Langevin Dynamics of a Single Particle
|
|
- Catherine Neal
- 5 years ago
- Views:
Transcription
1 Overview Langevin Dynamics of a Single Particle We consider a spherical particle of radius r immersed in a viscous fluid and suppose that the dynamics of the particle depend on two forces: A drag force arising from friction between the particle and the viscous fluid: F drag = γv where v is the velocity of the particle and γ is a friction constant. For a spherical particle, Stokes law gives γ = 6πrη, where η is the viscosity. A random force R(t) arising from the collisions of the solvent molecules with the particle. This is a phenomenological model in the sense that the solvent molecules are not modeled explicitly. Jay Taylor (ASU) APM Lecture 10 Fall / 29
2 Overview To make further progress, we need to specify R(t). Our first assumption is that on average the solvent fluctuations are non-directional: E [ R(t) ] = 0. Our second assumption concerns the covariance function of the solvent fluctuations: Cov ( R(t), R(t ) ) = κ 2 δ(t t )I, where κ is a parameter and I is the 3 3 identity matrix. This is motivated by two assumptions: Solvent fluctuations are isotropic and therefore uncorrelated along orthogonal directions. The particle is much heavier than the individual solvent molecules and so R(t) fluctuates much more rapidly than v. Jay Taylor (ASU) APM Lecture 10 Fall / 29
3 Overview Together, these assumptions imply that R(t) is a white noise process on R 3, i.e., R(t) = κẇ t where W t is a three-dimensional Brownian motion. Substituting the drag force and the white noise process into Newton s laws of motions leads to the following Langevin equation: which can be rewritten as a SDE: m v = γv + κẇt, ( γ ) dv t = v t dt + κ m m dw t ζv t + σdw t. Jay Taylor (ASU) APM Lecture 10 Fall / 29
4 Overview Furthermore, if we write v t = ( v (1) t, v (2) t, v (3) ) t W = ( W (1) t, W (2) t, W (3) ) t, then each component of v satisfies a SDE of the form dv (i) t = ζv (i) t dt + σdw (i) t, which we recognize as the Ornstein-Uhlenbeck equation. From the previous lecture, we know that the solution is a diffusion process given by the stochastic integral t v t = e ζt v 0 + σ e ζ(t s) dw t. 0 Jay Taylor (ASU) APM Lecture 10 Fall / 29
5 Overview Furthermore, if the initial velocity v 0 is prescribed, then the distribution of v t at time t is a three-dimensional Gaussian distribution with the following mean vector and covariance matrix: m(t) = v 0 e ζt Σ(t) = σ2 2ζ ( 1 e 2ζt) I. In particular, notice that as t, the limits m(t) 0 and Σ(t) (σ 2 /2ζ)I, are independent of the initial velocity. Since limits of sequences of Gaussian variables are also Gaussian, it follows that the stationary distribution of the velocity of the particle is just N(0, (σ 2 /2ζ)I). Jay Taylor (ASU) APM Lecture 10 Fall / 29
6 Overview This last result can be used to identify κ. If the solvent is held at constant temperature T, then in the limit t, the distribution of v t will tend to the Maxwellian distribution N(0, (k B T /m)i). Thus σ 2 2ζ = k BT m which along with ζ = γ/m implies that σ 2 = 2ζk BT = 2γk BT m m 2 κ 2 = m 2 σ 2 = 2γk B T. This is an example of the fluctuation-dissipation theorem: the energy imparted to the particle by solvent fluctuations is on average balanced by the energy lost to friction. Jay Taylor (ASU) APM Lecture 10 Fall / 29
7 Overview We can also solve for the trajectory of the particle by integrating the velocity process: t X t = X 0 + = X 0 + = X ζ = X ζ 0 t v s ds ( s ) e ζs v 0 + σ e ζ(s u) dw u ds 0 ( 1 e ζt) t t v 0 + σ e ζu dw u 0 ( 1 e ζt) v 0 + σ ζ 0 t 0 u e ζs ds ( 1 e ζ(t u)) dw u, where we have interchanged the order of integration in passing to the third line. Jay Taylor (ASU) APM Lecture 10 Fall / 29
8 Overview This shows that if the initial position X 0 = x 0 is fixed, then X t is Gaussian with mean vector and covariance matrix m(t) = x ζ { (σ ) 2 t Σ t = ζ where I is the 3 3 identity matrix. ( 1 e ζt) v 0 0 ( 1 e ζ(t s)) 2 ds } I = σ2 2ζ 3 [ 2ζt 3 + 4e ζt e 2ζt] I Jay Taylor (ASU) APM Lecture 10 Fall / 29
9 Overview This last result can be used to calculate the mean squared displacement of the particle in time t: [ E (X t x 0 ) 2] = ( ) 2 v0 ( 1 e ζt) 2 + ζ 3σ 2 [ 2ζ 3 2ζt 3 + 4e ζt e 2ζt]. When t is small, this is approximately [ E (X t x 0 ) 2] = v0 2 t 2 + O(t 3 ) which shows that the particle moves linearly over very short time intervals. Jay Taylor (ASU) APM Lecture 10 Fall / 29
10 Overview In contrast, for large t >> ζ 1, the mean squared displacement is approximately E [(X t x 0 ) 2] = 3σ2 ζ 2 t = 3k BT t γ which shows that the particle moves diffusively over long time intervals, with diffusion coefficient D = k BT γ = k BT 6πrη (Einstein-Stokes relation). Jay Taylor (ASU) APM Lecture 10 Fall / 29
11 Overview Brownian Dynamics of a Single Particle If the friction constant γ 1 is large, then the motion of the particle is approximately Brownian: X t = X ζ ( 1 e ζt) v 0 + σ t ζ ( ) kb T 1/2 X 0 + W t. γ 0 ( 1 e ζ(t u)) dw u This reflects the dominance of the stochastic forces acting on the particle over the inertial forces that lead to short-range linear motion. Jay Taylor (ASU) APM Lecture 10 Fall / 29
12 Langevin Dynamics Langevin Dynamics for Molecules Langevin dynamics have been used to address several needs in molecular dynamics simulations: as phenomenological models of solvent-macromolecular interactions; to enhance sampling of molecular conformations; to stabilize simulations using multiple timestep methods. Jay Taylor (ASU) APM Lecture 10 Fall / 29
13 Langevin Dynamics The Langevin equation for a molecule with potential energy function U is: where MẌ t = U(X t ) γmẋ t + R(t) M is the diagonal matrix of molecular masses; γ is a damping constant; Ṙ is a stationary Gaussian process with ] ) E [Ṙt = 0 and Cov (Ṙt, Ṙ s = 2γk B T Mδ(t s). Remark: The variance of the white noise process has been chosen to satisfy the fluctuation-dissipation theorem. Jay Taylor (ASU) APM Lecture 10 Fall / 29
14 Langevin Dynamics To put this on solid mathematical footing, we rewrite the Langevin Eqn. as a system of Itô SDEs: where MdV t = U(X t )dt γmv t dt + (2γk B T M) 1/2 dw t dx t = V t dt W t is a 3N-dimensional Brownian motion; M 1/2 is diagonal with elements M 1/2 ii. Remark: Because of the white noise and drag forces, this system is neither Hamiltonian nor symplectic. Jay Taylor (ASU) APM Lecture 10 Fall / 29
15 Langevin Dynamics One algorithm that has been used to numerically solve the Langevin equation for MD simulations is the Brooks-Brünger-Karplus (BBK) discretization: Here V n+1/2 = V n 1 t + M 2 [ U(Xn ) γmx n + R n ] X n+1 = X n + tv n+1/2 V n+1 = V n+1/2 1 t [ + M U(X n+1 ) γmx n+1 + R n+1]. 2 R 1, R 2, are IID Gaussian RVs with mean 0 and covariance matrix 2γk B T M/ t. This scheme reduces to the velocity Verlet method when γ = 0. Jay Taylor (ASU) APM Lecture 10 Fall / 29
16 Langevin Dynamics The Damping Constant The choice of γ determines the relative strengths of the inertial forces and the external stochastic forces. The choice of γ depends on the purposes of the simulation. When noise is added to stabilize MTS methods, γ is taken as small as possible, e.g., γ = 20 ps 1. When Langevin dynamics are used to model solvent effects, γ can be chosen according to Stokes law. For protein atoms exposed to water at room temperature, this gives γ 50 ps 1. In the diffusive limit, γ can be chosen to reproduce the measurable translation diffusion coefficient D t of the molecule: D t = k BT Mγ (M = molecular mass). Jay Taylor (ASU) APM Lecture 10 Fall / 29
17 Langevin Dynamics Brownian Dynamics Langevin dynamics can be simplified if γ is so large that inertial forces are negligible. In this case the momentum derivatives can be dropped: γẋ t = U(X t ) + Ṙ t which can be expressed as the following SDE: dx t = D k B T U(X t)dt + (2D) 1/2 dw t where D = k B T γ 1 is the diffusion coefficient. In this case, the molecular configuration X t is itself a diffusion process and the dynamics are known as Brownian dynamics. Jay Taylor (ASU) APM Lecture 10 Fall / 29
18 Langevin Dynamics More sophisticated BD models also account for the hydrodynamic interactions between the solvent and the macromolecule: where dx t = T(X t ) U(X t )dt + D(X t )dt + SdW t T = (T ij ) is the hydrodynamic tensor, which accounts for the transmission of frictional forces through the molecule. D = (k B T )T = SS T is the diffusion tensor. S can be calculated using Cholesky decomposition or Chebyshev approximations. Jay Taylor (ASU) APM Lecture 10 Fall / 29
19 Transcription Bubble Kinetics Alexandrov et al. (2009) Toward a Detailed Description of the Thermally Induced Dynamics of the Core Promoter. PLoS Comp. Biol. 5: e Background Eukaryotic protein-coding genes are transcribed from DNA to mrna by RNA polymerase II. Transcription initiation involves several steps: Basal transcription factors bind upstream of the gene; The basal apparatus recruits pol II to the promoter. pol II binds to the transcription start site (TSS). pol II requires ssdna at the TSS to initiate transcription. Thermal noise can cause spontaneous separation of dsdna. This study used Langevin dynamics to study the kinetics of transcription bubble formation in core promoter sequences. Jay Taylor (ASU) APM Lecture 10 Fall / 29
20 Transcription Bubble Kinetics Eukaryotic Transcription Complexes Jay Taylor (ASU) APM Lecture 10 Fall / 29
21 Transcription Bubble Kinetics Peyrard-Bishop-Dauxois (PBD) Model The PBD model is a one-dimensional Hamiltonian model of the transverse opening of dsdna with the following potential energy: U = where N n=1 { ( D n e a ny n 1 ) 2 k + (1 ) } + ρe β(yn+y n 1) (y n y n 1 ) 2 2 N is the number of base pairs (bp s); y n is the transverse displacement of the complementary bases of the n th bp; The first term is a Morse potential for each displacement The second term is a harmonic potential with a nonlinear coupling constant to account for stacking interactions between bp s. Jay Taylor (ASU) APM Lecture 10 Fall / 29
22 Transcription Bubble Kinetics Cooperative interactions between neighboring base pairs promotes bubble formation. Alexandrov et al., Fig. 2. Jay Taylor (ASU) APM Lecture 10 Fall / 29
23 Transcription Bubble Kinetics A Langevin PBD Model The effects of thermal noise on bubble formation can be investigated by incorporating the PBD potential energy function into the Langevin equation: mdv t = U(y)dt mγv t dt + mγdw t dy t = v t dt, where γ is the friction constant and W t is an N-dimensional Brownian motion. According to Alexandrov et al. (2006), γ = 0.05 ps 1 gives good agreement between the model and experiment (e.g., melting transitions and nuclease digestion). Jay Taylor (ASU) APM Lecture 10 Fall / 29
24 Transcription Bubble Kinetics For this study, 1000 independent 1 ns LD simulations were carried out for eight different mammalian pol II core promoters. Jay Taylor (ASU) APM Lecture 10 Fall / 29
25 Transcription Bubble Kinetics Fig. 6: Bubble formation in collagen promoter and intron This suggests that bubble formation occurs more readily in promoters than in nonpromoter sequences. Jay Taylor (ASU) APM Lecture 10 Fall / 29
26 Transcription Bubble Kinetics Fig. 3: Bubble probabilities by length and amplitude Jay Taylor (ASU) APM Lecture 10 Fall / 29
27 Transcription Bubble Kinetics Fig. 4: Bubble lifetimes (ps) by length Jay Taylor (ASU) APM Lecture 10 Fall / 29
28 Transcription Bubble Kinetics Conclusions pol II promoters are prone to bubble formation near the TSS. TSS bubbles are around 10 bp long and have mean lifetimes of 5-10 ps. Bubble formation is less prevalent in non-promoter sequences. Bubble formation is less concentrated along non-classical G/C rich promoters such as HSV-1 UL11 and snrna. However, this could be due to model inadequacy. Jay Taylor (ASU) APM Lecture 10 Fall / 29
29 References References Alexandrov, B.S., Wille, L.T., Rasmussen, K.O., Bishop, A.R. and Blagoev, K.B. (2006) Bubble statistics and dynamics in double-stranded DNA. Phys. Rev. E 74: Chaikin, P. M. and Lubensky, T. C. (1997) Principles of condensed matter physics. Cambridge University Press. Schlick, T. (2006) Molecular Modeling and Simulation. Springer. Jay Taylor (ASU) APM Lecture 10 Fall / 29
LANGEVIN THEORY OF BROWNIAN MOTION. Contents. 1 Langevin theory. 1 Langevin theory 1. 2 The Ornstein-Uhlenbeck process 8
Contents LANGEVIN THEORY OF BROWNIAN MOTION 1 Langevin theory 1 2 The Ornstein-Uhlenbeck process 8 1 Langevin theory Einstein (as well as Smoluchowski) was well aware that the theory of Brownian motion
More informationProtein Dynamics, Allostery and Function
Protein Dynamics, Allostery and Function Lecture 3. Protein Dynamics Xiaolin Cheng UT/ORNL Center for Molecular Biophysics SJTU Summer School 2017 1 Obtaining Dynamic Information Experimental Approaches
More informationBrownian Motion: Fokker-Planck Equation
Chapter 7 Brownian Motion: Fokker-Planck Equation The Fokker-Planck equation is the equation governing the time evolution of the probability density of the Brownian particla. It is a second order differential
More informationCHAPTER V. Brownian motion. V.1 Langevin dynamics
CHAPTER V Brownian motion In this chapter, we study the very general paradigm provided by Brownian motion. Originally, this motion is that a heavy particle, called Brownian particle, immersed in a fluid
More informationThe dynamics of small particles whose size is roughly 1 µmt or. smaller, in a fluid at room temperature, is extremely erratic, and is
1 I. BROWNIAN MOTION The dynamics of small particles whose size is roughly 1 µmt or smaller, in a fluid at room temperature, is extremely erratic, and is called Brownian motion. The velocity of such particles
More informationActive Matter Lectures for the 2011 ICTP School on Mathematics and Physics of Soft and Biological Matter Lecture 3: Hydrodynamics of SP Hard Rods
Active Matter Lectures for the 2011 ICTP School on Mathematics and Physics of Soft and Biological Matter Lecture 3: of SP Hard Rods M. Cristina Marchetti Syracuse University Baskaran & MCM, PRE 77 (2008);
More informationMD Thermodynamics. Lecture 12 3/26/18. Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky
MD Thermodynamics Lecture 1 3/6/18 1 Molecular dynamics The force depends on positions only (not velocities) Total energy is conserved (micro canonical evolution) Newton s equations of motion (second order
More informationMesoscale Simulation Methods. Ronojoy Adhikari The Institute of Mathematical Sciences Chennai
Mesoscale Simulation Methods Ronojoy Adhikari The Institute of Mathematical Sciences Chennai Outline What is mesoscale? Mesoscale statics and dynamics through coarse-graining. Coarse-grained equations
More informationModelling Biochemical Reaction Networks. Lecture 1: Overview of cell biology
Modelling Biochemical Reaction Networks Lecture 1: Overview of cell biology Marc R. Roussel Department of Chemistry and Biochemistry Types of cells Prokaryotes: Cells without nuclei ( bacteria ) Very little
More informationLangevin Methods. Burkhard Dünweg Max Planck Institute for Polymer Research Ackermannweg 10 D Mainz Germany
Langevin Methods Burkhard Dünweg Max Planck Institute for Polymer Research Ackermannweg 1 D 55128 Mainz Germany Motivation Original idea: Fast and slow degrees of freedom Example: Brownian motion Replace
More information08. Brownian Motion. University of Rhode Island. Gerhard Müller University of Rhode Island,
University of Rhode Island DigitalCommons@URI Nonequilibrium Statistical Physics Physics Course Materials 1-19-215 8. Brownian Motion Gerhard Müller University of Rhode Island, gmuller@uri.edu Follow this
More informationVilla et al. (2005) Structural dynamics of the lac repressor-dna complex revealed by a multiscale simulation. PNAS 102:
Villa et al. (2005) Structural dynamics of the lac repressor-dna complex revealed by a multiscale simulation. PNAS 102: 6783-6788. Background: The lac operon is a cluster of genes in the E. coli genome
More informationIntroduction to nonequilibrium physics
Introduction to nonequilibrium physics Jae Dong Noh December 18, 2016 Preface This is a note for the lecture given in the 2016 KIAS-SNU Physics Winter Camp which is held at KIAS in December 17 23, 2016.
More informationA path integral approach to the Langevin equation
A path integral approach to the Langevin equation - Ashok Das Reference: A path integral approach to the Langevin equation, A. Das, S. Panda and J. R. L. Santos, arxiv:1411.0256 (to be published in Int.
More informationSupplementary Figure S1: Numerical PSD simulation. Example numerical simulation of the power spectral density, S(f) from a trapped particle
Supplementary Figure S1: Numerical PSD simulation. Example numerical simulation of the power spectral density, S(f) from a trapped particle oscillating at Ω 0 /(2π) = f xy = 600Hz and subject to a periodic
More informationOnce Upon A Time, There Was A Certain Ludwig
Once Upon A Time, There Was A Certain Ludwig Statistical Mechanics: Ensembles, Distributions, Entropy and Thermostatting Srinivas Mushnoori Chemical & Biochemical Engineering Rutgers, The State University
More informationSolvation and Macromolecular Structure. The structure and dynamics of biological macromolecules are strongly influenced by water:
Overview Solvation and Macromolecular Structure The structure and dynamics of biological macromolecules are strongly influenced by water: Electrostatic effects: charges are screened by water molecules
More informationActive and Driven Soft Matter Lecture 3: Self-Propelled Hard Rods
A Tutorial: From Langevin equation to Active and Driven Soft Matter Lecture 3: Self-Propelled Hard Rods M. Cristina Marchetti Syracuse University Boulder School 2009 A Tutorial: From Langevin equation
More informationFORCE ENERGY. only if F = F(r). 1 Nano-scale (10 9 m) 2 Nano to Micro-scale (10 6 m) 3 Nano to Meso-scale (10 3 m)
What is the force? CEE 618 Scientific Parallel Computing (Lecture 12) Dissipative Hydrodynamics (DHD) Albert S. Kim Department of Civil and Environmental Engineering University of Hawai i at Manoa 2540
More informationAnomalous diffusion in biology: fractional Brownian motion, Lévy flights
Anomalous diffusion in biology: fractional Brownian motion, Lévy flights Jan Korbel Faculty of Nuclear Sciences and Physical Engineering, CTU, Prague Minisymposium on fundamental aspects behind cell systems
More informationLecture 7: Simple genetic circuits I
Lecture 7: Simple genetic circuits I Paul C Bressloff (Fall 2018) 7.1 Transcription and translation In Fig. 20 we show the two main stages in the expression of a single gene according to the central dogma.
More informationarxiv: v1 [cond-mat.stat-mech] 7 Mar 2019
Langevin thermostat for robust configurational and kinetic sampling Oded Farago, Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB EW, United Kingdom Department of Biomedical
More informationSome Basic Statistical Modeling Issues in Molecular and Ocean Dynamics
Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 1/2 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics Peter R. Kramer Department of Mathematical Sciences
More informationCollaborators: Aleksas Mazeliauskas (Heidelberg) & Derek Teaney (Stony Brook) Refs: , /25
2017 8 28 30 @ Collaborators: Aleksas Mazeliauskas (Heidelberg) & Derek Teaney (Stony Brook) Refs: 1606.07742, 1708.05657 1/25 1. Introduction 2/25 Ultra-relativistic heavy-ion collisions and the Bjorken
More informationLangevin Equation Model for Brownian Motion
Langevin Equation Model for Brownian Motion Friday, March 13, 2015 2:04 PM Reading: Gardiner Sec. 1.2 Homework 2 due Tuesday, March 17 at 2 PM. The friction constant shape of the particle. depends on the
More information4. The Green Kubo Relations
4. The Green Kubo Relations 4.1 The Langevin Equation In 1828 the botanist Robert Brown observed the motion of pollen grains suspended in a fluid. Although the system was allowed to come to equilibrium,
More informationStochastic Particle Methods for Rarefied Gases
CCES Seminar WS 2/3 Stochastic Particle Methods for Rarefied Gases Julian Köllermeier RWTH Aachen University Supervisor: Prof. Dr. Manuel Torrilhon Center for Computational Engineering Science Mathematics
More information15.3 The Langevin theory of the Brownian motion
5.3 The Langevin theory of the Brownian motion 593 Now, by virtue of the distribution (2), we obtain r(t) =; r 2 (t) = n(r,t)4πr 4 dr = 6Dt t, (22) N in complete agreement with our earlier results, namely
More informationBrownian motion and the Central Limit Theorem
Brownian motion and the Central Limit Theorem Amir Bar January 4, 3 Based on Shang-Keng Ma, Statistical Mechanics, sections.,.7 and the course s notes section 6. Introduction In this tutorial we shall
More informationThis is a Gaussian probability centered around m = 0 (the most probable and mean position is the origin) and the mean square displacement m 2 = n,or
Physics 7b: Statistical Mechanics Brownian Motion Brownian motion is the motion of a particle due to the buffeting by the molecules in a gas or liquid. The particle must be small enough that the effects
More informationEffects of Forcing Scheme on the Flow and the Relative Motion of Inertial Particles in DNS of Isotropic Turbulence
Effects of Forcing Scheme on the Flow and the Relative Motion of Inertial Particles in DNS of Isotropic Turbulence Rohit Dhariwal PI: Sarma L. Rani Department of Mechanical and Aerospace Engineering The
More informationHow DLS Works: Interference of Light
Static light scattering vs. Dynamic light scattering Static light scattering measures time-average intensities (mean square fluctuations) molecular weight radius of gyration second virial coefficient Dynamic
More informationTemperature and Pressure Controls
Ensembles Temperature and Pressure Controls 1. (E, V, N) microcanonical (constant energy) 2. (T, V, N) canonical, constant volume 3. (T, P N) constant pressure 4. (T, V, µ) grand canonical #2, 3 or 4 are
More informationNonlinear Single-Particle Dynamics in High Energy Accelerators
Nonlinear Single-Particle Dynamics in High Energy Accelerators Part 2: Basic tools and concepts Nonlinear Single-Particle Dynamics in High Energy Accelerators This course consists of eight lectures: 1.
More informationThe Smoluchowski-Kramers Approximation: What model describes a Brownian particle?
The Smoluchowski-Kramers Approximation: What model describes a Brownian particle? Scott Hottovy shottovy@math.arizona.edu University of Arizona Applied Mathematics October 7, 2011 Brown observes a particle
More informationTemporal Integrators for Langevin Equations with Applications to Fluctuating Hydrodynamics and Brownian Dynamics
Temporal Integrators for Langevin Equations with Applications to Fluctuating Hydrodynamics and Brownian Dynamics by Steven Delong A dissertation submitted in partial fulfillment of the requirements for
More informationBROWNIAN DYNAMICS SIMULATIONS WITH HYDRODYNAMICS. Abstract
BROWNIAN DYNAMICS SIMULATIONS WITH HYDRODYNAMICS Juan J. Cerdà 1 1 Institut für Computerphysik, Pfaffenwaldring 27, Universität Stuttgart, 70569 Stuttgart, Germany. (Dated: July 21, 2009) Abstract - 1
More informationOn the (multi)scale nature of fluid turbulence
On the (multi)scale nature of fluid turbulence Kolmogorov axiomatics Laurent Chevillard Laboratoire de Physique, ENS Lyon, CNRS, France Laurent Chevillard, Laboratoire de Physique, ENS Lyon, CNRS, France
More informationSolution structure and dynamics of biopolymers
Solution structure and dynamics of biopolymers Atomic-detail vs. low resolution structure Information available at different scales Mobility of macromolecules in solution Brownian motion, random walk,
More informationEffects of Forcing Scheme on the Flow and the Relative Motion of Inertial Particles in DNS of Isotropic Turbulence
Effects of Forcing Scheme on the Flow and the Relative Motion of Inertial Particles in DNS of Isotropic Turbulence Rohit Dhariwal and Vijaya Rani PI: Sarma L. Rani Department of Mechanical and Aerospace
More informationNUMERICAL METHODS FOR SECOND-ORDER STOCHASTIC DIFFERENTIAL EQUATIONS
SIAM J SCI COMPUT Vol 9, No, pp 45 64 c 7 Society for Industrial and Applied Mathematics NUMERICAL METHODS FOR SECOND-ORDER STOCHASTIC DIFFERENTIAL EQUATIONS KEVIN BURRAGE, IAN LENANE, AND GRANT LYTHE
More informationWhat is Classical Molecular Dynamics?
What is Classical Molecular Dynamics? Simulation of explicit particles (atoms, ions,... ) Particles interact via relatively simple analytical potential functions Newton s equations of motion are integrated
More informationMathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio ( )
Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio (2014-2015) Etienne Tanré - Olivier Faugeras INRIA - Team Tosca November 26th, 2014 E. Tanré (INRIA - Team Tosca) Mathematical
More informationF r (t) = 0, (4.2) F (t) = r= r a (t)
Chapter 4 Stochastic Equations 4.1 Langevin equation To explain the idea of introduction of the Langevin equation, let us consider the concrete example taken from surface physics: motion of an atom (adatom)
More informationComputing ergodic limits for SDEs
Computing ergodic limits for SDEs M.V. Tretyakov School of Mathematical Sciences, University of Nottingham, UK Talk at the workshop Stochastic numerical algorithms, multiscale modelling and high-dimensional
More informationStochastic Differential Equations
Chapter 5 Stochastic Differential Equations We would like to introduce stochastic ODE s without going first through the machinery of stochastic integrals. 5.1 Itô Integrals and Itô Differential Equations
More informationThe concentration of a drug in blood. Exponential decay. Different realizations. Exponential decay with noise. dc(t) dt.
The concentration of a drug in blood Exponential decay C12 concentration 2 4 6 8 1 C12 concentration 2 4 6 8 1 dc(t) dt = µc(t) C(t) = C()e µt 2 4 6 8 1 12 time in minutes 2 4 6 8 1 12 time in minutes
More informationPart III. Polymer Dynamics molecular models
Part III. Polymer Dynamics molecular models I. Unentangled polymer dynamics I.1 Diffusion of a small colloidal particle I.2 Diffusion of an unentangled polymer chain II. Entangled polymer dynamics II.1.
More informationCHE3935. Lecture 10 Brownian Dynamics
CHE3935 Lecture 10 Brownian Dynamics 1 What Is Brownian Dynamics? Brownian dynamics is based on the idea of the Brownian motion of particles So what is Brownian motion? Named after botanist Robert Brown
More informationGENE ACTIVITY Gene structure Transcription Transcript processing mrna transport mrna stability Translation Posttranslational modifications
1 GENE ACTIVITY Gene structure Transcription Transcript processing mrna transport mrna stability Translation Posttranslational modifications 2 DNA Promoter Gene A Gene B Termination Signal Transcription
More informationCEE 618 Scientific Parallel Computing (Lecture 12)
1 / 26 CEE 618 Scientific Parallel Computing (Lecture 12) Dissipative Hydrodynamics (DHD) Albert S. Kim Department of Civil and Environmental Engineering University of Hawai i at Manoa 2540 Dole Street,
More informationLecture 4: Ito s Stochastic Calculus and SDE. Seung Yeal Ha Dept of Mathematical Sciences Seoul National University
Lecture 4: Ito s Stochastic Calculus and SDE Seung Yeal Ha Dept of Mathematical Sciences Seoul National University 1 Preliminaries What is Calculus? Integral, Differentiation. Differentiation 2 Integral
More informationThe Kramers problem and first passage times.
Chapter 8 The Kramers problem and first passage times. The Kramers problem is to find the rate at which a Brownian particle escapes from a potential well over a potential barrier. One method of attack
More informationmacroscopic view (phenomenological) microscopic view (atomistic) computing reaction rate rate of reactions experiments thermodynamics
Rate Theory (overview) macroscopic view (phenomenological) rate of reactions experiments thermodynamics Van t Hoff & Arrhenius equation microscopic view (atomistic) statistical mechanics transition state
More informationChapter 6: The Rouse Model. The Bead (friction factor) and Spring (Gaussian entropy) Molecular Model:
G. R. Strobl, Chapter 6 "The Physics of Polymers, 2'nd Ed." Springer, NY, (1997). R. B. Bird, R. C. Armstrong, O. Hassager, "Dynamics of Polymeric Liquids", Vol. 2, John Wiley and Sons (1977). M. Doi,
More informationPolymer Dynamics and Rheology
Polymer Dynamics and Rheology 1 Polymer Dynamics and Rheology Brownian motion Harmonic Oscillator Damped harmonic oscillator Elastic dumbbell model Boltzmann superposition principle Rubber elasticity and
More informationLecture Notes for PHY 405 Classical Mechanics
Lecture Notes for PHY 405 Classical Mechanics From Thorton & Marion s Classical Mechanics Prepared by Dr. Joseph M. Hahn Saint Mary s University Department of Astronomy & Physics September 1, 2005 Chapter
More informationSTOCHASTIC REDUCTION METHOD FOR BIOLOGICAL CHEMICAL KINETICS USING TIME-SCALE SEPARATION
STOCHASTIC REDUCTION METHOD FOR BIOLOGICAL CHEMICAL KINETICS USING TIME-SCALE SEPARATION CHETAN D. PAHLAJANI 12, PAUL J. ATZBERGER 23, AND MUSTAFA KHAMMASH 3 Abstract. Many processes in cell biology encode
More information2. Mathematical descriptions. (i) the master equation (ii) Langevin theory. 3. Single cell measurements
1. Why stochastic?. Mathematical descriptions (i) the master equation (ii) Langevin theory 3. Single cell measurements 4. Consequences Any chemical reaction is stochastic. k P d φ dp dt = k d P deterministic
More informationBME 5742 Biosystems Modeling and Control
BME 5742 Biosystems Modeling and Control Lecture 24 Unregulated Gene Expression Model Dr. Zvi Roth (FAU) 1 The genetic material inside a cell, encoded in its DNA, governs the response of a cell to various
More informationSDE Coefficients. March 4, 2008
SDE Coefficients March 4, 2008 The following is a summary of GARD sections 3.3 and 6., mainly as an overview of the two main approaches to creating a SDE model. Stochastic Differential Equations (SDE)
More informationPolymer dynamics. Course M6 Lecture 5 26/1/2004 (JAE) 5.1 Introduction. Diffusion of polymers in melts and dilute solution.
Course M6 Lecture 5 6//004 Polymer dynamics Diffusion of polymers in melts and dilute solution Dr James Elliott 5. Introduction So far, we have considered the static configurations and morphologies of
More information16. Working with the Langevin and Fokker-Planck equations
16. Working with the Langevin and Fokker-Planck equations In the preceding Lecture, we have shown that given a Langevin equation (LE), it is possible to write down an equivalent Fokker-Planck equation
More informationTopics covered so far:
Topics covered so far: Chap 1: The kinetic theory of gases P, T, and the Ideal Gas Law Chap 2: The principles of statistical mechanics 2.1, The Boltzmann law (spatial distribution) 2.2, The distribution
More informationFast Brownian Dynamics for Colloidal Suspensions
Fast Brownian Dynamics for Colloidal Suspensions Aleksandar Donev, CIMS and collaborators: Florencio Balboa CIMS) Andrew Fiore and James Swan MIT) Courant Institute, New York University Modeling Complex
More informationA Nobel Prize for Molecular Dynamics and QM/MM What is Classical Molecular Dynamics? Simulation of explicit particles (atoms, ions,... ) Particles interact via relatively simple analytical potential
More informationBrownian Motion and Langevin Equations
1 Brownian Motion and Langevin Equations 1.1 Langevin Equation and the Fluctuation- Dissipation Theorem The theory of Brownian motion is perhaps the simplest approximate way to treat the dynamics of nonequilibrium
More informationA random perturbation approach to some stochastic approximation algorithms in optimization.
A random perturbation approach to some stochastic approximation algorithms in optimization. Wenqing Hu. 1 (Presentation based on joint works with Chris Junchi Li 2, Weijie Su 3, Haoyi Xiong 4.) 1. Department
More informationMolecular Dynamics Simulations
Molecular Dynamics Simulations Dr. Kasra Momeni www.knanosys.com Outline Long-range Interactions Ewald Sum Fast Multipole Method Spherically Truncated Coulombic Potential Speeding up Calculations SPaSM
More informationHydrodynamics: Viscosity and Diffusion Hydrodynamics is the study of mechanics in a liquid, where the frictional drag of the liquid cannot be ignored
Hydrodynamics: Viscosity and Diffusion Hydrodynamics is the study of mechanics in a liquid, where the frictional drag of the liquid cannot be ignored First let s just consider fluid flow, where the fluid
More informationTutorial School on Fluid Dynamics: Aspects of Turbulence Session I: Refresher Material Instructor: James Wallace
Tutorial School on Fluid Dynamics: Aspects of Turbulence Session I: Refresher Material Instructor: James Wallace Adapted from Publisher: John S. Wiley & Sons 2002 Center for Scientific Computation and
More informationDissipative nuclear dynamics
Dissipative nuclear dynamics Curso de Reacciones Nucleares Programa Inter universitario de Fisica Nuclear Universidad de Santiago de Compostela March 2009 Karl Heinz Schmidt Collective dynamical properties
More informationStochastic differential equation models in biology Susanne Ditlevsen
Stochastic differential equation models in biology Susanne Ditlevsen Introduction This chapter is concerned with continuous time processes, which are often modeled as a system of ordinary differential
More informationStatistical mechanics of random billiard systems
Statistical mechanics of random billiard systems Renato Feres Washington University, St. Louis Banff, August 2014 1 / 39 Acknowledgements Collaborators: Timothy Chumley, U. of Iowa Scott Cook, Swarthmore
More informationLecture 4: Introduction to stochastic processes and stochastic calculus
Lecture 4: Introduction to stochastic processes and stochastic calculus Cédric Archambeau Centre for Computational Statistics and Machine Learning Department of Computer Science University College London
More informationModeling of Micro-Fluidics by a Dissipative Particle Dynamics Method. Justyna Czerwinska
Modeling of Micro-Fluidics by a Dissipative Particle Dynamics Method Justyna Czerwinska Scales and Physical Models years Time hours Engineering Design Limit Process Design minutes Continious Mechanics
More informationFrom Gene to Protein
From Gene to Protein Gene Expression Process by which DNA directs the synthesis of a protein 2 stages transcription translation All organisms One gene one protein 1. Transcription of DNA Gene Composed
More informationExplaining and modelling the rheology of polymeric fluids with the kinetic theory
Explaining and modelling the rheology of polymeric fluids with the kinetic theory Dmitry Shogin University of Stavanger The National IOR Centre of Norway IOR Norway 2016 Workshop April 25, 2016 Overview
More informationANOMALOUS TRANSPORT IN RANDOM MEDIA: A ONE-DIMENSIONAL GAUSSIAN MODEL FOR ANOMALOUS DIFFUSION
THESIS FOR THE DEGREE OF MASTER OF SCIENCE ANOMALOUS TRANSPORT IN RANDOM MEDIA: A ONE-DIMENSIONAL GAUSSIAN MODEL FOR ANOMALOUS DIFFUSION ERIK ARVEDSON Department of Theoretical Physics CHALMERS UNIVERSITY
More informationFrom a Mesoscopic to a Macroscopic Description of Fluid-Particle Interaction
From a Mesoscopic to a Macroscopic Description of Fluid-Particle Interaction Carnegie Mellon University Center for Nonlinear Analysis Working Group, October 2016 Outline 1 Physical Framework 2 3 Free Energy
More informationReading Assignments. A. Genes and the Synthesis of Polypeptides. Lecture Series 7 From DNA to Protein: Genotype to Phenotype
Lecture Series 7 From DNA to Protein: Genotype to Phenotype Reading Assignments Read Chapter 7 From DNA to Protein A. Genes and the Synthesis of Polypeptides Genes are made up of DNA and are expressed
More informationStatistical Mechanics of Active Matter
Statistical Mechanics of Active Matter Umberto Marini Bettolo Marconi University of Camerino, Italy Naples, 24 May,2017 Umberto Marini Bettolo Marconi (2017) Statistical Mechanics of Active Matter 2017
More informationInverse Langevin approach to time-series data analysis
Inverse Langevin approach to time-series data analysis Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Universidade de Brasília Saratoga Springs, MaxEnt 2007 Outline 1 Motivation 2 Outline 1 Motivation
More informationProblem Set Number 01, MIT (Winter-Spring 2018)
Problem Set Number 01, 18.306 MIT (Winter-Spring 2018) Rodolfo R. Rosales (MIT, Math. Dept., room 2-337, Cambridge, MA 02139) February 28, 2018 Due Monday March 12, 2018. Turn it in (by 3PM) at the Math.
More informationEukaryotic Gene Expression
Eukaryotic Gene Expression Lectures 22-23 Several Features Distinguish Eukaryotic Processes From Mechanisms in Bacteria 123 Eukaryotic Gene Expression Several Features Distinguish Eukaryotic Processes
More informationComputational Physics
Computational Physics Molecular Dynamics Simulations E. Carlon, M. Laleman and S. Nomidis Academic year 015/016 Contents 1 Introduction 3 Integration schemes 4.1 On the symplectic nature of the Velocity
More informationIntroduction to Turbulence and Turbulence Modeling
Introduction to Turbulence and Turbulence Modeling Part I Venkat Raman The University of Texas at Austin Lecture notes based on the book Turbulent Flows by S. B. Pope Turbulent Flows Turbulent flows Commonly
More informationSmoluchowski Diffusion Equation
Chapter 4 Smoluchowski Diffusion Equation Contents 4. Derivation of the Smoluchoswki Diffusion Equation for Potential Fields 64 4.2 One-DimensionalDiffusoninaLinearPotential... 67 4.2. Diffusion in an
More informationThe Truth about diffusion (in liquids)
The Truth about diffusion (in liquids) Aleksandar Donev Courant Institute, New York University & Eric Vanden-Eijnden, Courant In honor of Berni Julian Alder LLNL, August 20th 2015 A. Donev (CIMS) Diffusion
More informationLecture 4: Numerical Solution of SDEs, Itô Taylor Series, Gaussian Approximations
Lecture 4: Numerical Solution of SDEs, Itô Taylor Series, Gaussian Approximations Simo Särkkä Aalto University, Finland November 18, 2014 Simo Särkkä (Aalto) Lecture 4: Numerical Solution of SDEs November
More informationTemperature and Pressure Controls
Ensembles Temperature and Pressure Controls 1. (E, V, N) microcanonical (constant energy) 2. (T, V, N) canonical, constant volume 3. (T, P N) constant pressure 4. (T, V, µ) grand canonical #2, 3 or 4 are
More informationGENERATION OF COLORED NOISE
International Journal of Modern Physics C, Vol. 12, No. 6 (2001) 851 855 c World Scientific Publishing Company GENERATION OF COLORED NOISE LORENZ BARTOSCH Institut für Theoretische Physik, Johann Wolfgang
More informationDiffusive Transport Enhanced by Thermal Velocity Fluctuations
Diffusive Transport Enhanced by Thermal Velocity Fluctuations Aleksandar Donev 1 Courant Institute, New York University & Alejandro L. Garcia, San Jose State University John B. Bell, Lawrence Berkeley
More informationStochastic Gradient Descent in Continuous Time
Stochastic Gradient Descent in Continuous Time Justin Sirignano University of Illinois at Urbana Champaign with Konstantinos Spiliopoulos (Boston University) 1 / 27 We consider a diffusion X t X = R m
More informationLocal vs. Nonlocal Diffusions A Tale of Two Laplacians
Local vs. Nonlocal Diffusions A Tale of Two Laplacians Jinqiao Duan Dept of Applied Mathematics Illinois Institute of Technology Chicago duan@iit.edu Outline 1 Einstein & Wiener: The Local diffusion 2
More informationDynamics of a tagged monomer: Effects of elastic pinning and harmonic absorption. Shamik Gupta
: Effects of elastic pinning and harmonic absorption Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris-Sud, France Joint work with Alberto Rosso Christophe Texier Ref.: Phys.
More informationName: SBI 4U. Gene Expression Quiz. Overall Expectation:
Gene Expression Quiz Overall Expectation: - Demonstrate an understanding of concepts related to molecular genetics, and how genetic modification is applied in industry and agriculture Specific Expectation(s):
More informationp 1 ( Y p dp) 1/p ( X p dp) 1 1 p
Doob s inequality Let X(t) be a right continuous submartingale with respect to F(t), t 1 P(sup s t X(s) λ) 1 λ {sup s t X(s) λ} X + (t)dp 2 For 1 < p
More informationStatistical Physics. Problem Set 4: Kinetic Theory
Statistical Physics xford hysics Second year physics course Dr A. A. Schekochihin and Prof. A. Boothroyd (with thanks to Prof. S. J. Blundell) Problem Set 4: Kinetic Theory PROBLEM SET 4: Collisions and
More informationUniversity of Washington Department of Chemistry Chemistry 453 Winter Quarter 2008
Lecture /1/8 University of Washington Department of Chemistry Chemistry 45 Winter Quarter 8 A. Analysis of Diffusion Coefficients: Friction Diffusion coefficients can be measured by a variety of methods
More information