PhysicsLogic / Jarzynski 1997 Sampling

Work Variance and Sampling Reliability

An exact equilibrium identity and small work variance can coexist with a failing finite estimator.

Definitions and assumptions

Scope and physical limitations

This independent two-state sudden quench uses beta = 1, with energies in k_B T units. The initial excited-state energy is log(999999); the protocol lowers it by 20. Every state retains positive canonical weight. The work variance is below 1/1000, yet for each independent sample count from 1 through 1000 the specified estimator fails with probability at least 999/1000. This is about the exponential-work estimator and the stated budget, not all estimators, impossibility at every finite count, or a contradiction of the source. No new production definitions are introduced.

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Examples and distinctions

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

Exact root statement

0 < ProbabilityTheory.variance work law ∧
  ProbabilityTheory.variance work law < 1 / 1000 ∧
    referenceMean = Real.exp (-freeEnergy) ∧
      ∀ (count : ℕ),
        0 < count →
          count ≤ 1000 →
            999 / 1000 ≤
              (sampleLaw count)
                {sample | sample ∈ failureEvent count ∧ Real.log 2 < |sampleFreeEnergy sample - freeEnergy|}
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