## 3 Analytical Methods  
### Statistical Representation of Voltage/Current Characteristics  
In order to observe the electrical behavior of the roller-rail contact, we used a statistical representation of current/voltage values proposed in previous studies [18]. This approach makes it possible to deduce the quality of the contact from the observation of the voltage-current characteristics (linearity, dominant resistance, maximum voltage reached, etc.). For this purpose, synchronized time measurements of (\(V_{contact}\)(t), \(V_{s}\)(t), and \(U_{position}\)(t)) are captured and organized into three-column matrices with \(n\) rows, where each row corresponds to a combination of (\(U_{contact}\)(V), \(I_{contact}\)(A), and \(position\)(cm)). The next step is to use the signal of position sensor to isolate the time segments directly linked to the movement of the roller in the zone of interest on the rail. These segments are then processed to exclude saturated periods when the voltage reaches the compliance of the current source (Keithley 6221 DC/AC Current Source, equipment from Tektronix, Les Ulis, France), which is 10 V as illustrated in Figure 4.  
The data is then used to create statistical maps illustrating the current-voltage relationship. The method consists in discretizing the contact voltage and current signals with steps \(\Delta U_{c}\) and \(\Delta I\) and estimating the density of occurrence of the different pairs \(\left(U_{cp},\ I_{q}\right)\). Figure 5 shows an example of this representation: the horizontal axes represent the two parameters \(U_{c}\) and \(I\), and the vertical axis \(z\) represents the density of occurrence, with a color scale where red represents the highest density. This density is expressed in logarithmic scale and corresponds to the number of points obtained in each discretization cell of dimensions \(\Delta I\cdot\Delta U_{c}\). An adjustment of +1 is applied to all occurrence densities to avoid values of zero (for which the logarithm would not be defined). Figure 3: Photographs of studied rail surfaces. (**a**) ‘Clean’ surfaces: with a microscopic oxide layer (left) and freshly stripped (right). (**b**) ‘Lightly oxidized surfaces’: three zones with different oxide thicknesses. For tests on oxidized rail, the current and voltage ranges were set at \(I_{min}=-0.11\) A and \(I_{max}=0.11\) A for current, and \(U_{min}=-9.8\) V and \(U_{max}=9.8\) V for voltage. This selection reflects the maximum current injected during testing, set at 0.1 A. For clean rail testing, the same current ranges were applied, while the voltage limits were \(U_{min}=-1\) V and \(U_{max}=1\) V, which include extreme voltage values reached during clean rail testing. These limit values were used to discretize the current-voltage space and to normalize the mappings. For fine resolution of the characteristics \(U-I\), the current-voltage space has been segmented into 401 \(\times\) 401 zones. The discretization is adapted according to the condition of the rail analyzed: for oxidized rails, the dimensions are 48.9 mV- 0.54 mA; whereas, for clean rails, they are reduced to 4.98 mV-0.54 mA. In fact, this differentiation offers greater finesse in the resolution of low voltages on clean rails. The probability density for each zone was then normalized by the total number of measurements. Figure 4: Example of a chronogram showing possible strong distortions in the current signal (-) associated with saturation of the contact voltage (-), reaching the compliance of the current source. Figure 5: Example of a statistical representation of a set of voltage–current characteristics. Areas of discretization are shown on the graph by gray lines. One can notice that the \(U-I\) characteristic is nonlinear. As outlined in [1], at the outset of a current cycle, the contact resistance is elevated due to the presence of the oxide film at the interface. As the current flows, localized Joule heating raises the temperature at the asperities [24], which in turn modifies the oxide layer and lowers its electrical resistivity [25].