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 Incoming links: The Fundamentals of Statistical Mechanics by David Tong

Past exam of the thermal and statistical physics course of the University of Cambridge / 2016 / Paper 1 / 1 / b / Solution 2026-10-20

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Take

μ > 0

to be the magnitude of either spin projection's magnetic moment, so the independent magnetic levels are

E_{\pm} = \mp μ B

. Their partition function and magnetic energy are

Z_{1} = 2 cosh x, Z = (2 cosh x)^{N}, U_{mag} = - \partial_{β} ln Z = - N μ B tanh x,

(1)

where

β = 1/ (k_{B} T)

and

x = β μ B

. The canonical entropy is consequently

S = k_{B} ln Z + \frac{U _{mag}}{T} = N k_{B} [ln (2 cosh x) - x tanh x] .

(2)

This independent-spin magnetic entropy counts the two orientation states of each independent moment; translational entropy and interactions are excluded.

Along the magnetic-entropy demagnetization adiabat, constant magnetic entropy fixes

x

, hence $B / T$ is constant. Lowering the field therefore gives the ideal adiabatic demagnetization relation

T_{f} = T_{i} \frac{B _{f}}{B _{i}} .

(3)

A useful cycle first magnetizes isothermally while releasing heat to a reservoir, then thermally isolates the sample and lowers the field. Cooling the entire sample requires its other heat capacities and spin-lattice equilibration to be accounted for. Magnetic interactions, residual fields and heat leaks prevent extrapolating the independent-moment formula to an attainable zero temperature.^[ref]

Figure 1.
Magnetic entropy decreases with B over T, while an ideal demagnetization adiabat has constant B over T
. **Source**. **Details**.

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Past exam of the thermal and statistical physics course of the University of Cambridge / 2016 / Paper 1 / 2 / a / Solution 2026-10-20

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An statistical ensemble specifies probabilities of microstates under particular macroscopic constraints. It represents statistical information about a system; the reservoir constraints of equilibrium ensembles specify what it can exchange with its surroundings.

The microcanonical ensemble has fixed $E, V, N$ . Accessible states in a narrow energy shell are equally probable, with $S = k_{B} ln Ω$ , where $Ω$ counts the states in that shell. It describes an isolated system.
The canonical ensemble has fixed $T, V, N$ and can exchange energy with a heat reservoir. A microstate has probability $p_{i} = e^{- β E_{i}} / Z$ , with $Z = \sum_{i} e^{- β E_{i}}$ and $F = - k_{B} T ln Z$ . The energy fluctuates; its mean is not imposed independently.
The grand canonical ensemble has fixed $T, V, μ_{ch}$ and exchanges both particles and energy. Its probabilities are proportional to $e^{- β (E_{i} - μ_{ch} N_{i})}$ , with $Ξ = \sum_{N} e^{β μ_{ch} N} Z_{N}$ and grand potential $Ω_{G} = - k_{B} T ln Ξ$ . For a homogeneous extensive equilibrium system, $Ω_{G} = - p V$ .

The thermodynamic potentials generate mean quantities and response functions. For example, canonical energy variance is

⟨(Δ E)^{2} ⟩ = k_{B} T^{2} C_{V}

. With short-range interactions and stable bulk phases, relative fluctuations usually vanish for large systems and these ensembles give equivalent bulk predictions after matching their mean variables. Finite systems, constraints and long-range interactions require greater care; the ensembles retain different fluctuating variables.^[ref]

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