Thermodynamic Uncertainty Relation

Precision costs entropy production: relative fluctuations × entropy rate ≥ 2kB — a fundamental tradeoff.

k₊ = 3.0
k₋ = 1.0
f = 0.50
Q = J²/Dσ = —  ≥  2.00 (TUR bound)
The Thermodynamic Uncertainty Relation (Barato & Seifert 2015): Q = J²/(D·σ̇) ≥ 2.
Here J = mean current, D = diffusivity, σ̇ = entropy production rate.
For a driven two-state motor with rates k₊e^{f/2} and k₋e^{-f/2}:
J = (k₊e^{f/2} − k₋e^{-f/2}) · π_ss, and Q saturates the bound in the linear regime. Motor positions shown above; plot shows Q vs driving force f (gold) vs bound 2 (blue).