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Equation 1 · Part 3 · Three Ways to Measure a System That Refuses to Settle Down

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

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

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 equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

Open the illustrated equality: what the equals sign claims guide →

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