Influence of noise on force measurements

Analysing Sources of Error in Total Internal Reflection Microscopy (TIRM) Experiments and Data Analysis

Many phenomena observed in synthetic and biological colloidal suspensions are dominated by the static interaction energies and the hydrodynamic interactions that act both between individual particles and also between colloids and macroscopic interfaces. This calls for methods that allow precise measurements of the corresponding forces. One method used for this purpose is total internal reflection microscopy (TIRM), which has been employed for around three decades to measure in particular the interactions between a single particle suspended in a liquid and a solid surface. However, given the importance of the observable variables, it is crucial to understand the possibilities and limitations of the method. In this paper, we investigate the influence of technically unavoidable noise effects and an inappropriate choice of particle size and sampling time on TIRM measurement results. Our main focus is on the measurement of diffusion coefficients and drift velocities, as the influence of error sources on dynamic properties has not been investigated so far. We find that detector shot noise and prolonged sampling times may cause erroneous results in the steep parts of the interaction potential where forces of the order of pico-Newtons or larger act on the particle, while the effect of background noise is negligible below certain thresholds. Furthermore, noise does not significantly affect dynamic data but we find that lengthy sampling times and/or probe particles with too small a radius will cause issues. Most importantly, we observe that dynamic results are very likely to differ from the standard hydrodynamic predictions for stick boundary conditions due to partial slip.

Ionic current driven by a viscosity gradient

Gradients of voltage, pressure, temperature, and salinity can transport objects in micro- and nanofluidic systems by well-known mechanisms. This paper explores the dynamics of particles in a viscosity gradient with numerical simulations. The different stochastic rules used to integrate the random motion of Brownian particles affect the steady-state distribution of particles in a diffusivity gradient. Importantly, the simulations illuminate the important role that the boundary conditions play, disallowing a steady-state flux when the boundary conditions mimic those of a closed container, but allowing flux when they mimic electrodes. These results provide an interpretation for measurements of a steady ionic current flowing between electrodes separated by a nanofluidic channel with a liquid viscosity gradient.

Tunable optical traps over nonreciprocal surfaces

We propose engineering optical traps over plasmonic surfaces and precisely controlling the trap position with an external bias by inducing in-plane nonreciprocity on the surface. The platform employs an incident Gaussian beam to polarize targeted nanoparticles, and exploits the interplay between nonreciprocal and spin-orbit lateral recoil forces to construct stable optical traps and manipulate their position within the surface. To model this process, we develop a theoretical framework based on the Lorentz force combined with nonreciprocal Green's functions and apply it to calculate the trapping potential. Rooted on this formalism, we explore the exciting possibilities offered by graphene to engineer stable optical traps using low-power laser beams in the mid-IR and to manipulate the trap position in a continuous manner by applying a longitudinal drift bias. Nonreciprocal metasurfaces may open new possibilities to trap, assemble and manipulate nanoparticles and overcome many challenges faced by conventional optical tweezers while dealing with nanoscale objects.

Optical Force of Bessel Pincer Light-Sheets Beam on a Dielectric Sphere of Arbitrary Size

In the framework of Generalized Lorenz-Mie theory (GLMT), based on the expansion results of electromagnetic field radiation components of Bessel pincer light sheets beam acting on dielectric particles of arbitrary size, the expression of radiation force components in a Cartesian coordinate system is obtained by using the Maxwell stress tensor method. On the one hand, the effects of the refractive index and the equivalent radius of spherical particles on the distribution of radiation force are discussed; On the other hand, the influence of beam scaling parameter and beam order of Bessel pincer light sheets beam on the distribution of radiation force are investigated. The results indicate that the changes of particle's refractive index and effective radius only affect the distribution of radiation force. However, the beam scaling parameter and beam order of Bessel pincer light sheets beam have a very sharp impact on the convergence position, distribution range and bending degree far away from the wave source of the radiation force. Single-beam optical tweezers using the self-focusing and self-bending Bessel pincer light-sheets beam are crucial for applications such as single molecule biophysics, optical manipulation and particle separation/clearing.

Efficiency of random search with space-dependent diffusivity

We address the problem of random search for a target in an environment with a space-dependent diffusion coefficient D(x). Considering a general form of the diffusion differential operator that includes Itô, Stratonovich, and Hänggi-Klimontovich interpretations of the associated stochastic process, we obtain and analyze the first-passage-time distribution and use it to compute the search efficiency E=〈1/t〉. For the paradigmatic power-law diffusion coefficient D(x)=D_{0}|x|^{α}, where x is the distance from the target and α<2, we show the impact of the different interpretations. For the Stratonovich framework, we obtain a closed-form expression for E, valid for arbitrary diffusion coefficient D(x). This result depends only on the distribution of diffusivity values and not on its spatial organization. Furthermore, the analytical expression predicts that a heterogeneous diffusivity profile leads to a lower efficiency than the homogeneous one with the same average level within the space between the target and the searcher initial position, but this efficiency can be exceeded for other interpretations.

Statistical physics of inhomogeneous transport: Unification of diffusion laws and inference from first-passage statistics

Characterization of composite materials, whose properties vary in space over microscopic scales, has become a problem of broad interdisciplinary interest. In particular, estimation of the inhomogeneous transport coefficients, e.g., the diffusion coefficient or the heat conductivity, which shape important processes in biology and engineering, is a challenging task. The analysis of such systems is further complicated because two alternative formulations of the inhomogeneous transport equations exist in the literature-the Smoluchowski and Fokker-Planck equations, which are also related to the so-called Ito-Stratonovich dilemma. Using the theory of statistical physics, we show that the two formulations, usually regarded as distinct models, are physically equivalent. From this result we develop efficient estimates for the transverse space-dependent diffusion coefficient in fluids near a phase boundary. Our method requires only measurements of escape probabilities and mean exit times of molecules leaving a narrow spatial region. We test our estimates in three case studies: (i) a Langevin model of a Büttikker-Landauer ratchet; atomistic molecular-dynamics simulations of liquid-water molecules in contact with (ii) vapor, and (iii) soap (surfactant) film which has promising applications in physical chemistry. Our analysis reveals that near the surfactant monolayer the mobility of water molecules is slowed down almost twice with respect to the bulk liquid. Moreover, the diffusion coefficient of water correlates with the transition from hydrophilic to hydrophobic parts of the film.

Consistent Hamiltonian models for space-momentum diffusion

We develop a unified Hamiltonian approach to the diffusion of a particle coupled to a dissipative environment, an archetypal model widely invoked to interpret condensed phase phenomena, such as polymerization and cold-atom diffusion in optical lattices. By appropriate choices of the coupling functions, we reformulate phenomenological diffusion models by adding otherwise ignored space-momentum terms. We thus numerically predict a variety of diffusion regimes, from diffusion saturation to superballistic diffusion. With reference to ultracold atoms in optical lattices, we also show that time correlated external noises prevent superdiffusion from exceeding Richardson's law. Some of these results are unexpected and call for experimental validation.

Stochastic Modeling of Particle Transport in Confined Geometries: Problems and Peculiarities

The equivalence between parabolic transport equations for solute concentrations and stochastic dynamics for solute particle motion represents one of the most fertile correspondences in statistical physics originating from the work by Einstein on Brownian motion. In this article, we analyze the problems and the peculiarities of the stochastic equations of motion in microfluidic confined systems. The presence of solid boundaries leads to tensorial hydrodynamic coefficients (hydrodynamic resistance matrix) that depend also on the particle position. Singularity issues, originating from the non-integrable divergence of the entries of the resistance matrix near a solid no-slip boundary, determine some mass-transport paradoxes whenever surface phenomena, such as surface chemical reactions at the walls, are considered. These problems can be overcome by considering the occurrence of non vanishing slippage. Added-mass effects and the influence of fluid inertia in confined geometries are also briefly addressed.

Breakdown of the Stokes-Einstein relation in supercooled water: the jump-diffusion perspective

Although water is the most ubiquitous liquid it shows many thermodynamic and dynamic anomalies. Some of the anomalies further intensify in the supercooled regime. While many experimental and theoretical studies have focused on the thermodynamic anomalies of supercooled water, fewer studies explored the dynamical anomalies very extensively. This is due to the intricacy of the experimental measurement of the dynamical properties of supercooled water. Violation of the Stokes-Einstein relation (SER), an important relation connecting the diffusion of particles with the viscosity of the medium, is one of the major dynamical anomalies. In absence of experimentally measured viscosity, researchers used to check the validity of SER indirectly using average translational relaxation time or α-relaxation time. Very recently, the viscosity of supercooled water was accurately measured at a wide range of temperatures and pressures. This allowed direct verification of the SER at different temperature-pressure thermodynamic state points. An increasing breakdown of the SER was observed with decreasing temperature. Increasing pressure reduces the extent of breakdown. Although some well-known theories explained the above breakdown, a detailed molecular mechanism was still elusive. Recently, a translational jump-diffusion (TJD) approach has been able to quantitatively explain the breakdown of the SER in pure supercooled water and an aqueous solution of methanol. The objective of this article is to present a detailed and state-of-the-art analysis of the past and present works on the breakdown of SER in supercooled water with a specific focus on the new TJD approach for explaining the breakdown of the SER.

Anomalous diffusion and nonergodicity for heterogeneous diffusion processes with fractional Gaussian noise

Heterogeneous diffusion processes (HDPs) feature a space-dependent diffusivity of the form D(x)=D_{0}|x|^{α}. Such processes yield anomalous diffusion and weak ergodicity breaking, the asymptotic disparity between ensemble and time averaged observables, such as the mean-squared displacement. Fractional Brownian motion (FBM) with its long-range correlated yet Gaussian increments gives rise to anomalous and ergodic diffusion. Here, we study a combined model of HDPs and FBM to describe the particle dynamics in complex systems with position-dependent diffusivity driven by fractional Gaussian noise. This type of motion is, inter alia, relevant for tracer-particle diffusion in biological cells or heterogeneous complex fluids. We show that the long-time scaling behavior predicted theoretically and by simulations for the ensemble- and time-averaged mean-squared displacements couple the scaling exponents α of HDPs and the Hurst exponent H of FBM in a characteristic way. Our analysis of the simulated data in terms of the rescaled variable y∼|x|^{1/(2/(2-α))}/t^{H} coupling particle position x and time t yields a simple, Gaussian probability density function (PDF), P_{HDP-FBM}(y)=e^{-y^{2}}/sqrt[π]. Its universal shape agrees well with theoretical predictions for both uni- and bimodal PDF distributions.

State-dependent diffusion in a bistable potential: Conditional probabilities and escape rates

We consider a simple model of a bistable system under the influence of multiplicative noise. We provide a path integral representation of the overdamped Langevin dynamics and compute conditional probabilities and escape rates in the weak noise approximation. The saddle-point solution of the functional integral is given by a diluted gas of instantons and anti-instantons, similar to the additive noise problem. However, in this case, the integration over fluctuations is more involved. We introduce a local time reparametrization that allows its computation in the form of usual Gaussian integrals. We found corrections to the Kramers escape rate produced by the diffusion function which governs the state-dependent diffusion for arbitrary values of the stochastic prescription parameter. Theoretical results are confirmed through numerical simulations.

Space-Time Inversion of Stochastic Dynamics

This article introduces the concept of space-time inversion of stochastic Langevin equations as a way of transforming the parametrization of the dynamics from time to a monotonically varying spatial coordinate. A typical physical problem in which this approach can be fruitfully used is the analysis of solute dispersion in long straight tubes (Taylor-Aris dispersion), where the time-parametrization of the dynamics is recast in that of the axial coordinate. This allows the connection between the analysis of the forward (in time) evolution of the process and that of its exit-time statistics. The derivation of the Fokker-Planck equation for the inverted dynamics requires attention: it can be deduced using a mollified approach of the Wiener perturbations “a-la Wong-Zakai” by considering a sequence of almost everywhere smooth stochastic processes (in the present case, Poisson-Kac processes), converging to the Wiener processes in some limit (the Kac limit). The mathematical interpretation of the resulting Fokker-Planck equation can be obtained by introducing a new way of considering the stochastic integrals over the increments of a Wiener process, referred to as stochastic Stjelties integrals of mixed order. Several examples ranging from stochastic thermodynamics and fractal-time models are also analyzed.

Statistical Tests for Force Inference in Heterogeneous Environments

We devise a method to detect and estimate forces in a heterogeneous environment based on experimentally recorded stochastic trajectories. In particular, we focus on systems modeled by the heterogeneous overdamped Langevin equation. Here, the observed drift includes a "spurious” force term when the diffusivity varies in space. We show how Bayesian inference can be leveraged to reliably infer forces by taking into account such spurious forces of unknown amplitude as well as experimental sources of error. The method is based on marginalizing the force posterior over all possible spurious force contributions. The approach is combined with a Bayes factor statistical test for the presence of forces. The performance of our method is investigated analytically, numerically and tested on experimental data sets. The main results are obtained in a closed form allowing for direct exploration of their properties and fast computation. The method is incorporated into TRamWAy, an open-source software platform for automated analysis of biomolecule trajectories.

Measurements of Particle-Surface Interactions in Both Equilibrium and Nonequilibrium Systems

Total internal reflection microscopy (TIRM) is a passive technique that measures colloidal interactions in aqueous solution. A traditional Boltzmann method requires that particles must fluctuate around equilibrium positions for a long time. A method based on multiparticle tracking and drift velocity method was developed to measure interactions in both equilibrium and nonequilibrium systems. This method relaxed the limitation of the traditional Boltzmann method and do not require any external force like optical tweezer. Theoretical predictions of particle sedimentation under the influence of various forces were investigated to determine the proper particle size and solution properties. We found that the polystyrene (PS) particle with a size of 2.1 μm took the longest time to finish sedimentation, and 5% (w/w) sucrose was chosen to suppress the Brownian motion. For single and ensemble particles in equilibrium, the experimental diffusion coefficients and potential energy profiles were consistent with the theoretical prediction. In nonequilibrium experiments, the van der Waals force between the bare/hybrid particles and flat surface was measured, and the silica shell acted to strengthen the van der Waals attraction. This method extends the application of TIRM to nonequilibrium systems without any active control. Moreover, the silica-coated PS core-shell hybrid particles facilitate surface modification with a variety of active chemicals. It would be a great advantage to measure all kinds of long-range interactions between surface-modified particles and surface in aqueous solution with TIRM.

Conditional probabilities in multiplicative noise processes

We address the calculation of transition probabilities in multiplicative noise stochastic differential equations using a path integral approach. We show the equivalence between the conditional probability and the propagator of a quantum particle with variable mass. Introducing a time reparametrization, we are able to transform the problem of multiplicative noise fluctuations into an equivalent additive one. We illustrate the method by showing the explicit analytic computation of the conditional probability of a harmonic oscillator in a nonlinear multiplicative environment.

Protein concentration gradients and switching diffusions

Morphogen gradients play a vital role in developmental biology by enabling embryonic cells to infer their spatial location and determine their developmental fate accordingly. The standard mechanism for generating a morphogen gradient involves a morphogen being produced from a localized source and subsequently degrading. While this mechanism is effective over the length and time scales of tissue development, it fails over typical subcellular length scales due to the rapid dissipation of spatial asymmetries. In a recent theoretical work, we found an alternative mechanism for generating concentration gradients of diffusing molecules, in which the molecules switch between spatially constant diffusivities at switching rates that depend on the spatial location of a molecule. Independently, an experimental and computational study later found that Caenorhabditis elegans zygotes rely on this mechanism for cell polarization. In this paper, we extend our analysis of switching diffusivities to determine its role in protein concentration gradient formation. In particular, we determine how switching diffusivities modifies the standard theory and show how space-dependent switching diffusivities can yield a gradient in the absence of a localized source. Our mathematical analysis yields explicit formulas for the intracellular concentration gradient which closely match the results of previous experiments and numerical simulations.

Polymer margination in uniform shear flows

We address the issue of polymer margination (migration towards surfaces) in uniform shear flows through extensive LBMD (lattice-Boltzmann molecular dynamics) simulations. In particular we consider the effect of monomer size, a on the chain's overall margination tendency for chains of length N = 16, 32 monomers in flows at multiple shear rates [small gamma, Greek, dot above]. We observed higher margination of chains with larger radii monomers in comparison to smaller radii monomer chains of the same length N. We quantify this effect by considering various measures such as the distribution of the maximum extent of the chain into the channel bulk, zm, distribution of its center of mass in the direction normal to the surface, zc and the distributions of the chain's radius of gyration in directions parallel and perpendicular to the surface i.e. Rx, Ry and Rz respectively.

Escape rate for nonequilibrium processes dominated by strong non-detailed balance force

Quantifying the escape rate from a meta-stable state is essential to understand a wide range of dynamical processes. Kramers' classical rate formula is the product of an exponential function of the potential barrier height and a pre-factor related to the friction coefficient. Although many applications of the rate formula focused on the exponential term, the prefactor can have a significant effect on the escape rate in certain parameter regions, such as the overdamped limit and the underdamped limit. There have been continuous interests to understand the effect of non-detailed balance on the escape rate; however, how the prefactor behaves under strong non-detailed balance force remains elusive. In this work, we find that the escape rate formula has a vanishing prefactor with decreasing friction strength under the strong non-detailed balance limit. We both obtain analytical solutions in specific examples and provide a derivation for more general cases. We further verify the result by simulations and propose a testable experimental system of a charged Brownian particle in electromagnetic field. Our study demonstrates that a special care is required to estimate the effect of prefactor on the escape rate when non-detailed balance force dominates.

Potential landscape of high dimensional nonlinear stochastic dynamics with large noise

Quantifying stochastic processes is essential to understand many natural phenomena, particularly in biology, including the cell-fate decision in developmental processes as well as the genesis and progression of cancers. While various attempts have been made to construct potential landscape in high dimensional systems and to estimate transition rates, they are practically limited to the cases where either noise is small or detailed balance condition holds. A general and practical approach to investigate real-world nonequilibrium systems, which are typically high-dimensional and subject to large multiplicative noise and the breakdown of detailed balance, remains elusive. Here, we formulate a computational framework that can directly compute the relative probabilities between locally stable states of such systems based on a least action method, without the necessity of simulating the steady-state distribution. The method can be applied to systems with arbitrary noise intensities through A-type stochastic integration, which preserves the dynamical structure of the deterministic counterpart dynamics. We demonstrate our approach in a numerically accurate manner through solvable examples. We further apply the method to investigate the role of noise on tumor heterogeneity in a 38-dimensional network model for prostate cancer, and provide a new strategy on controlling cell populations by manipulating noise strength.

Removing the effect of blooming from potential energy measurement by employing total internal reflection microscopy integrated with video microscopy

Total internal reflection microscopy (TIRM) measures the interactions between a colloidal particle and a flat surface in aqueous solution. Recently, TIRM has further integrated with video microscopy (VM) and enabled the simultaneous measurements of multi-particle colloid-surface interactions in the same ensemble. However, there still remain challenges about accurate image acquisition due to blooming. Blooming means the number of photons reaching the detector exceeds its maximum capacity, and the excess photons will either spill to adjacent pixels or not be counted, leading to an obstacle from precise determination of intensity. Our result shows that blooming gives rise to a deviation of the measured potential energy from the classical theory of Derjaguin, Landau, Verway, and Overbeek (DLVO). Therefore, a correction method was developed in this work to deduce the real intensity from the experimental measurement. The relationship between scattered light intensity and exposure time deviates from linearity when blooming occurs. A correction equation was developed to recover the real intensity, which was then confirmed by the accordance between the corresponding potential energy profiles and the DLVO theory. This correction method is suitable for VM systems of colloidal probes illuminated by scattered light, broadening the application of VM imaging to investigate colloidal interactions.

Hydrodynamic and entropic effects on colloidal diffusion in corrugated channels

In the absence of advection, confined diffusion characterizes transport in many natural and artificial devices, such as ionic channels, zeolites, and nanopores. While extensive theoretical and numerical studies on this subject have produced many important predictions, experimental verifications of the predictions are rare. Here, we experimentally measure colloidal diffusion times in microchannels with periodically varying width and contrast results with predictions from the Fick-Jacobs theory and Brownian dynamics simulation. While the theory and simulation correctly predict the entropic effect of the varying channel width, they fail to account for hydrodynamic effects, which include both an overall decrease and a spatial variation of diffusivity in channels. Neglecting such hydrodynamic effects, the theory and simulation underestimate the mean and standard deviation of first passage times by 40% in channels with a neck width twice the particle diameter. We further show that the validity of the Fick-Jacobs theory can be restored by reformulating it in terms of the experimentally measured diffusivity. Our work thus shows that hydrodynamic effects play a key role in diffusive transport through narrow channels and should be included in theoretical and numerical models.

Driven Anisotropic Diffusion at Boundaries: Noise Rectification and Particle Sorting

We study the diffusive dynamics of a Brownian particle in the proximity of a flat surface under nonequilibrium conditions, which are created by an anisotropic thermal environment with different temperatures being active along distinct spatial directions. By presenting the exact time-dependent solution of the Fokker-Planck equation for this problem, we demonstrate that the interplay between anisotropic diffusion and hard-core interaction with the plain wall rectifies the thermal fluctuations and induces directed particle transport parallel to the surface, without any deterministic forces being applied in that direction. Based on current micromanipulation technologies, we suggest a concrete experimental setup to observe this novel noise-induced transport mechanism. We furthermore show that it is sensitive to particle characteristics, such that this setup can be used for sorting particles of different sizes.

Hybrid colored noise process with space-dependent switching rates

A fundamental issue in the theory of continuous stochastic process is the interpretation of multiplicative white noise, which is often referred to as the Itô-Stratonovich dilemma. From a physical perspective, this reflects the need to introduce additional constraints in order to specify the nature of the noise, whereas from a mathematical perspective it reflects an ambiguity in the formulation of stochastic differential equations (SDEs). Recently, we have identified a mechanism for obtaining an Itô SDE based on a form of temporal disorder. Motivated by switching processes in molecular biology, we considered a Brownian particle that randomly switches between two distinct conformational states with different diffusivities. In each state, the particle undergoes normal diffusion (additive noise) so there is no ambiguity in the interpretation of the noise. However, if the switching rates depend on position, then in the fast switching limit one obtains Brownian motion with a space-dependent diffusivity of the Itô form. In this paper, we extend our theory to include colored additive noise. We show that the nature of the effective multiplicative noise process obtained by taking both the white-noise limit (κ→0) and fast switching limit (ε→0) depends on the order the two limits are taken. If the white-noise limit is taken first, then we obtain Itô, and if the fast switching limit is taken first, then we obtain Stratonovich. Moreover, the form of the effective diffusion coefficient differs in the two cases. The latter result holds even in the case of space-independent transition rates, where one obtains additive noise processes with different diffusion coefficients. Finally, we show that yet another form of multiplicative noise is obtained in the simultaneous limit ε,κ→0 with ε/κ^{2} fixed.

Temporal disorder as a mechanism for spatially heterogeneous diffusion

A fundamental issue in analyzing diffusion in heterogeneous media is interpreting the space dependence of the associated diffusion coefficient. This reflects the well-known Ito-Stratonovich dilemma for continuous stochastic processes with multiplicative noise. In order to resolve this dilemma it is necessary to introduce additional constraints regarding the underlying physical system. Here we introduce a mechanism for generating nonlinear Brownian motion based on a form of temporal disorder. Motivated by switching processes in molecular biology, we consider a Brownian particle that randomly switches between two distinct conformational states with different diffusivities. In each state the particle undergoes normal diffusion (additive noise) so there is no ambiguity in the interpretation of the noise. However, if the switching rates depend on position, then in the fast-switching limit one obtains Brownian motion with a space-dependent diffusivity. We show that the resulting multiplicative noise process is of the Ito form. In particular, we solve a first-passage time problem for finite switching rates and show that the mean first-passage time reduces to the Ito version in the fast-switching limit.

Influence of external potentials on heterogeneous diffusion processes

In this paper we consider heterogeneous diffusion processes with the power-law dependence of the diffusion coefficient on the position and investigate the influence of external forces on the resulting anomalous diffusion. The heterogeneous diffusion processes can yield subdiffusion as well as superdiffusion, depending on the behavior of the diffusion coefficient. We assume that not only the diffusion coefficient but also the external force has a power-law dependence on the position. We obtain analytic expressions for the transition probability in two cases: when the power-law exponent in the external force is equal to 2η-1, where 2η is the power-law exponent in the dependence of the diffusion coefficient on the position, and when the external force has a linear dependence on the position. We found that the power-law exponent in the dependence of the mean square displacement on time does not depend on the external force; this force changes only the anomalous diffusion coefficient. In addition, the external force having the power-law exponent different from 2η-1 limits the time interval where the anomalous diffusion occurs. We expect that the results obtained in this paper may be relevant for a more complete understanding of anomalous diffusion processes.

Isothermal Langevin dynamics in systems with power-law spatially dependent friction

We study the dynamics of Brownian particles in a heterogeneous one-dimensional medium with a spatially dependent diffusion coefficient of the form D(x)∼|x|^{c}, at constant temperature. The particle's probability distribution function (PDF) is calculated both analytically, by solving Fick's diffusion equation, and from numerical simulations of the underdamped Langevin equation. At long times, the PDFs calculated by both approaches yield identical results, corresponding to subdiffusion for c<0 and superdiffusion for 0<c<1. For c>1, the diffusion equation predicts that the particles accelerate. Here we show that this phenomenon, previously considered in several works as an illustration for the possible dramatic effects of spatially dependent thermal noise, is unphysical. We argue that in an isothermal medium, the motion cannot exceed the ballistic limit (〈x^{2}〉∼t^{2}). The ballistic limit is reached when the friction coefficient drops sufficiently fast at large distances from the origin and is correctly captured by Langevin's equation.

First-Passage Times in d-Dimensional Heterogeneous Media

Although there are many theoretical studies of the mean first-passage time (MFPT), most neglect the diffusive heterogeneity of real systems. We present exact analytical expressions for the MFPT and residence times of a pointlike particle diffusing in a spherically symmetric d-dimensional heterogeneous system composed of two concentric media with different diffusion coefficients with an absorbing inner boundary (target) and a reflecting outer boundary. By varying the convention, e.g., Itō, Stratonovich, or isothermal, chosen to interpret the overdamped Langevin equation with multiplicative noise describing the diffusion process, we find different predictions and counterintuitive results for the residence time in the outer region and hence for the MFPT, while the residence time in the inner region is independent of the convention. This convention dependence of residence times and the MFPT could provide insights about the heterogeneous diffusion in a cell or in a tumor, or for animal and insect searches inside their home range.

Brownian colloids in underdamped and overdamped regimes with nonhomogeneous temperature

The motion of Brownian particles when temperature is spatially dependent is studied by stochastic simulations and theoretical analysis. Nonequilibrium steady probability distributions P(st)(z,v) for both underdamped and overdamped regimes are analyzed. The existence of local kinetic energy equipartition theorem is also discussed. The transition between both regimes is characterized by a dimensionless friction parameter. This study is applied to three physical systems of colloidal particles.

The Raspberry model for hydrodynamic interactions revisited. II. The effect of confinement

The so-called "raspberry" model refers to the hybrid lattice-Boltzmann (LB) and Langevin molecular dynamics schemes for simulating the dynamics of suspensions of colloidal particles, originally developed by Lobaskin and Dünweg [New J. Phys. 6, 54 (2004)], wherein discrete surface points are used to achieve fluid-particle coupling. In this paper, we present a follow up to our study of the effectiveness of the raspberry model in reproducing hydrodynamic interactions in the Stokes regime for spheres arranged in a simple-cubic crystal [Fischer et al., J. Chem. Phys. 143, 084107 (2015)]. Here, we consider the accuracy with which the raspberry model is able to reproduce such interactions for particles confined between two parallel plates. To this end, we compare our LB simulation results to established theoretical expressions and finite-element calculations. We show that there is a discrepancy between the translational and rotational mobilities when only surface coupling points are used, as also found in Part I of our joint publication. We demonstrate that adding internal coupling points to the raspberry can be used to correct said discrepancy in confining geometries as well. Finally, we show that the raspberry model accurately reproduces hydrodynamic interactions between a spherical colloid and planar walls up to roughly one LB lattice spacing.

Summing over trajectories of stochastic dynamics with multiplicative noise

We demonstrate that previous path integral formulations for the general stochastic interpretation generate incomplete results exemplified by the geometric Brownian motion. We thus develop a novel path integral formulation for the overdamped Langevin equation with multiplicative noise. The present path integral leads to the corresponding Fokker-Planck equation, and naturally generates a normalized transition probability in examples. Our result solves the inconsistency of the previous path integral formulations for the general stochastic interpretation, and can have wide applications in chemical and physical stochastic processes.

Hydrodynamics and Brownian motions of a spheroid near a rigid wall

In this work, we study in detail the hydrodynamics and the Brownian motions of a spheroidal particle suspended in a Newtonian fluid near a flat rigid wall. We employ 3D Finite Element Method (FEM) simulations to compute how the mobility tensor of the spheroid varies with both the particle-wall separation distance and the particle orientation. We then study the Brownian motion of the spheroid by means of a discretized Langevin equation. We specifically focus on the additional drift terms arising from the position and orientational dependence of the mobility matrix. In this respect, we also propose a numerically convenient approximation of the orientational divergence of the mobility matrix that is required in the solution of the Langevin equation. Our results illustrate that both hydrodynamics and Brownian motions of a spheroidal particle near a confining wall display novel features from those of a sphere in the same type of confinement.

Pseudochemotactic drifts of artificial microswimmers

We numerically investigate the motion of active artificial microswimmers diffusing in a fuel concentration gradient. We observe that, in the steady state, their probability density accumulates in the low-concentration regions, whereas a tagged swimmer drifts with velocity depending in modulus and orientation on how the concentration gradient affects the self-propulsion mechanism. Under most experimentally accessible conditions, the particle drifts toward the high-concentration regions (pseudochemotactic drift). A correct interpretation of experimental data must account for such an "anti-Fickian" behavior.

Overdamped limit and inverse-friction expansion for Brownian motion in an inhomogeneous medium

We revisit the problem of the overdamped (large-friction) limit of the Brownian dynamics in an inhomogeneous medium characterized by a position-dependent friction coefficient and a multiplicative noise (local temperature) in one-dimensional space. Starting from the Kramers equation and analyzing it through the expansion in terms of eigenfunctions of a quantum harmonic oscillator, we derive analytically the corresponding Fokker-Planck equation in the overdamped limit. The result is fully consistent with the previous finding by Sancho, San Miguel, and Dürr [J. Stat. Phys. 28, 291 (1982)]. Our method allows us to generalize the Brinkman's hierarchy, and thus it would be straightforward to obtain higher-order corrections in a systematic inverse-friction expansion without any assumption. Our results are confirmed by numerical simulations for simple examples.

Langevin dynamics for vector variables driven by multiplicative white noise: A functional formalism

We discuss general multidimensional stochastic processes driven by a system of Langevin equations with multiplicative white noise. In particular, we address the problem of how time reversal diffusion processes are affected by the variety of conventions available to deal with stochastic integrals. We present a functional formalism to build up the generating functional of correlation functions without any type of discretization of the Langevin equations at any intermediate step. The generating functional is characterized by a functional integration over two sets of commuting variables, as well as Grassmann variables. In this representation, time reversal transformation became a linear transformation in the extended variables, simplifying in this way the complexity introduced by the mixture of prescriptions and the associated calculus rules. The stochastic calculus is codified in our formalism in the structure of the Grassmann algebra. We study some examples such as higher order derivative Langevin equations and the functional representation of the micromagnetic stochastic Landau-Lifshitz-Gilbert equation.

Anomalous diffusion in stochastic systems with nonhomogeneously distributed traps

The stochastic motion in a nonhomogeneous medium with traps is studied and diffusion properties of that system are discussed. The particle is subjected to a stochastic stimulation obeying a general Lévy stable statistics and experiences long rests due to nonhomogeneously distributed traps. The memory is taken into account by subordination of that process to a random time; then the subordination equation is position dependent. The problem is approximated by a decoupling of the medium structure and memory and exactly solved for a power-law position dependence of the memory. In the case of the Gaussian statistics, the density distribution and moments are derived: depending on geometry and memory parameters, the system may reveal both the subdiffusion and enhanced diffusion. The similar analysis is performed for the Lévy flights where the finiteness of the variance follows from a variable noise intensity near a boundary. Two diffusion regimes are found: in the bulk and near the surface. The anomalous diffusion exponent as a function of the system parameters is derived.

Controlling symmetry-breaking states by a hidden quantity in multiplicative noise

The inhomogeneity of multiplicative white noise leads to various coupling modes between deterministic and stochastic forces. We investigate the phase transition induced by the variation of the coupling mode through manipulating its characteristic parameter continuously. Even when the noise strength is fixed, an increase of this parameter can enhance or inhibit the symmetry-breaking state. We also propose a scheme to implement these phase transitions experimentally. Our result demonstrates that the coupling mode previously considered to be a mathematical convention serves as an additional quantity leading to physically observable phase transitions. This observation provides a mechanism to control the effect of noise without regulating the noise strength.

Nonequilibrium work relation beyond the Boltzmann-Gibbs distribution

The presence of multiplicative noise can alter measurements of forces acting on nanoscopic objects. Taking into account of multiplicative noise, we derive a series of nonequilibrium thermodynamical equalities as generalization of the Jarzynski equality, the detailed fluctuation theorem and the Hatano-Sasa relation. Our result demonstrates that the Jarzynski equality and the detailed fluctuation theorem remains valid only for systems with the Boltzmann-Gibbs distribution at the equilibrium state, but the Hatano-Sasa relation is robust with respect to different stochastic interpretations of multiplicative noise.

Time reversibility and nonequilibrium thermodynamics of second-order stochastic processes

Nonequilibrium thermodynamics of a general second-order stochastic system is investigated. We prove that at steady state, under inversion of velocities, the condition of time reversibility over the phase space is equivalent to the antisymmetry of spatial flux and the symmetry of velocity flux. Then we show that the condition of time reversibility alone cannot always guarantee the Maxwell-Boltzmann distribution. Comparing the two conditions together, we find that the frictional force naturally emerges as the unique odd term of the total force at thermodynamic equilibrium, and is followed by the Einstein relation. The two conditions respectively correspond to two previously reported different entropy production rates. In the case where the external force is only position dependent, the two entropy production rates become one. We prove that such an entropy production rate can be decomposed into two non-negative terms, expressed respectively by the conditional mean and variance of the thermodynamic force associated with the irreversible velocity flux at any given spatial coordinate. In the small inertia limit, the former term becomes the entropy production rate of the corresponding overdamped dynamics, while the anomalous entropy production rate originates from the latter term. Furthermore, regarding the connection between the first law and second law, we find that in the steady state of such a limit, the anomalous entropy production rate is also the leading order of the Boltzmann-factor weighted difference between the spatial heat dissipation densities of the underdamped and overdamped dynamics, while their unweighted difference always tends to vanish.

Interacting particles in a periodically moving potential: traveling wave and transport

We study a system of interacting particles in a periodically moving external potential, within the simplest possible description of paradigmatic symmetric exclusion process on a ring. The model describes diffusion of hardcore particles where the diffusion dynamics is locally modified at a uniformly moving defect site, mimicking the effect of the periodically moving external potential. The model, though simple, exhibits remarkably rich features in particle transport, such as polarity reversal and double peaks in particle current upon variation of defect velocity and particle density. By tuning these variables, the most efficient transport can be achieved in either direction along the ring. These features can be understood in terms of a traveling density wave propagating in the system. Our results could be experimentally tested, e.g., in a system of colloidal particles driven by a moving optical tweezer.

Langevin dynamics in inhomogeneous media: re-examining the Itô-Stratonovich dilemma

The diffusive dynamics of a particle in a medium with space-dependent friction coefficient is studied within the framework of the inertial Langevin equation. In this description, the ambiguous interpretation of the stochastic integral, known as the Itô-Stratonovich dilemma, is avoided since all interpretations converge to the same solution in the limit of small time steps. We use a newly developed method for Langevin simulations to measure the probability distribution of a particle diffusing in a flat potential. Our results reveal that both the Itô and Stratonovich interpretations converge very slowly to the uniform equilibrium distribution for vanishing time step sizes. Three other conventions exhibit significantly improved accuracy: (i) the "isothermal" (Hänggi) convention, (ii) the Stratonovich convention corrected by a drift term, and (iii) a newly proposed convention employing two different effective friction coefficients representing two different averages of the friction function during the time step. We argue that the most physically accurate dynamical description is provided by the third convention, in which the particle experiences a drift originating from the dissipation instead of the fluctuation term. This feature is directly related to the fact that the drift is a result of an inertial effect that cannot be well understood in the Brownian, overdamped limit of the Langevin equation.

Optical trapping and manipulation of nanostructures

Optical trapping and manipulation of micrometre-sized particles was first reported in 1970. Since then, it has been successfully implemented in two size ranges: the subnanometre scale, where light-matter mechanical coupling enables cooling of atoms, ions and molecules, and the micrometre scale, where the momentum transfer resulting from light scattering allows manipulation of microscopic objects such as cells. But it has been difficult to apply these techniques to the intermediate - nanoscale - range that includes structures such as quantum dots, nanowires, nanotubes, graphene and two-dimensional crystals, all of crucial importance for nanomaterials-based applications. Recently, however, several new approaches have been developed and demonstrated for trapping plasmonic nanoparticles, semiconductor nanowires and carbon nanostructures. Here we review the state-of-the-art in optical trapping at the nanoscale, with an emphasis on some of the most promising advances, such as controlled manipulation and assembly of individual and multiple nanostructures, force measurement with femtonewton resolution, and biosensors.

Self-consistent colloidal energy and diffusivity landscapes in macromolecular solutions

We report a dynamic analysis to simultaneously measure colloidal forces and hydrodynamic interactions in the presence of both adsorbed and unadsorbed macromolecules. A Bayesian inference method is used to self-consistently obtain the position-dependent potential energy (i.e., energy landscape) and diffusivity (i.e., diffusivity landscape) from measured colloidal trajectories normal to a wall. Measurements are performed for particles and surfaces with adsorbed polyethylene oxide (PEO) copolymer as a function of unadsorbed PEO homopolymer concentration. Energy landscapes are well described by a steric repulsion between adsorbed brushes and depletion attraction due to unadsorbed macromolecules. Diffusivity landscapes show agreement with predicted short-range permeable brush models and long-range mobilities determined by the bulk solution viscosity. Lower than expected mobilities in the vicinity of overlapping depletion layers are attributed to interactions of adsorbed and unadsorbed macromolecules altering nonconservative lubrication forces.

Drift in Diffusion Gradients

The longstanding problem of Brownian transport in a heterogeneous quasi one-dimensional medium with space-dependent self-diffusion coefficient is addressed in the overdamped (zero mass) limit. A satisfactory mesoscopic description is obtained in the Langevin equation formalism by introducing an appropriate drift term, which depends on the system macroscopic observables, namely the diffuser concentration and current. The drift term is related to the microscopic properties of the medium. The paradoxical existence of a finite drift at zero current suggests the possibility of designing a Maxwell demon operating between two equilibrium reservoirs at the same temperature.

Brownian motion in inhomogeneous suspensions

The Langevin description of Brownian motion in inhomogeneous suspensions is here revisited. Inhomogeneous suspensions are characterized by a position-dependent friction coefficient, which can significantly influence the dynamics of the suspended particles. Outstanding examples are suspensions in confinement or in the presence of a temperature gradient. The Langevin approach in inhomogeneous systems encounters a fundamental difficulty related to the interpretation of the multiplicative noise induced by the position-dependent friction. We show that the so-called Ito-Stratonovich dilemma is originated by the violation of the macroscopic force balance condition in the traditional procedure of eliminating the fast variables. Repairing this deficit, we rederive the extended overdamped Langevin equation directly from the infradamped Langevin equation. This is without invoking the Fokker-Planck formalism, such that the self-completeness of the Langevin framework is restored. Furthermore, we derive the generalized forms of the drift-force relation and the Smoluchowski equation for inhomogeneous suspensions in a straightforward manner.

Exploring a noisy van der Pol type oscillator with a stochastic approach

Based on conventional Ito or Stratonovich interpretation, zero-mean multiplicative noise can induce shifts of attractors or even changes of topology to a deterministic dynamics. Such phenomena usually introduce additional complications in analysis of these systems. We employ in this paper a new stochastic interpretation leading to a straightforward consequence: The steady state distribution is Boltzmann-Gibbs type with a potential function severing as a Lyapunov function for the deterministic dynamics. It implies that an attractor corresponds to the local extremum of the distribution function and the probability is equally distributed right on an attractor. We consider a prototype of nonequilibrium processes, noisy limit cycle dynamics. Exact results are obtained for a class of limit cycles, including a van der Pol type oscillator. These results provide a new angle for understanding processes without detailed balance and can be verified by experiments.

Dynamical behaviors determined by the Lyapunov function in competitive Lotka-Volterra systems

Dynamical behaviors of the competitive Lotka-Volterra system even for 3 species are not fully understood. In this paper, we study this problem from the perspective of the Lyapunov function. We construct explicitly the Lyapunov function using three examples of the competitive Lotka-Volterra system for the whole state space: (1) the general 2-species case, (2) a 3-species model, and (3) the model of May-Leonard. The basins of attraction for these examples are demonstrated, including cases with bistability and cyclical behavior. The first two examples are the generalized gradient system, where the energy dissipation may not follow the gradient of the Lyapunov function. In addition, under a new type of stochastic interpretation, the Lyapunov function also leads to the Boltzmann-Gibbs distribution on the final steady state when multiplicative noise is added.

Drift velocity in non-isothermal inhomogeneous systems

Drift velocity and driving force are not directly proportional in the case of inhomogeneous suspensions, where a space dependent mobility induces an additional contribution to the drift velocity. Similarly, particle flux and drift velocity are related not only by the gradient of density but also by an additional contribution given by the gradient of the self-diffusion coefficient. We provide quantitative support to this scenario in a non-equilibrium system by means of computer simulations with a temperature gradient. Moreover, our simulation results demonstrate that the temperature gradient-induced mass transport coefficient, namely thermal diffusion coefficient, is not directly proportional to the drift velocity so that the well-accepted relation of proportionality is just an approximation.