# What does the average mean in Jarzynski equality?

Jarzynski equality: $$\left<\exp(-\beta W)\right> = \exp(-\beta \Delta F)$$ In Jarzynski's paper on Physics Review Letter on arxiv, he just said canonical ensemble average, but he mean which system's canonical ensemble? And the Jarzynski equality is confirmed by experiment, how do we comprehend ensemble distribution in real life?

## 1 Answer

You can think of it as an average over trials.

On each trial, you do some work on the system by adjusting the external parameters. Each time you use the same protocol for changing the parameters. The amount of work will vary from trial to trial because of the probability distribution over initial states and the stochastic interactions with the heat bath (at least in later versions of the derivation).

This results in a probability distribution over $$W$$. You can average over that distribution, or equivalently average over an infinite number of trials. This is the average in the equality.

• Does the trails just have the same parameter change or the same parameter change process? ( that is, just $\lambda: A\to B$ or with the same $\lambda(t)$ function) Mar 30, 2019 at 2:49
• The same function Mar 30, 2019 at 15:26