The Fluctuation Theorems (FTs) of Evans & Searles and of Crooks are fundamental theorems of modern thermodynamics that have been suggested to be of practical use to scientists and engineers. Non-equilibrium processes with energy fluctuations on the order of thermal energy, κBT, are described by the FTs; examples include the stretching of a DNA molecule, the localisation of a colloidal particle in an optical trap of changing strength, and translation of an optically trapped colloidal particle. If the path or process is traversed over long times or the system is sufficiently large that it can be considered in the classical, thermodynamic limit, then, in principle, there is only one value of the energy characterising the path. However, for small systems, there exists a distribution of energy values and this distribution is associated with non-equilibrium fluctuations of the system that do not average out over short time. The FT of Evans & Searles, as well as the FT of Crooks (from which the Jarzynski relation is derived), describe the symmetry of this energy distribution about zero. This distribution is inherent to the dynamics of small systems, such as nano-machines and single molecular motors.
In this paper we present the FTs in a single unified language, considering that the work done on the system is either purely dissipative, achieves a change in thermodynamic state of the system, or a combination of these. We demonstrate this with a single colloidal particle in an optical trap and a single DNA molecule stretched in an OT experiment.