Stratonovich-to-Itô transition in noisy systems with multiplicative feedback
Giuseppe Pesce, Austin McDaniel, Scott Hottovy, Jan Wehr & Giovanni Volpe
Nature Communications 4, 2733 (2013)
Intrinsically noisy mechanisms drive most physical, biological and economic phenomena. Frequently, the system’s state influences the driving noise intensity (multiplicative feedback). These phenomena are often modelled using stochastic differential equations, which can be interpreted according to various conventions (for example, Itô calculus and Stratonovich calculus), leading to qualitatively different solutions. Thus, a stochastic differential equation–convention pair must be determined from the available experimental data before being able to predict the system’s behaviour under new conditions. Here we experimentally demonstrate that the convention for a given system may vary with the operational conditions: we show that a noisy electric circuit shifts from obeying Stratonovich calculus to obeying Itô calculus. We track such a transition to the underlying dynamics of the system and, in particular, to the ratio between the driving noise correlation time and the feedback delay time. We discuss possible implications of our conclusions, supported by numerics, for biology and economics.
Sorting of chiral microswimmers (Cover article)
Mite Mijalkov & Giovanni Volpe
Soft Matter 9(28), 6376—6381 (2013)
Microscopic swimmers, e.g., chemotactic bacteria and cells, are capable of directed motion by exerting a force on their environment. For asymmetric microswimmers, e.g., bacteria, spermatozoa and many artificial active colloidal particles, a torque is also present leading to circular motion (in two dimensions) and to helicoidal motion (in three dimensions) with a well-defined chirality. Here, we demonstrate with numerical simulations in two dimensions how the chirality of circular motion couples to chiral features present in the microswimmer environment. Levogyre and dextrogyre microswimmers as small as 50 nm can be separated and selectively trapped in chiral flowers of ellipses. Patterned microchannels can be used as funnels to rectify the microswimmer motion, as sorters to separate microswimmers based on their linear and angular velocities, and as sieves to trap microswimmers with specific parameters. We also demonstrate that these results can be extended to helicoidal motion in three dimensions.
Micron-sized self-propelled (active) particles can be considered as model systems for characterizing more complex biological organisms like swimming bacteria or motile cells. We produce asymmetric microswimmers by soft lithography and study their circular motion on a substrate and near channel boundaries. Our experimental observations are in full agreement with a theory of Brownian dynamics for asymmetric self-propelled particles, which couples their translational and orientational motion.
Simulation of a Brownian particle in an optical trap
Giorgio Volpe & Giovanni Volpe
American Journal of Physics 81(3), 224—230 (2013)
An optically trapped Brownian particle is a sensitive probe of molecular and nanoscopic forces. An understanding of its motion, which is caused by the interplay of random and deterministic contributions, can lead to greater physical insight into the behavior of stochastic phenomena. The modeling of realistic stochastic processes typically requires advanced mathematical tools. We discuss a finite difference algorithm to compute the motion of an optically trapped particle and the numerical treatment of the white noise term. We then treat the transition from the ballistic to the diffusive regime due to the presence of inertial effects on short time scales and examine the effect of an optical trap on the motion of the particle. We also outline how to use simulations of optically trapped Brownian particles to gain understanding of nanoscale force and torque measurements, and of more complex phenomena, such as Kramers transitions, stochastic resonant damping, and stochastic resonance.
Thermophoresis of Brownian particles driven by coloured noise
Scott Hottovy, Giovanni Volpe & Jan Wehr
EPL (Europhysics Letters) 99(6), 60002 (2012)
Brownian motion of microscopic particles is driven by collisions with surrounding fluid molecules. The resulting noise is not white, but coloured, due, e.g., to the presence of hydrodynamic memory. The noise characteristic time-scale is typically of the same order of magnitude as the inertial time-scale over which the particle’s kinetic energy is lost due to friction. We demonstrate theoretically that, in the presence of a temperature gradient, the interplay between these two characteristic time-scales can have measurable consequences on the particle’s long-time behaviour. Using homogenization theory, we analyse the infinitesimal generator of the stochastic differential equation describing the system in the limit where the two time-scales are taken to zero keeping their ratio constant and derive the thermophoretic transport coefficient, which, we find, can vary in both magnitude and sign, as observed in experiments. Studying the long-term stationary particle distribution, we show that particles accumulate towards the colder (positive thermophoresis) or the hotter (negative thermophoresis) regions depending on their physical parameters.
Active Brownian motion tunable by light
Ivo Buttinoni, Giovanni Volpe, Felix Kümmel, Giorgio Volpe & Clemens Bechinger
Journal of Physics: Condensed Matter 24(28), 284129 (2012)
Active Brownian particles are capable of taking up energy from their environment and converting it into directed motion; examples range from chemotactic cells and bacteria to artificial micro-swimmers. We have recently demonstrated that Janus particles, i.e. gold-capped colloidal spheres, suspended in a critical binary liquid mixture perform active Brownian motion when illuminated by light. In this paper, we investigate in more detail their swimming mechanism, leading to active Brownian motion. We show that the illumination-borne heating induces a local asymmetric demixing of the binary mixture, generating a spatial chemical concentration gradient which is responsible for the particle’s self-diffusiophoretic motion. We study this effect as a function of the functionalization of the gold cap, the particle size and the illumination intensity: the functionalization determines what component of the binary mixture is preferentially adsorbed at the cap and the swimming direction (towards or away from the cap); the particle size determines the rotational diffusion and, therefore, the random reorientation of the particle; and the intensity tunes the strength of the heating and, therefore, of the motion. Finally, we harness this dependence of the swimming strength on the illumination intensity to investigate the behavior of a micro-swimmer in a spatial light gradient, where its swimming properties are space-dependent.
Noise-induced drift in stochastic differential equations with arbitrary friction and diffusion in the Smoluchowski-Kramers limit
Scott Hottovy, Giovanni Volpe & Jan Wehr
Journal of Statistical Physics 146(4), 762—773 (2012)
We consider the dynamics of systems with arbitrary friction and diffusion. These include, as a special case, systems for which friction and diffusion are connected by Einstein fluctuation-dissipation relation, e.g. Brownian motion. We study the limit where friction effects dominate the inertia, i.e. where the mass goes to zero (Smoluchowski-Kramers limit). Using the Itô stochastic integral convention, we show that the limiting effective Langevin equations has different drift fields depending on the relation between friction and diffusion. Alternatively, our results can be cast as different interpretations of stochastic integration in the limiting equation, which can be parametrized by α∈ℝ. Interestingly, in addition to the classical Itô (α=0), Stratonovich (α=0.5) and anti-Itô (α=1) integrals, we show that position-dependent α=α(x), and even stochastic integrals with α∉[0,1] arise. Our findings are supported by numerical simulations.
Microswimmers in patterned environments
Giovanni Volpe, Ivo Buttinoni, Dominik Vogt, Hans-Jürgen Kümmerer & Clemens Bechinger
Soft Matter 7(19), 8810—8815 (2011)
Tiny self-propelled swimmers capable of autonomous navigation through complex environments provide appealing opportunities for localization, pick-up and delivery of microscopic and nanoscopic objects. Inspired by motile cells and bacteria, man-made microswimmers have been created and their motion in homogeneous environments has been studied. As a first step towards more realistic conditions under which such microswimmers will be employed, here we study, experimentally and with numerical simulations, their behavior in patterned surroundings that present complex spatial features where frequent encounters with obstacles become important. To study the microswimmers as a function of their swimming behavior, we develop a novel species of microswimmers whose active motion is due to the local demixing of a critical binary liquid mixture and can be easily tuned by illumination. We show that, when microswimmers are confined to a single pore whose diameter is comparable with their swimming length, the probability of finding them at the confinement walls significantly increases compared to Brownian particles. Furthermore, in the presence of an array of periodically arranged obstacles, microswimmers can steer even perpendicularly to an applied force. Since such behavior is very sensitive to the details of their specific swimming style, it can be employed to develop advanced sorting, classification and dialysis techniques.
Reply to comment on “Influence of noise on force measurements”
Giovanni Volpe, Laurent Helden, Thomas Brettschneider, Jan Wehr & Clemens Bechinger
Physical Review Letters 107(7), 078902 (2011)
Force measurement in the presence of Brownian noise: Equilibrium distribution method vs. drift method
Thomas Brettschneider, Giovanni Volpe, Laurent Helden, Jan Wehr & Clemens Bechinger
Physical Review E 83(4), 041113 (2011)
The study of microsystems and the development of nanotechnologies require alternative techniques to measure piconewton and femtonewton forces at microscopic and nanoscopic scales. Among the challenges is the need to deal with the ineluctable thermal noise, which, in the typical experimental situation of a spatial diffusion gradient, causes a spurious drift. This leads to a correction term when forces are estimated from drift measurements [G. Volpe, L. Helden, T. Brettschneider, J. Wehr, and C. Bechinger, Phys. Rev. Lett. 104, 170602 (2010)]. Here we provide a systematic study of such an effect by comparing the forces acting on various Brownian particles derived from equilibrium-distribution and drift measurements. We discuss the physical origin of the correction term, its dependence on wall distance and particle radius, and its relation to the convention used to solve the respective stochastic integrals. Such a correction term becomes more significant for smaller particles and is predicted to be on the order of several piconewtons for particles the size of a biomolecule.