Jarzynski’s equality

Searching for relations between different physical processes is one of the most successful ways of doing science. Usually these relations are not only surprising but also useful as they offer us a different approach to certain problems. In statistical mechanics such a relation was showed by Christopher Jarzynski [1] who found an equality connecting equilibrium information with non-equilibrium measurements for finite classical systems.

From thermodynamics we know that in the case of quasi-static or, in other words, processes which take an infinitely long time the work done on the system is equal to the difference of Helmholtz free energy of the system. However, for finite time processes some proportion of the work can be dissipated and so in general the total work may surpass the difference of Helmholtz free energy.  Jarzynski’s result shows it still possible to find the value of that difference from finite time measurements which actually makes the whole thing accessible experimentally [2].

To give you some taste how it is done, Jarzynski simply uses the special method of averaging the exponential function of work. This approach provides better weights for different work trajectories to get at the end the mean value of the exponent of dissipated work equaled to one.  It occurs that from Jarzynski’s equality it is easy to derive the fluctuation-dissipation relation as well as the already mentioned inequality between average work and the difference of the system’s free energy.

One interesting question recently asked by researchers was whether there are quantum analogues of this result.  Reported positive answers only prove the general importance of Jarzynski’s equality.

[1] C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997).

[2] J. Liphard et al., Science 296, 1832 (2002).