Jarza: Jarzynski Equality
Thermodynamics & Statistical Physics arc, page 47 of 193. Back to The Pixelated Physics Universe
Who they are
Jarza pulls a molecular rope again and again. Some pulls cost less than the free energy. Average them the right way and the answer comes out exact.
Jarza is a character of the Thermodynamics & Statistical Physics arc: drawn in pixels, ruled by one piece of real physics, and never allowed to break it.
The physics
Pulling a small system out of equilibrium, the work W varies from run to run. The Jarzynski equality says the exponential average of that work gives the equilibrium free-energy difference exactly. Individual pulls can come in below ΔF; on average W ≥ ΔF still holds. C. Jarzynski published it in 1997, G. Crooks generalised it in 1999, and Liphardt et al. tested it by pulling single RNA molecules (Science, 2002).
$$ \begin{gathered} \langle e^{-\beta W}\rangle = e^{-\beta\Delta F}, \qquad \beta = \frac{1}{k_B T} \\ \frac{P_F(W)}{P_R(-W)} = e^{\beta(W - \Delta F)} \end{gathered} $$
| Symbol | Meaning | SI unit |
|---|---|---|
| $W$ | work done on the system in one run | J |
| $\langle\cdots\rangle$ | average over many runs started from equilibrium | 1 |
| $\Delta F$ | equilibrium (Helmholtz) free-energy difference | J |
| $\beta$ | inverse thermal energy | 1/J |
| $k_B$ | Boltzmann constant, 1.380 649 × 10⁻²³ J/K (exact) | J/K |
| $T$ | bath temperature | K |
| $P_F(W)$ | probability density of work in forward runs | 1/J |
| $P_R(-W)$ | probability density of work −W in time-reversed runs | 1/J |
Honest caveat: It is not a second-law violation, and it needs the system to start in equilibrium.
Sources
- Jarzynski (1997) Nonequilibrium Equality for Free Energy Differences, Phys. Rev. Lett. 78, 2690
- Crooks (1999) Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences, Phys. Rev. E 60, 2721
- Liphardt et al. (2002) Equilibrium Information from Nonequilibrium Measurements in an Experimental Test of Jarzynski's Equality, Science 296, 1832
Crossovers
Jarza meets Erasa (Forgetting Costs Heat). Erasa's bound is an average. Jarza's equality says that individual runs can beat it, with work below the free-energy change, but the exponential average of e^(−W/k_BT) comes out exactly e^(−ΔF/k_BT).
Jarza meets Flukessa (Fluctuation Theorem Entropy). Jarza and Flukessa share a family: both are exact statements about fluctuations far from equilibrium. On average, the second law still holds for each of them.
Wear the lore
Concept keywords: Jarzynski Equality, Jarza, Thermodynamics & Statistical Physics, physics lore, Pixelated Physics