macroscopic view (phenomenological) microscopic view (atomistic) computing reaction rate rate of reactions experiments thermodynamics

Size: px
Start display at page:

Download "macroscopic view (phenomenological) microscopic view (atomistic) computing reaction rate rate of reactions experiments thermodynamics"

Transcription

1 Rate Theory (overview) macroscopic view (phenomenological) rate of reactions experiments thermodynamics Van t Hoff & Arrhenius equation microscopic view (atomistic) statistical mechanics transition state theories Eyring theory effect of environment static: potential of mean force dynamic: Kramer s theory computing reaction rate optimizating transition states normal mode analysis simulating barrier crossing practical next week

2 Chromophore in water p-hydroxybenzylidene acetone (pck) CASSCF(6,6)/3-21G//SPCE molecular dynamics resonance structures

3 Rate of photoisomerization of double bond uni-molecular process, initiated by photon absorption

4 Rate of photoisomerization of double bond uni-molecular process, initiated by photon absorption

5 measuring reaction rate simulation & pump-probe fluorescence

6 kinetics & thermodynamics approaching equilibrium unimolecular process A k + k B d[a] dt d[b] dt = k + [A]+k [B] =+k + [A] k [B] conservation law so that [A]+[B] =[A] 0 d[a] dt = k + [A]+k ([A] 0 [A]) = (k + + k )[A]+k [A] 0 solution of the differential equations [A] = k + k + e (k +k )t k + + k [A] 0

7 kinetics & thermodynamics approaching equilibrium eventually... lim [A] = t k k + + k [A] 0 lim [B] =[A] 0 [A] = t equilibrium constant & reaction free energy K = [B] = k + =exp G [A] k RT k + k + + k [A] 0

8 temperature dependence of reaction rates Gibbs-Helmholtz relation G = H TS S = H G T G T G p = S = G H T T p G T = H T

9 temperature dependence of reaction rates Gibbs-Helmholtz relation G = H TS S = H G T G T G p = S = G H T T p G T = H T G T T T p = H T

10 temperature dependence of reaction rates Gibbs-Helmholtz relation G = H TS S = H G T G T G p = S = G H T T p G T = H T G G T T T p = H T T T p = H T 2

11 temperature dependence of reaction rates Gibbs-Helmholtz relation G = H TS S = H G T G T G p = S = G H T T p G T = H T G G T T T p = H T T T p = H T 2

12 temperature dependence of reaction rates Van t Hoff equation equilibrium constant d ln K dt ln K = G = 1 R d dt RT Gibbs-Helmholtz predicts effect of temperature on equilibrium constant G T p = H RT 2 d ln K d1/t = H R

13 temperature dependence of reaction rates Van t Hof equation equilibrium constant d ln K dt ln K = G = 1 R d dt RT Gibbs-Helmholtz predicts effect of temperature on equilibrium constant K = k + k G T p = H RT 2 relation between equilibrium and rate constant d ln K d dt ln k + d dt ln k = H RT 2 d1/t = H R

14 temperature dependence of reaction rates Van t Hof equation equilibrium constant d ln K dt ln K = G = 1 R d dt RT Gibbs-Helmholtz predicts effect of temperature on equilibrium constant K = k + k G T p = H RT 2 relation between equilibrium and rate constant therefore d d1/t ln k = E R + a d ln K d dt ln k + d dt ln k = H RT 2 d1/t = H R

15 temperature dependence of reaction rates Arrhenius equation activated state A A B K = [A ] [A]

16 temperature dependence of reaction rates Arrhenius equation activated state A A B K = [A ] [A] d ln K d1/t = H R k = a exp E a RT ln k =lna E a R 1 T

17 microscopic picture statistical mechanics partition function K = p B p A = Q B Q A = B exp[ βh]dpdq A exp[ βh]dpdq β = 1 k B T

18 microscopic picture statistical mechanics partition function K = p B p A = Q B Q A = Hamiltonian H = T + V B exp[ βh]dpdq A exp[ βh]dpdq β = 1 k B T H = i p 2 i 2m i + V (q 1,q 2,..,q n )

19 microscopic picture statistical mechanics partition function K = p B p A = Q B Q A = Hamiltonian H = T + V B exp[ βh]dpdq A exp[ βh]dpdq β = 1 k B T H = i p 2 i 2m i + V (q 1,q 2,..,q n ) integrate over momenta K = exp[ βv ]dq B exp[ βv ]dq A equilibrium determined solely by potential energy surface

20 microscopic picture compute rates from simulations rare event τ rxn τ eq k =1/τ rxn

21 microscopic picture compute rates from simulations rare event τ rxn τ eq basic assumptions initial rate stationary conditions dρ(p, q) =0 dt k =1/τ rxn

22 microscopic picture compute rates from simulations rare event τ rxn τ eq basic assumptions initial rate stationary conditions flux dρ(p, q) dt =0 J = kc A c A = Θ(x x) k =1/τ rxn

23 microscopic picture compute rates from simulations rare event τ rxn τ eq basic assumptions initial rate stationary conditions flux dρ(p, q) dt =0 k =1/τ rxn J = kc A c A = Θ(x x) sampling problem... ρ(x exp[ βv (x)]δ(x x )dx )= exp[ βv (x)]dx

24 Eyring theory assumptions classical dynamics no recrossing molecules at barrier in thermal equilibrium with molecules in reactant well

25 Eyring theory δl observations barrier is flat: f(x )= du dx x=x =0

26 Eyring theory δl observations barrier is flat: f(x )= du dx x=x =0 there are δnmolecules in δl

27 Eyring theory δl observations barrier is flat: f(x )= du dx x=x =0 there are reaction if δnmolecules in v> δl δt δl

28 Eyring theory δl observations barrier is flat: f(x )= du dx x=x =0 there are δnmolecules in δl reaction if v> δl δt the number of molecules passing TST in dt N rxn = δn vdt δl

29 Eyring theory δl observations barrier is flat: f(x )= du dx x=x =0 there are δnmolecules in δl reaction if v> δl δt the number of molecules passing TST in dt N rxn = δn vdt δl reaction rate k + = N rxn Ndt = δn N v δl

30 Eyring theory δl observations barrier is flat: f(x )= du dx x=x =0 there are δnmolecules in δl reaction if v> δl δt the number of molecules passing TST in dt N rxn = δn vdt δl reaction rate k + = N rxn Ndt = δn N v δl δn N = q q A partition function k + = q q A v δl

31 Eyring theory partition function of TST q = 1 h δl exp[ β( p2 2m + V (x )]dp q = δl h exp[ β p2 2m ]dp exp[ βv (x )] q = δl h 2mkB T π exp[ V (x ) k B T ]

32 Eyring theory partition function of TST q = 1 h δl exp[ β( p2 2m + V (x )]dp q = δl h exp[ β p2 2m ]dp exp[ βv (x )] q = δl h 2mkB T π exp[ V (x ) k B T ] only positive velocities contribute v + = v + = vθ(v)exp[ β p2 2m ]dp exp[ β p2 2m ]dp 1 1 m 2 2mk BT kb 2mkB T π = T 2πm

33 Eyring theory taking together to express rate k + = δl h 2mkB T π δlq A kb T 2πm exp V (x ) k B T k + = k BT hq A exp V (x ) k B T

34 Eyring theory taking together to express rate k + = δl h 2mkB T π δlq A kb T 2πm exp V (x ) k B T k + = k BT hq A exp V (x ) k B T partition function of A q A = 1 h x exp V (x) k B T dx exp β p2 2m dp q A = 1 h 2πmkB T x exp V (x) k B T dx

35 Eyring theory taking together to express rate k + = δl h 2mkB T π δlq A kb T 2πm exp V (x ) k B T k + = k BT hq A exp V (x ) k B T partition function of A q A = 1 h x exp V (x) k B T dx exp β p2 2m dp q A = 1 h 2πmkB T x exp V (x) k B T dx

36 Eyring theory harmonic approximation V (x) 1 2 k f (x x A ) 2 V (x) 1 2 mω2 A(x x A ) 2 ω A = kf m

37 Eyring theory harmonic approximation V (x) 1 2 k f (x x A ) 2 V (x) 1 2 mω2 A(x x A ) 2 ω A = kf m partition function q A = h 1 2πmkB T 2k B T mω 2 A π q A = 1 h 2πk BT 1 ω A

38 Eyring theory harmonic approximation V (x) 1 2 k f (x x A ) 2 V (x) 1 2 mω2 A(x x A ) 2 ω A = kf m partition function q A = 1 h 2πk BT 1 ω A Final result: Eyring equation k + = ω A 2π exp V (x ) k B T

39 Eyring theory harmonic approximation V (x) 1 2 k f (x x A ) 2 V (x) 1 2 mω2 A(x x A ) 2 ω A = kf m partition function q A = 1 h 2πk BT 1 ω A Final result: Eyring equation attempt frequency k + = ω A 2π exp V (x ) k B T probability to be at barrier (Boltzmann factor)

40 static solvent effects: potential of mean force oops

41 Dynamic solvent effects: Kramers Theory coupling between reaction coordinate and other coordinates friction due to interactions dv dt = g v = γv m v = v 0 e γt thermal noise (Brownian motion): Langevin dynamics dv dt = γv + F R(t) noise properties F R =0 F R (t 1 )F R (t 2 ) = φ(t 2 t 1 ) fδ(t 2 t 1 ) solution v = v 0 exp [ γt]+exp[ γt] t 0 exp [γt] F R (τ)dτ

42 Dynamic solvent effects: Kramers Theory coupling between reaction coordinate and other coordinates solution v = v 0 exp [ γt]+exp[ γt] properties v = v 0 exp [ γt] t 0 exp [γt] F R (τ)dτ v 2 = v 2 0 exp [ 2γt]+ f 2γ (1 exp [ 2γt]) lim v =0 t lim t v2 = f 2γ equipartition theorem: 1 2 mv2 = 1 2 k BT F R (t)f R (0) = δ(t)2γk B T/m f =2γk B T/m

43 Dynamic solvent effects: Kramers Theory Fokker-Planck equation P (r, v; t) t = v P (r, v; t) r + 1 M U r P (r, v; t) v + γ v (vp(r, v; t)) + γk BT M 2 P (r, v; t) v 2 probability P (r, v; t)drdv to find a particle at r, r + dr with velocity v, v + dv

44 Dynamic solvent effects: Kramers Theory Fokker-Planck equation P (r, v; t) t = v P (r, v; t) r + 1 M U r P (r, v; t) v + γ v (vp(r, v; t)) + γk BT M 2 P (r, v; t) v 2 stationary solution P t =0 mv 2 P (r, v) = 1 Q exp 2 + V (r) /k B T Boltzmann distribution

45 Dynamic solvent effects: Kramers Theory Fokker-Planck equation P (r, v; t) t = v P (r, v; t) r + 1 M U r P (r, v; t) v + γ v (vp(r, v; t)) + γk BT M 2 P (r, v; t) v 2 steady state solution mv 2 P (r, v) =Y (r, v) 1 Q exp boundary conditions 2 + V (r) /k B T r r A Y (r, v) =1 r r C Y (r, v) =0

46 Dynamic solvent effects: Kramers Theory free energy surface surface U(r) =U(r A )+ 1 2 mω2 A(r r A ) 2 U(r) =U(r B ) 1 2 mω2 B(r r B ) 2

47 Dynamic solvent effects: Kramers Theory free energy surface surface U(r) =U(r A )+ 1 2 mω2 A(r r A ) 2 U(r) =U(r B ) 1 2 mω2 B(r r B ) 2 stationary solution to Fokker-Planck equation k + = ω A γ 2 2πω B 4 + ω2 B γ exp [ (U(r B ) U(r A )) /k B T ] 2

48 Dynamic solvent effects: Kramers Theory free energy surface surface U(r) =U(r A )+ 1 2 mω2 A(r r A ) 2 U(r) =U(r B ) 1 2 mω2 B(r r B ) 2 stationary solution to F-P equation γ 2 k + = ω A 2πω B 4 + ω2 B γ 2 exp [ (U(r B ) U(r A )) /k B T ]

49 Dynamic solvent effects: Kramers Theory free energy surface surface U(r) =U(r A )+ 1 2 mω2 A(r r A ) 2 U(r) =U(r B ) 1 2 mω2 B(r r B ) 2 stationary solution to F-P equation γ 2 k + = ω A 2πω B 4 + ω2 B γ 2 exp [ (U(r B ) U(r A )) /k B T ] limiting cases γ/2 ω B k + = ω Aω B 2πγ exp U /k B T γ/2 ω B k + = ω A 2π exp U /k B T high friction low friction

50 Dynamic solvent effects: Kramers Theory free energy surface surface U(r) =U(r A )+ 1 2 mω2 A(r r A ) 2 U(r) =U(r B ) 1 2 mω2 B(r r B ) 2 stationary solution to F-P equation γ 2 k + = ω A 2πω B 4 + ω2 B γ 2 exp [ (U(r B ) U(r A )) /k B T ] limiting cases γ/2 ω B k + = ω Aω B 2πγ exp U /k B T γ/2 ω B k + = ω A 2π exp U /k B T transmission coefficient k + = κk TST + high friction low friction

macroscopic view (phenomenological) microscopic view (atomistic) computing reaction rate rate of reactions experiments thermodynamics

macroscopic view (phenomenological) microscopic view (atomistic) computing reaction rate rate of reactions experiments thermodynamics Rate heory (overview) macroscopic view (phenomenological) rate of reactions experiments thermodynamics Van t Hoff & Arrhenius equation microscopic view (atomistic) statistical mechanics transition state

More information

The Kramers problem and first passage times.

The 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 information

Major Concepts Langevin Equation

Major Concepts Langevin Equation Major Concepts Langevin Equation Model for a tagged subsystem in a solvent Harmonic bath with temperature, T Friction & Correlated forces (FDR) Markovian/Ohmic vs. Memory Chemical Kinetics Master equation

More information

Foundations of Chemical Kinetics. Lecture 12: Transition-state theory: The thermodynamic formalism

Foundations of Chemical Kinetics. Lecture 12: Transition-state theory: The thermodynamic formalism Foundations of Chemical Kinetics Lecture 12: Transition-state theory: The thermodynamic formalism Marc R. Roussel Department of Chemistry and Biochemistry Breaking it down We can break down an elementary

More information

16. Working with the Langevin and Fokker-Planck equations

16. 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 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

Reaction Dynamics (2) Can we predict the rate of reactions?

Reaction Dynamics (2) Can we predict the rate of reactions? Reaction Dynamics (2) Can we predict the rate of reactions? Reactions in Liquid Solutions Solvent is NOT a reactant Reactive encounters in solution A reaction occurs if 1. The reactant molecules (A, B)

More information

Solutions of a Focker-Planck Equation

Solutions of a Focker-Planck Equation Solutions of a Focker-Planck Equation W. C. Troy March 31, 006 1. Overview.. The Langevin Equation for Brownian Motion. 3. The Focker-Planck equation. 4. The steady state equation. 1 Overview. In these

More information

Express the transition state equilibrium constant in terms of the partition functions of the transition state and the

Express the transition state equilibrium constant in terms of the partition functions of the transition state and the Module 7 : Theories of Reaction Rates Lecture 33 : Transition State Theory Objectives After studying this Lecture you will be able to do the following. Distinguish between collision theory and transition

More information

Brownian Motion: Fokker-Planck Equation

Brownian 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 information

F r (t) = 0, (4.2) F (t) = r= r a (t)

F 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 information

Advanced Physical Chemistry CHAPTER 18 ELEMENTARY CHEMICAL KINETICS

Advanced Physical Chemistry CHAPTER 18 ELEMENTARY CHEMICAL KINETICS Experimental Kinetics and Gas Phase Reactions Advanced Physical Chemistry CHAPTER 18 ELEMENTARY CHEMICAL KINETICS Professor Angelo R. Rossi http://homepages.uconn.edu/rossi Department of Chemistry, Room

More information

Thermodynamics and Kinetics

Thermodynamics and Kinetics Thermodynamics and Kinetics C. Paolucci University of Notre Dame Department of Chemical & Biomolecular Engineering What is the energy we calculated? You used GAMESS to calculate the internal (ground state)

More information

Major Concepts Kramers Turnover

Major Concepts Kramers Turnover Major Concepts Kramers Turnover Low/Weak -Friction Limit (Energy Diffusion) Intermediate Regime bounded by TST High/Strong -Friction Limit Smoluchovski (Spatial Diffusion) Fokker-Planck Equation Probability

More information

LANGEVIN THEORY OF BROWNIAN MOTION. Contents. 1 Langevin theory. 1 Langevin theory 1. 2 The Ornstein-Uhlenbeck process 8

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 information

Statistical Mechanics of Active Matter

Statistical 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 information

PHYS 352 Homework 2 Solutions

PHYS 352 Homework 2 Solutions PHYS 352 Homework 2 Solutions Aaron Mowitz (, 2, and 3) and Nachi Stern (4 and 5) Problem The purpose of doing a Legendre transform is to change a function of one or more variables into a function of variables

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

08. Brownian Motion. University of Rhode Island. Gerhard Müller University of Rhode Island,

08. 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 information

7. Kinetics controlled by fluctuations: Kramers theory of activated processes

7. Kinetics controlled by fluctuations: Kramers theory of activated processes 7. Kinetics controlled by fluctuations: Kramers theory of activated processes Macroscopic kinetic processes (time dependent concentrations) Elementary kinetic process Reaction mechanism Unimolecular processes

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

Foundations of Chemical Kinetics. Lecture 30: Transition-state theory in the solution phase

Foundations of Chemical Kinetics. Lecture 30: Transition-state theory in the solution phase Foundations of Chemical Kinetics Lecture 30: Transition-state theory in the solution phase Marc R. Roussel Department of Chemistry and Biochemistry Transition-state theory in solution We revisit our original

More information

Non equilibrium thermodynamics: foundations, scope, and extension to the meso scale. Miguel Rubi

Non equilibrium thermodynamics: foundations, scope, and extension to the meso scale. Miguel Rubi Non equilibrium thermodynamics: foundations, scope, and extension to the meso scale Miguel Rubi References S.R. de Groot and P. Mazur, Non equilibrium Thermodynamics, Dover, New York, 1984 J.M. Vilar and

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

Evaluation of the rate constant and deposition velocity for the escape of Brownian particles over potential barriers

Evaluation of the rate constant and deposition velocity for the escape of Brownian particles over potential barriers arxiv:1411.0692v1 [cond-mat.stat-mech] 22 Oct 2014 Evaluation of the rate constant and deposition velocity for the escape of Brownian particles over potential barriers Michael W Reeks School of Mechanical

More information

Ideal gases. Asaf Pe er Classical ideal gas

Ideal gases. Asaf Pe er Classical ideal gas Ideal gases Asaf Pe er 1 November 2, 213 1. Classical ideal gas A classical gas is generally referred to as a gas in which its molecules move freely in space; namely, the mean separation between the molecules

More information

Lecture 12. Temperature Lidar (1) Overview and Physical Principles

Lecture 12. Temperature Lidar (1) Overview and Physical Principles Lecture 2. Temperature Lidar () Overview and Physical Principles q Concept of Temperature Ø Maxwellian velocity distribution & kinetic energy q Temperature Measurement Techniques Ø Direct measurements:

More information

Time-Dependent Transition State Theory to Determine Dividing Surfaces and Reaction Rates in Multidimensional Systems

Time-Dependent Transition State Theory to Determine Dividing Surfaces and Reaction Rates in Multidimensional Systems Time-Dependent Transition State Theory to Determine Dividing Surfaces and Reaction Rates in Multidimensional Systems Master s thesis of Robin Bardakcioglu April 5th, 018 First Examiner: Second Examiner:

More information

Lecture 27 Thermodynamics: Enthalpy, Gibbs Free Energy and Equilibrium Constants

Lecture 27 Thermodynamics: Enthalpy, Gibbs Free Energy and Equilibrium Constants Physical Principles in Biology Biology 3550 Fall 2017 Lecture 27 Thermodynamics: Enthalpy, Gibbs Free Energy and Equilibrium Constants Wednesday, 1 November c David P. Goldenberg University of Utah goldenberg@biology.utah.edu

More information

Introduction to nonequilibrium physics

Introduction 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 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

UNIVERSITY OF SOUTHAMPTON

UNIVERSITY OF SOUTHAMPTON UNIVERSITY OF SOUTHAMPTON PHYS1013W1 SEMESTER 2 EXAMINATION 2014-2015 ENERGY AND MATTER Duration: 120 MINS (2 hours) This paper contains 8 questions. Answers to Section A and Section B must be in separate

More information

1 Particles in a room

1 Particles in a room Massachusetts Institute of Technology MITES 208 Physics III Lecture 07: Statistical Physics of the Ideal Gas In these notes we derive the partition function for a gas of non-interacting particles in a

More information

1. Introduction to Chemical Kinetics

1. Introduction to Chemical Kinetics 1. Introduction to Chemical Kinetics objectives of chemical kinetics 1) Determine empirical rate laws H 2 + I 2 2HI How does the concentration of H 2, I 2, and HI change with time? 2) Determine the mechanism

More information

Quiz 3 for Physics 176: Answers. Professor Greenside

Quiz 3 for Physics 176: Answers. Professor Greenside Quiz 3 for Physics 176: Answers Professor Greenside True or False Questions ( points each) For each of the following statements, please circle T or F to indicate respectively whether a given statement

More information

A path integral approach to the Langevin equation

A 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 information

Nanoscale simulation lectures Statistical Mechanics

Nanoscale simulation lectures Statistical Mechanics Nanoscale simulation lectures 2008 Lectures: Thursdays 4 to 6 PM Course contents: - Thermodynamics and statistical mechanics - Structure and scattering - Mean-field approaches - Inhomogeneous systems -

More information

STOCHASTIC CHEMICAL KINETICS

STOCHASTIC CHEMICAL KINETICS STOCHASTIC CHEICAL KINETICS Dan Gillespie GillespieDT@mailaps.org Current Support: Caltech (NIGS) Caltech (NIH) University of California at Santa Barbara (NIH) Past Support: Caltech (DARPA/AFOSR, Beckman/BNC))

More information

A Brief Introduction to Statistical Mechanics

A Brief Introduction to Statistical Mechanics A Brief Introduction to Statistical Mechanics E. J. Maginn, J. K. Shah Department of Chemical and Biomolecular Engineering University of Notre Dame Notre Dame, IN 46556 USA Monte Carlo Workshop Universidade

More information

The Microcanonical Approach. (a) The volume of accessible phase space for a given total energy is proportional to. dq 1 dq 2 dq N dp 1 dp 2 dp N,

The Microcanonical Approach. (a) The volume of accessible phase space for a given total energy is proportional to. dq 1 dq 2 dq N dp 1 dp 2 dp N, 8333: Statistical Mechanics I Problem Set # 6 Solutions Fall 003 Classical Harmonic Oscillators: The Microcanonical Approach a The volume of accessible phase space for a given total energy is proportional

More information

BAE 820 Physical Principles of Environmental Systems

BAE 820 Physical Principles of Environmental Systems BAE 820 Physical Principles of Environmental Systems Estimation of diffusion Coefficient Dr. Zifei Liu Diffusion mass transfer Diffusion mass transfer refers to mass in transit due to a species concentration

More information

Diffusion in the cell

Diffusion in the cell Diffusion in the cell Single particle (random walk) Microscopic view Macroscopic view Measuring diffusion Diffusion occurs via Brownian motion (passive) Ex.: D = 100 μm 2 /s for typical protein in water

More information

arxiv:physics/ v2 [physics.ed-ph] 7 May 1999

arxiv:physics/ v2 [physics.ed-ph] 7 May 1999 Notes on Brownian motion and related phenomena Deb Shankar Ray Department of Physical Chemistry, Indian Association for the Cultivation of Science, Jadavpur, Calcutta 70003, India. (February, 008) arxiv:physics/9903033v

More information

Supplementary 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 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 information

Protein Dynamics, Allostery and Function

Protein 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 information

ChemE Chemical Kinetics & Reactor Design Solutions to Exercises for Calculation Session 3

ChemE Chemical Kinetics & Reactor Design Solutions to Exercises for Calculation Session 3 ChemE 3900 - Chemical Kinetics & Reactor Design Solutions to Exercises for Calculation Session 3. It is useful to begin by recalling the criteria for the steady-state approximation (on B), the pre-equilibrium

More information

Microcanonical Ensemble

Microcanonical Ensemble Entropy for Department of Physics, Chungbuk National University October 4, 2018 Entropy for A measure for the lack of information (ignorance): s i = log P i = log 1 P i. An average ignorance: S = k B i

More information

Two recent works on molecular systems out of equilibrium

Two recent works on molecular systems out of equilibrium Two recent works on molecular systems out of equilibrium Frédéric Legoll ENPC and INRIA joint work with M. Dobson, T. Lelièvre, G. Stoltz (ENPC and INRIA), A. Iacobucci and S. Olla (Dauphine). CECAM workshop:

More information

Electron-proton transfer theory and electrocatalysis Part I

Electron-proton transfer theory and electrocatalysis Part I Electron-proton transfer theory and electrocatalysis Part I Marc Koper ELCOREL Workshop Herman Boerhaave Outline Molecular theory of electrode reactions Reaction rate theory - Marcus theory ion transfer

More information

Active 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: 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 information

Topics covered so far:

Topics 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 information

Handout 10. Applications to Solids

Handout 10. Applications to Solids ME346A Introduction to Statistical Mechanics Wei Cai Stanford University Win 2011 Handout 10. Applications to Solids February 23, 2011 Contents 1 Average kinetic and potential energy 2 2 Virial theorem

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

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

Stochastic equations for thermodynamics

Stochastic equations for thermodynamics J. Chem. Soc., Faraday Trans. 93 (1997) 1751-1753 [arxiv 1503.09171] Stochastic equations for thermodynamics Roumen Tsekov Department of Physical Chemistry, University of Sofia, 1164 Sofia, ulgaria The

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

Phase Equilibria and Molecular Solutions Jan G. Korvink and Evgenii Rudnyi IMTEK Albert Ludwig University Freiburg, Germany

Phase Equilibria and Molecular Solutions Jan G. Korvink and Evgenii Rudnyi IMTEK Albert Ludwig University Freiburg, Germany Phase Equilibria and Molecular Solutions Jan G. Korvink and Evgenii Rudnyi IMTEK Albert Ludwig University Freiburg, Germany Preliminaries Learning Goals Phase Equilibria Phase diagrams and classical thermodynamics

More information

Smoluchowski Diffusion Equation

Smoluchowski 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 information

where R = universal gas constant R = PV/nT R = atm L mol R = atm dm 3 mol 1 K 1 R = J mol 1 K 1 (SI unit)

where R = universal gas constant R = PV/nT R = atm L mol R = atm dm 3 mol 1 K 1 R = J mol 1 K 1 (SI unit) Ideal Gas Law PV = nrt where R = universal gas constant R = PV/nT R = 0.0821 atm L mol 1 K 1 R = 0.0821 atm dm 3 mol 1 K 1 R = 8.314 J mol 1 K 1 (SI unit) Standard molar volume = 22.4 L mol 1 at 0 C and

More information

Transition Theory Abbreviated Derivation [ A - B - C] # E o. Reaction Coordinate. [ ] # æ Æ

Transition Theory Abbreviated Derivation [ A - B - C] # E o. Reaction Coordinate. [ ] # æ Æ Transition Theory Abbreviated Derivation A + BC æ Æ AB + C [ A - B - C] # E A BC D E o AB, C Reaction Coordinate A + BC æ æ Æ æ A - B - C [ ] # æ Æ æ A - B + C The rate of reaction is the frequency of

More information

The First Principle Calculation of Green Kubo Formula with the Two-Time Ensemble Technique

The First Principle Calculation of Green Kubo Formula with the Two-Time Ensemble Technique Commun. Theor. Phys. (Beijing, China 35 (2 pp. 42 46 c International Academic Publishers Vol. 35, No. 4, April 5, 2 The First Principle Calculation of Green Kubo Formula with the Two-Time Ensemble Technique

More information

Thermodynamics (Lecture Notes) Heat and Thermodynamics (7 th Edition) by Mark W. Zemansky & Richard H. Dittman

Thermodynamics (Lecture Notes) Heat and Thermodynamics (7 th Edition) by Mark W. Zemansky & Richard H. Dittman Thermodynamics (Lecture Notes Heat and Thermodynamics (7 th Edition by Mark W. Zemansky & Richard H. Dittman 2 Chapter 1 Temperature and the Zeroth Law of Thermodynamics 1.1 Macroscopic Point of View If

More information

Linear Response and Onsager Reciprocal Relations

Linear Response and Onsager Reciprocal Relations Linear Response and Onsager Reciprocal Relations Amir Bar January 1, 013 Based on Kittel, Elementary statistical physics, chapters 33-34; Kubo,Toda and Hashitsume, Statistical Physics II, chapter 1; and

More information

9.1 System in contact with a heat reservoir

9.1 System in contact with a heat reservoir Chapter 9 Canonical ensemble 9. System in contact with a heat reservoir We consider a small system A characterized by E, V and N in thermal interaction with a heat reservoir A 2 characterized by E 2, V

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

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

Major Concepts Lecture #11 Rigoberto Hernandez. TST & Transport 1

Major Concepts Lecture #11 Rigoberto Hernandez. TST & Transport 1 Major Concepts Onsager s Regression Hypothesis Relaxation of a perturbation Regression of fluctuations Fluctuation-Dissipation Theorem Proof of FDT & relation to Onsager s Regression Hypothesis Response

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

Inverse Langevin approach to time-series data analysis

Inverse 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 information

G : Statistical Mechanics

G : Statistical Mechanics G25.2651: Statistical Mechanics Notes for Lecture 15 Consider Hamilton s equations in the form I. CLASSICAL LINEAR RESPONSE THEORY q i = H p i ṗ i = H q i We noted early in the course that an ensemble

More information

Foundations of Chemical Kinetics. Lecture 17: Unimolecular reactions in the gas phase: Lindemann-Hinshelwood theory

Foundations of Chemical Kinetics. Lecture 17: Unimolecular reactions in the gas phase: Lindemann-Hinshelwood theory Foundations of Chemical Kinetics Lecture 17: Unimolecular reactions in the gas phase: Lindemann-Hinshelwood theory Marc R. Roussel Department of Chemistry and Biochemistry The factorial The number n(n

More information

4. Systems in contact with a thermal bath

4. Systems in contact with a thermal bath 4. Systems in contact with a thermal bath So far, isolated systems microcanonical methods 4.1 Constant number of particles:kittelkroemer Chap. 3 Boltzmann factor Partition function canonical methods Ideal

More information

Chemical Kinetics. Topic 7

Chemical Kinetics. Topic 7 Chemical Kinetics Topic 7 Corrosion of Titanic wrec Casón shipwrec 2Fe(s) + 3/2O 2 (g) + H 2 O --> Fe 2 O 3.H 2 O(s) 2Na(s) + 2H 2 O --> 2NaOH(aq) + H 2 (g) Two examples of the time needed for a chemical

More information

Major Concepts. Brownian Motion & More. Chemical Kinetics Master equation & Detailed Balance Relaxation rate & Inverse Phenomenological Rate

Major Concepts. Brownian Motion & More. Chemical Kinetics Master equation & Detailed Balance Relaxation rate & Inverse Phenomenological Rate Major Conceps Brownian Moion & More Langevin Equaion Model for a agged subsysem in a solven Harmonic bah wih emperaure, T Fricion & Correlaed forces (FDR) Markovian/Ohmic vs. Memory Fokker-Planck Equaion

More information

Minimum Bias Events at ATLAS

Minimum Bias Events at ATLAS Camille Bélanger-Champagne Lehman McGill College University City University of New York Thermodynamics Charged Particle and Correlations Statistical Mechanics in Minimum Bias Events at ATLAS Statistical

More information

Lecture 6 Free Energy

Lecture 6 Free Energy Lecture 6 Free Energy James Chou BCMP21 Spring 28 A quick review of the last lecture I. Principle of Maximum Entropy Equilibrium = A system reaching a state of maximum entropy. Equilibrium = All microstates

More information

VIII.B Equilibrium Dynamics of a Field

VIII.B Equilibrium Dynamics of a Field VIII.B Equilibrium Dynamics of a Field The next step is to generalize the Langevin formalism to a collection of degrees of freedom, most conveniently described by a continuous field. Let us consider the

More information

Gaussian processes for inference in stochastic differential equations

Gaussian processes for inference in stochastic differential equations Gaussian processes for inference in stochastic differential equations Manfred Opper, AI group, TU Berlin November 6, 2017 Manfred Opper, AI group, TU Berlin (TU Berlin) inference in SDE November 6, 2017

More information

STRUCTURE OF MATTER, VIBRATIONS AND WAVES, AND QUANTUM PHYSICS

STRUCTURE OF MATTER, VIBRATIONS AND WAVES, AND QUANTUM PHYSICS Imperial College London BSc/MSci EXAMINATION June 2008 This paper is also taken for the relevant Examination for the Associateship STRUCTURE OF MATTER, VIBRATIONS AND WAVES, AND QUANTUM PHYSICS For 1st-Year

More information

Non-equilibrium phenomena and fluctuation relations

Non-equilibrium phenomena and fluctuation relations Non-equilibrium phenomena and fluctuation relations Lamberto Rondoni Politecnico di Torino Beijing 16 March 2012 http://www.rarenoise.lnl.infn.it/ Outline 1 Background: Local Thermodyamic Equilibrium 2

More information

T(K) k(cm 3 /molecule s) 7.37 x x x x x 10-12

T(K) k(cm 3 /molecule s) 7.37 x x x x x 10-12 CHM 5423 Atmospheric Chemistry Problem Set 3 Due date: Tuesday, February 19 th. The first hour exam is on Thursday, February 21 st. It will cover material from the first four handouts for the class. Do

More information

Stochastic Particle Methods for Rarefied Gases

Stochastic 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 information

Radiation Damping. 1 Introduction to the Abraham-Lorentz equation

Radiation Damping. 1 Introduction to the Abraham-Lorentz equation Radiation Damping Lecture 18 1 Introduction to the Abraham-Lorentz equation Classically, a charged particle radiates energy if it is accelerated. We have previously obtained the Larmor expression for the

More information

Appendix 4. Appendix 4A Heat Capacity of Ideal Gases

Appendix 4. Appendix 4A Heat Capacity of Ideal Gases Appendix 4 W-143 Appendix 4A Heat Capacity of Ideal Gases We can determine the heat capacity from the energy content of materials as a function of temperature. The simplest material to model is an ideal

More information

STRUCTURE OF MATTER, VIBRATIONS & WAVES and QUANTUM PHYSICS

STRUCTURE OF MATTER, VIBRATIONS & WAVES and QUANTUM PHYSICS UNIVERSITY OF LONDON BSc/MSci EXAMINATION June 2007 for Internal Students of Imperial College of Science, Technology and Medicine This paper is also taken for the relevant Examination for the Associateship

More information

Mesoscale Simulation Methods. Ronojoy Adhikari The Institute of Mathematical Sciences Chennai

Mesoscale 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 information

Gravitational-like interactions in a cloud of cold atoms?

Gravitational-like interactions in a cloud of cold atoms? Gravitational-like interactions in a cloud of cold atoms? J. Barré (1), B. Marcos (1), A. Olivetti (1), D. Wilkowski (2), M. Chalony (2) (+ R. Kaiser (2) ) 1 Laboratoire JA Dieudonné, U. de Nice-Sophia

More information

CHAPTER 21: Reaction Dynamics

CHAPTER 21: Reaction Dynamics CHAPTER 21: Reaction Dynamics I. Microscopic Theories of the Rate Constant. A. The Reaction Profile (Potential Energy diagram): Highly schematic and generalized. A---B-C B. Collision Theory of Bimolecular

More information

CHEMICAL ENGINEERING THERMODYNAMICS. Andrew S. Rosen

CHEMICAL ENGINEERING THERMODYNAMICS. Andrew S. Rosen CHEMICAL ENGINEERING THERMODYNAMICS Andrew S. Rosen SYMBOL DICTIONARY 1 TABLE OF CONTENTS Symbol Dictionary... 3 1. Measured Thermodynamic Properties and Other Basic Concepts... 5 1.1 Preliminary Concepts

More information

Lecture 3: Optical Properties of Insulators, Semiconductors, and Metals. 5 nm

Lecture 3: Optical Properties of Insulators, Semiconductors, and Metals. 5 nm Metals Lecture 3: Optical Properties of Insulators, Semiconductors, and Metals 5 nm Course Info Next Week (Sept. 5 and 7) no classes First H/W is due Sept. 1 The Previous Lecture Origin frequency dependence

More information

Introduction to thermodynamics

Introduction to thermodynamics Chapter 6 Introduction to thermodynamics Topics First law of thermodynamics Definitions of internal energy and work done, leading to du = dq + dw Heat capacities, C p = C V + R Reversible and irreversible

More information

arxiv:cond-mat/ v4 [cond-mat.stat-mech] 14 Jan 2007

arxiv:cond-mat/ v4 [cond-mat.stat-mech] 14 Jan 2007 Exact analytical evaluation of time dependent transmission coefficient from the method of reactive flux for an inverted parabolic barrier arxiv:cond-mat/79v4 [cond-mat.stat-mech] 4 Jan 7 Raarshi Chakrabarti

More information

3. Stochastic Processes

3. Stochastic Processes 3. Stochastic Processes We learn in kindergarten about the phenomenon of Brownian motion, the random jittery movement that a particle suffers when it is placed in a liquid. Famously, it is caused by the

More information

Advanced Topics of Theoretical Physics II. The statistical properties of matter. Peter E. Blöchl with a guest chapter by Tom Kirchner

Advanced Topics of Theoretical Physics II. The statistical properties of matter. Peter E. Blöchl with a guest chapter by Tom Kirchner Advanced Topics of Theoretical Physics II The statistical properties of matter Peter E. Blöchl with a guest chapter by Tom Kirchner Caution! This is an unfinished draft version Errors are unavoidable!

More information

On different notions of timescales in molecular dynamics

On different notions of timescales in molecular dynamics On different notions of timescales in molecular dynamics Origin of Scaling Cascades in Protein Dynamics June 8, 217 IHP Trimester Stochastic Dynamics out of Equilibrium Overview 1. Motivation 2. Definition

More information

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

MD 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 information

Solution. For one question the mean grade is ḡ 1 = 10p = 8 and the standard deviation is 1 = g

Solution. For one question the mean grade is ḡ 1 = 10p = 8 and the standard deviation is 1 = g Exam 23 -. To see how much of the exam grade spread is pure luck, consider the exam with ten identically difficult problems of ten points each. The exam is multiple-choice that is the correct choice of

More information

TOPIC 6: Chemical kinetics

TOPIC 6: Chemical kinetics TOPIC 6: Chemical kinetics Reaction rates Reaction rate laws Integrated reaction rate laws Reaction mechanism Kinetic theories Arrhenius law Catalysis Enzimatic catalysis Fuente: Cedre http://loincognito.-iles.wordpress.com/202/04/titanic-

More information

ANSWERS TO PROBLEM SET 1

ANSWERS TO PROBLEM SET 1 CHM1485: Molecular Dynamics and Chemical Dynamics in Liquids ANSWERS TO PROBLEM SET 1 1 Properties of the Liouville Operator a. Give the equations of motion in terms of the Hamiltonian. Show that they

More information

Classical Statistical Mechanics: Part 1

Classical Statistical Mechanics: Part 1 Classical Statistical Mechanics: Part 1 January 16, 2013 Classical Mechanics 1-Dimensional system with 1 particle of mass m Newton s equations of motion for position x(t) and momentum p(t): ẋ(t) dx p =

More information