Jarzynski equation

The Jarzynski equation is an equation of statistical mechanics . It links the difference in free energy with the work that is done on a system in non-equilibrium processes. ${\displaystyle F}$ ${\displaystyle W}$

In general, the inequality applies to deterministic thermodynamic non-equilibrium processes between the equilibrium states A and B.

${\displaystyle \Delta F=F_{\mathrm {B} }-F_{\mathrm {A} }\leq \langle W\rangle .}$

The Jarzynski equation

${\displaystyle e^{-\beta \Delta F}=\left\langle e^{-\beta W}\right\rangle }$

applies to any process regardless of the speed of process control, with

• of energy normalization ${\displaystyle \beta ={\frac {1}{k_{\mathrm {B} }T}}}$
• the Boltzmann constant ${\displaystyle k_{\mathrm {B} }}$
• the absolute temperature ${\displaystyle T}$
• the classic ensemble mean . ${\displaystyle \langle \cdots \rangle }$

It follows from this that there must be processes that occur with a probability of occurrence greater than zero, for which the following applies:

${\displaystyle \Delta F=F_{\mathrm {B} }-F_{\mathrm {A} }>W.}$

The Jarzynski equation becomes more plausible when one considers the thermal fluctuations in the ensemble on which work is being done. Sometimes they bring the system further from the initial to the final state. This rarely occurs, but is heavily influenced by the negative exponent in the ensemble mean.

literature

• C. Jarzynski: Nonequilibrium Equality for Free Energy Differences . In: Physical Review Letters . tape 78 , no. 14 , March 7, 1997, p. 2690 , doi : 10.1103 / PhysRevLett.78.2690 .