Forced thermal ratchets.
TL;DR: In this paper, the authors considered a Brownian particle in a periodic potential under heavy damping and showed that if the particle is subject to an external force having time correlations, detailed balance is lost and the particle can exhibit a nonzero net drift speed.
Abstract: We consider a Brownian particle in a periodic potential under heavy damping. The second law forbids it from displaying any net drift speed, even if the symmetry of the potential is broken. But if the particle is subject to an external force having time correlations, detailed balance is lost and the particle can exhibit a nonzero net drift speed. Thus, broken symmetry and time correlations are sufficient ingredients for transport.
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TL;DR: A review of the physics of small volumes (nanoliters) of fluids is presented, as parametrized by a series of dimensionless numbers expressing the relative importance of various physical phenomena as mentioned in this paper.
Abstract: Microfabricated integrated circuits revolutionized computation by vastly reducing the space, labor, and time required for calculations. Microfluidic systems hold similar promise for the large-scale automation of chemistry and biology, suggesting the possibility of numerous experiments performed rapidly and in parallel, while consuming little reagent. While it is too early to tell whether such a vision will be realized, significant progress has been achieved, and various applications of significant scientific and practical interest have been developed. Here a review of the physics of small volumes (nanoliters) of fluids is presented, as parametrized by a series of dimensionless numbers expressing the relative importance of various physical phenomena. Specifically, this review explores the Reynolds number Re, addressing inertial effects; the Peclet number Pe, which concerns convective and diffusive transport; the capillary number Ca expressing the importance of interfacial tension; the Deborah, Weissenberg, and elasticity numbers De, Wi, and El, describing elastic effects due to deformable microstructural elements like polymers; the Grashof and Rayleigh numbers Gr and Ra, describing density-driven flows; and the Knudsen number, describing the importance of noncontinuum molecular effects. Furthermore, the long-range nature of viscous flows and the small device dimensions inherent in microfluidics mean that the influence of boundaries is typically significant. A variety of strategies have been developed to manipulate fluids by exploiting boundary effects; among these are electrokinetic effects, acoustic streaming, and fluid-structure interactions. The goal is to describe the physics behind the rich variety of fluid phenomena occurring on the nanoliter scale using simple scaling arguments, with the hopes of developing an intuitive sense for this occasionally counterintuitive world.
4,044 citations
Cites background from "Forced thermal ratchets."
...…and Austin, 1998; Ertas ’ , 1998; Chou et al., 1999; Austin et al., 2002; Cabodi et al., 2002; Huang, Silberzan, et al., 2002 , related to theoretical ideas involving “ratchet” potentials Ajdari and Prost, 1992; Magnasco, 1993; Astumian and Bier, 1994; Prost et al., 1994; Bier and Astumian, 1996 ....
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TL;DR: In this paper, the basic laws describing the essential aspects of collective motion are reviewed and a discussion of the various facets of this highly multidisciplinary field, including experiments, mathematical methods and models for simulations, are provided.
Abstract: We review the observations and the basic laws describing the essential aspects of collective motion -- being one of the most common and spectacular manifestation of coordinated behavior Our aim is to provide a balanced discussion of the various facets of this highly multidisciplinary field, including experiments, mathematical methods and models for simulations, so that readers with a variety of background could get both the basics and a broader, more detailed picture of the field The observations we report on include systems consisting of units ranging from macromolecules through metallic rods and robots to groups of animals and people Some emphasis is put on models that are simple and realistic enough to reproduce the numerous related observations and are useful for developing concepts for a better understanding of the complexity of systems consisting of many simultaneously moving entities As such, these models allow the establishing of a few fundamental principles of flocking In particular, it is demonstrated, that in spite of considerable differences, a number of deep analogies exist between equilibrium statistical physics systems and those made of self-propelled (in most cases living) units In both cases only a few well defined macroscopic/collective states occur and the transitions between these states follow a similar scenario, involving discontinuity and algebraic divergences
2,120 citations
TL;DR: In this paper, the main emphasis is put on directed transport in so-called Brownian motors (ratchets), i.e. a dissipative dynamics in the presence of thermal noise and some prototypical perturbation that drives the system out of equilibrium without introducing a priori an obvious bias into one or the other direction of motion.
2,098 citations
TL;DR: In this paper, the constructive role of Brownian motion is exemplified for various physical and technological setups, which are inspired by the cellular molecular machinery: the working principles and characteristics of stylized devices are discussed to show how fluctuations, either thermal or extrinsic, can be used to control diffusive particle transport.
Abstract: In systems possessing spatial or dynamical symmetry breaking, Brownian motion combined with unbiased external input signals, deterministic and random alike, can assist directed motion of particles at submicron scales. In such cases, one speaks of ``Brownian motors.'' In this review the constructive role of Brownian motion is exemplified for various physical and technological setups, which are inspired by the cellular molecular machinery: the working principles and characteristics of stylized devices are discussed to show how fluctuations, either thermal or extrinsic, can be used to control diffusive particle transport. Recent experimental demonstrations of this concept are surveyed with particular attention to transport in artificial, i.e., nonbiological, nanopores, lithographic tracks, and optical traps, where single-particle currents were first measured. Much emphasis is given to two- and three-dimensional devices containing many interacting particles of one or more species; for this class of artificial motors, noise rectification results also from the interplay of particle Brownian motion and geometric constraints. Recently, selective control and optimization of the transport of interacting colloidal particles and magnetic vortices have been successfully achieved, thus leading to the new generation of microfluidic and superconducting devices presented here. The field has recently been enriched with impressive experimental achievements in building artificial Brownian motor devices that even operate within the quantum domain by harvesting quantum Brownian motion. Sundry akin topics include activities aimed at noise-assisted shuttling other degrees of freedom such as charge, spin, or even heat and the assembly of chemical synthetic molecular motors. This review ends with a perspective for future pathways and potential new applications.
1,319 citations
Cites background from "Forced thermal ratchets."
...The early prediction of this transport effect, based on simple perturbation arguments (Bartussek et al., 1994; Doering et al., 1994; Luczka et al., 1995; Magnasco, 1993; Millonas and Dykman, 1994), kindled a widespread interest in the thermodynamics of molecular motors....
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...Most notably, Ajdari and Prost (1992); Astumian and Bier (1994); Bartussek et al. (1994); Chauwin et al. (1994); Doering et al. (1994); Magnasco (1993); Millonas and Dykman (1994); Prost et al. (1994); Rousselet et al. (1994) helped ignite a tumultuous activity in this topical area which kept…...
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TL;DR: In this paper, a comprehensive approach to classify all components of the yeast NPC (nucleoporins) was taken, which involved identifying all the proteins present in a highly enriched NPC fraction, determining which of these proteins were nucleopors, and localizing each nucleoporin within the NPC.
Abstract: An understanding of how the nuclear pore complex (NPC) mediates nucleocytoplasmic exchange requires a comprehensive inventory of the molecular components of the NPC and a knowledge of how each component contributes to the overall structure of this large molecular translocation machine. Therefore, we have taken a comprehensive approach to classify all components of the yeast NPC (nucleoporins). This involved identifying all the proteins present in a highly enriched NPC fraction, determining which of these proteins were nucleoporins, and localizing each nucleoporin within the NPC. Using these data, we present a map of the molecular architecture of the yeast NPC and provide evidence for a Brownian affinity gating mechanism for nucleocytoplasmic transport.
1,319 citations
References
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Book•
01 Jan 1981
TL;DR: In this article, the authors introduce the Fokker-planck equation, the Langevin approach, and the diffusion type of the master equation, as well as the statistics of jump events.
Abstract: Preface to the first edition. Preface to the second edition. Abbreviated references. I. Stochastic variables. II. Random events. III. Stochastic processes. IV. Markov processes. V. The master equation. VI. One-step processes. VII. Chemical reactions. VIII. The Fokker-Planck equation. IX. The Langevin approach. X. The expansion of the master equation. XI. The diffusion type. XII. First-passage problems. XIII. Unstable systems. XIV. Fluctuations in continuous systems. XV. The statistics of jump events. XVI. Stochastic differential equations. XVII. Stochastic behavior of quantum systems.
7,858 citations
01 Jan 1992
Abstract: Preface to the first edition. Preface to the second edition. Abbreviated references. I. Stochastic variables. II. Random events. III. Stochastic processes. IV. Markov processes. V. The master equation. VI. One-step processes. VII. Chemical reactions. VIII. The Fokker-Planck equation. IX. The Langevin approach. X. The expansion of the master equation. XI. The diffusion type. XII. First-passage problems. XIII. Unstable systems. XIV. Fluctuations in continuous systems. XV. The statistics of jump events. XVI. Stochastic differential equations. XVII. Stochastic behavior of quantum systems.
6,887 citations
Book•
01 Jan 1986
TL;DR: Molecular cell biology, Molecular cell biology , مرکز فناوری اطلاعات و اصاع رسانی, کδاوρزی
Abstract: Molecular cell biology , Molecular cell biology , مرکز فناوری اطلاعات و اطلاع رسانی کشاورزی
6,754 citations
Book•
07 Feb 2012
TL;DR: In this paper, the Fokker-Planck Equation for N Variables (FPE) was extended to N = 1 variable and N = 2 variables, where N is the number of variables in the system.
Abstract: 1. Introduction.- 1.1 Brownian Motion.- 1.1.1 Deterministic Differential Equation.- 1.1.2 Stochastic Differential Equation.- 1.1.3 Equation of Motion for the Distribution Function.- 1.2 Fokker-Planck Equation.- 1.2.1 Fokker-Planck Equation for One Variable.- 1.2.2 Fokker-Planck Equation for N Variables.- 1.2.3 How Does a Fokker-Planck Equation Arise?.- 1.2.4 Purpose of the Fokker-Planck Equation.- 1.2.5 Solutions of the Fokker-Planck Equation.- 1.2.6 Kramers and Smoluchowski Equations.- 1.2.7 Generalizations of the Fokker-Planck Equation.- 1.3 Boltzmann Equation.- 1.4 Master Equation.- 2. Probability Theory.- 2.1 Random Variable and Probability Density.- 2.2 Characteristic Function and Cumulants.- 2.3 Generalization to Several Random Variables.- 2.3.1 Conditional Probability Density.- 2.3.2 Cross Correlation.- 2.3.3 Gaussian Distribution.- 2.4 Time-Dependent Random Variables.- 2.4.1 Classification of Stochastic Processes.- 2.4.2 Chapman-Kolmogorov Equation.- 2.4.3 Wiener-Khintchine Theorem.- 2.5 Several Time-Dependent Random Variables.- 3. Langevin Equations.- 3.1 Langevin Equation for Brownian Motion.- 3.1.1 Mean-Squared Displacement.- 3.1.2 Three-Dimensional Case.- 3.1.3 Calculation of the Stationary Velocity Distribution Function.- 3.2 Ornstein-Uhlenbeck Process.- 3.2.1 Calculation of Moments.- 3.2.2 Correlation Function.- 3.2.3 Solution by Fourier Transformation.- 3.3 Nonlinear Langevin Equation, One Variable.- 3.3.1 Example.- 3.3.2 Kramers-Moyal Expansion Coefficients.- 3.3.3 Ito's and Stratonovich's Definitions.- 3.4 Nonlinear Langevin Equations, Several Variables.- 3.4.1 Determination of the Langevin Equation from Drift and Diffusion Coefficients.- 3.4.2 Transformation of Variables.- 3.4.3 How to Obtain Drift and Diffusion Coefficients for Systems.- 3.5 Markov Property.- 3.6 Solutions of the Langevin Equation by Computer Simulation.- 4. Fokker-Planck Equation.- 4.1 Kramers-Moyal Forward Expansion.- 4.1.1 Formal Solution.- 4.2 Kramers-Moyal Backward Expansion.- 4.2.1 Formal Solution.- 4.2.2 Equivalence of the Solutions of the Forward and Backward Equations.- 4.3 Pawula Theorem.- 4.4 Fokker-Planck Equation for One Variable.- 4.4.1 Transition Probability Density for Small Times.- 4.4.2 Path Integral Solutions.- 4.5 Generation and Recombination Processes.- 4.6 Application of Truncated Kramers-Moyal Expansions.- 4.7 Fokker-Planck Equation for N Variables.- 4.7.1 Probability Current.- 4.7.2 Joint Probability Distribution.- 4.7.3 Transition Probability Density for Small Times.- 4.8 Examples for Fokker-Planck Equations with Several Variables.- 4.8.1 Three-Dimensional Brownian Motion without Position Variable.- 4.8.2 One-Dimensional Brownian Motion in a Potential.- 4.8.3 Three-Dimensional Brownian Motion in an External Force.- 4.8.4 Brownian Motion of Two Interacting Particles in an External Potential.- 4.9 Transformation of Variables.- 4.10 Covariant Form of the Fokker-Planck Equation.- 5. Fokker-Planck Equation for One Variable Methods of Solution.- 5.1 Normalization.- 5.2 Stationary Solution.- 5.3 Ornstein-Uhlenbeck Process.- 5.4 Eigenfunction Expansion.- 5.5 Examples.- 5.5.1 Parabolic Potential.- 5.5.2 Inverted Parabolic Potential.- 5.5.3 Infinite Square Well for the Schrudinger Potential.- 5.5.4 V-Shaped Potential for the Fokker-Planck Equation.- 5.6 Jump Conditions.- 5.7 A Bistable Model Potential.- 5.8 Eigenfunctions and Eigenvalues of Inverted Potentials.- 5.9 Approximate and Numerical Methods for Determining Eigenvalues and Eigenfunctions.- 5.9.1 Variational Method.- 5.9.2 Numerical Integration.- 5.9.3 Expansion into a Complete Set.- 5.10 Diffusion Over a Barrier.- 5.10.1 Kramers' Escape Rate.- 5.10.2 Bistable and Metastable Potential.- 6. Fokker-Planck Equation for Several Variables Methods of Solution.- 6.1 Approach of the Solutions to a Limit Solution.- 6.2 Expansion into a Biorthogonal Set.- 6.3 Transformation of the Fokker-Planck Operator, Eigenfunction Expansions.- 6.4 Detailed Balance.- 6.5 Ornstein-Uhlenbeck Process.- 6.6 Further Methods for Solving the Fokker-Planck Equation.- 6.6.1 Transformation of Variables.- 6.6.2 Variational Method.- 6.6.3 Reduction to an Hermitian Problem.- 6.6.4 Numerical Integration.- 6.6.5 Expansion into Complete Sets.- 6.6.6 Matrix Continued-Fraction Method.- 6.6.7 WKB Method.- 7. Linear Response and Correlation Functions.- 7.1 Linear Response Function.- 7.2 Correlation Functions.- 7.3 Susceptibility.- 8. Reduction of the Number of Variables.- 8.1 First-Passage Time Problems.- 8.2 Drift and Diffusion Coefficients Independent of Some Variables.- 8.2.1 Time Integrals of Markovian Variables.- 8.3 Adiabatic Elimination of Fast Variables.- 8.3.1 Linear Process with Respect to the Fast Variable.- 8.3.2 Connection to the Nakajima-Zwanzig Projector Formalism.- 9. Solutions of Tridiagonal Recurrence Relations, Application to Ordinary and Partial Differential Equations.- 9.1 Applications and Forms of Tridiagonal Recurrence Relations.- 9.1.1 Scalar Recurrence Relation.- 9.1.2 Vector Recurrence Relation.- 9.2 Solutions of Scalar Recurrence Relations.- 9.2.1 Stationary Solution.- 9.2.2 Initial Value Problem.- 9.2.3 Eigenvalue Problem.- 9.3 Solutions of Vector Recurrence Relations.- 9.3.1 Initial Value Problem.- 9.3.2 Eigenvalue Problem.- 9.4 Ordinary and Partial Differential Equations with Multiplicative Harmonic Time-Dependent Parameters.- 9.4.1 Ordinary Differential Equations.- 9.4.2 Partial Differential Equations.- 9.5 Methods for Calculating Continued Fractions.- 9.5.1 Ordinary Continued Fractions.- 9.5.2 Matrix Continued Fractions.- 10. Solutions of the Kramers Equation.- 10.1 Forms of the Kramers Equation.- 10.1.1 Normalization of Variables.- 10.1.2 Reversible and Irreversible Operators.- 10.1.3 Transformation of the Operators.- 10.1.4 Expansion into Hermite Functions.- 10.2 Solutions for a Linear Force.- 10.2.1 Transition Probability.- 10.2.2 Eigenvalues and Eigenfunctions.- 10.3 Matrix Continued-Fraction Solutions of the Kramers Equation.- 10.3.1 Initial Value Problem.- 10.3.2 Eigenvalue Problem.- 10.4 Inverse Friction Expansion.- 10.4.1 Inverse Friction Expansion for K0(t), G0,0(t) and L0(t).- 10.4.2 Determination of Eigenvalues and Eigenvectors.- 10.4.3 Expansion for the Green's Function Gn,m(t).- 10.4.4 Position-Dependent Friction.- 11. Brownian Motion in Periodic Potentials.- 11.1 Applications.- 11.1.1 Pendulum.- 11.1.2 Superionic Conductor.- 11.1.3 Josephson Tunneling Junction.- 11.1.4 Rotation of Dipoles in a Constant Field.- 11.1.5 Phase-Locked Loop.- 11.1.6 Connection to the Sine-Gordon Equation.- 11.2 Normalization of the Langevin and Fokker-Planck Equations.- 11.3 High-Friction Limit.- 11.3.1 Stationary Solution.- 11.3.2 Time-Dependent Solution.- 11.4 Low-Friction Limit.- 11.4.1 Transformation to E and x Variables.- 11.4.2 'Ansatz' for the Stationary Distribution Functions.- 11.4.3 x-Independent Functions.- 11.4.4 x-Dependent Functions.- 11.4.5 Corrected x-Independent Functions and Mobility.- 11.5 Stationary Solutions for Arbitrary Friction.- 11.5.1 Periodicity of the Stationary Distribution Function.- 11.5.2 Matrix Continued-Fraction Method.- 11.5.3 Calculation of the Stationary Distribution Function.- 11.5.4 Alternative Matrix Continued Fraction for the Cosine Potential.- 11.6 Bistability between Running and Locked Solution.- 11.6.1 Solutions Without Noise.- 11.6.2 Solutions With Noise.- 11.6.3 Low-Friction Mobility With Noise.- 11.7 Instationary Solutions.- 11.7.1 Diffusion Constant.- 11.7.2 Transition Probability for Large Times.- 11.8 Susceptibilities.- 11.8.1 Zero-Friction Limit.- 11.9 Eigenvalues and Eigenfunctions.- 11.9.1 Eigenvalues and Eigenfunctions in the Low-Friction Limit.- 12. Statistical Properties of Laser Light.- 12.1 Semiclassical Laser Equations.- 12.1.1 Equations Without Noise.- 12.1.2 Langevin Equation.- 12.1.3 Laser Fokker-Planck Equation.- 12.2 Stationary Solution and Its Expectation Values.- 12.3 Expansion in Eigenmodes.- 12.4 Expansion into a Complete Set Solution by Matrix Continued Fractions.- 12.4.1 Determination of Eigenvalues.- 12.5 Transient Solution.- 12.5.1 Eigenfunction Method.- 12.5.2 Expansion into a Complete Set.- 12.5.3 Solution for Large Pump Parameters.- 12.6 Photoelectron Counting Distribution.- 12.6.1 Counting Distribution for Short Intervals.- 12.6.2 Expectation Values for Arbitrary Intervals.- Appendices.- A1 Stochastic Differential Equations with Colored Gaussian Noise.- A2 Boltzmann Equation with BGK and SW Collision Operators.- A3 Evaluation of a Matrix Continued Fraction for the Harmonic Oscillator.- A4 Damped Quantum-Mechanical Harmonic Oscillator.- A5 Alternative Derivation of the Fokker-Planck Equation.- A6 Fluctuating Control Parameter.- S. Supplement to the Second Edition.- S.1 Solutions of the Fokker-Planck Equation by Computer Simulation (Sect. 3.6).- S.2 Kramers-Moyal Expansion (Sect. 4.6).- S.3 Example for the Covariant Form of the Fokker-Planck Equation (Sect. 4.10).- S.4 Connection to Supersymmetry and Exact Solutions of the One Variable Fokker-Planck Equation (Chap. 5).- S.5 Nondifferentiability of the Potential for the Weak Noise Expansion (Sects. 6.6 and 6.7).- S.6 Further Applications of Matrix Continued-Fractions (Chap. 9).- S.7 Brownian Motion in a Double-Well Potential (Chaps. 10 and 11).- S.8 Boundary Layer Theory (Sect. 11.4).- S.9 Calculation of Correlation Times (Sect. 7.12).- S.10 Colored Noise (Appendix A1).- S.11 Fokker-Planck Equation with a Non-Positive-Definite Diffusion Matrix and Fokker-Planck Equation with Additional Third-Order-Derivative Terms.- References.
2,582 citations