Questions: The Thermodynamic Limit and Extensivity

Question 1

A theorist simulates a magnetic material with exactly 100 spins and tries to locate the ferromagnetic phase transition by looking for a non-analytic point in the free energy. What will they find, and why?

• A. They will find a sharp phase transition at the critical temperature, because 100 particles is enough for thermodynamics to apply
• B. They will find a smooth crossover rather than a sharp transition, because the partition function of a finite system is analytic everywhere
• C. They will find a transition, but it will be shifted to a slightly different temperature due to finite-size effects
• D. Phase transitions occur at any system size; the thermodynamic limit only affects the sharpness of the transition

Answer: (B)

A finite system has a partition function Z = Σ exp(−βEᵢ) that is a finite sum of smooth exponentials. Since log Z is analytic everywhere — no poles, no branch cuts — the free energy per spin has no non-analytic points. Phase transitions are defined by non-analyticities (discontinuities or divergences in derivatives of free energy), so they cannot exist in a finite system. Instead, there is a smooth crossover that sharpens as N increases but only becomes a true sharp transition in the N → ∞ thermodynamic limit.

Question 2

In a canonical ensemble simulation at finite N, the energy fluctuates around its mean. In the microcanonical ensemble, energy is fixed. Why can physicists freely switch between ensembles without worrying about which is 'correct'?

• A. The ensembles always give exactly the same results, regardless of system size
• B. In the thermodynamic limit, the canonical distribution concentrates so sharply around the mean energy that it becomes effectively equivalent to the microcanonical
• C. The microcanonical ensemble is always more accurate; the canonical ensemble is used for computational convenience only
• D. Ensemble equivalence holds only for ideal gases, not for interacting systems

Answer: (B)

In a finite system, the two ensembles give genuinely different predictions: the canonical ensemble allows energy fluctuations while the microcanonical fixes energy exactly. But in the thermodynamic limit, the relative energy fluctuation σ_E/⟨E⟩ ~ 1/√N vanishes. The canonical distribution becomes so sharply peaked around the mean energy that it is effectively microcanonical. This ensemble equivalence is a consequence of the thermodynamic limit, not a fundamental feature of all system sizes, and it is why physicists can choose whichever ensemble is mathematically convenient.

Question 3

The thermodynamic limit is just an approximation for large systems; real materials with 10²³ particles have 'nearly' sharp phase transitions.

• T. True
• F. False

Answer: False

False — or at minimum, deeply misleading. The thermodynamic limit is not merely a convenient approximation: it is the mathematical setting in which phase transitions actually exist as well-defined objects. A finite partition function is strictly analytic; non-analyticities only appear in the infinite-N limit. We do observe apparently sharp transitions in real materials, but this sharpness is an extreme approximation justified by the immense value of N (10²³). The conceptual point is that 'phase transition' is a mathematical idealization, not a physical fact about finite systems — it lives in the limit.

Question 4

In the thermodynamic limit, relative fluctuations in extensive quantities become negligible compared to their mean values.

• T. True
• F. False

Answer: True

True. For an extensive quantity like total energy E, absolute fluctuations scale as √N (standard deviation grows with system size), but the mean ⟨E⟩ scales as N. The relative fluctuation σ/⟨E⟩ ~ √N/N = 1/√N → 0 as N → ∞. This is why thermodynamic quantities like temperature and pressure are deterministic in everyday experience — the probability of observing a macroscopic fluctuation is exponentially suppressed in N. For 10²³ particles, spontaneous large deviations are so improbable they effectively never occur.

Question 5

Why does taking N → ∞ allow phase transitions to exist, when a finite system cannot have them?

Model answer: A finite system's partition function is a finite sum of terms of the form exp(−βEᵢ), each smooth in β (inverse temperature). A finite sum of smooth functions is itself smooth — the free energy log Z is analytic, meaning it has derivatives of all orders everywhere. Phase transitions require non-analyticities: a first-order transition is a discontinuity in the first derivative (latent heat), a continuous transition is a divergence in the second derivative. These features require infinitely many terms in the sum, which is only achieved in the thermodynamic limit N → ∞. Only then can the free energy per particle develop the sharp features we observe as phase transitions.

This result — due to the analysis of partition functions as complex functions — explains why phase transitions are fundamentally a collective, large-N phenomenon. No matter how strong the interactions, a small system will always exhibit a smooth crossover rather than a sharp transition. The thermodynamic limit is not a crutch but the precise mathematical statement of what 'phase transition' means: a non-analytic point in the free energy of an infinite system. Finite-size scaling theory then tells you how real (finite) systems approach this ideal as N grows.