## 2 Concepts and Theory  
### Principles of FCS  
The conceptual basis and theory of FCS are based on the seminal papers by [PERSON] and [PERSON] published in 1974 [11; 13]. The method is based on measuring the rate of decay of spontaneous concentration fluctuations of fluorophores present in a confined volume generating data conveying information at first hand that relates to the molecular diffusion but potentially also concerning chemical reaction kinetics. The theory is based on the fact that even at equilibrium the local concentration of molecules will constantly fluctuate due to thermodynamic processes. As stated in the original paper [13], \"Observation of these spontaneous fluctuations can yield a complete kinetic description of the system equivalent to that obtained by conventional means\". This means that these microscopically defined spontaneous fluctuations at equilibrium can be interpreted using the same type of theory as applied to nonequilibrium systems at the macroscopic scale. The trick is to study the fluctuations in a small and well-defined volume containing a limited number of molecules rather than the complete sample. In the early FCS experiments [11; 13], the volume was defined by focusing an Argon laser into a cuvette with a fluorescent solution resulting in an elongated detection volume as illustrated in Figure 1a. With the introduction of confocal optics using high NA objective lenses and pinhole detection the detection volume could be substantially reduced matching the point-spread-function (PSF) of the optical system [14] (Figure 1b). Since the detection volume when applying FCS in combination with MPM is corresponding to the ellipsoid shape of excitation volume, being related to PSF, this conceptual view will be used throughout this paper rather than the elongated case as discussed by [PERSON] and [PERSON]. Even so, the fundamental concepts and theory will be the same. According to [PERSON] and [PERSON] [13], the local concentration of a chemical compound at a specific location (**r**) and time (**t**) in the sample can be defined as \(\mathrm{C}(\mathbf{r},\mathbf{t})\). For simplicity, we limit the case to diffusion of one single chemical component in this guideline, why the indices \(j\) and \(l\) are omitted. The thermodynamic mean concentration is given by the macroscopic equilibrium concentration, which is independent of position and time, so that \(\langle\mathrm{C}(\mathbf{r},\mathbf{t})\rangle=\overline{\mathrm{C}}\), where \(\langle\,\rangle\) denotes the ensemble average. This means that the local fluctuation, i.e., the deviation from equilibrium concentration is defined as \(\mathrm{\delta C}(\mathbf{r},\mathbf{t})=\mathrm{C}(\mathbf{r},\mathbf{t})- \overline{\mathrm{C}}\), and the concentration correlation function at location \(\mathbf{r}\) can simplified be expressed as:  
\[\Phi(\tau)=\langle\mathrm{\delta C}(\mathbf{t})\mathrm{\delta C}(\mathbf{t}+ \tau)\rangle \tag{1}\]  
where the variable \(\tau\), also called the lag-time, denotes a time different from the absolute time t. This equation has three mathematical implications:  
1. The concentration correlation is independent on absolute time t and will only depend on the lag-time \(\tau\). 2. At short time intervals, typically orders of magnitude shorter than the diffusion time of the molecules, the value of the correlation function will approach the equilibrium state, i.e., \(\phi(\tau)\rightarrow\phi(0)\)
3.