## Jarzynksi on Non-Equilibrium Statistical Mechanics

Here at the Santa Fe Institute we’re having a workshop on Statistical Physics, Information Processing and Biology. Unfortunately the talks are not being videotaped, so it’s up to me to spread the news of what’s going on here.

Christopher Jarzynski is famous for discovering the Jarzynski equality. It says

$\displaystyle{ e^ { -\Delta F / k T} = \langle e^{ -W/kT } \rangle }$

where $k$ is Boltzmann’s consstant and $T$ is the temperature of a system that’s in equilibrium before some work is done on it. $\Delta F$ is the change in free energy, $W$ is the amount of work, and the angle brackets represent an average over the possible options for what takes place—this sort of process is typically nondeterministic.

We’ve seen a good quick explanation of this equation here on Azimuth:

• Eric Downes, Crooks’ Fluctuation Theorem, Azimuth, 30 April 2011.

We’ve also gotten a proof, where it was called the ‘integral fluctuation theorem’:

• Matteo Smerlak, The mathematical origin of irreversibility, Azimuth, 8 October 2012.

It’s a fundamental result in nonequilibrium statistical mechanics—a subject where inequalities are so common that this equation is called an ‘equality’.

Two days ago, Jarzynski gave an incredibly clear hour-long tutorial on this subject, starting with the basics of thermodynamics and zipping forward to modern work. With his permission, you can see the slides here:

• Christopher Jarzynski, A brief introduction to the delights of non-equilibrium statistical physics.

Also try this review article:

• Christopher Jarzynski, Equalities and inequalities: irreversibility and the Second Law of thermodynamics at the nanoscale, Séminaire Poincaré XV Le Temps (2010), 77–102.

### 12 Responses to Jarzynksi on Non-Equilibrium Statistical Mechanics

1. DoubleDuce says:

Great post. Have you seen this:

• Amanda Morris, Reconfiguring active particles into dynamic patterns, Phys.org, 11 July 2016.

and

• Natalie Wolchover, A new physics theory of life, Quanta, 22 January 2014.

2. David Lyon says:

When I first encountered the Jarzynski equality and the fluctuation theorem, my first thought was “wow, these are amazing, why didn’t I learn them as an undergraduate?” and immediately following that was “wait, these were only discovered in the 1990s? Why was there a 63 year break in statistical mechanics progress between Nyquist & Onsager in 1928-1931 and Denis Evans & Christopher Jarzynski in 1994-1997?”

• John Baez says:

The lesson, I think, is that there’s a lot left to learn—and it’s wise to identify the regions where basic results haven’t been found yet!

• tomate says:

Because there wasn’t.

• John Baez says:

Yes, there certainly wasn’t a “63 year break in statistical mechanics progress”. A lot occurred in that interval! I’ll just mention two earth-shakers:

1) The renormalization group analysis of second-order phase transitions, linking these phase transitions to quantum field theory.

2) Prigogine’s work on minimal entropy production for steady states near equilibrium, and his work on structure formation far from equilibrium.

3. domenico says:

It is interesting workshop.
I am reading the Christopher Jarzynkski slides, and I am thinking that the basic process in thermodynamic is a piston moving in a heat engine, so that if someone simulate with a software a moving piston in a nanoscale engine (with some million of equal particles) using the collision of the molecules with an arbitrary velocity and an arbitrary acceleration of the piston (for a nonequilibrium state), then it could be possible to write an exact statistical law for this system using different initial conditions, and it could be possible to write the statistical law for a cycle based on the dynamic of the piston (a law from the parameter of the trajectory of the piston).

• John Baez says:

It sounds like maybe you’re talking about ‘stochastic thermodynamics’

4. If Bob Dylan can’t accept the Nobel, maybe he can have Joan Baez accept it for him? Makes a little sense…

• John Baez says:

Maybe Joan can accept the task and delegate it to her cousin. At this point, that’s my best shot at getting my hands on a Nobel.

5. In Jarzynski’s context there is a description of the probability for a forward process relative to the probability of the reverse process (Eq. 31 in his Seminaire Poincare XV paper). That should have an immediate translation to your and Blake’s quantropy formalism. Right?

(Check spelling of Jarzynski in the title.)

6. […] John Baez’s excellent coverage of Jarzynksi. […]