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Understanding Molecular Simulations Molsim 2012 Why Molecular Simulations Molsim 2012 Why Molecular Simulations Paul Dirac, after completing his formalism of quantum mechanics: Molsim 2012 Why Molecular Simulations Paul Dirac, after completing his formalism of quantum mechanics: “The rest is chemistry…”. Molsim 2012 Why Molecular Simulations Paul Dirac, after completing his formalism of quantum mechanics: “The rest is chemistry…”. This is a heavy burden the shoulders of “chemistry”: Molsim 2012 Why Molecular Simulations Paul Dirac, after completing his formalism of quantum mechanics: “The rest is chemistry…”. This is a heavy burden the shoulders of “chemistry”: The “rest” amounts to the quantitative description of the world around us and the prediction of all every-day phenomena ranging from the chemical reactions of small molecules to the integrated description of living organisms. Molsim 2012 Concept of Molecular Simulations for a given intermolecular potential “exactly” predict the thermodynamic and transport properties of the system Molsim 2012 Concept of Molecular Simulations for a given intermolecular potential “exactly” predict the thermodynamic and transport properties of the system We assume the interactions between the particles are known! Molsim 2012 Concept of Molecular Simulations Exact = in the limit of infinitely long simulations the error bars can be made infinitely small for a given intermolecular potential “exactly” predict the thermodynamic and transport properties of the system Molsim 2012 Concept of Molecular Simulations for a given intermolecular potential “exactly” predict the thermodynamic and transport properties of the system Pressure Heat capacity Heat of adsorption Structure …. Molsim 2012 Concept of Molecular Simulations for a given intermolecular potential “exactly” predict the thermodynamic and transport properties of the system Diffusion coefficient Viscosity … Molsim 2012 Concept of Molecular Simulations If one could envision an experimental system of these particles that interact potential with the potential. for a given intermolecular “exactly” predict the thermodynamic and transport properties of the system Molsim 2012 Concept of Molecular Simulations for a given intermolecular potential “exactly” predict the thermodynamic and transport properties of the system Molsim 2012 THE question: Molsim 2012 “Can we predict the macroscopic properties of (classical) many-body systems?” THE question: “Can we predict the macroscopic properties of (classical) many-body systems?” NEWTON says: yes, my F=ma gives the future time evolution of the system Molsim 2012 THE question: “Can we predict the macroscopic properties of (classical) many-body systems?” NEWTON says: yes, my F=ma gives the future time evolution of the system LAPLACE says: in principle yes, provided that we know the position, velocity and interaction of all molecules, then the future behavior is predictable,.. Molsim 2012 THE question: “Can we predict the macroscopic properties of (classical) many-body systems?” NEWTON says: yes, my F=ma gives the future time evolution of the system LAPLACE says: in principle yes, provided that we know the position, velocity and interaction of all molecules, then the future behavior is predictable,.. BOLZMANN says: yes, just solve my phase space integral (for static properties) Molsim 2012 THE question: “Can we predict the macroscopic properties of (classical) many-body systems?” NEWTON says: yes, my F=ma gives the future time evolution of the system LAPLACE says: in principle yes, provided that we know the position, velocity and interaction of all molecules, then the future behavior is predictable,.. BOLZMANN says: yes, just solve my phase space integral (for static properties) both approaches should lead to the same results: Molsim 2012 THE question: “Can we predict the macroscopic properties of (classical) many-body systems?” NEWTON says: yes, my F=ma gives the future time evolution of the system LAPLACE says: in principle yes, provided that we know the position, velocity and interaction of all molecules, then the future behavior is predictable,.. BOLZMANN says: yes, just solve my phase space integral (for static properties) both approaches should lead to the same results: Molsim 2012 But...... .... there are so many molecules.... Molsim 2012 .... there are so many molecules.... ...making these approaches untractable. What was the alternative at the time ? Molsim 2012 .... there are so many molecules.... ...making these approaches untractable. What was the alternative at the time ? Molsim 2012 .... there are so many molecules.... ...making these approaches untractable. What was the alternative at the time ? 1. Smart tricks (“theory”) Only works in special cases: the Ising model: the ideal gas, etc etc Molsim 2012 .... there are so many molecules.... ...making these approaches untractable. What was the alternative at the time ? 1. Smart tricks (“theory”) Only works in special cases: the Ising model: the ideal gas, etc etc 2. Constructing a model (“molecular lego”). Molsim 2012 J.D. Bernal’s ball-bearing model of an atomic liquid Watson and Crick’s model of DNA double helix The computer age The computer age (1953…) Berni Alder Mary-Ann Mansigh Tom Wainwright With computers we can follow the behavior of hundreds to Withhundreds computers can follow the behavior of hundreds to hundreds of we millions of molecules. of millions of molecules. Molsim 2012 Increment of power since 1950’s Molsim 2012 Super computer performance development Performance Development !"%%4'()*+,% 100 Pflop/s 100000000 10 Pflop/s 2#3%4'()*+,% 10000000 1 Pflop/s 1000000 100 Tflop/s 56/% 100000 -0%1'()*+,% 10 Tflop/s 10000 1 Tflop/s 1000 780% 0#0$%1'()*+,% 6-8 years 100 Gflop/s 100 78!..% !"#$%&'()*+,% My Laptop (12 Gflop/s) 10 Gflop/s 10 My iPad2 & iPhone 4s (1.02 Gflop/s) 1 Gflop/s 1 100 Mflop/s 0.1 Molsim 2012 -..%/'()*+,% Speed of MD simulations Psec/atom/CPU h 1012 106 1 1950 Molsim 2012 1970 1990 2010 Speed of MD simulations Psec/atom/CPU h 1012 ab-initio MD is invented 106 1 1950 Molsim 2012 1970 1990 2010 Uses of Molecular Simulations Molsim 2012 Uses of Molecular Simulations • Mimic the real world: – Predicting properties of (new) materials – Computer ‘experiments’ at extreme conditions – Understanding phenomena on a molecular scale Molsim 2012 Uses of Molecular Simulations • Mimic the real world: – Predicting properties of (new) materials – Computer ‘experiments’ at extreme conditions – Understanding phenomena on a molecular scale • Model systems – test theory using same simple model – explore consequences of model – explain poorly understood phenomena in terms of essential physics Molsim 2012 Properties of materials Critical properties of long chain hydrocarbons Molsim 2012 Properties of materials Critical properties of long chain hydrocarbons To predict the thermodynamic properties (boiling points) of the hydrocarbon mixtures it is convenient to know the critical points of the hydrocarbons. Molsim 2012 Hydrocarbons intermolecular potential United-atom model – – – – Molsim 2012 Fixed bond length Bond-bending Torsion Non-bonded: Lennard-Jones CH2 CH2 CH3 CH2 CH3 Vapour-liquid equilibria 13 Molsim 2012 Vapour-liquid equilibria Computational issues: 13 Molsim 2012 Vapour-liquid equilibria Computational issues: • How to compute vapour-liquid equilibrium? 13 Molsim 2012 Vapour-liquid equilibria Computational issues: • How to compute vapour-liquid equilibrium? • How to deal with long chain hydrocarbons? 13 Molsim 2012 Vapour-liquid equilibria Course on Free Energies and Phase Equilibrium Computational issues: • How to compute vapour-liquid equilibrium? • How to deal with long chain hydrocarbons? 13 Molsim 2012 Vapour-liquid equilibria Course on Free Energies and Phase Equilibrium Computational issues: • How to compute vapour-liquid equilibrium? • How to deal with long chain hydrocarbons? Course on CBMC: configurational-bias Monte Carlo 13 Molsim 2012 Properties at extreme conditions • • Carbon phase behavior at very high pressure and temperature Empirical pair potential depending on carbon coordination coordination number Molsim 2012 Properties at extreme conditions • • Carbon phase behavior at very high pressure and temperature Empirical pair potential depending on carbon coordination coordination number Molsim 2012 Understand molecular processes • • Molsim 2012 Protein conformational change with Molecular dynamics Empirical potential, including bonds, angles dihedrals Understand molecular processes • • Protein conformational change with Molecular dynamics Empirical potential, including bonds, angles dihedrals Course on MD Molsim 2012 Understand molecular processes Molsim 2012 • • Protein conformational change with Molecular dynamics Empirical potential, including bonds, angles dihedrals • Transition path sampling Understand molecular processes • • Protein conformational change with Molecular dynamics Empirical potential, including bonds, angles dihedrals • Transition path sampling Course on rare events Molsim 2012 Understand molecular processes Molsim 2012 • • Protein conformational change with Molecular dynamics Empirical potential, including bonds, angles dihedrals • Transition path sampling Understand molecular processes Molsim 2012 • • Protein conformational change with Molecular dynamics Empirical potential, including bonds, angles dihedrals • Transition path sampling Understand molecular processes • • Protein conformational change with Molecular dynamics Empirical potential, including bonds, angles dihedrals • Transition path sampling Leads to insight and new hypotheses Molsim 2012 Testing theories with simple models Molsim 2012 Testing theories with simple models In the 1950‘s an important question arose: Is an attractive interaction always required to form a solid phase? Molsim 2012 Testing theories with simple models In the 1950‘s an important question arose: Is an attractive interaction always required to form a solid phase? YES: – Because theories (up to then) predict that attractions are needed for vapourliquid equilibrium and thus why not for solids Molsim 2012 Testing theories with simple models In the 1950‘s an important question arose: Is an attractive interaction always required to form a solid phase? YES: – Because theories (up to then) predict that attractions are needed for vapourliquid equilibrium and thus why not for solids MAYBE NOT: – These theories do not apply to solids Molsim 2012 Testing theories with simple models In the 1950‘s an important question arose: Is an attractive interaction always required to form a solid phase? YES: – Because theories (up to then) predict that attractions are needed for vapourliquid equilibrium and thus why not for solids MAYBE NOT: – These theories do not apply to solids HOWEVER: – There no are molecules with only repulsive interactions Molsim 2012 Testing theories with simple models In the 1950‘s an important question arose: Is an attractive interaction always required to form a solid phase? YES: – Because theories (up to then) predict that attractions are needed for vapourliquid equilibrium and thus why not for solids MAYBE NOT: – These theories do not apply to solids HOWEVER: – There no are molecules with only repulsive interactions So how to test the hypothesis that molecules with only repulsive interaction can freeze? Molsim 2012 Testing theories with simple models In the 1950‘s an important question arose: Is an attractive interaction always required to form a solid phase? YES: – Because theories (up to then) predict that attractions are needed for vapourliquid equilibrium and thus why not for solids MAYBE NOT: – These theories do not apply to solids HOWEVER: – There no are molecules with only repulsive interactions So how to test the hypothesis that molecules with only repulsive interaction can freeze? Molsim 2012 Testing theories with simple models In the 1950‘s an important question arose: Is an attractive interaction always required to form a solid phase? YES: – Because theories (up to then) predict that attractions are needed for vapourliquid equilibrium and thus why not for solids MAYBE NOT: – These theories do not apply to solids HOWEVER: – There no are molecules with only repulsive interactions So how to test the hypothesis that molecules with only repulsive Your hypothesis is interaction can freeze? WRONG it disagrees with the experiments Molsim 2012 Testing theories with simple models My hypothesis is In the 1950‘s an important question arose: Is an attractive interaction RIGHT: but this always required to form a solid phase? experimentalist refuses YES: to usearemolecules that – Because theories (up to then) predict that attractions needed for vapourliquid equilibrium and thus why not for solids do not have any MAYBE NOT: attractive interactions – These theories do not apply to solids HOWEVER: – There no are molecules with only repulsive interactions So how to test the hypothesis that molecules with only repulsive Your hypothesis is interaction can freeze? WRONG it disagrees with the experiments Molsim 2012 Simulation of hard spheres An older Alder Molsim 2012 Simulation of hard spheres • Bernie Alder et al. carried out Molecular Dynamics simulations of the freezing of hard spheres An older Alder Molsim 2012 Simulation of hard spheres • Bernie Alder et al. carried out Molecular Dynamics simulations of the freezing of hard spheres • But, …. did the scientific community accept this computer results as evidence …? An older Alder Molsim 2012 Simulation of hard spheres • Bernie Alder et al. carried out Molecular Dynamics simulations of the freezing of hard spheres • But, …. did the scientific community accept this computer results as evidence …? • … during a New Jersey conference in 1957 it was proposed to vote on it … An older Alder Molsim 2012 Simulation of hard spheres • Bernie Alder et al. carried out Molecular Dynamics simulations of the freezing of hard spheres • But, …. did the scientific community accept this computer results as evidence …? • … during a New Jersey conference in 1957 it was proposed to vote on it … • … and it was voted against the results of Alder! An older Alder Molsim 2012 Experiments are now possible .. not on molecules but on colloids from :Yethiraj and van Blaaderen Nature 421, 513-517 (2003) and show that hard spheres indeed crystallize at high density Molsim 2012 Explore consequences of model Molsim 2012 • Trans membrane peptides • Coarse-grained model of a membrane to study the interactions between peptides in a membrane Explore consequences of model • Trans membrane peptides • Coarse-grained model of a membrane to study the interactions between peptides in a membrane Course on coarse graining Molsim 2012 Explore consequences of model Molsim 2012 • Trans membrane peptides • Coarse-grained model of a membrane to study the interactions between peptides in a membrane Explore consequences of model Molsim 2012 • Trans membrane peptides • Coarse-grained model of a membrane to study the interactions between peptides in a membrane Understanding in terms of essential physics Molsim 2012 Understanding in terms of essential physics • Molsim 2012 Protein folding is complex: explicit MD intractable Understanding in terms of essential physics • Protein folding is complex: explicit MD intractable • Make simple lattice model with essential physics of proteins – polymer connectivity – heteropolymer sequence – attraction/repulsion Molsim 2012 Understanding in terms of essential physics • Protein folding is complex: explicit MD intractable • Make simple lattice model with essential physics of proteins – polymer connectivity – heteropolymer sequence – attraction/repulsion Molsim 2012 Understanding in terms of essential physics • Protein folding is complex: explicit MD intractable • Make simple lattice model with essential physics of proteins – polymer connectivity – heteropolymer sequence – attraction/repulsion • Molsim 2012 MC simulation yields understanding Understanding in terms of essential physics • Protein folding is complex: explicit MD intractable • Make simple lattice model with essential physics of proteins – polymer connectivity – heteropolymer sequence – attraction/repulsion • MC simulation yields understanding Course on lattice models Molsim 2012 The limits of Molecular Simulation Molsim 2012 • Brute-force simulations can never bridge all the scales between microscopic (nanometers/picoseconds) and macroscopic (cells, humans, planets). • Need different levels of description (“coarse graining”) - and we need input from experiments at many different levels to validate our models. The limits of experiments Molsim 2012 The limits of experiments • Molsim 2012 Increasingly, experiments generate far more data than humans can digest. The limits of experiments Molsim 2012 • Increasingly, experiments generate far more data than humans can digest. • Result: “Experulation” or “Simuriment”. The limits of experiments Molsim 2012 • Increasingly, experiments generate far more data than humans can digest. • Result: “Experulation” or “Simuriment”. • Simulations are becoming an integral part of the analysis of experimental data. Understanding Molecular Simulation Molecular simulations are based on the framework of statistical mechanics/thermodynamics Hence: Molsim 2012 Understanding Molecular Simulation Molecular simulations are based on the framework of statistical mechanics/thermodynamics Hence: An introduction (or refresher) of Statistical Thermodynamics Molsim 2012 Outline • Basic Assumption – micro-canonical ensemble – relation to thermodynamics • Canonical ensemble – free energy – thermodynamic properties • Other ensembles – constant pressure – grand-canonical ensemble Molsim 2012 Statistical Thermodynamics: the basics Molsim 2012 Statistical Thermodynamics: the basics • Nature is quantum-mechanical Molsim 2012 Statistical Thermodynamics: the basics • Nature is quantum-mechanical • Consequence: – Systems have discrete quantum states. – For finite “closed” systems, the number of states is finite (but usually very large) Molsim 2012 Statistical Thermodynamics: the basics • Nature is quantum-mechanical • Consequence: – Systems have discrete quantum states. – For finite “closed” systems, the number of states is finite (but usually very large) • Hypothesis: In a closed system, every state is equally likely to be observed. Molsim 2012 Statistical Thermodynamics: the basics • Nature is quantum-mechanical • Consequence: – Systems have discrete quantum states. – For finite “closed” systems, the number of states is finite (but usually very large) • Hypothesis: In a closed system, every state is equally likely to be observed. • Consequence: ALL of equilibrium Statistical Mechanics and Thermodynamics Molsim 2012 Simpler example: standard statistics First: Simpler example (standard statistics) Draw N balls from an infinite vessel that contains an equal Draw of N red ballsand from an balls infinite vessel that contains an number blue equal number of red and blue balls ! Molsim 2012 ! 1.0 0.8 0.6 0.4 0.2 0.2 Molsim 2012 0.4 0.6 NR/N 0.8 1.0 1.0 0.8 N=10 0.6 0.4 0.2 0.2 Molsim 2012 0.4 0.6 NR/N 0.8 1.0 1.0 0.8 N=10 N=100 0.6 0.4 0.2 0.2 Molsim 2012 0.4 0.6 NR/N 0.8 1.0 N=1000 1.0 0.8 N=10 N=100 0.6 0.4 0.2 0.2 Molsim 2012 0.4 0.6 NR/N 0.8 1.0 N=1000 N=10000 1.0 0.8 N=10 N=100 0.6 0.4 0.2 0.2 Molsim 2012 0.4 0.6 NR/N 0.8 1.0 Molecular consequence of basic assumption Molsim 2012 Molecular consequence of basic assumption Each individual microstate is equally probable Molsim 2012 …, but there are not many Molecular consequence of microstates that give these basic assumption extreme results Each individual microstate is equally probable Molsim 2012 …, but there are not many Molecular consequence of microstates that give these basic assumption extreme results Each individual microstate is equally probable If the number of particles is large (>100) the probability functions for these states are sharply peaked Molsim 2012 Does the basis assumption lead to something that is consistent with classical thermodynamics? Molsim 2012 Does the basis assumption lead to something that is consistent with classical thermodynamics? Systems 1 and 2 are weakly coupled such that they can exchange energy. What will be E1? Molsim 2012 Does the basis assumption lead to something that is consistent with classical thermodynamics? Systems 1 and 2 are weakly coupled such that they can exchange energy. What will be E1? Molsim 2012 Does the basis assumption lead to something that is consistent with classical thermodynamics? Systems 1 and 2 are weakly coupled such that they can exchange energy. What will be E1? BA: each configuration is equally probable; but the number of states that give an energy E1 is not (yet) known. Molsim 2012 Does the basis assumption lead to something that is consistent with classical thermodynamics? Systems 1 and 2 are weakly coupled such that they can exchange energy. What will be E1? BA: each configuration is equally probable; but the number of states that give an energy E1 is not (yet) known. The most likely E1, is the E1 that maximizes Ω1(E1)x Ω2(E-E1) Molsim 2012 Molsim 2012 Molsim 2012 Molsim 2012 Molsim 2012 ( ) ( ) ⎛ ∂ ln Ω1 E1 ⎞ ⎛ ∂ ln Ω2 E − E1 ⎞ + ⎜ ⎜ ⎟ ⎟ ∂E1 ∂E1 ⎝ ⎠ N ,V ⎝ ⎠ N 1 Molsim 2012 1 =0 2 ,V2 Energy is conserved! dE1=-dE2 ( ) ( ) ⎛ ∂ ln Ω1 E1 ⎞ ⎛ ∂ ln Ω2 E − E1 ⎞ + ⎜ ⎜ ⎟ ⎟ ∂E1 ∂E1 ⎝ ⎠ N ,V ⎝ ⎠ N 1 Molsim 2012 1 =0 2 ,V2 Energy is conserved! dE1=-dE2 ( ) ( ) ⎛ ∂ ln Ω1 E1 ⎞ ⎛ ∂ ln Ω2 E − E1 ⎞ + ⎜ ⎜ ⎟ ⎟ ∂E1 ∂E1 ⎝ ⎠ N ,V ⎝ ⎠ N 1 Molsim 2012 1 =0 2 ,V2 Energy is conserved! dE1=-dE2 ( ) ( ) ⎛ ∂ ln Ω1 E1 ⎞ ⎛ ∂ ln Ω2 E − E1 ⎞ + ⎜ ⎜ ⎟ ⎟ ∂E1 ∂E1 ⎝ ⎠ N ,V ⎝ ⎠ N 1 1 =0 2 ,V2 This can be seen as an equilibrium condition Molsim 2012 Energy is conserved! dE1=-dE2 ( ) ( ) ⎛ ∂ ln Ω1 E1 ⎞ ⎛ ∂ ln Ω2 E − E1 ⎞ + ⎜ ⎜ ⎟ ⎟ ∂E1 ∂E1 ⎝ ⎠ N ,V ⎝ ⎠ N 1 1 =0 2 ,V2 This can be seen as an equilibrium condition Indeed this is the condition for thermal equilibrium: “no spontaneous heat flow between 1 and 2” Molsim 2012 Normally, thermal equilibrium means: equal temperatures Let us define: Molsim 2012 Normally, thermal equilibrium means: equal temperatures Let us define: Molsim 2012 Normally, thermal equilibrium means: equal temperatures Let us define: Then, thermal equilibrium is equivalent to: This suggests that β is a function of T. Molsim 2012 Relation to thermodynamics Conjecture: Almost right. Good features: – Extensivity – Third law of thermodynamics comes for free Bad feature: – It assumes that entropy is dimensionless but (for unfortunate, historical reasons, it is not…) Molsim 2012 Relation to thermodynamics Conjecture: Almost right. Good features: – Extensivity – Third law of thermodynamics comes for free Bad feature: – It assumes that entropy is dimensionless but (for unfortunate, historical reasons, it is not…) Molsim 2012 We have to live with the past, therefore With kB= 1.380662 10-23 J/K Molsim 2012 We have to live with the past, therefore With kB= 1.380662 10-23 J/K In thermodynamics, the absolute (Kelvin) temperature scale was defined such that Molsim 2012 We have to live with the past, therefore With kB= 1.380662 10-23 J/K In thermodynamics, the absolute (Kelvin) temperature scale was defined such that Molsim 2012 We have to live with the past, therefore With kB= 1.380662 10-23 J/K In thermodynamics, the absolute (Kelvin) temperature scale was defined such that n dE = TdS-pdV + ∑ µi dN i i =1 Molsim 2012 We have to live with the past, therefore With kB= 1.380662 10-23 J/K In thermodynamics, the absolute (Kelvin) temperature scale was defined such that Molsim 2012 We have to live with the past, therefore With kB= 1.380662 10-23 J/K In thermodynamics, the absolute (Kelvin) temperature scale was defined such that But we found (defined): Molsim 2012 And this gives the “statistical” definition of temperature: with 1 β= kB T Molsim 2012 And this gives the “statistical” definition of temperature: with 1 β= kB T In short: Entropy and temperature are both related to the fact that we can COUNT states. Molsim 2012 Number of configurations 35 Molsim 2012 Number of configurations How large is Ω? 35 Molsim 2012 Number of configurations How large is Ω? For macroscopic systems, super-astronomically large. 35 Molsim 2012 Number of configurations How large is Ω? For macroscopic systems, super-astronomically large. For instance, for a glass of water at room temperature: 35 Molsim 2012 Number of configurations How large is Ω? For macroscopic systems, super-astronomically large. For instance, for a glass of water at room temperature: 35 Molsim 2012 Number of configurations How large is Ω? For macroscopic systems, super-astronomically large. For instance, for a glass of water at room temperature: Macroscopic deviations from the second law of thermodynamics are not forbidden, but they are extremely unlikely. 35 Molsim 2012 Deviation from the 2nd law? What is the probability that 1 mole of argon gas spontaneously lowers it entropy by 0.00000001% (=10-10) ? standard molar entropy of argon : Sargon =307.2 J/K mol-1 slightly lower entropy Sargon,low =307.2 (1 - 10-10) J/K mol-1 probability of occurrence is Molsim 2012 Deviation from the 2nd law? What is the probability that 1 mole of argon gas spontaneously lowers it entropy by 0.00000001% (=10-10) ? standard molar entropy of argon : Sargon =307.2 J/K mol-1 slightly lower entropy Sargon,low =307.2 (1 - 10-10) J/K mol-1 probability of occurrence is � � � −8 Ωargon,low Sargon,low − Sargon −3.07 × 10 P ≈ = exp = exp Ωargon kB 1.38 × 10−23 Molsim 2012 � −1015 ≈ 10 Deviation from the 2nd law? What is the probability that 1 mole of argon gas spontaneously lowers it entropy by 0.00000001% (=10-10) ? standard molar entropy of argon : Sargon =307.2 J/K mol-1 slightly lower entropy Sargon,low =307.2 (1 - 10-10) J/K mol-1 probability of occurrence is � � � −8 Ωargon,low Sargon,low − Sargon −3.07 × 10 P ≈ = exp = exp Ωargon kB 1.38 × 10−23 A mathematical relation : 10-1015 ≠ 0 A physical relation : 10-1015 = 0 Molsim 2012 � −1015 ≈ 10 Canonical ensemble Consider a small system that can exchange heat with a big reservoir Molsim 2012 Canonical ensemble Consider a small system that can exchange heat with a big reservoir Molsim 2012 Canonical ensemble Consider a small system that can exchange heat with a big reservoir Molsim 2012 1/kBT Canonical ensemble Consider a small system that can exchange heat with a big reservoir Molsim 2012 Canonical ensemble Consider a small system that can exchange heat with a big reservoir Molsim 2012 Canonical ensemble Consider a small system that can exchange heat with a big reservoir Hence, the probability to find Ei: Molsim 2012 Canonical ensemble Consider a small system that can exchange heat with a big reservoir Hence, the probability to find Ei: Molsim 2012 Canonical ensemble Consider a small system that can exchange heat with a big reservoir Hence, the probability to find Ei: Partition� function: Q= exp(−Ej /kB T ) j Molsim 2012 Canonical ensemble Consider a small system that can exchange heat with a big reservoir Hence, the probability to find Ei: Molsim 2012 Canonical ensemble Consider a small system that can exchange heat with a big reservoir Hence, the probability to find Ei: Molsim 2012 Canonical ensemble Consider a small system that can exchange heat with a big reservoir Hence, the probability to find Ei: Boltzmann distribution Molsim 2012 Canonical ensemble Consider a small system that can exchange heat with a big reservoir Hence, the probability to find Ei: “Low energies are more likely than high energies” Molsim 2012 Thermodynamics What is the average energy of the system? Molsim 2012 Thermodynamics What is the average energy of the system? Molsim 2012 Thermodynamics What is the average energy of the system? Molsim 2012 Thermodynamics What is the average energy of the system? Molsim 2012 Thermodynamics What is the average energy of the system? Compare: Molsim 2012 Thermodynamics Thermo recall What is the average energy ofequation the system? Fundamental Helmholtz Free energy: Compare: Molsim 2012 Thermodynamics What is the average energy of the system? Compare: Hence: Molsim 2012 The classical limit We have assume quantum mechanics (discrete states) but we are interested in the classical limit N Particles are indistinguishable N “volume of phase space” Why does planck constant appear? Molsim 2012 Appearance of Plank’s constant? Easiest to look at translation partition function Qx of particle in 1D box 1 2 p2 "trans = mv = 2 2m ! Molsim 2012 p= momentum m=mass Postulate quantum mechanics: p=h/λ met h Planck’s constant Particle is standing wave with length Appearance of Plank’s constant? Easiest to look at translation partition function Qx of particle in 1D box p= momentum m=mass 1 2 p2 "trans = mv = 2 2m ! Postulate quantum mechanics: p=h/λ met h Planck’s constant Particle is standing wave with length "= 2L n ! ! Molsim 2012 2 p 2 ( h / #) n 2h 2 "trans = = = 2m 2m 8mL2 Appearance of Plank’s constant? Easiest to look at translation partition function Qx of particle in 1D box p= momentum m=mass 1 2 p2 "trans = mv = 2 2m ! Postulate quantum mechanics: p=h/λ met h Planck’s constant Particle is standing wave with length "= 2L n 2 p 2 ( h / #) n 2h 2 "trans = = = 2m 2m 8mL2 $ ! n2h2 ' Qx = # exp(!!"trans ) = # exp & ! 2 ) 8mL % ( n=1 n=1 " ! ! Molsim 2012 " Appearance of Plank’s constant? Easiest to look at translation partition function Qx of particle in 1D box p= momentum m=mass 1 2 p2 "trans = mv = 2 2m ! Postulate quantum mechanics: p=h/λ met h Planck’s constant Particle is standing wave with length "= 2 p 2 ( h / #) n 2h 2 "trans = = = 2m 2m 8mL2 2L n $ ! n2h2 ' Qx = # exp(!!"trans ) = # exp & ! 2 ) 8mL % ( n=1 n=1 " " ! ! Qx = Molsim 2012 # " 1 dne ! ! n 2 h 2 /(8mL2 ) = L 2" m h ! Appearance of Plank’s constant? Easiest to look at translation partition function Qx of particle in 1D box p= momentum m=mass 1 2 p2 "trans = mv = 2 2m ! Postulate quantum mechanics: p=h/λ met h Planck’s constant Particle is standing wave with length "= 2 p 2 ( h / #) n 2h 2 "trans = = = 2m 2m 8mL2 2L n $ ! n2h2 ' Qx = # exp(!!"trans ) = # exp & ! 2 ) 8mL % ( n=1 n=1 " " ! ! Qx = # " 1 dne ! ! n 2 h 2 /(8mL2 ) = L 2" m h ! now compare � � to classical integration without Planck’s constant � 2 Qx = p dpdr exp(−β + U (r)) = L 2m factor of h is missing for each degree of freedom. Molsim 2012 2 p dp exp(−β )=L 2m � 2πm β The classical limit We have assume quantum mechanics (discrete states) but we are interested in the classical limit N Particles are indistinguishable N Molsim 2012 Volume of phase space (particle in a box) The classical limit We have assume quantum mechanics (discrete states) but we are interested in the classical limit N Particles are indistinguishable N Volume of phase space (particle in a box) Integration over the momenta can be carried out for most systems: Molsim 2012 The classical limit Molsim 2012 The classical limit Define de Broglie wavelength: Molsim 2012 The classical limit Define de Broglie wavelength: Partition function: Molsim 2012 Example: ideal gas Molsim 2012 Example: ideal gas Molsim 2012 Example: ideal gas Molsim 2012 Example: ideal gas Free energy: ! N ln " 3 # N lnV + N ln N # N Molsim 2012 Example: ideal gas Free energy: ! N ln " 3 # N lnV + N ln N # N ! F = N ( ln( !! 3 ) "1) Molsim 2012 with density ρ=N/V Example: ideal gas Thermo recall Helmholtz Free energy: Free energy: Pressure ! N ln " 3 # N lnV + N ln N # N 3 ! F = NEnergy: ln( ! ! ) "1) ( Molsim 2012 with density ρ=N/V Example: ideal gas Free energy: ! N ln " 3 # N lnV + N ln N # N ! F = N ( ln( !! 3 ) "1) Molsim 2012 with density ρ=N/V Example: ideal gas Free energy: ! N ln " 3 # N lnV + N ln N # N ! F = N ( ln( !! 3 ) "1) Pressure: Molsim 2012 with density ρ=N/V Example: ideal gas Free energy: ! N ln " 3 # N lnV + N ln N # N ! F = N ( ln( !! 3 ) "1) Pressure: Molsim 2012 with density ρ=N/V Energy: Ideal gas (2) Chemical potential: Molsim 2012 ⎛ ∂F ⎞ µi = ⎜ ⎟ ∂N ⎝ i ⎠ T ,V , N j Ideal gas (2) ⎛ ∂F ⎞ µi = ⎜ ⎟ ∂N ⎝ i ⎠ T ,V , N j Chemical potential: ! F = N ( ln( !! ) "1) 3 Molsim 2012 with density ρ=N/V Ideal gas (2) ⎛ ∂F ⎞ µi = ⎜ ⎟ ∂N ⎝ i ⎠ T ,V , N j Chemical potential: ! F = N ( ln( !! ) "1) 3 !µ = ln ! 3 + ln " Molsim 2012 with density ρ=N/V Ideal gas (2) ⎛ ∂F ⎞ µi = ⎜ ⎟ ∂N ⎝ i ⎠ T ,V , N j Chemical potential: ! F = N ( ln( !! ) "1) 3 !µ = ln ! 3 + ln " βµ Molsim 2012 IG 0 = βµ + ln ρ with density ρ=N/V Heat capacity from energy fluctuation ! ! E $ ! !" $ !! E $ CV = # & = # & # & " ! T %V ,N " !" %V ,N " ! T % Molsim 2012 Heat capacity from energy fluctuation ! ! E $ ! !" $ !! E $ CV = # & = # & # & " ! T %V ,N " !" %V ,N " ! T % Molsim 2012 "! E % kBT CV = ! $ ' # !" &V ,N 2 Heat capacity from energy fluctuation ! ! E $ ! !" $ !! E $ CV = # & = # & # & " ! T %V ,N " !" %V ,N " ! T % "! E % kBT CV = ! $ ' # !" &V ,N 2 " E exp(!! E ) = " exp(!! E ) i E i i i i Molsim 2012 Heat capacity from energy fluctuation "! E % kBT CV = ! $ ' # !" &V ,N ! ! E $ ! !" $ !! E $ CV = # & = # & # & " ! T %V ,N " !" %V ,N " ! T % E exp(!" E ) " ! =! !" " exp(!" E ) i kBT 2CV 2 i i i i " E exp(!" Ei ) #% " Ei exp(!" Ei ) &( = i !% i ( exp(! " E ) exp(! " E ) " i i % " ( $ i ' i 2 i = E Molsim 2012 2 ! E 2 = (E ! E ) 2 2 Heat capacity from energy fluctuation "! E % kBT CV = ! $ ' # !" &V ,N ! ! E $ ! !" $ !! E $ CV = # & = # & # & " ! T %V ,N " !" %V ,N " ! T % E exp(!" E ) " ! =! !" " exp(!" E ) i kBT 2CV 2 i i i i " E exp(!" Ei ) #% " Ei exp(!" Ei ) &( = i !% i ( exp(! " E ) exp(! " E ) " i i % " ( $ i ' i 2 i = E Molsim 2012 2 ! E 2 = (E ! E ) 2 P(E) fluctuation in E grows as 1/√N 2 !" Computing the pressure ∂F P =− ∂V F = −kT ln Q Introduce “scaled” coordinates: Introduce scaled coordinates Molsim 2012 Computing the pressure Molsim 2012 Molsim 2012 Free energy derivative can be written as ensemble average! Molsim 2012 For pairwise additive forces: Then Molsim 2012 i and j are dummy variable hence: And we can write Molsim 2012 But as action equals reaction (Newton’s 3rd law): And hence Inserting this in our expression for the pressure, we get: Where This is known as the virial expression Molsim 2012 Whatdo to do youif cannot useuse the the virial expression? What youif do you can’t virial expression? This will be the case e.g. with discontinuous potentials. Molsim 2012 Summary: micro-canonical ensemble (N,V,E) Molsim 2012 Summary: micro-canonical ensemble (N,V,E) Partition function: Molsim 2012 Summary: micro-canonical ensemble (N,V,E) Partition function: Probability to find a particular configuration Molsim 2012 Summary: micro-canonical ensemble (N,V,E) Partition function: Probability to find a particular configuration Entropy S = kB lnQN,V ,E Molsim 2012 Summary: Canonical ensemble (N,V,T) Molsim 2012 Summary: Canonical ensemble (N,V,T) Partition function: Molsim 2012 Summary: Canonical ensemble (N,V,T) Partition function: Probability to find a particular configuration Molsim 2012 Summary: Canonical ensemble (N,V,T) Partition function: Probability to find a particular configuration Free energy Molsim 2012 Ensemble averages For properties that only depend on the configurational part Molsim 2012 Ensemble averages For properties that only depend on the configurational part Probability to find a configuration: Molsim 2012 Ensemble averages For properties that only depend on the configurational part Probability to find a configuration: 1 P(!) = exp ["!U(!)] Q Molsim 2012 Ensemble averages For properties that only depend on the configurational part Probability to find a configuration: 1 P(!) = exp ["!U(!)] Q Ensemble average: A = Molsim 2012 " d!A(!)P(!) Ergodicity theorem suppose we have an ensemble average of a system defined by U(Γ) obtained by MC Now suppose we have a NVT molecular dynamics trajectory for the same system A time average over the trajectory is simply Molsim 2012 Ergodicity theorem suppose we have an ensemble average of a system defined by U(Γ) obtained by MC Now suppose we have a NVT molecular dynamics trajectory for the same system A time average over the trajectory is simply Ergodicity theorem states that for an ‘ergodic system’ Molsim 2012 Ergodicity theorem suppose we have an ensemble average of a system defined by U(Γ) obtained by MC Now suppose we have a NVT molecular dynamics trajectory for the same system A time average over the trajectory is simply Ergodicity theorem states that for an ‘ergodic system’ MC and MD give the same averages Molsim 2012 Other ensembles? Molsim 2012 Other ensembles? In the thermodynamic limit the thermodynamic properties are independent of the ensemble: so buy a bigger computer … Molsim 2012 Other ensembles? In the thermodynamic limit the thermodynamic properties are independent of the ensemble: so buy a bigger computer … However, it is most of the times much better to think and to carefully select an appropriate ensemble. Molsim 2012 Other ensembles? In the thermodynamic limit the thermodynamic properties are independent of the ensemble: so buy a bigger computer … However, it is most of the times much better to think and to carefully select an appropriate ensemble. For this it is important to know how to simulate in the various ensembles. Molsim 2012 Other ensembles? In the thermodynamic limit the thermodynamic properties are Course on independent of the ensemble: so buy a bigger MDcomputer and MC in… different ensembles However, it is most of the times much better to think and to carefully select an appropriate ensemble. For this it is important to know how to simulate in the various ensembles. Molsim 2012 Other ensembles? In the thermodynamic limit the thermodynamic properties are independent of the ensemble: so buy a bigger computer … However, it is most of the times much better to think and to carefully select an appropriate ensemble. For this it is important to know how to simulate in the various ensembles. Molsim 2012 Other ensembles? In the thermodynamic limit the thermodynamic properties are independent of the ensemble: so buy a bigger computer … However, it is most of the times much better to think and to carefully select an appropriate ensemble. For this it is important to know how to simulate in the various ensembles. But for doing this we need to know the Statistical Thermodynamics of the various ensembles. Molsim 2012 Constant pressure simulations: N,P,T ensemble Consider a small system that can exchange volume and energy with a big reservoir Molsim 2012 Constant pressure simulations: N,P,T ensemble Consider a small system that can exchange volume and energy with a big reservoir Molsim 2012 Constant pressure simulations: N,P,T ensemble 1/kBT p/kBT Consider a small system that can exchange volume and energy with a big reservoir Molsim 2012 Constant pressure simulations: N,P,T ensemble Thermo recall 1/kBT p/kBT Fundamental equation Consider a small system that can exchange volume and energy with a big reservoir Hence and Molsim 2012 Constant pressure simulations: N,P,T ensemble 1/kBT p/kBT Consider a small system that can exchange volume and energy with a big reservoir Molsim 2012 Constant pressure simulations: N,P,T ensemble Consider a small system that can exchange volume and energy with a big reservoir Molsim 2012 Constant pressure simulations: N,P,T ensemble Consider a small system that can exchange volume and energy with a big reservoir Molsim 2012 Constant pressure simulations: N,P,T ensemble Consider a small system that can exchange volume and energy with a big reservoir Hence, the probability to find Ei,Vi: Molsim 2012 Constant pressure simulations: N,P,T ensemble Consider a small system that can exchange volume and energy with a big reservoir Hence, the probability to find Ei,Vi: Molsim 2012 N,P,T ensemble (2) In the classical limit, the partition function becomes Molsim 2012 N,P,T ensemble (2) In the classical limit, the partition function becomes Molsim 2012 N,P,T ensemble (2) In the classical limit, the partition function becomes The probability to find a particular configuration: Molsim 2012 N,P,T ensemble (2) In the classical limit, the partition function becomes The probability to find a particular configuration: Molsim 2012 Grand-canonical simulations: µ,V,T ensemble Consider a small system that can exchange particles and energy with a big reservoir Molsim 2012 Grand-canonical simulations: µ,V,T ensemble Consider a small system that can exchange particles and energy with a big reservoir Molsim 2012 Grand-canonical simulations: µ,V,T ensemble 1/kBT Consider a small system that can exchange particles and energy with a big reservoir Molsim 2012 Grand-canonical simulations: µ,V,T ensemble -µ/kBT 1/kBT Consider a small system that can exchange particles and energy with a big reservoir Molsim 2012 Grand-canonical simulations: µ,V,T ensemble Thermo recall -µ/kBT 1/kBT Fundamental equation Consider a small system that can exchange particles and energy with a big reservoir Hence and Molsim 2012 Grand-canonical simulations: µ,V,T ensemble -µ/kBT 1/kBT Consider a small system that can exchange particles and energy with a big reservoir Molsim 2012 Grand-canonical simulations: µ,V,T ensemble Consider a small system that can exchange particles and energy with a big reservoir Molsim 2012 Grand-canonical simulations: µ,V,T ensemble Consider a small system that can exchange particles and energy with a big reservoir Molsim 2012 Grand-canonical simulations: µ,V,T ensemble Consider a small system that can exchange particles and energy with a big reservoir Hence, the probability to find Ei,Ni: Molsim 2012 Grand-canonical simulations: µ,V,T ensemble Consider a small system that can exchange particles and energy with a big reservoir Hence, the probability to find Ei,Ni: Molsim 2012 µ,V,T ensemble (2) In the classical limit, the partition function becomes Molsim 2012 µ,V,T ensemble (2) In the classical limit, the partition function becomes Molsim 2012 µ,V,T ensemble (2) In the classical limit, the partition function becomes The probability to find a particular configuration: Molsim 2012 Summary Molsim 2012 Summary • Molsim 2012 Molecular simulation is firmly rooted in equilibrium statistical mechanics/thermodynamics Summary Molsim 2012 • Molecular simulation is firmly rooted in equilibrium statistical mechanics/thermodynamics • Basis of statistical thermodynamics: configurations with same energy are equally likely: microcanonical ensemble Summary Molsim 2012 • Molecular simulation is firmly rooted in equilibrium statistical mechanics/thermodynamics • Basis of statistical thermodynamics: configurations with same energy are equally likely: microcanonical ensemble • At constant temperature Boltzmann distribution follows: canonical ensemble Summary Molsim 2012 • Molecular simulation is firmly rooted in equilibrium statistical mechanics/thermodynamics • Basis of statistical thermodynamics: configurations with same energy are equally likely: microcanonical ensemble • At constant temperature Boltzmann distribution follows: canonical ensemble • other ensembles are isobaric and grand canonical ensembles Summary Molsim 2012 • Molecular simulation is firmly rooted in equilibrium statistical mechanics/thermodynamics • Basis of statistical thermodynamics: configurations with same energy are equally likely: microcanonical ensemble • At constant temperature Boltzmann distribution follows: canonical ensemble • other ensembles are isobaric and grand canonical ensembles • MD and MC are two roads to the same equilibrium answer: ergodicity Summary Molsim 2012 • Molecular simulation is firmly rooted in equilibrium statistical mechanics/thermodynamics • Basis of statistical thermodynamics: configurations with same energy are equally likely: microcanonical ensemble • At constant temperature Boltzmann distribution follows: canonical ensemble • other ensembles are isobaric and grand canonical ensembles • MD and MC are two roads to the same equilibrium answer: ergodicity • MD also gives dynamical properties (viscosity, diffusion etc) Summary Molsim 2012 • Molecular simulation is firmly rooted in equilibrium statistical mechanics/thermodynamics • Basis of statistical thermodynamics: configurations with same energy are equally likely: microcanonical ensemble • At constant temperature Boltzmann distribution follows: canonical ensemble • other ensembles are isobaric and grand canonical ensembles • MD and MC are two roads to the same equilibrium answer: ergodicity • MD also gives dynamical properties (viscosity, diffusion etc) • Rest of the week: MC and MD in depth. The end Molsim 2012