1.4 Thermodynamics of small systems

The thermodynamics of small systems (also known as mesoscopic dynamics) is a subtopic of physics that deals with systems that have an intermediate scale between the microscopic and the macroscopic systems [3]. At this intermediate scale, the typical number of microscopic components of the system is much larger than 1 but much smaller than Avogadro's number. A characteristic property of these systems is that the energy exchanged between the system and the environment is of the same order of magnitude than the energy fluctuations. As a result, the energy fluctuations of the system contain valuable and meaningful information.

According to the classical statistical mechanics [52], the relative fluctuations of observable magnitudes decrease as $ \sim1/\sqrt{N}$, where $ N$ is the number of particles of the system. Traditionally, the fluctuations of macroscopic thermodynamic systems (gas, magnet) have been hardly observed (except for the critical points), due to the large number of particles involved in these systems ( $ N\sim10^{23}$). However, the boom of single-molecule techniques have provided a new collection of experiments that exhibit significant fluctuations. Indeed, the biomolecules used in these experiments have few degrees of freedom (thousands of atoms). This has been very useful for the physicists interested in the equilibrium and non-equilibrium thermodynamics of small systems. In fact, some non-equilibrium theorems have been firstly tested on single-molecule experiments [53,54].

Theorists devoted to the thermodynamics of small systems have focused their attention on the Fluctuation Theorems (FT). The FT relate the equilibrium properties of a system with the work performed on this system along irreversible processes. This was initially stated by Jarzynski [5] and generalized by Crooks [4]. The FT allowed to establish generic results (i.e., non-system dependent) in systems out of equilibrium. Nowadays, the application of FT in biophysics is quite frequent in order to obtain the free energies of formation of biomolecules. The experiments with biomolecules have also contributed to expand the FT. In particular, FT have been extended to include the partial equilibrium, which allows to infer the free energy of formation of different conformational states of biomolecules [55].

One of the more interesting topics in the thermodynamics of small systems are molecular motors of the cell. These systems are strongly affected by the environment in which they perform their functions (transportation, polymerization). In fact, thermal fluctuations dominate the behavior of such motors. In spite of the fluctuations, these motors are still extremely efficient. How can these systems work properly embedded in such wild conditions? The single-molecule micro-manipulation techniques shed light into this question [56].

Figure 1.4: Artistic view of a kinesin at work. Kinesin (in red) is a molecular motor that carries vesicles of substances (yellow) by walking along the microtubules (violet) located within the cell.
\includegraphics[width=10cm]{figs/chapter1/kinesin.eps}

JM Huguet 2014-02-12