← All parts of this equation

Equation 1 · Part 5 · Three Ways to Measure a System That Refuses to Settle Down

superscript

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

What this part means

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

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…

Read this part in the article →

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.

Open the illustrated exponents: repeated multiplication and powers guide →

The article lists its research sources here.