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The microcanonical ensemble assigns equal probability to accessible quantum states in a narrow energy shell of an isolated system with fixed energy, particle number and external parameters. To derive the canonical ensemble, put a small system in weak thermal contact with a much larger isolated bath. If the small-system state has energy $E_i$, its probability is proportional to the number ${\Omega }_b(E_{\mathrm{total}}-E_i)$ of bath states. Expanding the bath entropy gives $S_b(E_{\mathrm{total}}-E_i)=S_b(E_{\mathrm{total}})-E_i/T+\cdots$, since $\partial S_b/\partial E=1/T$. Negligible interaction energy and an effectively constant bath temperature therefore give $p_i=e^{-E_i/(k_{B}T)}/Z$. The microcanonical ensemble suits an isolated fixed-energy system; the canonical ensemble suits a system exchanging energy with a thermostat at fixed temperature, with particle number still fixed.

For one atom, summing the finite geometric series over $m=-J,\dots ,J$ gives the partition functionAt $x=0$ the ratio is understood by continuity as $2J+1$. Logarithmic differentiation gives the mean magnetic dipole momentThis is odd in $x$, has small-$x$ expansion ${\mu }_{B}J(J+1)x/3+O(x^3)$, and saturates at $\pm J{\mu }_B$. The requested common graph is plotted in units of ${\mu }_B$: its ordinate is $⟨{\mu }_z⟩/(J{\mu }_B)$, so multiplication by ${\mu }_B$ gives $⟨{\mu }_z⟩/J$. The normalized slopes at zero are $(J+1)/3$ and the limits are $\pm 1$.

The magnetic susceptibility isEquivalently it is $N{\mu }_B^2\mathrm{Var}(m)/(k_{B}T)\ge 0$. For $k_{B}T\gg {\mu }_B|B|$, the apparent singular terms cancel, yielding Curie lawFor fixed nonzero $B$, $T\to 0$ and integer $J\ge 1$, the first excited magnetic level is separated from the aligned ground state by ${\mu }_B|B|$. Its exponentially small occupation givesThe moments are saturated, so changing the magnetic field hardly changes their alignment. This is a fixed-nonzero-field low-temperature limit. At exactly $B=0$, differentiating first gives the Curie expression for every $T>0$, which diverges as $T\to 0$; the two limits do not commute in this noninteracting model. For $J=0$ both moment and susceptibility vanish identically.

In the microcanonical ensemble, the microcanonical energy-shell entropy is the Boltzmann entropyFor a macroscopic system, changing the small energy resolution by a fixed factor changes $\log \Omega$ only by a nonextensive additive term, while $S=O(N{k}_B)$. Its relative effect therefore vanishes in the thermodynamic limit.

The thermodynamic temperature, pressure, and chemical potential are defined in the entropy representation byThusor, equivalently, the first law of thermodynamics in fundamental form is

The microcanonical ensemble fixes energy, volume and particle number; the canonical ensemble fixes temperature, volume and particle number; and the grand canonical ensemble fixes temperature, volume and chemical potential while allowing particle exchange. For macroscopic systems with short-range interactions, away from phase coexistence and other nonconcave regimes, relative fluctuations vanish in the thermodynamic limit and the ensembles give equivalent local thermodynamics.

The microcanonical ensemble describes an isolated system with fixed energy, volume, and particle number. It assigns equal probabilities to the accessible microstates in the chosen narrow energy shell. The canonical ensemble describes a system in thermal contact with a large heat reservoir: its temperature, volume, and particle number are fixed, while its energy can fluctuate.

For the Gibbs entropyfirst maximize subject only to ${\sum }_n{p}_n=1$ over $\Omega$ accessible microstates. A Lagrange multiplier givesso every $p_n$ is equal. Normalization yields the microcanonical ensemble

For the canonical ensemble, impose both normalization and the mean-energy constraint ${\sum }_n{p}_n{E}_n=E$. Stationarity ofgives $p_n=C{e}^{-\beta E_n}$. The canonical partition function fixes $C$, soThermodynamic consistency identifies $\beta =(k_{B}T{)}^{-1}$. This is the maximum-entropy derivation of equilibrium ensembles.

The microcanonical ensemble describes an isolated system at fixed energy $E$, volume $V$, and particle number $N$, assigning equal weight to accessible states in a narrow energy shell.

The canonical ensemble describes a system that can exchange energy with a heat reservoir at fixed temperature $T$, while $V$ and $N$ remain fixed. A microstate $s$ has probability

The grand canonical ensemble allows exchange of both energy and particles at fixed $T$, $V$, and chemical potential $\mu$. Its state weights are

The choice is irrelevant for bulk thermodynamic observables in the thermodynamic limit when the system is additive, has sufficiently short-range interactions, and lies away from singular phase-coexistence regimes. Then relative energy and particle-number fluctuations vanish as the system size grows. These are the standard conditions for equivalence of statistical ensembles.