Single Paramagnet — Negative Temperature

N independent dipoles sit in a fixed field B₀, each pointing along the field (↑, energy −μB₀) or against it (↓, energy +μB₀). Unlike an Einstein solid's energy quanta, the number of up-spins N_↑ is capped between 0 and N — so entropy is a symmetric hump in energy, not a monotonic climb, and temperature can come out negative once more than half the dipoles point up. Drag "Inspect N↑" below to walk across every energy the system can have and watch it happen.

Multiplicity, Energy, Entropy

$$\Omega(N,N_\uparrow) = \binom{N}{N_\uparrow}, \qquad \frac{U}{\mu B_0} = N-2N_\uparrow, \qquad \frac{S}{k_B} = \ln\Omega$$

Ω(N,N↑) is the number of ways to choose which N↑ of the N dipoles point along the field. S(U) is a symmetric hump, zero at both U=−NμB₀ (all up) and U=+NμB₀ (all down), peaking at U=0 (half up, half down).

Temperature & Heat Capacity

$$\frac{1}{T} = \frac{\partial S}{\partial U} \approx \frac{S(N_\uparrow{+}1)-S(N_\uparrow{-}1)}{U(N_\uparrow{+}1)-U(N_\uparrow{-}1)}$$

This table uses a centered difference (both neighbors), not a forward one — it's what correctly places T = ±∞ exactly at the entropy peak instead of smearing it across a step. This matches how Schroeder's Table 3.2 itself is computed, which is why the numbers here line up with it exactly at N=100.

$$C = \frac{\partial U}{\partial T} \approx \frac{U(N_\uparrow{+}1)-U(N_\uparrow{-}1)}{T(N_\uparrow{+}1)-T(N_\uparrow{-}1)}$$

Blank ("—") wherever this would divide across the T=±∞ discontinuity (the two neighbors land on opposite branches) or at the very edges (N_↑=0 or N), where one side has no neighbor at all — exactly where Table 3.2 leaves it blank too.

Single Paramagnet: Negative Temperature

Inspected State

Multiplicity, Energy, Entropy

Temperature & Heat Capacity

Entropy vs. Energy (Fig. 3.8)

Temperature vs. Energy (Fig. 3.9)

Table 3.2-Style Data