Negative specific heat - 2D caloric curve

Why the heat capacity can be negative

Every particle pair of the long-range gas attracts with the softened potential u(r) = −G m²/√(r²+ε²). For a bound system in equilibrium the virial theorem gives 2⟨KE⟩ + ⟨PE⟩ ≈ 0, so E = KE + PE ≈ −KE. The temperature is T = (2/3) KE / N (k_B = 1), hence E ≈ −(3N/2) T and the heat capacity C = dE/dT is negative: removing energy makes the cloud contract, the potential energy falls by twice what the system lost, and the kinetic energy rises.

The short-range gas

With a cutoff and a repulsive core, energy is additive: E ≈ (3N/2) T and C = +3N/2 > 0, so the curve rises monotonically. The long-range gas has a non-additive energy (it scales as N²), which is why the canonical and microcanonical ensembles disagree and why globular clusters and stars can have negative specific heat.

Honest simplifications: N = 84 particles, a softened potential, a spherical reflecting wall, and a velocity-rescaling reservoir that only nudges the energy towards the target, so phase transitions and fluctuations are smeared and the virial ratio is below 1 because of the softening. Same physics module as the 3D page; the 2D view projects the cloud on the x-y plane.

• Press Cool the long-range gas, then slowly lower the energy budget: T rises while E falls and the core fraction jumps.
• Press Auto sweep and watch the caloric curve draw a backward-bending branch.
• Press Short-range gas and sweep: T just follows E.