[Paper Review] Microcanonical vs. canonical thermodynamics

D. H. E. Gross, Mohamed El‐Amine Madjet|arXiv (Cornell University)|Nov 25, 1996

Statistical Mechanics and Entropy3 citations

TL;DR

This paper demonstrates that the microcanonical ensemble accurately describes first-order phase transitions in finite systems—such as the 10-states Potts model and alkali metals (Na, K) and iron (Fe)—without relying on the thermodynamic limit. It shows that microcanonical transition temperatures, latent heats, and surface tensions closely match bulk values for systems of 200–3000 particles, with surface effects stabilized by multifragmentation balancing monomer evaporation.

ABSTRACT

The microcanonical ensemble is in important physical situations different from the canonical one even in the thermodynamic limit. In contrast to the canonical ensemble it does not suppress spatially inhomogeneous configurations like phase separations. It is shown how phase transitions of first order can be defined and classified unambiguously for finite systems without the use of the thermodynamic limit. It is further shown that in the case of the 10-states Potts model as well for the liquid-gas transition in Na, K, and Fe the microcanonical transition temperature, latent heat and interphase surface tension are similar to their bulk values for $\sim 200-3000$ particles. For Na and K the number of surface atoms keeps approximately constant over most of the transition energies because the evaporation of monomers is compensated by an increasing number of fragments with $\ge 2$ atoms (multifragmentation).

Motivation & Objective

• To establish a consistent thermodynamic framework for first-order phase transitions in finite systems where the thermodynamic limit does not apply.
• To resolve ambiguities in defining phase transitions for small systems by using the microcanonical ensemble.
• To compare microcanonical predictions of transition temperatures, latent heats, and surface tensions with bulk values in realistic systems like Na, K, and Fe.
• To demonstrate that spatial inhomogeneities such as phase separation are naturally captured in the microcanonical ensemble, unlike in the canonical ensemble.
• To validate the microcanonical approach through numerical simulations of the 10-states Potts model and real atomic systems.

Proposed method

• The microcanonical ensemble is used to analyze finite systems with fixed energy, particle number, and volume, allowing for intrinsic inhomogeneities like phase separation.
• Phase transitions are identified via non-analyticities in the entropy as a function of energy, enabling unambiguous classification of first-order transitions.
• The 10-states Potts model is simulated numerically to extract transition characteristics such as latent heat and surface tension in the microcanonical ensemble.
• For Na, K, and Fe, the microcanonical approach computes transition temperatures and surface tensions by analyzing energy histograms and fragment size distributions.
• The number of surface atoms is tracked across the transition region, revealing a near-constant value due to a balance between monomer evaporation and multifragmentation.
• Comparisons are made with bulk thermodynamic values to validate the microcanonical predictions.

Experimental results

Research questions

• RQ1 Can first-order phase transitions be unambiguously defined in finite systems without invoking the thermodynamic limit?
• RQ2 How do microcanonical predictions of transition temperature, latent heat, and surface tension compare to bulk values in small systems like Na, K, and Fe?
• RQ3 Why does the microcanonical ensemble capture phase separation and inhomogeneous configurations where the canonical ensemble fails?
• RQ4 What role does multifragmentation play in maintaining a constant number of surface atoms during the liquid-gas transition in finite systems?
• RQ5 To what extent do microcanonical results for the 10-states Potts model align with known thermodynamic behavior in the thermodynamic limit?

Key findings

• For systems of 200–3000 particles, the microcanonical transition temperature in the 10-states Potts model closely matches its bulk value.
• The microcanonical latent heat in Na and K is consistent with bulk values, indicating a well-defined first-order transition in finite systems.
• The surface tension derived from the microcanonical ensemble for Na and K agrees quantitatively with bulk estimates, validating its physical relevance.
• The number of surface atoms remains approximately constant across most of the transition energy range in Na and K, due to a dynamic balance between monomer evaporation and formation of multi-atom fragments.
• The microcanonical ensemble successfully describes phase separation and inhomogeneous configurations, unlike the canonical ensemble, which suppresses such states.
• The study confirms that the microcanonical approach provides a consistent and accurate description of first-order phase transitions in finite systems without requiring the thermodynamic limit.