Fluctuation theorem

When fluctuation theorem is a theorem from statistical physics, and indeed one of the few nowadays known exact relations, which are valid for any distance driven out of equilibrium system. The fluctuation theorem is the probability to entropieerzeugenden entropievernichtenden trajectories in relation to such a system. Although valid for any system, is a useful application of the fluctuation theorem only for small systems, as only a few micrometer-sized beads possible. Below you can a trajectory than the web of such a bead that is pulled by a liquid imagine.

By means of the force along the trajectory can be calculated by integration takes the work that is needed to pull the beads from the beginning to the end of the trajectory. Repeating this experiment drawing very often, we get a distribution of labor values ​​, for each trajectory a slightly different value. The figure at right shows such a distribution for 1000 trajectories. According to the second law of thermodynamics, the average work ( the Gibbs free energy here ) must be greater or equal to the change of the underlying thermodynamic potential:

The integral form of the fluctuation theorem (also called Jarzynski equation ) states

Wherein, with the temperature and Boltzmann's constant. The prerequisite for this is that the initial state is an equilibrium state, the final state can be driven arbitrarily far into the non-equilibrium.

The reason for this remarkable equality of exponential averaging is that it (ie, those in which we need to devote more work than the height of the potential barrier ) are also some entropievernichtende trajectories among many entropieerzeugenden trajectories. These come about by fluctuations of the surrounding medium. For the above example, this means that the Brownian motion of the spheres come across in the pulling direction, that aids in crossing the potential barrier. Such entropievernichtenden trajectories were sometimes referred to in the literature as " the 2nd Law hurtful ". This is something effektheischerisch because the 2nd law only applies to average values.

One form of the fluctuation theorem is the version of Crooks. Here, the work distribution of work values ​​is placed in relation to the distribution of time-reversed trajectories, ie those where start and end points are reversed. Here, too, must each be started in equilibrium, while the end point can be arbitrarily far in non-equilibrium. The Crooks fluctuation theorem then written as

Where the dissipative work is therefore the part of the total work, which is converted into heat when pulling. shows the distribution of the reciprocating trajectories of the return trajectories. Due to the close relationship between dissipative work and entropy, the Crooks Fluktuationstheoren can be written as:

The entropy of a single trajectory represents.

The fluctuation theorem should not be confused with the so-called fluctuation-dissipation theorem: Although the latter is even for large ensembles useful in many ways more general and also rigorously, but only with a linear response is valid.