On the equivalence of the microcanonical and the canonical ensembles: a geometrical approach

In this paper, we consider the volume enclosed by the microcanonical ensemble in phase space as a statistical ensemble. This can be interpreted as an intermediate image between the microcanonical and the canonical pictures. By maintaining the ergodic hypothesis over this ensemble, that is, the equiprobability of all its accessible states, the equivalence of this ensemble in the thermodynamic limit with the microcanonical and the canonical ensembles is suggested by means of geometrical arguments. The Maxwellian and the Boltzmann-Gibbs distributions are obtained from this formalism. In the appendix, the derivation of the Boltzmann factor from a new microcanonical image of the canonical ensemble is also given.

💡 Research Summary

The paper introduces a novel statistical ensemble – the “microcanonical volume ensemble” – and uses it to provide a geometric proof of the equivalence between the traditional microcanonical and canonical ensembles in the thermodynamic limit.
In the standard microcanonical picture, a system with fixed energy E, volume V, and particle number N is represented by the hypersurface in phase space defined by H(p,q)=E. All points on this (2dN‑1)-dimensional surface are assumed equally probable (the ergodic hypothesis). The canonical ensemble, on the other hand, describes a system in thermal contact with a heat bath at temperature T, assigning each microstate a weight proportional to exp(‑βE) with β=1/k_BT. While it is well known that the two ensembles give identical macroscopic predictions when N→∞, the usual derivations rely on either the Laplace transform of the density of states or on large‑deviation arguments. This work proposes a more intuitive geometric route.
The author defines V(E) as the phase‑space volume enclosed by the energy surface, i.e. the integral over all states with H≤E. This volume includes the microcanonical surface Ω(E)=∂V/∂E as a thin shell but also contains all lower‑energy states. The key hypothesis is that the ergodic principle applies to the whole volume: every point inside the energy‑bounded region is equally accessible. Consequently the entropy can be defined as S_V(E)=k_B ln V(E). By differentiating, one obtains ∂S_V/∂E = k_B Ω(E)/V(E). In the high‑dimensional limit the ratio Ω/V tends to a constant that can be identified with 1/T, because both Ω and V grow exponentially with N and share the same leading exponent. Hence S_V and the usual microcanonical entropy S(E)=k_B ln Ω(E) generate identical thermodynamic relations, establishing the equivalence of the two definitions of entropy.
The paper then shows how the Maxwell‑Boltzmann velocity distribution follows from the volume ensemble. For an ideal gas, the kinetic energy is a quadratic form in the momenta, so the set of states with total kinetic energy ≤E corresponds to a 3N‑dimensional sphere of radius √(2mE). The uniform distribution inside this sphere leads, after integrating over all but one momentum component, to the familiar f(v)∝v^{d‑1} exp(‑mv²/2k_BT) law, where d is the spatial dimension. This derivation mirrors the standard canonical result but originates solely from the geometry of the volume.
To connect with the canonical ensemble, the author introduces a Lagrange multiplier λ that enforces the energy constraint on the volume ensemble. Maximizing the entropy under the constraint ⟨H⟩=E yields a probability density P(Γ)∝exp(‑λH(Γ)). The multiplier is identified as λ=∂S_V/∂E, which, as shown above, equals β. Therefore the Boltzmann factor emerges naturally from the volume‑based description, confirming that the canonical ensemble is just the constrained version of the microcanonical volume ensemble.
The appendix provides a complementary construction: the canonical ensemble is obtained by embedding the small system into a huge “heat reservoir” that itself is treated microcanonically. The total phase‑space volume of the combined system is V_total(E_total). Because the reservoir possesses an astronomically larger number of degrees of freedom, the ratio of the combined volume to the reservoir’s volume reduces to exp(‑βE) for the subsystem, reproducing the Boltzmann factor without invoking any external temperature parameter. This reinforces the geometric interpretation of temperature as the derivative of the reservoir’s entropy with respect to energy.
Overall, the paper offers a clear geometric picture: the microcanonical surface, the surrounding volume, and the canonical weight are three manifestations of the same underlying phase‑space structure. By treating the enclosed volume as a legitimate ensemble and applying the ergodic hypothesis, the author demonstrates that in the thermodynamic limit the three ensembles become indistinguishable. The approach not only reproduces the Maxwell‑Boltzmann and Boltzmann‑Gibbs distributions but also provides a pedagogically appealing route to understand why temperature, entropy, and probability weights are intimately linked to the geometry of high‑dimensional phase space. This perspective may prove valuable for extending statistical‑mechanical reasoning to systems with constraints, long‑range interactions, or non‑equilibrium settings where traditional ensemble definitions become ambiguous.