Abstract
The motion of selfpropelled massive particles through a gaseous medium is dominated by inertial effects. Examples include vibrated granulates, activated complex plasmas and flying insects. However, inertia is usually neglected in standard models. Here, we experimentally demonstrate the significance of inertia on macroscopic selfpropelled particles. We observe a distinct inertial delay between orientation and velocity of particles, originating from the finite relaxation times in the system. This effect is fully explained by an underdamped generalisation of the Langevin model of active Brownian motion. In stark contrast to passive systems, the inertial delay profoundly influences the longtime dynamics and enables new fundamental strategies for controlling selfpropulsion in active matter.
Introduction
Newton’s first law states that because of inertia, a massive object resists any change of momentum. Before this groundbreaking idea, the dominant theory of motion was based on Aristotelian physics, which posits that objects come to rest unless propelled by a driving force. In retrospect, this perception is unsurprising, as the motions of everyday objects are influenced significantly by friction. In microscopic systems such as colloids, inertial forces are completely overwhelmed by viscous friction. In fact, in the absence of inertia, particles cannot move by reciprocal shape deformations due to kinematic reversibility. Biological organisms such as bacteria must therefore selfpropel by implementing nonreciprocal motion^{1}.
However, any finitely massive object performs ballistic motion, even if only on minuscule time and length scales. For example, colloidal particles undertake ballistic motion below 1 Å for approximately 100 ns. Experimental verification of this motion requires high accuracy measurements and has been achieved only for passive colloids^{2,3,4}. In contrast, for macroscopic selfpropelled particles, such as animals and robots, the magnitude of inertial forces can be comparable to that of the propulsion forces and influence the dynamics on large time scales.
A particularly simple example of a macroscopic selfpropelled particle is a minimalistic robot called a vibrobot, which converts vibrational energy into directed motion using its tilted elastic legs^{5}. Collectives of such particles exhibit novel nonequilibrium dynamics^{6,7,8,9}, selforganisation^{10}, clustering^{5,11} and swarming^{12,13,14,15}. Along with animals^{16}, artificial and biological microswimmers^{17,18,19}, vibrobots belong to the class of active soft matter.
Here, we demonstrate that the inertia of selfpropelled particles causes a significant delay between their orientation and velocity and increases the longtime diffusion coefficient through persistent correlations in the underdamped rotational motion. Standard models, such as the Vicsekmodel^{20} and active Brownian motion^{21} cannot explain this behaviour as they neglect inertia. Instead, the dynamics can be understood in terms of underdamped Langevin equations with a selfpropulsion term that couples the rotational and translational degrees of freedom. Using the mean squared displacements (MSDs) and velocity distributions, fitted by numerical and analytical results, we extract a unique set of parameters for the model. We derive analytic solutions for the short and longtime behaviour of the MSD and prove that the longtime diffusion coefficient explicitly depends on the moment of inertia.
Results
Experimental observation of inertial effects
Our experimental particles are 3Dprinted vibrobots driven by sinusoidal vibrations from an electromagnetic shaker. To investigate a wide range of parameter combinations, we varied the leg inclination, mass and moment of inertia of the particles (see Fig. 1a–d). The excitation frequency and amplitude were fixed to f = 80 Hz and A = 66 μm, respectively, which ensures stable quasitwodimensional motion of the particles.
The mechanism is illustrated in Fig. 1e and Supplementary Movie 1. The vibrobots move by a ratcheting mechanism driven by repeated collisions of their tilted elastic legs on the vibrating surface. Their propulsion velocity depends on the excitation frequency, amplitude, leg inclination and material properties such as the elasticity and friction coefficients^{5,22,23,24}. Longtime random motions are induced by microscopic surface inhomogeneities and (under sufficiently strong driving) a bouncing ball instability^{24}, that causes the particle’s legs to jump asynchronously and perform a tiny but very irregular precession, which in turn leads to random reorientations of the particle. Thereby, the vibrobot motion is considered as a macroscopic realisation of active Brownian motion^{12,13,25,26}. Figure 1f shows three representative trajectories of particles with different average propulsion velocities (see also Supplementary Movie 2). The persistence length is noticeably shorter for slower particles than for faster particles, as generally expected for selfpropelled particles^{19}.
However, the significance of inertial forces is an important difference between motile granulates and microswimmers^{11,27}. Massive particles do not move instantaneously, but accelerate from rest when the vibration is started. The time dependence of the initial velocity (averaged over up to 165 runs per particle) is shown in Fig. 2a. The particles noticeably accelerated up to the steady state on a time scale of 10^{−1} s, one order of magnitude larger than the inverse excitation frequency and the relaxation time of the shaker. When perturbed by an external force, vibrationally driven particles approach their steady state on a similar time scale^{10}. The relaxation process is well fitted by an exponential function, as expected for inertial relaxation. Inertia also influences the dynamical behaviour of the particles’ orientation relative to their velocity. The orientation (red arrows in Fig. 2b) systematically deviates from the movement direction (black arrows in Fig. 2b). Particularly, during sharp turns the orientation deviates towards the centre of the curve, whereas the velocity is obviously tangential to the trajectory. We compare the angle of orientation ϕ to the angle of velocity \({\mathrm{\Theta }} = {\mathrm{atan2}}(\dot Y,\dot X)\) in Fig. 2b and find that Θ systematically pursues ϕ with an inertial delay of order 10^{−1} s. A slowmotion recording of one particle in Supplementary Movie 3 illustrates the dynamic delay between motion and orientation. The particle quickly reorients, but its previous direction is retained by inertia. Consequently, the particle drifts around the corner, mimicking the wellknown intentional oversteering of racing cars.
Underdamped Langevin model
Despite the complex nonlinear dynamics of the vibrobots^{5,23,24,28,29}, our observations can be fully described by a generalised active Brownian motion model with explicit inertial forces. The dynamics are characterised by the centreofmass position R(t) = (X(t), Y(t)) and the orientation n(t) = (cosϕ(t), sinϕ(t)), where ϕ(t) defines the direction of the propulsion force. The coupled equations of motion for R(t) and ϕ(t), describing the force balance between the inertial, viscous and random forces, are given by
Here, M and J are the mass and moment of inertia, respectively, and ξ and ξ_{r} denote the translational and rotational friction coefficients. The translational and rotational Brownian fluctuations are quantified by their respective shorttime diffusion coefficients D and D_{r}. The random forces f_{st}(t) and torque τ_{st}(t) are white noise terms with zero mean and correlation functions \(\left\langle {{\bf{f}}_{{\mathrm{st}}}(t) \otimes {\bf{f}}_{{\mathrm{st}}}(t\prime )} \right\rangle = \delta \left( {t  t\prime } \right)\mathbf{1}\) and 〈τ_{st}(t)τ_{st}(t′)〉 = δ(t − t′), respectively, where 〈⋯〉 denotes the ensemble average and 1 is the unit matrix. Owing to the strong nonequilibrium nature of the system, the diffusion and damping constants are not related by the Stokes−Einstein relation^{30}. Moreover, as typical particles are not perfectly symmetrical, they tend to perform circular motions on intermediate time scales. To capture this behaviour, we applied an external torque τ_{0} that induces circular movement with average velocity ω = τ_{0}/ξ_{r}^{31,32}. Similar models applied in the literature have typically neglected the moment of inertia or have only been solved numerically^{11,33,34,35,36,37}. The motion of a particle governed by Eqs. (1) and (2) is determined by different time scales given by the friction rates ξ/M = τ^{−1} and \(\xi _{\mathrm{r}}{\mathrm{/}}J = \tau _{\mathrm{r}}^{  1}\), the rotational diffusion rate D_{r}, the angular frequency ω and the crossover times \(2D{\mathrm{/}}V_{\mathrm{p}}^2\) and \(2D_{\mathrm{r}}{\mathrm{/}}\tau _0^2\). In the limit of vanishing M and J the model is equivalent to the wellknown active Brownian motion formulation^{21}.
The trajectories obtained by numerically integrating the Langevin model compare well with the experimental observations. As show by the representative trajectory in Fig. 2d, e, the model reproduces the delay between the orientation and velocity, when the friction is sufficiently weaker than the inertia. The model can be analytically solved by averaging and integration. The orientational correlation
where 〈⋯〉_{T} is the time average, quantifies the temporal evolution of the active noise term. The periodic cosine term results from the external torque and captures the induced circular motion. The rotational noise, quantified by D_{r}, decorrelates the orientation on longtime scales. This decorrelation is described by the exponential term in Eq. (3). The double exponential reflects the additional orientation correlation on short time scales imposed by the inertial damping rate \(\tau _{\mathrm{r}}^{  1}\). Consequently, the particle dynamics nontrivially depend on the orientation, even in the short and longtime limits. In the shorttime limit the MSD is given by
with
The first term is the equilibrium solution for a passive particle, and the second term arises from the active motion term. The latter is proportional to \(V_{\mathrm{p}}^2\), i.e. the kinetic energy injected by the propulsion. This contribution is quantified by the ratio of competing time scales, i.e. the dimensionless delay numbers
through the function
where Re denotes the real part and γ is the lower incomplete gamma function. The longtime behaviour of the motion is diffusive, with the longtime diffusion coefficient
In Eq. (8), the first term is the passive diffusion coefficient and the second term represents the contribution from the driving force with persistence time given by
Equation (8) is similar to the active Brownian motion model, where the persistence time 1/D_{r} is replaced by Eq. (9). The longtime diffusion coefficient is therefore a function of the inertial correlations introduced by J through \({\frak D}_0\). This starkly contrasts with passive Brownian motion, which assumes an inertiaindependent diffusion coefficient.
Comparison between model and measurement
Equations (5) and (8) depend nontrivially on six independent parameters. They are determined by fitting the MSD given by Eq. (4) and the linear and absolute velocity distributions, obtained by numerically solving Eqs. (1) and (2), to the measurements. The measurements and fitting curves for the four different particle types are summarised in Fig. 3. The angular MSDs in Fig. 3a–d show a ballistic shorttime regime and a diffusive longtime regime (dashed lines) from which we can determine τ_{r} and D_{r}, respectively. The \(\dot \phi\)distribution in Fig. 3e–h is a shifted Gaussian. The minor deviations at small velocities are caused by the finite tracking accuracy. The first moment of this distribution gives the mean angular velocity ω. The parameters τ, D and V_{p} are extracted from the linear velocity distributions \(P\left( {v_{{\mathrm{lin}}}} \right) = P\left( {\dot X} \right) = P\left( {\dot Y} \right)\) (Fig. 3i–p) and the translational MSDs (Fig. 3q–t), which can be directly fitted by Eqs. (4) and (5). The linear velocity distribution is not a simple Gaussian, but shows a double peak related to the activity. The absolute velocity distribution also clearly deviates from the twodimensional Maxwell−Boltzmann distribution of passive particles, especially, the maximum is shifted by the propulsion force. The translational MSD mainly depicts the ballistic shorttime behaviour, because the persistence length of our particles is of the order of the system size. To test the parameters on an independent quantity, we systematically compared the model with the measured inertial delay. We define the correlation function
i.e. the average difference between the projection of the orientation on the initial velocity and projection of the velocity on the initial orientation. This function starts at zero and reapproaches zero in the limit t → ∞. In overdamped systems, Eq. (10) is zero at all times. In the underdamped case the velocity direction pursues the orientation and C(\({\dot{\bf R}}\)(t), n(t)) reaches its maximum after a specific delay. Pronounced peaks, related to the decay numbers and τ_{r} are observed in Fig. 4a–d. The measurements and theoretical predictions using the parameters determined from Fig. 3 are consistent. Some deviations above the statistical error are visible, in particular for the tug particle (dotted line in Fig. 4c), due to overfitting the parameters, when the delay function is not explicitly taken into account. This is confirmed by a more general fit, which minimises the total mean squared error for the curves from Figs. 3 and 4. All curves obtained from this agree with the measurements within the statistical error (see dashed magenta lines in Fig. 4a–d).
Inertial dependence
Strikingly, both the short and longtime particle dynamics in our system depend on the delay number \({\frak D}_0\). The fundamental reason is the additional orientational correlation in Eq. (3), which is delayed by the rotational friction rate \(\tau _{\mathrm{r}}^{  1}\). The exponent in this expression represents the MSD of ϕ, which is dominated by order t^{2} at short times and order t at long times. Consequently, neglecting external torque, this function follows a Gaussian decay at short times and an exponential decay at long times. The significance of the inertial delay is quantified by \({\frak D}_0\). For small \({\frak D}_0\), the correlation approaches the overdamped result and for large \({\frak D}_0\) the correlation time is significantly delayed by τ_{r}. To confirm this prediction, we compare the measured correlation functions and the solutions of Eq. (3). The results are consistent, as shown in Fig. 5a.
The numerical and analytical dependence of the ballistic and diffusive regimes on the moment of inertia are displayed in Fig. 5b, c, which show that \(\left\langle {{\dot{\bf R}}^{\mathrm{2}}} \right\rangle\) and D_{L} increase with J. The effects of finite J can be simply demonstrated mathematically by expanding Eqs. (5) and (8) in the limit J → 0, ∞. As J vanishes, we find that
which agrees with the results reported in ref. ^{34}. For infinitely large J we obtain
which simply corresponds to the sum of the thermal and injected kinetic energies. For the longtime diffusion constant the asymptotic behaviour for small moments of inertia is
which intuitively demonstrates, how, the leading order J increases the persistence time (namely by a linear term proportional to (ξ_{r}/J)^{−1}). The dependence of D_{L} on \({\frak D}_0\) has no upper bound, and its asymptotic behaviour is described by
The origin of this dependence can be intuitively understood by considering the turnaround manoeuvre of a simple noisefree active particle. When a torque is applied perpendicularly to the velocity, the particle will turn around at point P and eventually approach circular motion. As the moment of inertia quantifies the resistance of a particle to changing its angular momentum, a particle with low J will turn faster than one with high J, as shown in Fig. 5d. This applies only to the transient states, where \(\ddot \phi \ne 0\). In the steady state, the radius r of the final circle is independent of J. The angular momentum of an active particle with random reorientations is constantly changing. Its inertia resists these changes and modifies the distribution of reorientations directly opposing the effect of rotational noise.
Discussion
Our observations demonstrate the profound influence of inertia on the long and shorttime dynamics of selfpropelled particles. Considering the relevance of inertia^{30}, our model is applicable to various systems, such as levitating^{38,39} and floating^{40} granular particles and dusty plasmas^{41}. It is straightforward to extend the model to elongated particles^{5,11,12,14,37,42} and it was shown numerically that collective motion of rodlike particles is well described by similar equations of motion^{37}. Qualitatively, in our system, rodlike particles show an inertial delay as well (see Supplementary Fig. 2). In a more general framework, diffusion and friction coefficient could be tensorial and additional nonlinear force terms, such as a selfaligning torque reported in refs. ^{27,33}, might be added to the force balance. Our model predicts that microswimmers perform a shorttime ballistic motion like passive particles, but in practice, their motion also depends on their specific propulsion mechanism^{43,44} and hydrodynamic effects^{45,46}. Generally, the inertial effects will depend on the corresponding time scales in the system. In numerical experiments, this can be demonstrated by gradually reducing the density of hypothetical particles, retaining all other parameters as constants. At very low densities, the MSD exhibits four different regimes: shorttime ballistic, shorttime diffusive, active ballistic and longtime diffusive regime (see Fig. 6).
The longtime diffusion coefficient of passive particles is independent of inertia and is related to the friction coefficient via the Stokes−Einstein relation. However, for actively moving particles we find an explicit dependence on the moment of inertia (with no explicit dependence on the total mass M). This finding illustrates the importance of J for macroscopic selfpropelled particles. While mass distribution and shape are generally important for efficient motion of animals^{47,48,49,50} and adaptation to the environment^{51,52}, our results suggest that J can be exploited in novel control strategies for active matter. Biological organisms cannot rapidly vary their mass, but they can change J by moving their limbs. For instance, cheetahs use tail motion to stabilise fast turns^{53}. While the effect on the longtime diffusivity of vibrobots is a few percent, our theory predicts that for flying and floating particles these changes are more dramatic. For similar sized particles flying in air (e.g. insects) we can expect that friction is about two orders of magnitude smaller. Also, biological organisms can vary their moment of inertia dynamically for up to almost two orders of magnitude, depending on the position and the axis of rotation^{54}. In this case, from Eq. (8), the longtime diffusion coefficient changes up to a factor of about three per order of magnitude in J. By increasing J, animals can then faster explore a large area. Conversely, by decreasing J, they can more easily dodge obstacles or predators and counteract sensorial^{55} and behavioural delay^{16}. Even under conditions, where animals cannot control their rotational deflections, such as aerodynamic turbulence, or during random collisions with neighbours^{56}, they could stabilise their movements through variations of J.
Methods
Particle fabrication
Four particle types were designed and printed: The generic particle consists of a cylindrical core (diameter 9 mm, length 4 mm) topped by a cylindrical cap (diameter 15 mm, length 2 mm). Beneath the cap, seven tilted cylindrical legs (each of diameter 0.8 mm) were attached in parallel in a regular heptagon around the core. The legs lift the bottom of the body by 1 mm above the surface. The typical mass was about m = 0.76 g. From the mass and shape of the particle the moment of inertia was approximated as J = 1.64×10^{−8} kg m^{2}. To vary the propulsion velocity of the particles, we printed five types with different leg inclination angles 0°, 2°, 4°, 6° and 8°.
The carrier particle was fabricated with the same core as generic, but its cap was topped with a 1 mm tall, 8.5 mm diameter cylinder. The carrier socket held two galvanised steel washers, each with an outer diameter of 16 mm and a mass of 1.6 g. The leg inclination of carrier particles was fixed at 2°, and mass and moment of inertia were m = 4.07 g, J = 1.46×10^{−7} kg m^{2}, respectively.
The tug particle was a generic with a fixed leg inclination of 2° and thinner core (diameter 4 mm). This core held a hexagonal M5 threaded galvanised steel nut with a short diagonal and height of 8 and 3.75 mm, respectively. The mass and moment of inertia were m = 1.57 g and J = 2.54×10^{−8} kg m^{2}, respectively.
The ring particle had a leg inclination of 4° and a ringshaped cap with a hole (diameter 9 mm) in the middle. The mass and moment of inertia were m = 0.33 g and J = 1.26×10^{−8} kg m^{2}, respectively.
All particles were labelled with a simple high contrast image allowing the detection software to identify the particle’s position and orientation. The particles were printed from a proprietary methacrylatebased photopolymer (FormLabs Grey V3, FLGPGR03) of typical density 1.11(1) kg/L at a precision of 0.05 mm. They were subsequently cleaned in high purity (>97%) isopropyl alcohol in a still bath, followed by an ultrasound bath, then hardened by three 10min bursts under four 9 W UVA bulbs. Finally, irregularities were manually filed away and the label sticker was attached.
Experimental setup
The vibrobots were excited by vertical vibrations generated by a circular acrylic baseplate (diameter 300 mm, thickness 15 mm) attached to an electromagnetic shaker (Tira TV 51140) and surrounded by a barrier to confine the particles. The tilt of the plate was adjusted with an accuracy of 0.01°. The vibration frequency and amplitude was set to f = 80 Hz and A = 66(4) μm, respectively, guaranteeing stable excitation with peak accelerations of 1.7(1) g (measured by four LIS3DH accelerometers). To ensure homogeneous excitation, the acceleration amplitude was measured at different radial and azimuthal positions in steps of 3 cm and 45° respectively, at constant frequency. The variation of amplitudes at a mean acceleration of 1.7 g is below 5% (see Fig. 7a, b). To ensure that no other factors significantly affect the isotropy of the system, the average particle velocity was measured as a function of the radial distance to the centre (Fig. 7c). The resulting fluctuations are small compared to the mean (standard deviation lies between 1.8 and 3.6% of the respective mean). Experiments were recorded using a highspeed camera system (Allied Vision MakoU130B) operating at up to 152 fps with a spatial resolution of 1024 × 1024 pixels. Single particles were tracked to subpixel accuracy using standard image recognition methods. The tracking accuracy was determined from test measurements of a particle rigidly attached to the plate at different locations. A bivariate Gaussian distribution was fitted to the positions, from which the covariance matrix was obtained. The accuracy 2 · σ_{max} is defined from the maximum of the diagonal entries in this covariance matrices \(\sigma _{{\mathrm{max}}}^2\) (see Fig. 8a). For the angular position, the error is directly obtained from the 95.4% confidence interval, since the distribution is nonnormal due to pixel locking effects (see Fig. 8b). The resulting accuracy is ±4.7×10^{−4} m and ±0.013 rad. Multiple single trajectories were recorded for each particle, until 10 min of data were acquired. Events involving particleborder collisions were discarded.
Analytic results
The rotational behaviour of the particle was obtained by stochastic integration^{57} of Eq. (2). The angular frequency and angular coordinate we obtained as
and
respectively. Here, ϕ_{0} and \(\dot \varphi _0\) are initial angle and angular velocity, respectively, and the initial time was set to zero. As \(\dot \phi (t)\) and ϕ(t) are both linear combinations of Gaussian variables, the corresponding probability distributions are also Gaussian. Thus, by calculating the mean
and the variance
one obtains the angular probability distribution
At times much longer than the reorientation time scale 1/D_{r} and the rotational friction rate J/ξ_{r}, the variance of the angular distribution far exceeds 2π, while the mean cycles between 0 and 2π. This behaviour converges to the stationary state with a uniform distribution of ϕ. At times much longer than the rotational friction rate J/ξ_{r}, the stationary distribution of the angular velocity reduces to
The width of this distribution is inversely proportional to the moment of inertia.
From the translational equation of motion i.e. Eq. (1), the velocity in the laboratory frame of reference is obtained as
where the initial velocity is denoted by \({\dot{\mathbf R}}_0\). The centreofmass position of a particle beginning its motion from the origin is calculated as
The mean square displacement 〈R^{2}〉 is obtained in the following integral form:
where 〈n(t)〉 = e^{−μ(t)/2} (cos〈ϕ(t)〉, sin〈ϕ(t)〉) and 〈n(t_{1})⋅n(t_{2})〉 is defined by
The inertial delay correlation function Eq. (10) is given by
In the stationary case the Fokker−Planck equation of our model projected into one dimension reduces to
One can anticipate the linear velocity distribution to be in the following form:
where q and W are functions of D, D_{r}, τ, τ_{r}, V_{p} and ω. Analytic approximations for q and W are obtained from comparing the second and fourth moments to the solution of the Langevin equation(see Supplementary Methods).
Parameter matching
All parameters are obtained from fitting analytic and numeric results to the measurements of MSDs, velocity distributions and eventually the delay function, which describes a crosscorrelation between orientation and velocity. The velocity in experiments is defined from the displacement of successive positions of the particle v(t) = (r(t + Δt) − r(t))/Δt, where Δt is the the time between two frames. Correspondingly the angular velocity is defined as \(\dot \phi\) = (ϕ(t + Δt) − ϕ(t))/Δt. The recording frame rate is 152 Hz, which corresponds to a minimal Δt_{0} = 0.0066 s. When Δt is small enough, i.e. below τ, such that the ballistic motion of the particle can be captured accurately, the distribution of v and \(\dot \phi\) will approach the stationary state. In our experiment we find τ is on the order of 0.1 s and Δt = Δt_{0} is noticeably smaller. However, to ensure that the choice of Δt does not significantly alter the parameters, fits with Δt = 2, 3, 4Δt_{0} were checked and show no significant difference within the error bars of the parameters. The distribution and delay functions are provided for Δt = 1, 2, 3, 4Δt_{0} in Supplementary Fig. 1 as reference. Note that for the experimental linear velocity distribution, i.e. the distribution of the components of the velocity vector, each trajectory is numerically rotated by a random angle to reduce anisotropy of the distribution that arises from the initial conditions, where each particle, at start, points towards the plate centre.
Initial parameters can be obtained from analytic results of the model directly. The parameters τ_{r}, D_{r} and τ_{0} are straightforwardly determined from fitting the wellknown solution to Eq. (2) (Ornstein−Uhlenbeck process). The first moment of the angular velocity distribution gives τ_{0} = ωγ_{r}. Angular diffusion coefficient and relaxation time are determined from the fit to the angular MSD.
The determination of the remaining parameters D, τ and V_{p} is more sophisticated. The analytic solution for the initial slope of the translational MSD is given by Eq. (5) and an analytic approximation of the linear velocity distribution is obtained from Eq. (27). The function \({\frak f}\left( {{\frak D}_0,{\frak D}_1,{\frak D}_2} \right)\) in Eq. (5) starts from zero and goes asymptotically to 1 as \({\frak D}_2\) grows large, such that it is confined to the interval [0, 1]. This gives upper and lower bounds for V_{p}, namely \(V_{\mathrm{p}} \in \left[ {\sqrt {\left\langle {{\dot{\mathbf R}}^2} \right\rangle  2D\tau ^{  1}} ,\sqrt {{\textstyle{{\left\langle {{\dot{\mathbf R}}^2} \right\rangle  2D\tau ^{  1}} \over {\frak f}}}} } \right]\). Accordingly, the following iterative procedure is used to determine a set of parameters.
The iteration starts with the initial guess \(V_{\mathrm{p}} = \sqrt {\left\langle {{\dot{\mathbf R}}^2} \right\rangle }\) and \({\frak f} = 1\). The analytic approximation of the linear velocity distribution, Eq. (27), is fitted to the measurement to estimate τ and D. After this initial stage, there are two different choices for the postiterations; either keeping \({\frak f} = 1\) and changing the value of V_{p} to \(\sqrt {\left\langle {{\dot{\mathbf R}}^2} \right\rangle  2D\tau ^{  1}}\), or computing the value of \({\frak f}\) with respect to the estimated value for τ from the preiteration, such that \(V_{\mathrm{p}} = \sqrt {{\textstyle{{\left\langle {{\dot{\mathbf R}}^2} \right\rangle  2D\tau ^{  1}} \over {\frak f}}}}\). This leads to two estimates for τ and D via fitting the linear velocity distribution to the outcomes of the experiment.
By comparing the agreement between the resulting τ and D from both choices and the measurement through taking MSD and absolute velocity distribution into account, the estimate with the worst agreement gets discarded. The iteration continues with the accepted estimate for the set of values of the parameters until the resulting curves agree below the standard error.
The resulting set of parameters fit the experimental curves in Fig. 3 with high accuracy. Note that the delay function is not explicitly fitted in this scheme, but used as a crosscheck of our parameters. However, this can potentially overfit the parameters, such as for the tug particle (see Fig. 4c). We additionally implemented a numerical optimisation routine (Nelder−Mead optimisation^{58}), which fits the numerical solution of the model to all experiment curves (MSDs, velocity distributions and delay function, where velocities are defined such that they match the experimental time scale Δt_{0}), by minimising the weighted sum of the mean squared errors. For the generic, carrier and ring particle only minor improvements were found. In the case of the tug particle a significantly better agreement for the delay function can be found by slightly sacrificing the agreement of the other curves. In particular, τ and D are sensitive to small variations. This is in accordance with our model, where only the product of τ and D enters in dominating terms. Nevertheless, the deviation between parameters without and with taking the delay function into account vary in the worst case by about 50% (tug particle) in τ and D and much less for all other parameters. Both sets of parameter values are shown in Supplementary Tables 1 and 2 for comparison. In the latter case an error estimate was obtained from the parameter variation that quadruples the mean squared error.
Code availability
The custom code that supports the findings of this study is available from the corresponding authors upon reasonable request.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
We acknowledge funding by the German Research Foundation (Grant Nos. SCHO 1700/11 and LO 418/231).
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C.S. designed the experimental setup. C.S. and A.L. carried out the experiments. C.S., S.J., and A.L. analysed the measurements. S.J. and C.S. wrote the simulation code. S.J. and C.S. performed and analysed the simulations. S.J. developed the theoretical results. All authors discussed the results and wrote the manuscript.
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Scholz, C., Jahanshahi, S., Ldov, A. et al. Inertial delay of selfpropelled particles. Nat Commun 9, 5156 (2018). https://doi.org/10.1038/s4146701807596x
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