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Two independent two-level sites give a product thermal law

Physical scope

A calculated two-site finite product model with additive energy and no interaction; it is not an observation or a claim about interacting sites.

Example

Let each site have states $0, 1$ with energies $0, ε$ , and take $T > 0$ with $ε = k_{B} T \log 2$ . The composite has states $(n_{1}, n_{2}) \in {0, 1}^{2}$ and additive energy $E = ε (n_{1} + n_{2})$ .

Facts & Assumptions

Given: Two finite sites, additive energy, common positive temperature, and canonical sampling.

[F1]

Finite statistical microstate model: the composite state set is the Cartesian product and additivity of energy is explicit model data.

[F2]

Finite canonical partition, energy, variance, entropy and free-energy identities: each finite canonical law has probabilities proportional to $e^{- β E}$ .

[F3]

Partition, Gibbs-entropy and free-energy factorization for independent finite systems: under the explicit product and additive-energy assumptions, partition sums factor and the joint canonical law is the product of the two subsystem laws.

[A1]

Set $β ε = \log 2$ , hence the single-site excited-state weight is $q = 1 / 2$ .

Verification

Given: The model data and $q = 1 / 2$ .

1.1F1F2F3A1algebra

Each site has $Z_{1} = 1 + q = 3 / 2$ and probabilities $(2 / 3, 1 / 3)$ . By [F3], $Z_{12} = Z_{1}^{2} = 9 / 4$ and the joint probabilities of $(00, 01, 10, 11)$ are $(4 / 9, 2 / 9, 2 / 9, 1 / 9)$ .

2.1F2F3step 1.1algebra∎

The expected total occupation is the sum of the two marginal means, $1 / 3 + 1 / 3 = 2 / 3$ , so the mean energy is $2 ε / 3$ . This conclusion follows for the declared noninteracting product model only.

Two independent two-level sites give a product thermal law

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources