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Finite-mode non-interacting thermal ensembles: two fermions in three modes and two bosons in two modes

Physical scope

Two exact finite-mode thermal computations in the declared non-interacting model: the canonical partition sums, ground energies and occupation means of two spinless fermions in three modes with energies 0, epsilon, 2 epsilon, and of two spinless bosons in two modes with energies 0 and epsilon, at assigned temperature. The examples are model computations in declared finite-dimensional idealizations; they assert no physical particle species, no measured occupation and no thermodynamic limit.

Example

Adopt P1, P2 and P6 in the readings of The declared finite-mode model and its occupation readouts and Assigned sharp observables and the outcome law of the framework, work in the finite-mode realization of the declared fixed- $N$ model of The declared fixed-N many-body model: species sector, one-particle data, interaction and the sector energy observable, and adopt the canonical equilibrium postulate The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature at assigned temperature $T > 0$ with $β = 1 / (k_{B} T)$ . Fix $ε > 0$ in joules.

Fermionic model. One-particle space $K = C^{3}$ with modes $f_{1}, f_{2}, f_{3}$ , fermionic statistics $σ = -$ , $N = 2$ , zero interaction $W = 0$ , and the declared one-particle operator $h : = ε (0 \cdot T_{f_{1}, f_{1}} + 1 \cdot T_{f_{2}, f_{2}} + 2 \cdot T_{f_{3}, f_{3}})$ .

Bosonic model. One-particle space $K = C^{2}$ with modes $g_{1}, g_{2}$ , bosonic statistics $σ = +$ , $N = 2$ , zero interaction $W = 0$ , and $h^{'} : = ε (0 \cdot T_{g_{1}, g_{1}} + 1 \cdot T_{g_{2}, g_{2}})$ .

Then:

(a) in the fermionic model the antisymmetric sector has orthonormal eigenbasis $e_{{1, 2}}^{\land}, e_{{1, 3}}^{\land}, e_{{2, 3}}^{\land}$ with energies $ε, 2 ε, 3 ε$ ; the partition sum and the exact occupation means are $Z_{F} = e^{- β ε} + e^{- 2 β ε} + e^{- 3 β ε},$ $⟨ N_{1} ⟩_{β} = \frac{e^{- β ε} + e^{- 2 β ε}}{Z_{F}}, ⟨ N_{2} ⟩_{β} = \frac{e^{- β ε} + e^{- 3 β ε}}{Z_{F}}, ⟨ N_{3} ⟩_{β} = \frac{e^{- 2 β ε} + e^{- 3 β ε}}{Z_{F}},$ with $⟨ N_{1} + N_{2} + N_{3} ⟩_{β} = 2$ and ground energy $ε$ (configuration ${1, 2}$ );

(b) in the bosonic model the symmetric sector has orthogonal eigenbasis $m_{(2, 0)}, m_{(1, 1)}, m_{(0, 2)}$ with energies $0, ε, 2 ε$ ; the partition sum and the exact occupation means are $Z_{B} = 1 + e^{- β ε} + e^{- 2 β ε}, ⟨ N_{1} ⟩_{β} = \frac{2 + e^{- β ε}}{Z_{B}}, ⟨ N_{2} ⟩_{β} = \frac{e^{- β ε} + 2 e^{- 2 β ε}}{Z_{B}},$ with $⟨ N_{1} + N_{2} ⟩_{β} = 2$ and ground energy $0 = N ε_{1}$ ;

(c) scope: the computations are exact statements about the declared finite-mode models at the assigned temperature; they assert no physical species, no measured occupation, no thermalisation and no thermodynamic limit.

Verification

[L1] Diagonalisation theorem. In the non-interacting finite-mode model the labelled operator $\sum_{i} h^{(i)}$ is diagonal on the product basis with eigenvalue $\sum_{k} ε_{i_{k}}$ , the Slater vectors $e_{J}^{\land}$ and the symmetric vectors $m_{λ}$ are eigenvectors of the sector operator with eigenvalues $\sum_{j \in J} ε_{j}$ and $\sum_{j} λ_{j} ε_{j}$ , and the partition sum is the corresponding finite sum (Diagonalisation of the non-interacting finite-mode sector: occupation eigenvalues, partition sum and ground energy clauses (a)-(c)).

[L2] Occupation operators. The occupation operators $N_{j}$ act on the sector basis by $N_{j} e_{J}^{\land} = 1_{J} (j) e_{J}^{\land}$ and $N_{j} m_{λ} = λ_{j} m_{λ}$ , and the sector dimensions are three in both the bosonic and fermionic cases; the occupations are assigned sharp observables with finite means in every preparation (Slater and symmetric occupation-number bases of the finite-dimensional permutation sectors clauses (c), (d), Finite-mode occupation configurations: fermionic exclusion counts, bosonic counts and truncated dimensions clauses (a), (b), Mean, second moment and variance of an assigned observable in a preparation).

Proof technique: direct; list the occupation configurations, read their energies from the diagonalisation theorem, and compute the weighted means in the canonical weights.

Given: the declared fermionic and bosonic finite-mode models, the energies $0, ε, 2 ε$ , the assigned temperature data $β > 0$ , and [L1], [L2].

1.1L1L2algebra

[Fermionic configuration sum] The two-element subsets of ${1, 2, 3}$ are ${1, 2}, {1, 3}, {2, 3}$ with energies $0 + ε = ε$ , $0 + 2 ε = 2 ε$ , $ε + 2 ε = 3 ε$ by [L1]; the canonical weights are $q_{J} = e^{- β λ_{J}} / Z_{F}$ , and the occupation means are $\sum_{J} q_{J} 1_{J} (j)$ by [L2], giving the three displayed formulas. The total occupation is $\sum_{J} q_{J} \cdot 2 = 2$ because every configuration has two occupied modes, and the smallest energy is that of ${1, 2}$ , namely $ε$ , by [L1] clause (c). This proves (a).

1.2L1L2algebra

[Bosonic configuration sum] The occupation vectors of total two in two modes are $(2, 0), (1, 1), (0, 2)$ with energies $0, ε, 2 ε$ by [L1]; the canonical weights are $q_{λ} = e^{- β λ_{λ}} / Z_{B}$ and the occupation means are $\sum_{λ} q_{λ} λ_{j}$ by [L2], giving $⟨ N_{1} ⟩ = (2 \cdot 1 + 1 \cdot e^{- β ε} + 0) / Z_{B}$ and $⟨ N_{2} ⟩ = (0 + 1 \cdot e^{- β ε} + 2 \cdot e^{- 2 β ε}) / Z_{B}$ , and the total occupation is $2$ because every configuration has $\sum_{j} λ_{j} = 2$ ; the ground energy is $0 = N ε_{1}$ by [L1] clause (c). This proves (b).

2.1L1L2step 1.1step 1.2algebra∎

[Checks and scope] In both models $Z$ is a finite sum of positive terms, hence $0 < Z < \infty$ ; the displayed occupation sums equal the particle number $N = 2$ identically in $β$ by steps 1.1-1.2, and the ground energies agree with the general formulas of [L1] clause (c) for the sorted energy lists $(0, ε, 2 ε)$ and $(0, ε)$ . Every quantity is a spectral datum of the declared finite-dimensional model; no physical species, measured occupation, thermalisation or thermodynamic limit entered, which is (c).

Remarks

Dimension counts. The fermionic sector is three-dimensional by exclusion and the bosonic sector is three-dimensional by the configuration count; the two examples are therefore directly comparable finite checks of the diagonalisation theorem at different statistics.
Exactness. The formulas are exact rational functions of $e^{- β ε}$ ; no expansion in $ε / (k_{B} T)$ is used, and the dilute-limit approximations are treated separately on this page.

Finite-mode non-interacting thermal ensembles: two fermions in three modes and two bosons in two modes

Example

Verification

Remarks

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