Fluctuation theorem

This article describes the behavior of middle work in relation to free energy. The fluctuation theorem should not be confused with the so-called fluctuation-dissipation theorem : the latter is useful even for large ensembles, in various respects somewhat more general and also rigorous, but only applies to linear response behavior.

The fluctuation theorem is a theorem from statistical physics , namely one of the few exact relations known today that are valid for systems that are driven out of equilibrium at will . For such a system, the fluctuation theorem relates the probability of entropy- generating trajectories to entropy-destroying trajectories . This is particularly important for microscopic systems.

In macroscopic systems, on the other hand, typical trajectories correspond to large changes in entropy, and the theorem provides a vanishingly small probability for entropy-destroying trajectories (in accordance with the 2nd law of thermodynamics ).

Non-equilibrium systems

Work distribution of 1000 trajectories.

The subject of the fluctuation theorem is systems in contact with a heat bath, which leave the equilibrium state by changing parameters. The parameters change in the same way each time the experiment is repeated. a. the work to be done . ${\displaystyle W}$

A good example is a micrometer-sized bead that is pulled through a liquid by a force field . The force field is switched on at a given point in time and acts over a time interval of a given duration, i.e. H. the parameter here is the strength of the force field. The work to be done results as the integral of the force along the trajectory. If the experiment is repeated, more or less different values ​​result due to thermal fluctuations - the result is a distribution of work values. The figure on the right shows such a distribution for 1000 trajectories. In addition to many entropy-generating trajectories (i.e. those in which friction work has to be applied) there are also some entropy-destroying trajectories. These are caused by fluctuations in the surrounding medium. For the example this means that the Brownian molecular motion pushes the bead randomly in the pulling direction. Such entropy-destroying trajectories were sometimes referred to in the literature as "violating the 2nd law". This is wrong because the 2nd law only applies to mean values. ${\displaystyle F\left({\textbf {x}}\right)}$

Crooks fluctuation theorem

Crooks' theorem links the probability of a trajectory to the probability of the time-reversed trajectory (with the start and end point reversed). It must always be started in equilibrium, while the end point can be as far as desired in non-equilibrium. The Crooks fluctuation theorem reads ${\displaystyle P_{F}(W)}$ ${\displaystyle P_{R}(-W)}$

${\displaystyle {\frac {P_{F}(W)}{P_{R}(-W)}}=e^{\beta W_{diss}}=e^{\Delta S}}$

where is the dissipative work, i.e. the part of the total work that is converted into heat while changing the parameters. The symbol stands for , with the temperature and the Boltzmann constant . ${\displaystyle W_{diss}=W-\Delta F}$ ${\displaystyle \beta }$ ${\displaystyle \beta =1/k_{B}T}$ ${\displaystyle T}$ ${\displaystyle k_{B}}$

The second equal sign is trivially based on the proportionality of dissipative work and entropy production . ${\displaystyle \Delta S}$

The Jarzynski equation is obtained with the help of the integral

${\displaystyle \int dWP_{F}(W)e^{-\beta W}=\langle e^{-\beta W}\rangle .}$

Jarzynski equation

The Jarzynski equation follows from the canonical probability distribution without further assumptions, but can also be derived from the Crooks fluctuation theorem. According to the 2nd law of thermodynamics , the mean work must be greater than or equal to the change in the underlying thermodynamic potential (here the free energy ): ${\displaystyle F}$

${\displaystyle \Delta F\leq \langle W\rangle }$

The Jarzynski equation says beyond that

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

The prerequisite here is that the initial state is a state of equilibrium, the end state can be driven into non-equilibrium as far as desired.

Individual evidence

1. ^ Gavin E. Crooks: Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences . In: Physical Review E . 60, 1999, pp. 2721-2726. doi : 10.1103 / PhysRevE.60.2721 .
2. ↑ 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 .