PhysicsLogic / Callen-Welton 1951

Thermal Noise and Incomplete Probes

Identical noise and response can conceal a non-Gibbs state of the same Hamiltonian.

Definitions and assumptions

Scope and physical limitations

The independent four-level system has energies i log 2 and beta = hbar = 1. Two strictly positive diagonal states give identical records for one Hermitian transition probe, although the alternative is not Gibbs at any real inverse temperature. Even all three separately resolved probes miss a positive state's off-diagonal coherence: their autocorrelations and retarded commutator records match Gibbs for every time, but a cross correlation distinguishes the states and free evolution changes the coherent state. Commutation with this nondegenerate Hamiltonian removes every off-diagonal entry; normalization and all adjacent balances then determine the Gibbs density. A separate direct decoder uses calibrated noise at zero and response at pi/(2 log 2) for probes 0 and 2. These four entries determine exactly the diagonal without an assumed balance law; with commutation they identify the Gibbs matrix. Equal-time records alone fail even for stationary states. These are ideal exact records, not full quantum-state tomography, finite measured estimates, bath-induced equilibration, a continuum Fourier density, or Callen-Welton's force-noise and impedance calculation.

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Examples and distinctions

Constructed examples of this model; each answer includes the Lean statement.

Exact root statement

¬recordAccess.RecordConstrainsValueAt () (recordAccess.recordAt () gibbs) fun population =>
    population = ensemble.density
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