## 2 Materials and Methods  
### Physical Model  
If we consider a spherical magnetic particle with radius \(R\), mass density \(\rho_{\rm P}\), and norm of magnetic moment \(\mu_{\rm P}\), initially perpendicularly rotated to an external homogeneous magnetic field with flux density \(\vec{B}_{0}\equiv B_{0}\vec{i}\), i.e., \(\vec{\mu}_{\rm P}(0)\equiv\mu_{\rm P}\vec{j}\), located in a viscous ambient fluid with mass density \(\rho\), dynamic viscosity \(\eta\), and kinematic viscosity \(\
u\equiv\eta/\rho\), then the rotational movement of the particle (alignment to the direction of the external magnetic field) can be described with a system of differential equations (DEs):  
\[\vec{\vec{I}}_{\rm P}\cdot\frac{\mathrm{d}\vec{\omega}(t)}{\mathrm{d}t}=\vec{ \mu}_{\rm P}\times\vec{B}_{0}+\vec{\vec{T}}_{\rm v}(t)+\vec{\vec{T}}_{\rm B}(t), \tag{1}\]  
where \(\vec{\vec{I}}_{\rm p}\) is the moment of inertia tensor of the spherical particle, which for simple rotation about the particle axis reduces to the scalar \(I_{\rm P}=\frac{2}{8}m_{\rm P}R^{2}\) and \(\vec{\omega}\equiv\omega\vec{k}\) is the angular velocity of the rotational movement of the spherical particle. Unit vectors \(\vec{i}\), \(\vec{j}\), and \(\vec{k}\) form the basis of a Cartesian coordinate system. \(m_{\rm P}=V_{\rm P}\rho_{\rm P}\) is the weight of the magnetic particle and \(V_{\rm P}\) is its volume. The first term on the right-hand side of Equation (1) is the magnetic torque exerted by the external homogeneous magnetic field \(\vec{B}_{0}\) on the particle with magnetic moment \(\vec{\mu}_{\rm P}=\mu_{\rm P}(\cos\varphi\vec{i}-\sin\varphi\vec{j})\). The second term in Equation (1) is the viscous torque of the viscous ambient fluid on the rotating spherical particle. The final term in Equation (1) is the random Brownian torque exerted on the spherical particle. Due to the axial symmetry of the rotational movement of the spherical particle (\(\vec{\omega}\parallel\vec{k}\)), the system of DEs (1) becomes a scalar differential equation:  
\[\underbrace{I_{\rm P}\frac{\mathrm{d}\omega(t)}{\mathrm{d}t}}_{\rm inertial}= \underbrace{\mu_{\rm P}B_{0}\sin\varphi}_{\rm magnetic}+\underbrace{T_{\rm v }(t)}_{\rm viscous}+\underbrace{T_{\rm B}(t)}_{\rm Brownian}, \tag{2}\]  
which together with:  
\[\frac{\mathrm{d}\varphi(t)}{\mathrm{d}t}=-\omega(t), \tag{3}\]  
and initial conditions:  
\[\varphi(0)=\frac{\pi}{2}\,\mathrm{rad},\hskip 56.905512 pt\omega(0)=0\, \mathrm{rad}/\mathrm{s}, \tag{4}\]  
define the rotational alignment of the spherical particle with magnetic moment \(\mu_{\rm P}\) in the external homogeneous magnetic field with flux density \(B_{0}\) and viscous fluid with thermodynamic temperature \(T\). The viscous torque \(\vec{T}_{\rm v}(t)=T_{\rm v}(t)\vec{k}\) exerted on the spherical particle for arbitrary angular velocity \(\omega(t)\) can be expressed according to [30]:  
\[T_{\rm v}(t)\equiv\underbrace{-\gamma\omega(t)}_{\rm quasi-steady}\underbrace{- \frac{\gamma}{3\sqrt{\pi}}\int\limits_{-\infty}^{t}K(t-\tau)\frac{\mathrm{d} \omega(\tau)}{\mathrm{d}\tau}\,\mathrm{d}\tau}_{\rm history\ acceleration\ torque}, \tag{5}\]where \(\gamma\equiv 8\pi\eta R^{3}\) is the Stokes coefficient of the viscous torque and:  
\[K(t-\tau)\equiv\frac{R}{\sqrt{v(t-\tau)}}-\sqrt{\pi}\exp\left(\frac{\
u(t-\tau)} {R^{2}}\right)\,\mathrm{erfc}\left(\frac{\sqrt{v(t-\tau)}}{R}\right) \tag{6}\]  
is a kernel function. This brings a challenge to the solution of the system of DEs (2)-(3) with initial conditions (4), due to the history integral term in equation (5) over the whole evolution of the angular acceleration of the particle in time, known as the history acceleration torque.