepsilon = -0.30\ \mathrm{eV}\). The surface is in contact with a gas that fixes \(T = 300\ \mathrm{K}\) and \(\mu = -0.25\ \mathrm{eV}\). Find the fraction of occupied sites, the mean number adsorbed, and the size of the fluctuations.
- Grand partition function of one site: \(\xi = 1 + e^{-\beta(\varepsilon - \mu)}\), with \(\beta = 1/k_BT = 1/(8.617 \times 10^{-5} \times 300) = 38.68\ \mathrm{eV^{-1}}\).
- Exponent: \(\beta(\varepsilon - \mu) = 38.68 \times (-0.30 + 0.25) = -1.934\), so \(e^{-\beta(\varepsilon-\mu)} = e^{1.934} = 6.92\).
- Occupation probability: \(\theta = \dfrac{6.92}{1 + 6.92} = \dfrac{1}{e^{\beta(\varepsilon-\mu)} + 1} = \dfrac{1}{0.1445 + 1} = 0.874\).
- Mean number adsorbed: \(\langle N \rangle = 1000\,\theta = 874\). Because the sites are independent, \(\langle(\Delta N)^2\rangle = 1000\,\theta(1-\theta) = 110\), so \(\sigma_N = 10.5\).
- Relative fluctuation \(10.5/874 = 1.2\%\); for \(10^{20}\) sites it would be about \(10^{-10}\). Check with the general formula: \(k_BT\,\partial\langle N \rangle/\partial\mu = 1000\,\theta(1-\theta)\), the same 110.
Units check. \(\beta\) in eV-1 times energies in eV gives a pure number; \(\theta\) and \(N\) are dimensionless.
Three quick tests
- What is the relationship between the grand partition function \( \Xi \) and the grand potential \( \Phi \)?
Answer
\( \Phi = -kT \ln \Xi \) - How does the average particle number \( \langle N \rangle \) depend on the chemical potential \( \mu \)?
Answer
\( \langle N \rangle \) increases as \( \mu \) increases, following \( \langle N \rangle = \frac{1}{\beta} \frac{\partial \ln \Xi}{\partial \mu} \) - In the thermodynamic limit, what happens to the relative fluctuations in particle number?
Answer
They vanish, i.e., \( \sqrt{\langle (\Delta N)^2 \rangle} / \langle N \rangle \to 0 \)
Problem set for PU-203 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes checked line by line by the editor; every number in the worked example was recomputed.