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. Analysis of the model results of the simulation
Collection of MD Data DCD trajectory file coordinates for each atom velocities for each atom Output file global energies temperature, pressure, Analysis of MD Data. Structural properties. Equilibrium properties 3. Non-equilibrium properties Can be studied via both equilibrium and non-equilibrium MD simulations
Equilibrium (hermodynamic Properties MD simulation microscopic information Phase space trajectory Γ[r(t,p(t] Statistical Mechanics macroscopic properties Ensemble average over probability density ρ(γ Statistical Ensemble Collection of large number of replicas (on a macroscopic level of the system Each replica is characterized by the same macroscopic parameters (e.g., N, NP he microscopic state of each replica (at a given time is determined by Γ in phase space 3
ime vs Ensemble Average For t, Γ(t generates an ensemble with ρ( Γ dγ = lim dτ / t t Ergodic Hypothesis: ime and Ensemble averages are equivalent, i.e., ime average: Ensemble average: A( r, p = A( Γ A t t = 0 ρ dt A[ r( t, p( t] A = dγρ( Γ A( Γ hermodynamic Properties from MD Simulations hermodynamic (equilibrium averages can be calculated via time averaging of MD simulation time series A hermodynamic average N N i = A( t i MD simulation time series Finite simulation time means incomplete sampling! 4
Common Statistical Ensembles. Microcanonical (N,,E: ρ ( Γ δ[ H ( Γ E] NE. Canonical (N,,: ρ ( Γ = exp{[ F H ( Γ]/ k } N 3. Isothermal-isobaric (N,p, ρ ( Γ = exp{[ G H ( Γ]/ k } NP Newton s eq. of motion Langevin dynamics Nose-Hoover method Different simulation protocols [Γ(t Γ(t+δt ] sample different statistical ensembles Examples of hermodynamic Observables Energies (kinetic, potential, internal, emperature [equipartition theorem] Pressure [virial theorem] hermodynamic derivatives are related to mean square fluctuations of thermodynamic quantities Specific heat capacity C v and C P hermal expansion coefficient α P Isothermal compressibility β hermal pressure coefficient γ 5
otal (internal energy: Kinetic energy: Potential energy: Mean Energies N E = E( t i N i= N M p j K = N i= j= U = E K ( t Note: You can conveniently use namdplot to graph the time evolution of different energy terms (as well as, P, during simulation m i j OAL KINEIC OND ANGLE DIHED IMPRP ELEC DW From the equipartition theorem emperature = K 3Nk pk H/ pk = k Instantaneous kinetic temperature = K 3Nk namdplot EMP vs S Note: in the NP ensemble N N-N c, with N c =3 6
From the virial theorem P he virial is defined as with W = 3 M j= Pressure rk H/ rk = k = Nk + W r j f w ( r = r dv( r / dr j = 3 i, j> i ij w( r pairwise interaction Instantaneous pressure function (not unique! P = ρk + W / hermodynamic Fluctuations (F δa N [ A( t i A ] N i= Mean Square Fluctuations (MSF δa = ( A A = A A According to Statistical Mechanics, the probability distribution of thermodynamic fluctuations is ρ fluct δp δ δ exp k δs 7
F in N Ensemble In MD simulations distinction must be made between properly defined mechanical quantities (e.g., energy E, kinetic temperature, instantaneous pressure P and thermodynamic quantities, e.g.,, P, For example: ut: Other useful formulas: C = ( E / γ = ( P / δe = δh = k C δp δp = k / β δk δu = = k δu δp = k 3N ( k ( C ( γ 3Nk ρk / F in NP Ensemble δ = k β δ( H + P = δ δ(h k + P = C P k α P y definition: α = ( / C = ( E / P P P ; β = ( / P 8
How to Calculate C?. From definition C = ( E / Perform multiple simulations to determine E E then calculate the derivative of E( with respect to as a function of,. From the MSF of the total energy E C = δe / k with δe = E E 9