Equivalence of Ensembles and the Thermodynamic Limit

The Core Idea in Plain Language

Imagine a large box of gas. If you isolate it, its energy is fixed, but the individual molecules exchange energy constantly. If you put it in contact with a huge heat bath, the box's energy can fluctuate slightly as it exchanges heat with the bath. For a tiny system, these two situations would look very different. However, for a macroscopic system with a huge number of particles, the energy fluctuations in the canonical case are incredibly small compared to the total energy. The system spends almost all its time at a single, specific energy value. This means that whether you calculate properties assuming fixed energy or fixed temperature, you get the same answer. The "sharpness" of the energy distribution is the key to this equivalence.

The Laplace Transform Connection

The mathematical link between the two ensembles is the Laplace transform. The canonical partition function \( Z(\beta) \) is related to the microcanonical density of states \( \Omega(E) \) by the integral:

Here, \( \beta = 1/k_B T \). The density of states \( \Omega(E) \) grows exponentially with energy, while the Boltzmann factor \( e^{-\beta E} \) decays exponentially. The product of these two terms creates a sharp peak. This peak occurs at the energy \( E^* \) where the microcanonical temperature of the system matches the temperature of the bath. This is the point of maximum contribution to the integral.

Steepest Descent and the Gaussian Peak

To evaluate the integral for large systems, we use the method of steepest descent. We expand the logarithm of the integrand around the peak \( E^* \). Let \( f(E) = \ln \Omega(E) - \beta E \). The first derivative vanishes at \( E^* \), and the second derivative determines the width of the peak. For a system with extensive entropy and positive heat capacity \( C \), the second derivative is negative, ensuring a maximum. Since \( \partial^2 \ln\Omega/\partial E^2 = -1/(k_B T^2 C) \), the integral is then approximated by a Gaussian centred at \( E^* \) with variance \( k_B T^2 C \).

The width of this Gaussian, \( \sigma_E \), is proportional to \( \sqrt{k_B T^2 C} \). Since the heat capacity \( C \) is extensive (proportional to \( N \)), the relative width \( \sigma_E / E^* \) scales as \( 1/\sqrt{N} \). As \( N \to \infty \), this relative width vanishes, meaning the energy distribution becomes a delta function. This is the mathematical statement of ensemble equivalence.

Free Energy and the Legendre Transform

From the steepest descent approximation, we can extract the canonical free energy \( F(T) \). Taking the logarithm of \( Z(\beta) \) and using \( F = -k_B T \ln Z \), we find:

The dominant term \( E^* - T S(E^*) \) is precisely the Legendre transform of the microcanonical entropy \( S(E) \) with respect to energy. This shows that the canonical free energy is not an independent quantity but is derived directly from the microcanonical entropy. The correction term \( \mathcal{O}(k_B T \ln N) \) represents the finite-size effects due to the finite width of the energy distribution. These corrections vanish in the thermodynamic limit \( N \to \infty \), confirming the equivalence of the ensembles.