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Equation 1 · Part 2 · Three Ways to Measure a System That Refuses to Settle Down

Symbol e^-β Δ F

⟨e−βW⟩=e−βΔF\left\langle e^{-\beta W} \right\rangle = e^{-\beta \Delta F}
e−βΔFe^{-\beta \Delta F}

What this part means

e−e^-β Δ F is one of the signed contributions combined to compute the quantity on the left.

Its job in the formula

e−e^-β Δ F is one of the signed contributions combined to compute the quantity on the left.

The passage around this formula

Because these three frameworks share vocabulary, it is worth being explicit about the one place where a formal expression clarifies rather than decorates the comparison: the Jarzynski equality itself. Written compactly, it states ⟨e−βW⟩=e−βΔF\left\langle e^{-\beta W} \right\rangle = e^{-\beta \Delta F}. where W is the work performed on the system along a single nonequilibrium trajectory, Δ\Delta F is the equilibrium free-energy difference between the initial and final states, and β\beta is inverse temperature. The average on the left is taken over repeated realizations of the same driving protocol, each of which can dissipate a different amount of work due to thermal fluctuations. The identity holds exactly regardless of how fast or far from…

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Learn the underlying idea

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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