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Singh P. Extreme value statistics and arcsine laws for heterogeneous diffusion processes. Phys Rev E 2022; 105:024113. [PMID: 35291128 DOI: 10.1103/physreve.105.024113] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/22/2021] [Accepted: 01/26/2022] [Indexed: 06/14/2023]
Abstract
Heterogeneous diffusion with a spatially changing diffusion coefficient arises in many experimental systems such as protein dynamics in the cell cytoplasm, mobility of cajal bodies, and confined hard-sphere fluids. Here, we showcase a simple model of heterogeneous diffusion where the diffusion coefficient D(x) varies in a power-law way, i.e., D(x)∼|x|^{-α} with the exponent α>-1. This model is known to exhibit anomalous scaling of the mean-squared displacement (MSD) of the form ∼t^{2/2+α} and weak ergodicity breaking in the sense that ensemble averaged and time averaged MSDs do not converge. In this paper, we look at the extreme value statistics of this model and derive, for all α, the exact probability distributions of the maximum spatial displacement M(t) and arg-maximum t_{m}(t) (i.e., the time at which this maximum is reached) till duration t. In the second part of our paper, we analyze the statistical properties of the residence time t_{r}(t) and the last-passage time t_{ℓ}(t) and compute their distributions exactly for all values of α. Our study unravels that the heterogeneous version (α≠0) displays many rich and contrasting features compared to that of the standard Brownian motion (BM). For example, while for BM (α=0), the distributions of t_{m}(t),t_{r}(t), and t_{ℓ}(t) are all identical (á la "arcsine laws" due to Lévy), they turn out to be significantly different for nonzero α. Another interesting property of t_{r}(t) is the existence of a critical α (which we denote by α_{c}=-0.3182) such that the distribution exhibits a local maximum at t_{r}=t/2 for α<α_{c} whereas it has minima at t_{r}=t/2 for α≥α_{c}. The underlying reasoning for this difference hints at the very contrasting natures of the process for α≥α_{c} and α<α_{c} which we thoroughly examine in our paper. All our analytical results are backed by extensive numerical simulations.
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Affiliation(s)
- Prashant Singh
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
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Serov AS, Laurent F, Floderer C, Perronet K, Favard C, Muriaux D, Westbrook N, Vestergaard CL, Masson JB. Statistical Tests for Force Inference in Heterogeneous Environments. Sci Rep 2020; 10:3783. [PMID: 32123194 PMCID: PMC7052274 DOI: 10.1038/s41598-020-60220-1] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/20/2019] [Accepted: 02/05/2020] [Indexed: 01/22/2023] Open
Abstract
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.
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Affiliation(s)
- Alexander S Serov
- Decision and Bayesian Computation, USR 3756 (C3BI/DBC) & Neuroscience department CNRS UMR 3751, Institut Pasteur, CNRS, Paris, France.
| | - François Laurent
- Decision and Bayesian Computation, USR 3756 (C3BI/DBC) & Neuroscience department CNRS UMR 3751, Institut Pasteur, CNRS, Paris, France
| | - Charlotte Floderer
- Infectious Disease Research Institute of Montpellier, CNRS UMR 9004, University of Montpellier, Montpellier, France
| | - Karen Perronet
- Laboratoire Charles Fabry, Université Paris-Saclay, Institut d'Optique Graduate School, CNRS UMR8501, 91127, Palaiseau Cedex, France
| | - Cyril Favard
- Infectious Disease Research Institute of Montpellier, CNRS UMR 9004, University of Montpellier, Montpellier, France
| | - Delphine Muriaux
- Infectious Disease Research Institute of Montpellier, CNRS UMR 9004, University of Montpellier, Montpellier, France
| | - Nathalie Westbrook
- Laboratoire Charles Fabry, Université Paris-Saclay, Institut d'Optique Graduate School, CNRS UMR8501, 91127, Palaiseau Cedex, France
| | - Christian L Vestergaard
- Decision and Bayesian Computation, USR 3756 (C3BI/DBC) & Neuroscience department CNRS UMR 3751, Institut Pasteur, CNRS, Paris, France.
| | - Jean-Baptiste Masson
- Decision and Bayesian Computation, USR 3756 (C3BI/DBC) & Neuroscience department CNRS UMR 3751, Institut Pasteur, CNRS, Paris, France.
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Niether D, Wiegand S. Thermophoresis of biological and biocompatible compounds in aqueous solution. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2019; 31:503003. [PMID: 31491783 DOI: 10.1088/1361-648x/ab421c] [Citation(s) in RCA: 33] [Impact Index Per Article: 6.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/20/2023]
Abstract
With rising popularity of microscale thermophoresis for the characterisation of protein-ligand binding reactions and possible applications in microfluidic devices, there is a growing interest in considering thermodiffusion in the context of life sciences. But although the understanding of thermodiffusion in non-polar mixtures has grown rapidly in recent years, predictions for associated mixtures like aqueous solutions remain challenging. This review aims to give an overview of the literature on thermodiffusion in aqueous systems, show the difficulties in theoretical description that arise from the non-ideal behaviour of water-mixtures, and highlight the relevance of thermodiffusion in a biological context. We find that the thermodiffusion in aqueous systems is dominated by contributions from heat of transfer, hydrogen bond interactions and charge effects. However, the separation of these effects is often difficult, especially in case of biological systems where a systematic exclusion of contributions may not be feasible.
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Affiliation(s)
- D Niether
- ICS-3 Soft Condensed Matter, Forschungszentrum Jülich GmbH, D-52428 Jülich, Germany
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Semenov SN, Schimpf ME. Thermoosomosis in microfluidic devices containing a temperature gradient normal to the channel walls. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2019; 42:141. [PMID: 31720868 DOI: 10.1140/epje/i2019-11914-8] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/21/2019] [Accepted: 10/11/2019] [Indexed: 06/10/2023]
Abstract
We analyze a microfluidic pump from the literature that utilizes a flat channel with boundary walls at different temperatures and tilted elongated pillars within in order to construct an adequate theory for designing devices in which the temperature gradient between channel walls is transformed into a longitudinal temperature gradient along the channel length. The action of the device is based on thermoosmosis in the secondary longitudinal temperature gradient associated with the specific geometry of the device, which can be described using physicochemical hydrodynamics without invoking the concept of thermophoretic force. We also describe a rotating drive device based on the same principle and design.
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Abstract
Colloidal migration in a temperature gradient is referred to as thermophoresis. In contrast to particles with a spherical shape, we show that elongated colloids may have a thermophoretic response that varies with the colloid orientation. Remarkably, this can translate into a non-vanishing thermophoretic force in the direction perpendicular to the temperature gradient. Opposite to the friction force, the thermophoretic force of a rod oriented with the temperature gradient can be larger or smaller than when oriented perpendicular to it. The precise anisotropic thermophoretic behavior clearly depends on the colloidal rod aspect ratio, and also on its surface details, which provides an interesting tunability to the devices constructed based on this principle. By means of mesoscale hydrodynamic simulations, we characterize this effect for different types of rod-like colloids.
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Affiliation(s)
- Zihan Tan
- Theoretical Soft-Matter and Biophysics, Institute of Complex Systems, Forschungszentrum Jülich, 52425 Jülich, Germany.
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Pieprzyk S, Heyes DM, Brańka AC. Spatially dependent diffusion coefficient as a model for pH sensitive microgel particles in microchannels. BIOMICROFLUIDICS 2016; 10:054118. [PMID: 27795750 PMCID: PMC5065575 DOI: 10.1063/1.4964935] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/12/2016] [Accepted: 10/04/2016] [Indexed: 05/23/2023]
Abstract
Solute transport and intermixing in microfluidic devices is strongly dependent on diffusional processes. Brownian Dynamics simulations of pressure-driven flow of model microgel particles in microchannels have been carried out to explore these processes and the factors that influence them. The effects of a pH-field that induces a spatial dependence of particle size and consequently the self-diffusion coefficient and system thermodynamic state were focused on. Simulations were carried out in 1D to represent some of the cross flow dependencies, and in 2D and 3D to include the effects of flow and particle concentration, with typical stripe-like diffusion coefficient spatial variations. In 1D, the mean square displacement and particle displacement probability distribution function agreed well with an analytically solvable model consisting of infinitely repulsive walls and a discontinuous pH-profile in the middle of the channel. Skew category Brownian motion and non-Gaussian dynamics were observed, which follows from correlations of step lengths in the system, and can be considered to be an example of so-called "diffusing diffusivity." In Poiseuille flow simulations, the particles accumulated in regions of larger diffusivity and the largest particle concentration throughput was found when this region was in the middle of the channel. The trends in the calculated cross-channel diffusional behavior were found to be very similar in 2D and 3D.
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Affiliation(s)
- S Pieprzyk
- Institute of Molecular Physics, Polish Academy of Sciences , M. Smoluchowskiego 17, 60-179 Poznań, Poland
| | - D M Heyes
- Department of Physics, Royal Holloway, University of London , Egham, Surrey TW20 0EX, United Kingdom
| | - A C Brańka
- Institute of Molecular Physics, Polish Academy of Sciences , M. Smoluchowskiego 17, 60-179 Poznań, Poland
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Ghosh PK, Li Y, Marchesoni F, Nori F. Pseudochemotactic drifts of artificial microswimmers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:012114. [PMID: 26274132 DOI: 10.1103/physreve.92.012114] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/22/2015] [Indexed: 06/04/2023]
Abstract
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.
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Affiliation(s)
- Pulak K Ghosh
- Department of Chemistry, Presidency University, Kolkata 700073, India
| | - Yunyun Li
- Center for Phononics and Thermal Energy Science, School of Physics Science and Engineering, Tongji University, Shanghai 200092, People's Republic of China
| | - Fabio Marchesoni
- Center for Phononics and Thermal Energy Science, School of Physics Science and Engineering, Tongji University, Shanghai 200092, People's Republic of China
- Dipartimento di Fisica, Università di Camerino, I-62032 Camerino, Italy
| | - Franco Nori
- CEMS, RIKEN, Saitama, 351-0198, Japan
- Physics Department, University of Michigan, Ann Arbor, Michigan 48109, USA
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Cherstvy AG, Metzler R. Nonergodicity, fluctuations, and criticality in heterogeneous diffusion processes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:012134. [PMID: 25122278 DOI: 10.1103/physreve.90.012134] [Citation(s) in RCA: 56] [Impact Index Per Article: 5.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/12/2014] [Indexed: 06/03/2023]
Abstract
We study the stochastic behavior of heterogeneous diffusion processes with the power-law dependence D(x) ∼ |x|(α) of the generalized diffusion coefficient encompassing sub- and superdiffusive anomalous diffusion. Based on statistical measures such as the amplitude scatter of the time-averaged mean-squared displacement of individual realizations, the ergodicity breaking and non-Gaussianity parameters, as well as the probability density function P(x,t), we analyze the weakly nonergodic character of the heterogeneous diffusion process and, particularly, the degree of irreproducibility of individual realizations. As we show, the fluctuations between individual realizations increase with growing modulus |α| of the scaling exponent. The fluctuations appear to diverge when the critical value α = 2 is approached, while for even larger α the fluctuations decrease, again. At criticality, the power-law behavior of the mean-squared displacement changes to an exponentially fast growth, and the fluctuations of the time-averaged mean-squared displacement do not converge for increasing number of realizations. From a systematic comparison we observe some striking similarities of the heterogeneous diffusion process with the familiar subdiffusive continuous time random walk process with power-law waiting time distribution and diverging characteristic waiting time.
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Affiliation(s)
- A G Cherstvy
- Institute for Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - R Metzler
- Institute for Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany and Department of Physics, Tampere University of Technology, 33101 Tampere, Finland
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Hannaoui R, Galliero G, Hoang H, Boned C. Influence of confinement on thermodiffusion. J Chem Phys 2014; 139:114704. [PMID: 24070302 DOI: 10.1063/1.4821128] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
This work focuses on a possible influence of a nanoporous medium on the thermodiffusion of a fluid "isotopic" mixture. To do so, we performed molecular dynamics simulations of confined Lennard-Jones binary equimolar mixtures using grand-canonical like and non-equilibrium approaches in sub- and super-critical conditions. The study was conducted in atomistic slit pore of three adsorbent natures for various widths (from 5 to 35 times the size of a molecule). The simulation results indicate that for all thermodynamic conditions and whatever the pore characteristics, the confinement has a negligible effect on the thermal diffusion factor/Soret coefficient. However, when considered separately, the mass diffusion and thermodiffusion coefficients have been found to be largely influenced by the pore characteristics. These two coefficients decrease noticeably when adsorption is stronger and pore width smaller, a behavior that is consistent with a simple hydrodynamic explanation.
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Affiliation(s)
- Rachid Hannaoui
- LFC-R (UMR5150 with CNRS and TOTAL), Université de Pau et des Pays de l'Adour, BP 1155, F-64013 Pau Cedex, France
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Yang M, Ripoll M. Brownian motion in inhomogeneous suspensions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:062110. [PMID: 23848630 DOI: 10.1103/physreve.87.062110] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/08/2012] [Indexed: 06/02/2023]
Abstract
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.
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Affiliation(s)
- Mingcheng Yang
- Theoretical Soft-Matter and Biophysics, Institute of Complex Systems, Forschungszentrum Jülich, 52425 Jülich, Germany.
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