## 2 Results  
### Optical Forces  
In order to analyze particle dynamics, we have solved the overdamped [PERSON] equation [15]:  
\[\vec{F}(\vec{R}(t))-\gamma\frac{d\vec{R}}{dt}+\vec{\mathcal{F}}(t)=0 \tag{1}\]where \(\vec{F}(\vec{R}(t))\) is the optical force, \(\vec{R}(t)\) is the position vector of the particle at time \(t\), \(-\gamma\dfrac{d\vec{R}}{dt}\) is the frictional force of a particle, and \(\vec{\mathcal{F}}(t)\) is a random function force with time that we have assumed null (the Brownian effect is not taken into account). Coefficient \(\gamma=6\pi\eta_{\
u}a\), where \(\eta_{\
u}\) is the viscosity of the medium (water in our case) and parameter \(a\) is the particle radius. We consider small chiral particles with radius \(a\) much lower than the wavelength of the electromagnetic field \(\vec{E}\), \(\vec{H}\), inside a dielectric medium of refractive index \(n\). The optical force on the chiral sphere (CS) can be deduced by using the dipole approximation for which CS is modeled as an induced magnetic (\(\vec{m}\)) and electric dipole (\(\vec{p}\)) given by [16]:  
\[\vec{m}=4\pi(-i\dfrac{\alpha_{em}}{\eta}\vec{E}+\alpha_{mm}\vec{H})\] \[\vec{p}=4\pi\epsilon_{0}(\alpha_{ee}\vec{E}+i\,\alpha_{em}\eta \vec{H}) \tag{2}\]  
where \(\vec{H}\) and \(\vec{E}\) are the magnetic and electric fields, \(\alpha_{ee}\), \(\alpha_{mm}\), and \(\alpha_{em}\) are the electric, magnetic, and chiral polarizability, respectively, \(\eta\) is the vacuum impedance, and \(\epsilon_{0}\) is the permittivity of free space. In the case of Rayleigh particles with a small chirality parameter \(\delta\) (it is assumed that the refractive index of the chiral particle is given by \(n_{LR}=np\pm\delta\), where \(n_{L}=np+\delta\) defines the refractive index for the left-handed enantiomer and \(n_{R}=np+\delta\) for the right-handed one), the electric, magnetic, and chiral polarizability can be written (in absence of absorption) as [3,16]:  
\[\alpha_{ee}=n^{2}\,a^{3}\,\dfrac{n_{p}^{2}-n^{2}}{n_{p}^{2}+2n^{2}}\] \[\alpha_{mm}=0 \tag{3}\] \[\alpha_{em}=-a^{3}\,\dfrac{(\pm\,\delta\,n^{2})}{n_{p}^{2}+2n^{2}}\]  
These expressions have been obtained by taking into account that the chirality parameter \(|\delta|<<1\), which implies that in the dipolar limit, all magnetic parts (\(\alpha_{mm}\) is proportional to \(\delta^{2}\)) of force can be disregarded [2]. The electric energy density (\(U_{e}\)) and the helicity density (\(U_{h}\)) can be defined as:  
\[U_{e}=\dfrac{\epsilon_{0}\epsilon_{r}}{4}|\vec{E}|^{2} \tag{4}\] \[U_{h}=\dfrac{\epsilon_{0}\mu_{0}}{2\,k_{0}}\Im(\vec{E}.\vec{H}^ {*}) \tag{5}\]  
where \(\Im\) and \(*\) denote the imaginary part and the complex conjugate respectively, \(\epsilon_{r}\) is the relative permittivity of the medium in which the particle is immersed, and \(k_{0}\) is the wavenumber. Due to the electric density energy \(U_{e}\) and helicity density \(U_{h}\), the particle is subject to a potential energy W given by:  
\[W=\pi\epsilon_{0}\alpha_{ee}|\vec{E}|^{2}-2\pi\epsilon_{0}\eta\alpha_{em}\Im (\vec{E}.\vec{H}^{*}) \tag{6}\]  
In the Rayleigh regime, the optical force \(\vec{F}(\vec{R}(t))\), described by Equation (1), exerted upon the chiral spheres by the electromagnetic field, can be approximated to a gradient force [4,17]:  
\[\vec{F}(\vec{R}(t))=\
abla W \tag{7}\]