MD Thermodynamics. Lecture 12 3/26/18. Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky

Size: px
Start display at page:

Download "MD Thermodynamics. Lecture 12 3/26/18. Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky"

Transcription

1 MD Thermodynamics Lecture 1 3/6/18 1

2 Molecular dynamics The force depends on positions only (not velocities) Total energy is conserved (micro canonical evolution) Newton s equations of motion (second order diff. eq.) for all particles are integrated to predict time evolution of the system i N N i i i r r r V r r F dt r d m ),, ( ),, ( 1 1 ) ( ), ( ) ( ), ( t t t t t t i i i i v r v r

3 More accurate integration: Verlet algorithm Expand Taylor series to higher order: Adding the two expressions gives Odd power terms vanish and leading error is only 4 th order 3

4 MD workflow Requires atomic position and velocities at the beginning Step through discrete time by integrating Newton s equations Repeated calculation of atomic forces at each step 4

5 Maxwell Boltzmann statistics The temperature and the distribution of atomic velocities in a system are related through the Maxwell Boltzmann equation n( v) m k 3 mv v exp BT kbt v rms 3kBT m H O at room T: 500 m/s Initial velocities are generated from the Gaussian distribution of v x, v y, and v z using a random number generator Total momentum should be 0 5

6 Limitations of accuracy Classical nuclei Energy and force models Time scales Length scales 6

7 Classical nuclei Maxwell Boltzmann statistics, no zero point vibrational motion Specific heat is affected by statistics (even HO at 300K) Also, light atoms at low temperature can tunnel classical quantum specific heat of a solid Einstein classical 7

8 Ab initio molecular dynamics How to describe bond breaking, reactions in MD? Born Oppenheimer ab initio molecular dynamics At each point on the trajectory, the KS DFT wavefunctions are minimized to their ground state Then forces are computed using Hellman Feynman theorem If not in an exact ground state, can get large noise in forces which integrator is best to use? mr dvˆ F dr i i i i 8

9 Length scales Fluctuations at phase transitions, long range effects, mechanics Periodic boundaries can help D. S. Ivanov and L. V. Zhigilei, Phys. Rev. Lett. 91, , 003. A. Movahedi Rad, R. Alizadeh, J Mod Phys Vol.5 No.8 (014) 9

10 Time scales Atomic vibrations versus conformation change Coarse graining, statistics of configurations rare hops 10

11 Sampling the potential energy landscape The choice of initial conditions (and temperature) determines the total energy of the system This determines the region of the potential energy landscape that will be sampled 11

12 Quantity averaging Ergodic hypothesis: time average of a thermodynamic quantity along a long trajectory is equal to the ensemble average over the phase space (large number of independent systems) A A e d d E( rp, ) (, rp) r p 1 e E( rp, ) drp d T T 0 At () dt Role of MD equilibration phase is to generate low energy (high probability) configurations are visited more frequently Not all regions of space may be accessible Need to explicitly sample distinct initial configurations 1

13 Temperature in MD Equipartition energy theorem relates temperature to the average kinetic energy of the system Instantaneous temperature is related to the instantaneous velocities KE N miv i1 i 1 N f k B T ( t) 3 Nk B T ( t) Temperature is an average over degrees of freedom To obtain macroscopic temperature, can also do a time average Tt 1 N t N t i1 Tt i 13

14 Energy and temperature fluctuations Simplest MD is performed in constant energy NVE (microcanonical) ensemble Kinetic energy (and T) will fluctuate, equal and opposite to potential energy Fluctuation of T(t) : Tt kt NC B () For large systems, fluctuation of T is negligible Total energy should fluctuate much less (at least orders of magnitude less than KE) V total kinetic potential 14

15 Energy and temperature fluctuations Hard to specify the temperature from initial conditions Temperature can drift, depending on dynamics and time step Need ability to regulate the temperature of the system : canonical ensemble In the canonical ensemble (NVT), temperature is constant by definition, but total energy fluctuates E 1 E H H NkBT0 C V 0 E N For large systems, canonical ensemble (NVT) is equivalent to microcanonical ensemble (NVE) 15

16 Thermostats Temperature cannot be held fixed in a NVE MD simulation Algorithm needs to be modified T must be given as an input and monitored during simulation Thermostat is a feedback control algorithm acting on the average kinetic energy of the system It must periodically act on the velocities Q: Which integrator can be used? 16

17 Control of system variables Strong coupling methods scale system variable to give exact preset derived value Weak coupling methods scale system variable in direction of desired derived value Stochastic methods constrain a system variable to preset distribution function Extended system dynamics extend degrees of freedom to include temperature or pressure terms 17

18 Velocity rescaling Temperature and velocity are related as T v Scale the velocities to reach desired temperature T o Tt Desired temperature Instantaneous temperature Can be easily used in NVE simulations (which integrator?) Disadvantage: Strong coupling affects the natural dynamics 18

19 Berendsen thermostat Weakly coupled, reformulated velocity scaling Scale velocities at each step such that, dt dt t 1 T target T t Thermal bath is introduced with a coupling parameter Scale factor: dt T T t target 1 1 Value of has to be chosen carefully; typically = ps Large : sampling unphysical ensemble, too slow Small poor coupling to the thermostat, too aggressive Kinetic energy fluctuations do not match canonical ensemble For = dt, this is nothing but velocity scaling 19

20 Andresen thermostat Stochastic: influence the temperature by reassigning the velocity of a random particle (a collision ) Probability of collision The new velocity is from the Maxwell Boltzmann distribution corresponding to the desired T. Langevin thermostat: apply friction and random forces to momenta Sample a micro canonical ensemble between collisions It can be shown that overall the canonical ensemble is preserved Should not be used while computing transport and nonequilibrium properties: rates, diffusion. 0

21 Nosé Hoover thermostat It is possible to rigorously maintain the canonical ensemble Extended system method: introduced by Nosé and later developed by Hoover Principle: Introduce a heat bath as an integral part of the system, as an additional degree of freedom in the system This is done by modifying the Lagrangian of the system by introducing a fictitious variable s associated with a mass Q and velocity ds/dt v t sr st 1

22 Nosé Hoover thermostat An extra degree of freedom is introduced, which represents friction which slows down or accelerates particles until the temperature reaches target value T Combined system evolves microcanonically, but energy is exchanged between the real system and bath, sampling the true canonical statistics for the real system

23 Harmonic oscillator (NVE) 3

24 Harmonic oscillator Nosé Hoover thermostat on oscillator 4

25 Example: LJ fluid 5

26 Pressure in MD The pressure is defined as the average force on the container wall due to the physical system enclosed therein. Thermodynamically, it is defined as P H V H is the Helmholtz free energy V is the volume of the system Microscopic estimator can be expressed from Virial theorem P NT V 1 3V i r i F i 1 3V i i p m i 1 i j r ij F ij F i is the force on particle i 6

27 Berendsen barostat Pressure control by rescaling positions and volume r i r i 1 t ( P target P( t)) 1/3 Instantaneous pressure vs time in an MD simulation of 56 particles using the Berendsen barostat to impose an instantaneous pressure jump from 1.0 to 6.0. Each curve corresponds to a different value of the time constant τ 7

28 Thermostat summary Method Pro Con Berendsen Nosé Hoover stochastic (Langevin, Andersen) Fast, smooth first order approach to equilibrium, good for non equilibrium Canonical ensemble. Considered most reliable for simulation at equilibrium Canonical ensemble Non canonical, less reliable for simulation at equilibrium slow, second order approach to equilibrium, affects dynamics affects dynamics, not reliable for kinetics 8

29 Equilibration How to test for equilibration? Drop longer and longer initial segments of your dynamical trajectory, when accumulating averages Transient part of the trajectory can be long, depends on dynamics Equilibration Production 9

30 Accumulating averages Potential + kinetic = total energy (conserved) divided by N Energy conservation needs to be monitored in every simulation Temperature (from kinetic energy) T(t) 1 N N i1 1 mv i (t) v i (t) Pressure Potential energy: caloric curve E(T), latent heat of fusion Radial (pair) distribution function local structure Mean square displacements (diffusion) 30

31 Error bars Averages are computed as: A 1 n n i1 A i Statistical error is approximated by the variance ( ) n( n 1) Error decreases with sample size n Actually this is strictly true only for uncorrelated results With MD it s often good to average over ensemble of trajectories started from different structures ( A i i A A ) 31

32 Caloric curve in Pb slab (100) 1 M time steps 6400 atoms Surface melting transition 3

33 Radial distribution function The probability density p(r) of finding a particle at a given distance, (between r and r+dr) increases with r, since the shell volume is 4r dr Define a normalized distribution function counting the probability of finding two particles at a given distance r g( r) ( N 1) p( r) r N V d 3 rg( r) ( N 1) 33

34 RDF: measure of local structure RDF: ratio of the average number density (r) at a distance r from any given atom and the density at a distance r from an atom in an uniform gas at the same overall density Any deviation in g(r) from 1 reflects correlations between particles due to the intermolecular interactions 34

35 Water with ab initio MD (3 molecules) Can tell the size required to simulate infinite sample Does not freeze into an ordered crystal, T m is off Nuclear quantum effects are important 35

Javier Junquera. Statistical mechanics

Javier Junquera. Statistical mechanics Javier Junquera Statistical mechanics From the microscopic to the macroscopic level: the realm of statistical mechanics Computer simulations Thermodynamic state Generates information at the microscopic

More information

Ab initio molecular dynamics. Simone Piccinin CNR-IOM DEMOCRITOS Trieste, Italy. Bangalore, 04 September 2014

Ab initio molecular dynamics. Simone Piccinin CNR-IOM DEMOCRITOS Trieste, Italy. Bangalore, 04 September 2014 Ab initio molecular dynamics Simone Piccinin CNR-IOM DEMOCRITOS Trieste, Italy Bangalore, 04 September 2014 What is MD? 1) Liquid 4) Dye/TiO2/electrolyte 2) Liquids 3) Solvated protein 5) Solid to liquid

More information

What is Classical Molecular Dynamics?

What 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 information

Ab initio molecular dynamics and nuclear quantum effects

Ab initio molecular dynamics and nuclear quantum effects Ab initio molecular dynamics and nuclear quantum effects Luca M. Ghiringhelli Fritz Haber Institute Hands on workshop density functional theory and beyond: First principles simulations of molecules and

More information

Energy and Forces in DFT

Energy and Forces in DFT Energy and Forces in DFT Total Energy as a function of nuclear positions {R} E tot ({R}) = E DF T ({R}) + E II ({R}) (1) where E DF T ({R}) = DFT energy calculated for the ground-state density charge-density

More information

Computer simulation methods (1) Dr. Vania Calandrini

Computer simulation methods (1) Dr. Vania Calandrini Computer simulation methods (1) Dr. Vania Calandrini Why computational methods To understand and predict the properties of complex systems (many degrees of freedom): liquids, solids, adsorption of molecules

More information

Gear methods I + 1/18

Gear methods I + 1/18 Gear methods I + 1/18 Predictor-corrector type: knowledge of history is used to predict an approximate solution, which is made more accurate in the following step we do not want (otherwise good) methods

More information

Temperature and Pressure Controls

Temperature 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 information

Ab Ini'o Molecular Dynamics (MD) Simula?ons

Ab Ini'o Molecular Dynamics (MD) Simula?ons Ab Ini'o Molecular Dynamics (MD) Simula?ons Rick Remsing ICMS, CCDM, Temple University, Philadelphia, PA What are Molecular Dynamics (MD) Simulations? Technique to compute statistical and transport properties

More information

Understanding Molecular Simulation 2009 Monte Carlo and Molecular Dynamics in different ensembles. Srikanth Sastry

Understanding Molecular Simulation 2009 Monte Carlo and Molecular Dynamics in different ensembles. Srikanth Sastry JNCASR August 20, 21 2009 Understanding Molecular Simulation 2009 Monte Carlo and Molecular Dynamics in different ensembles Srikanth Sastry Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore

More information

A 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 information

Ab initio Molecular Dynamics Born Oppenheimer and beyond

Ab initio Molecular Dynamics Born Oppenheimer and beyond Ab initio Molecular Dynamics Born Oppenheimer and beyond Reminder, reliability of MD MD trajectories are chaotic (exponential divergence with respect to initial conditions), BUT... With a good integrator

More information

Temperature and Pressure Controls

Temperature 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 information

Introduction to Simulation - Lectures 17, 18. Molecular Dynamics. Nicolas Hadjiconstantinou

Introduction to Simulation - Lectures 17, 18. Molecular Dynamics. Nicolas Hadjiconstantinou Introduction to Simulation - Lectures 17, 18 Molecular Dynamics Nicolas Hadjiconstantinou Molecular Dynamics Molecular dynamics is a technique for computing the equilibrium and non-equilibrium properties

More information

Classical Molecular Dynamics

Classical Molecular Dynamics Classical Molecular Dynamics Matt Probert Condensed Matter Dynamics Group Department of Physics, University of York, U.K. http://www-users.york.ac.uk/~mijp1 Overview of lecture n Motivation n Types of

More information

Statistical Thermodynamics and Monte-Carlo Evgenii B. Rudnyi and Jan G. Korvink IMTEK Albert Ludwig University Freiburg, Germany

Statistical Thermodynamics and Monte-Carlo Evgenii B. Rudnyi and Jan G. Korvink IMTEK Albert Ludwig University Freiburg, Germany Statistical Thermodynamics and Monte-Carlo Evgenii B. Rudnyi and Jan G. Korvink IMTEK Albert Ludwig University Freiburg, Germany Preliminaries Learning Goals From Micro to Macro Statistical Mechanics (Statistical

More information

A Study of the Thermal Properties of a One. Dimensional Lennard-Jones System

A Study of the Thermal Properties of a One. Dimensional Lennard-Jones System A Study of the Thermal Properties of a One Dimensional Lennard-Jones System Abstract In this study, the behavior of a one dimensional (1D) Lennard-Jones (LJ) system is simulated. As part of this research,

More information

Introduction to molecular dynamics

Introduction to molecular dynamics 1 Introduction to molecular dynamics Yves Lansac Université François Rabelais, Tours, France Visiting MSE, GIST for the summer Molecular Simulation 2 Molecular simulation is a computational experiment.

More information

Scientific Computing II

Scientific Computing II Scientific Computing II Molecular Dynamics Simulation Michael Bader SCCS Summer Term 2015 Molecular Dynamics Simulation, Summer Term 2015 1 Continuum Mechanics for Fluid Mechanics? Molecular Dynamics the

More information

Kinetic theory. Collective behaviour of large systems Statistical basis for the ideal gas equation Deviations from ideality

Kinetic theory. Collective behaviour of large systems Statistical basis for the ideal gas equation Deviations from ideality Kinetic theory Collective behaviour of large systems Statistical basis for the ideal gas equation Deviations from ideality Learning objectives Describe physical basis for the kinetic theory of gases Describe

More information

Brownian Motion and Langevin Equations

Brownian 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 information

CHAPTER V. Brownian motion. V.1 Langevin dynamics

CHAPTER 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 information

Analysis of MD Results Using Statistical Mechanics Methods. Molecular Modeling

Analysis of MD Results Using Statistical Mechanics Methods. Molecular Modeling Analysis of MD Results Using Statistical Mechanics Methods Ioan Kosztin eckman Institute University of Illinois at Urbana-Champaign Molecular Modeling. Model building. Molecular Dynamics Simulation 3.

More information

Lecture 25 Goals: Chapter 18 Understand the molecular basis for pressure and the idealgas

Lecture 25 Goals: Chapter 18 Understand the molecular basis for pressure and the idealgas Lecture 5 Goals: Chapter 18 Understand the molecular basis for pressure and the idealgas law. redict the molar specific heats of gases and solids. Understand how heat is transferred via molecular collisions

More information

Energy Barriers and Rates - Transition State Theory for Physicists

Energy Barriers and Rates - Transition State Theory for Physicists Energy Barriers and Rates - Transition State Theory for Physicists Daniel C. Elton October 12, 2013 Useful relations 1 cal = 4.184 J 1 kcal mole 1 = 0.0434 ev per particle 1 kj mole 1 = 0.0104 ev per particle

More information

UNIVERSITY OF OSLO FACULTY OF MATHEMATICS AND NATURAL SCIENCES

UNIVERSITY OF OSLO FACULTY OF MATHEMATICS AND NATURAL SCIENCES UNIVERSITY OF OSLO FCULTY OF MTHEMTICS ND NTURL SCIENCES Exam in: FYS430, Statistical Mechanics Day of exam: Jun.6. 203 Problem :. The relative fluctuations in an extensive quantity, like the energy, depends

More information

Heat Transport in Glass-Forming Liquids

Heat Transport in Glass-Forming Liquids Heat Transport in Glass-Forming Liquids by VINAY VAIBHAV The Institute of Mathematical Sciences CIT Campus, Taramani, Chennai 600 113, India. A thesis submitted in partial fulfillment of requirements for

More information

Foundations of Chemical Kinetics. Lecture 19: Unimolecular reactions in the gas phase: RRKM theory

Foundations of Chemical Kinetics. Lecture 19: Unimolecular reactions in the gas phase: RRKM theory Foundations of Chemical Kinetics Lecture 19: Unimolecular reactions in the gas phase: RRKM theory Marc R. Roussel Department of Chemistry and Biochemistry Canonical and microcanonical ensembles Canonical

More information

Physics 207 Lecture 25. Lecture 25, Nov. 26 Goals: Chapter 18 Understand the molecular basis for pressure and the idealgas

Physics 207 Lecture 25. Lecture 25, Nov. 26 Goals: Chapter 18 Understand the molecular basis for pressure and the idealgas Lecture 25, Nov. 26 Goals: Chapter 18 Understand the molecular basis for pressure and the idealgas law. redict the molar specific heats of gases and solids. Understand how heat is transferred via molecular

More information

Citation for published version (APA): Hess, B. (2002). Stochastic concepts in molecular simulation Groningen: s.n.

Citation for published version (APA): Hess, B. (2002). Stochastic concepts in molecular simulation Groningen: s.n. University of Groningen Stochastic concepts in molecular simulation Hess, Berk IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check

More information

Molecular Dynamics Simulations. Dr. Noelia Faginas Lago Dipartimento di Chimica,Biologia e Biotecnologie Università di Perugia

Molecular Dynamics Simulations. Dr. Noelia Faginas Lago Dipartimento di Chimica,Biologia e Biotecnologie Università di Perugia Molecular Dynamics Simulations Dr. Noelia Faginas Lago Dipartimento di Chimica,Biologia e Biotecnologie Università di Perugia 1 An Introduction to Molecular Dynamics Simulations Macroscopic properties

More information

Molecular Dynamics Simulations

Molecular 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 information

Statistical Mechanics

Statistical Mechanics 42 My God, He Plays Dice! Statistical Mechanics Statistical Mechanics 43 Statistical Mechanics Statistical mechanics and thermodynamics are nineteenthcentury classical physics, but they contain the seeds

More information

4. The Green Kubo Relations

4. 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 information

Thermodynamics of nuclei in thermal contact

Thermodynamics of nuclei in thermal contact Thermodynamics of nuclei in thermal contact Karl-Heinz Schmidt, Beatriz Jurado CENBG, CNRS/IN2P3, Chemin du Solarium B.P. 120, 33175 Gradignan, France Abstract: The behaviour of a di-nuclear system in

More information

THERMOSTATS AND THERMAL TRANSPORT IN SOLIDS

THERMOSTATS AND THERMAL TRANSPORT IN SOLIDS Hands-On Tutorial Workshop, July 19 th 2011 THERMOSTATS AND THERMAL TRANSPORT IN SOLIDS Christian Carbogno FRITZ-HABER-INSTITUT DER MAX-PLANCK-GESELLSCHAFT, BERLIN - GERMANY Hands-On Tutorial Workshop,

More information

1. Thermodynamics 1.1. A macroscopic view of matter

1. Thermodynamics 1.1. A macroscopic view of matter 1. Thermodynamics 1.1. A macroscopic view of matter Intensive: independent of the amount of substance, e.g. temperature,pressure. Extensive: depends on the amount of substance, e.g. internal energy, enthalpy.

More information

Introduction Statistical Thermodynamics. Monday, January 6, 14

Introduction Statistical Thermodynamics. Monday, January 6, 14 Introduction Statistical Thermodynamics 1 Molecular Simulations Molecular dynamics: solve equations of motion Monte Carlo: importance sampling r 1 r 2 r n MD MC r 1 r 2 2 r n 2 3 3 4 4 Questions How can

More information

(# = %(& )(* +,(- Closed system, well-defined energy (or e.g. E± E/2): Microcanonical ensemble

(# = %(& )(* +,(- Closed system, well-defined energy (or e.g. E± E/2): Microcanonical ensemble Recall from before: Internal energy (or Entropy): &, *, - (# = %(& )(* +,(- Closed system, well-defined energy (or e.g. E± E/2): Microcanonical ensemble & = /01Ω maximized Ω: fundamental statistical quantity

More information

5. Systems in contact with a thermal bath

5. Systems in contact with a thermal bath 5. Systems in contact with a thermal bath So far, isolated systems (micro-canonical methods) 5.1 Constant number of particles:kittel&kroemer Chap. 3 Boltzmann factor Partition function (canonical methods)

More information

Introduction. Chapter The Purpose of Statistical Mechanics

Introduction. Chapter The Purpose of Statistical Mechanics Chapter 1 Introduction 1.1 The Purpose of Statistical Mechanics Statistical Mechanics is the mechanics developed to treat a collection of a large number of atoms or particles. Such a collection is, for

More information

A (short) practical introduction to kinetic theory and thermodynamic properties of gases through molecular dynamics

A (short) practical introduction to kinetic theory and thermodynamic properties of gases through molecular dynamics A (short) practical introduction to kinetic theory and thermodynamic properties of gases through molecular dynamics Miguel A. Caro mcaroba@gmail.com March 28, 2018 Contents 1 Preface 3 2 Review of thermodynamics

More information

ChE 503 A. Z. Panagiotopoulos 1

ChE 503 A. Z. Panagiotopoulos 1 ChE 503 A. Z. Panagiotopoulos 1 STATISTICAL MECHANICAL ENSEMLES 1 MICROSCOPIC AND MACROSCOPIC ARIALES The central question in Statistical Mechanics can be phrased as follows: If particles (atoms, molecules,

More information

Basics of Statistical Mechanics

Basics of Statistical Mechanics Basics of Statistical Mechanics Review of ensembles Microcanonical, canonical, Maxwell-Boltzmann Constant pressure, temperature, volume, Thermodynamic limit Ergodicity (see online notes also) Reading assignment:

More information

Introduction to Computer Simulations of Soft Matter Methodologies and Applications Boulder July, 19-20, 2012

Introduction to Computer Simulations of Soft Matter Methodologies and Applications Boulder July, 19-20, 2012 Introduction to Computer Simulations of Soft Matter Methodologies and Applications Boulder July, 19-20, 2012 K. Kremer Max Planck Institute for Polymer Research, Mainz Overview Simulations, general considerations

More information

Rate of Heating and Cooling

Rate of Heating and Cooling Rate of Heating and Cooling 35 T [ o C] Example: Heating and cooling of Water E 30 Cooling S 25 Heating exponential decay 20 0 100 200 300 400 t [sec] Newton s Law of Cooling T S > T E : System S cools

More information

Dissipative nuclear dynamics

Dissipative 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 information

Statistical Mechanics in a Nutshell

Statistical Mechanics in a Nutshell Chapter 2 Statistical Mechanics in a Nutshell Adapted from: Understanding Molecular Simulation Daan Frenkel and Berend Smit Academic Press (2001) pp. 9-22 11 2.1 Introduction In this course, we will treat

More information

Simulation of molecular systems by molecular dynamics

Simulation of molecular systems by molecular dynamics Simulation of molecular systems by molecular dynamics Yohann Moreau yohann.moreau@ujf-grenoble.fr November 26, 2015 Yohann Moreau (UJF) Molecular Dynamics, Label RFCT 2015 November 26, 2015 1 / 35 Introduction

More information

Biomolecular modeling I

Biomolecular modeling I 2015, December 15 Biomolecular simulation Elementary body atom Each atom x, y, z coordinates A protein is a set of coordinates. (Gromacs, A. P. Heiner) Usually one molecule/complex of interest (e.g. protein,

More information

to satisfy the large number approximations, W W sys can be small.

to satisfy the large number approximations, W W sys can be small. Chapter 12. The canonical ensemble To discuss systems at constant T, we need to embed them with a diathermal wall in a heat bath. Note that only the system and bath need to be large for W tot and W bath

More information

Accelerated Quantum Molecular Dynamics

Accelerated Quantum Molecular Dynamics Accelerated Quantum Molecular Dynamics Enrique Martinez, Christian Negre, Marc J. Cawkwell, Danny Perez, Arthur F. Voter and Anders M. N. Niklasson Outline Quantum MD Current approaches Challenges Extended

More information

424 Index. Eigenvalue in quantum mechanics, 174 eigenvector in quantum mechanics, 174 Einstein equation, 334, 342, 393

424 Index. Eigenvalue in quantum mechanics, 174 eigenvector in quantum mechanics, 174 Einstein equation, 334, 342, 393 Index After-effect function, 368, 369 anthropic principle, 232 assumptions nature of, 242 autocorrelation function, 292 average, 18 definition of, 17 ensemble, see ensemble average ideal,23 operational,

More information

Brownian motion and the Central Limit Theorem

Brownian 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 information

MOLECULAR DYNAMIC SIMULATION OF WATER VAPOR INTERACTION WITH VARIOUS TYPES OF PORES USING HYBRID COMPUTING STRUCTURES

MOLECULAR DYNAMIC SIMULATION OF WATER VAPOR INTERACTION WITH VARIOUS TYPES OF PORES USING HYBRID COMPUTING STRUCTURES MOLECULAR DYNAMIC SIMULATION OF WATER VAPOR INTERACTION WITH VARIOUS TYPES OF PORES USING HYBRID COMPUTING STRUCTURES V.V. Korenkov 1,3, a, E.G. Nikonov 1, b, M. Popovičová 2, с 1 Joint Institute for Nuclear

More information

Computational Physics

Computational 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 information

Statistical Mechanics

Statistical Mechanics Franz Schwabl Statistical Mechanics Translated by William Brewer Second Edition With 202 Figures, 26 Tables, and 195 Problems 4u Springer Table of Contents 1. Basic Principles 1 1.1 Introduction 1 1.2

More information

Statistical methods in atomistic computer simulations

Statistical methods in atomistic computer simulations Statistical methods in atomistic computer simulations Prof. Michele Ceriotti, michele.ceriotti@epfl.ch This course gives an overview of simulation techniques that are useful for the computational modeling

More information

KINETIC THEORY OF GASES

KINETIC THEORY OF GASES LECTURE 8 KINETIC THEORY OF GASES Text Sections 0.4, 0.5, 0.6, 0.7 Sample Problems 0.4 Suggested Questions Suggested Problems Summary None 45P, 55P Molecular model for pressure Root mean square (RMS) speed

More information

ChE 210B: Advanced Topics in Equilibrium Statistical Mechanics

ChE 210B: Advanced Topics in Equilibrium Statistical Mechanics ChE 210B: Advanced Topics in Equilibrium Statistical Mechanics Glenn Fredrickson Lecture 1 Reading: 3.1-3.5 Chandler, Chapters 1 and 2 McQuarrie This course builds on the elementary concepts of statistical

More information

Chapter 4: Going from microcanonical to canonical ensemble, from energy to temperature.

Chapter 4: Going from microcanonical to canonical ensemble, from energy to temperature. Chapter 4: Going from microcanonical to canonical ensemble, from energy to temperature. All calculations in statistical mechanics can be done in the microcanonical ensemble, where all copies of the system

More information

Organization of NAMD Tutorial Files

Organization of NAMD Tutorial Files Organization of NAMD Tutorial Files .1.1. RMSD for individual residues Objective: Find the average RMSD over time of each residue in the protein using VMD. Display the protein with the residues colored

More information

Speed Distribution at CONSTANT Temperature is given by the Maxwell Boltzmann Speed Distribution

Speed Distribution at CONSTANT Temperature is given by the Maxwell Boltzmann Speed Distribution Temperature ~ Average KE of each particle Particles have different speeds Gas Particles are in constant RANDOM motion Average KE of each particle is: 3/2 kt Pressure is due to momentum transfer Speed Distribution

More information

Modeling of Micro-Fluidics by a Dissipative Particle Dynamics Method. Justyna Czerwinska

Modeling 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 information

Statistical Mechanics

Statistical Mechanics Statistical Mechanics Newton's laws in principle tell us how anything works But in a system with many particles, the actual computations can become complicated. We will therefore be happy to get some 'average'

More information

Ab initio molecular dynamics

Ab initio molecular dynamics Ab initio molecular dynamics Molecular dynamics Why? allows realistic simulation of equilibrium and transport properties in Nature ensemble averages can be used for statistical mechanics time evolution

More information

Set the initial conditions r i. Update neighborlist. r i. Get new forces F i

Set the initial conditions r i. Update neighborlist. r i. Get new forces F i Set the initial conditions r i t 0, v i t 0 Update neighborlist Get new forces F i r i Solve the equations of motion numerically over time step t : r i t n r i t n + v i t n v i t n + Perform T, P scaling

More information

Turning up the heat: thermal expansion

Turning up the heat: thermal expansion Lecture 3 Turning up the heat: Kinetic molecular theory & thermal expansion Gas in an oven: at the hot of materials science Here, the size of helium atoms relative to their spacing is shown to scale under

More information

LECTURE 6 : BASICS FOR MOLECULAR SIMULATIONS - Historical perspective - Skimming over Statistical Mechanics - General idea of Molecular Dynamics -

LECTURE 6 : BASICS FOR MOLECULAR SIMULATIONS - Historical perspective - Skimming over Statistical Mechanics - General idea of Molecular Dynamics - LECTURE 6 : BASICS FOR MOLECULAR SIMULATIONS - Historical perspective - Skimming over Statistical Mechanics - General idea of Molecular Dynamics - Force calculations, structure of MD, equations of motion

More information

Constant Pressure Langevin Dynamics: Theory and Application to the Study of Phase Behaviour in Core-Softened Systems.

Constant Pressure Langevin Dynamics: Theory and Application to the Study of Phase Behaviour in Core-Softened Systems. Constant Pressure Langevin Dynamics: Theory and Application to the Study of Phase Behaviour in Core-Softened Systems. David Quigley A thesis submitted for the degree of Doctor of Philosophy University

More information

Statistical thermodynamics for MD and MC simulations

Statistical thermodynamics for MD and MC simulations Statistical thermodynamics for MD and MC simulations knowing 2 atoms and wishing to know 10 23 of them Marcus Elstner and Tomáš Kubař 22 June 2016 Introduction Thermodynamic properties of molecular systems

More information

The micro-properties of [hmpy+] [Tf 2 N-] Ionic liquid: a simulation. study. 1. Introduction

The micro-properties of [hmpy+] [Tf 2 N-] Ionic liquid: a simulation. study. 1. Introduction ISBN 978-1-84626-081-0 Proceedings of the 2010 International Conference on Application of Mathematics and Physics Volume 1: Advances on Space Weather, Meteorology and Applied Physics Nanjing, P. R. China,

More information

CH 240 Chemical Engineering Thermodynamics Spring 2007

CH 240 Chemical Engineering Thermodynamics Spring 2007 CH 240 Chemical Engineering Thermodynamics Spring 2007 Instructor: Nitash P. Balsara, nbalsara@berkeley.edu Graduate Assistant: Paul Albertus, albertus@berkeley.edu Course Description Covers classical

More information

From Molecular Dynamics to hydrodynamics a novel Galilean invariant thermostat

From Molecular Dynamics to hydrodynamics a novel Galilean invariant thermostat To appear in: Journal of Chemical Physics, 08 APR 005 From Molecular Dynamics to hydrodynamics a novel Galilean invariant thermostat Simeon D. Stoyanov and Robert D. Groot Unilever Research Vlaardingen,

More information

On the Asymptotic Convergence. of the Transient and Steady State Fluctuation Theorems. Gary Ayton and Denis J. Evans. Research School Of Chemistry

On the Asymptotic Convergence. of the Transient and Steady State Fluctuation Theorems. Gary Ayton and Denis J. Evans. Research School Of Chemistry 1 On the Asymptotic Convergence of the Transient and Steady State Fluctuation Theorems. Gary Ayton and Denis J. Evans Research School Of Chemistry Australian National University Canberra, ACT 0200 Australia

More information

Molecular Dynamics Simulation Study of Transport Properties of Diatomic Gases

Molecular Dynamics Simulation Study of Transport Properties of Diatomic Gases MD Simulation of Diatomic Gases Bull. Korean Chem. Soc. 14, Vol. 35, No. 1 357 http://dx.doi.org/1.51/bkcs.14.35.1.357 Molecular Dynamics Simulation Study of Transport Properties of Diatomic Gases Song

More information

Ab initio molecular dynamics: Basic Theory and Advanced Met

Ab initio molecular dynamics: Basic Theory and Advanced Met Ab initio molecular dynamics: Basic Theory and Advanced Methods Uni Mainz October 30, 2016 Bio-inspired catalyst for hydrogen production Ab-initio MD simulations are used to learn how the active site

More information

Advanced sampling. fluids of strongly orientation-dependent interactions (e.g., dipoles, hydrogen bonds)

Advanced sampling. fluids of strongly orientation-dependent interactions (e.g., dipoles, hydrogen bonds) Advanced sampling ChE210D Today's lecture: methods for facilitating equilibration and sampling in complex, frustrated, or slow-evolving systems Difficult-to-simulate systems Practically speaking, one is

More information

APMA 2811T. By Zhen Li. Today s topic: Lecture 2: Theoretical foundation and parameterization. Sep. 15, 2016

APMA 2811T. By Zhen Li. Today s topic: Lecture 2: Theoretical foundation and parameterization. Sep. 15, 2016 Today s topic: APMA 2811T Dissipative Particle Dynamics Instructor: Professor George Karniadakis Location: 170 Hope Street, Room 118 Time: Thursday 12:00pm 2:00pm Dissipative Particle Dynamics: Foundation,

More information

MP203 Statistical and Thermal Physics. Jon-Ivar Skullerud and James Smith

MP203 Statistical and Thermal Physics. Jon-Ivar Skullerud and James Smith MP203 Statistical and Thermal Physics Jon-Ivar Skullerud and James Smith October 3, 2017 1 Contents 1 Introduction 3 1.1 Temperature and thermal equilibrium.................... 4 1.1.1 The zeroth law of

More information

IV. Classical Molecular Dynamics

IV. Classical Molecular Dynamics IV. Classical Molecular Dynamics Basic Assumptions: 1. Born-Oppenheimer Approximation 2. Classical mechanical nuclear motion Unavoidable Additional Approximations: 1. Approximate potential energy surface

More information

Time-Dependent Statistical Mechanics 5. The classical atomic fluid, classical mechanics, and classical equilibrium statistical mechanics

Time-Dependent Statistical Mechanics 5. The classical atomic fluid, classical mechanics, and classical equilibrium statistical mechanics Time-Dependent Statistical Mechanics 5. The classical atomic fluid, classical mechanics, and classical equilibrium statistical mechanics c Hans C. Andersen October 1, 2009 While we know that in principle

More information

fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES

fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Content-Thermodynamics & Statistical Mechanics 1. Kinetic theory of gases..(1-13) 1.1 Basic assumption of kinetic theory 1.1.1 Pressure exerted by a gas 1.2 Gas Law for Ideal gases: 1.2.1 Boyle s Law 1.2.2

More information

Molecular Dynamics at Constant Pressure: Allowing the System to Control Volume Fluctuations via a Shell Particle

Molecular Dynamics at Constant Pressure: Allowing the System to Control Volume Fluctuations via a Shell Particle Entropy 2013, 15, 3941-3969; doi:10.3390/e15093941 Review OPEN ACCESS entropy ISSN 1099-4300 www.mdpi.com/journal/entropy Molecular Dynamics at Constant Pressure: Allowing the System to Control Volume

More information

Statistical Mechanics. Atomistic view of Materials

Statistical Mechanics. Atomistic view of Materials Statistical Mechanics Atomistic view of Materials What is statistical mechanics? Microscopic (atoms, electrons, etc.) Statistical mechanics Macroscopic (Thermodynamics) Sample with constrains Fixed thermodynamics

More information

Computer simulations as concrete models for student reasoning

Computer simulations as concrete models for student reasoning Computer simulations as concrete models for student reasoning Jan Tobochnik Department of Physics Kalamazoo College Kalamazoo MI 49006 In many thermal physics courses, students become preoccupied with

More information

The Failure of Classical Mechanics

The Failure of Classical Mechanics Chapter 1 The Failure of Classical Mechanics Classical mechanics, erected by Galileo and Newton, with enormous contributions from many others, is remarkably successful. It enables us to calculate celestial

More information

Table of Contents [ttc]

Table of Contents [ttc] Table of Contents [ttc] 1. Equilibrium Thermodynamics I: Introduction Thermodynamics overview. [tln2] Preliminary list of state variables. [tln1] Physical constants. [tsl47] Equations of state. [tln78]

More information

Basics of Statistical Mechanics

Basics of Statistical Mechanics Basics of Statistical Mechanics Review of ensembles Microcanonical, canonical, Maxwell-Boltzmann Constant pressure, temperature, volume, Thermodynamic limit Ergodicity (see online notes also) Reading assignment:

More information

Molecular dynamics simulation of Aquaporin-1. 4 nm

Molecular dynamics simulation of Aquaporin-1. 4 nm Molecular dynamics simulation of Aquaporin-1 4 nm Molecular Dynamics Simulations Schrödinger equation i~@ t (r, R) =H (r, R) Born-Oppenheimer approximation H e e(r; R) =E e (R) e(r; R) Nucleic motion described

More information

Kinetic theory of the ideal gas

Kinetic theory of the ideal gas Appendix H Kinetic theory of the ideal gas This Appendix contains sketchy notes, summarizing the main results of elementary kinetic theory. The students who are not familiar with these topics should refer

More information

Enabling constant pressure hybrid Monte Carlo simulations using the GROMACS molecular simulation package

Enabling constant pressure hybrid Monte Carlo simulations using the GROMACS molecular simulation package Enabling constant pressure hybrid Monte Carlo simulations using the GROMACS molecular simulation package Mario Fernández Pendás MSBMS Group Supervised by Bruno Escribano and Elena Akhmatskaya BCAM 18 October

More information

The dynamics of small particles whose size is roughly 1 µmt or. smaller, in a fluid at room temperature, is extremely erratic, and is

The 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 information

Van Gunsteren et al. Angew. Chem. Int. Ed. 45, 4064 (2006)

Van Gunsteren et al. Angew. Chem. Int. Ed. 45, 4064 (2006) Van Gunsteren et al. Angew. Chem. Int. Ed. 45, 4064 (2006) Martini Workshop 2015 Coarse Graining Basics Alex de Vries Every word or concept, clear as it may seem to be, has only a limited range of applicability

More information

Multiscale Materials Modeling

Multiscale Materials Modeling Multiscale Materials Modeling Lecture 02 Capabilities of Classical Molecular Simulation These notes created by David Keffer, University of Tennessee, Knoxville, 2009. Outline Capabilities of Classical

More information

Molecular Dynamics. Park City June 2005 Tully

Molecular Dynamics. Park City June 2005 Tully Molecular Dynamics John Lance Natasa Vinod Xiaosong Dufie Priya Sharani Hongzhi Group: August, 2004 Prelude: Classical Mechanics Newton s equations: F = ma = mq = p Force is the gradient of the potential:

More information

1 Phase Spaces and the Liouville Equation

1 Phase Spaces and the Liouville Equation Phase Spaces and the Liouville Equation emphasize the change of language from deterministic to probablistic description. Under the dynamics: ½ m vi = F i ẋ i = v i with initial data given. What is the

More information

2. Thermodynamics. Introduction. Understanding Molecular Simulation

2. Thermodynamics. Introduction. Understanding Molecular Simulation 2. Thermodynamics Introduction Molecular Simulations Molecular dynamics: solve equations of motion r 1 r 2 r n Monte Carlo: importance sampling r 1 r 2 r n How do we know our simulation is correct? Molecular

More information

Contents. 1 Introduction and guide for this text 1. 2 Equilibrium and entropy 6. 3 Energy and how the microscopic world works 21

Contents. 1 Introduction and guide for this text 1. 2 Equilibrium and entropy 6. 3 Energy and how the microscopic world works 21 Preface Reference tables Table A Counting and combinatorics formulae Table B Useful integrals, expansions, and approximations Table C Extensive thermodynamic potentials Table D Intensive per-particle thermodynamic

More information

Molecular Dynamics. A very brief introduction

Molecular Dynamics. A very brief introduction Molecular Dynamics A very brief introduction Sander Pronk Dept. of Theoretical Physics KTH Royal Institute of Technology & Science For Life Laboratory Stockholm, Sweden Why computer simulations? Two primary

More information