Javier Junquera

1. Statistical mechanics Javier Junquera

2. From the microscopic to the macroscopic level: the realm of statistical mechanics Thermodynamicstate Computersimulations Generatesinformation at themicroscopiclevel Definedby a small set of macroscopicparameters Atomic positions Pressure Temperature Momenta Number of particles Can be thought of as coordinates of a multidimensional spaceof dimensions: thephasespace Otherthermodynamicpropertiesmay be derivedthroughtheknowledge of theequation of state and the fundamental equations of thermodynamics

3. Statistical sampling from ensembles A particular point in phasespace A givenproperty (forinstance, thepotentialenergy) Instantaneousvalue of thepropertywhenthesystemis at point Itisreasonabletoassumethattheexperimentally observable macroscopicpropertyisreallythe time average of takenover a long time interval Theequationsgoverningthis time evolution (Newton'sequation of motion in a simple classicalsystem) are wellknown

4. Statistical sampling from ensembles Theequationsgoverningthis time evolution (Newton'sequation of motion in a simple classicalsystem) are wellknown • Butwe can not hope toextendtheintegration of thedynamicalequations: • For a trulymacroscopicnumber 1023 • Toinfinite time • Wemight be satisfiedtoaverageover: • A system of theorder of thousand of atoms(length and sizescale) • Over a longbutfinite time (time scale) Thatiswhatwe do in MD simulations: Theequations of motionissolvedstepbystep a largenumber of steps

5. Practical questions regarding the validity of the method Whetherornot a sufficientregion of phasespaceisexploredbythesystemtrajectorywithin a feasibleamount of computer time Whetherthermodynamicconsistency can be attainedbetweensimulationwithidenticalmacroscopicparameters (density, energy,…) butdifferentinitialconditions Suchsimulationruns are withinthepower of moderncomputers

6. Ergodicity • In MD we want to replace a full sampling on the appropriate statistical ensemble by a SINGLE very long trajectory. • This is OK only if system is ergodic. • Ergodic Hypothesis: a phase point for any isolated system passes in succession through every point compatible with the energy of the system before finally returning to its original position in phase space. This journey takes a Poincare cycle. • In other words, Ergodichypothesis: each state consistent with our knowledge is equally “likely”. • Implies the average value does not depend on initial conditions. • <A>time= <A>ensemble , so <Atime> = (1/NMD) = ∑t=1,N At is good estimator. • Are systems in nature really ergodic? Not always! • Non-ergodic examples are glasses, folding proteins (in practice) and harmonic crystals (in principle).

7. Ergodicity isthephasespacedensity

8. Different aspects of ergodicity The system relaxes on a “reasonable” time scale towards a unique equilibrium state (microcanonical state) Trajectories wander irregularly through the energy surface eventually sampling all of accesible phase space. Trajectories initially close together separate rapidily (Sensitivity to initial conditions). Ergodicbehavior makes possible the use of statistical methods on MD of small system. Small round-off errors and other mathematical approximations should not matter.

9. Particle in a smooth/rough circle From J.M. Haile: MD Simulations

10. Simple thermodynamic averages: mechanical energy Thekinetic, potential, and total energiesmight be calculatedusingthephasefunctionsalreadyseen Average of potentialenergy Sum overallpairs, triplets,… dependingonthecomplexity of thepotentialenergyfunction Average of kineticenergy Sum of contributionfrom individual particlemomenta

11. Simple thermodynamic averages: virial theorem in the form of “generalized equipartition”

12. Simple thermodynamic averages: temperature Applyingtheprevious formula to compute theaveragekineticenergy in theatomic case Familiar equipartitionenergy: anaverageenergy of per degree of freedom

13. Simple thermodynamic averages: instantaneous temperature We can define an “instantaneouskineticenergyfunction” fromthemomenta of theatoms at a given time step (notaveraged) whoseaverageisequalto For a system of atomssubjecttointernal molecular constraints, thenumber of degrees of freedomwill be , whereisthe total number of independentinternalconstraints In weshouldincludethefixed bond lengths, angles, oradditional global constraints (onthe center of massmotion, forinstance)

14. Simple thermodynamic averages: pressure Letusassumethatwe are usingCartesiancoordinates, and we use theHamilton’sequation of motion

15. Simple thermodynamic averages: pressure Letusassumethatwe are usingCartesiancoordinates, and we use theHamilton’sequation of motion representsthe sum of internal molecular forces and externalforces

16. Simple thermodynamic averages: pressure Theexternalforces are relatedtotheexternalpressure. Consideringtheeffect of thecontainerwallsonthesystem Ifwe define the “internalvirial”, wherewenowrestrictourattentiontothe intermolecular forces, then So, finally

17. Simple thermodynamic averages: instantaneous pressure As ithappenedwiththetemperature, we can define theinstantaneouspressurefunction as whoseaverageissimply Thisdefinition of theinstantaneouspressureisnotunique. Forinstance, for a constanttemperatureensemblewe can define Bothdefinitions of theinstantaneouspressuregivethesamewhenaveraged, buttheirfluctuations in anyensemblewould be different Althoughthesystemssimulatedwe use periodicboundaryconditions (and thereforethere are no walls), theresults are thesame

18. Simple thermodynamic averages: instantaneous pressure Forpairwiseinteraction, we can express in a formthatisexplicitlyindependent of theorigin of coordinates Theindices and are equivalent Newton’sthirdlawisusedtoswitchtheforceindices where

19. Simple thermodynamic averages: instantaneous pressure In a simulationthat uses periodicboundarycondition, itisessentialto use wherethe intermolecular pairvirialfunction Likeislimitedbytherange of theinteractions, and henceshould be a wellbehaved, ensemble-independentfunction in most cases

20. Simple thermodynamic averages: number of molecules and volume Averagenumber of particles Averagenumber of particles Are easilyevaluated in thesimulation of ensembles in whichthesequantitiesvary, and derivedfunctionssuch as theenthalpy are easilycalculatedfromtheabove

21. Constant temperature molecular dynamics: the canonical ensemble Thereservoirissimplyincludedbyanextra degree of freedom, , withmomentum SYSTEM T Reservoir A physicalpicture of a systemcorrespondingtothe canonical ensemble Involvesweakinteractionsbetweenmolecules of thesystem and theparticles of a heatbath at a specifictemperature Simulatean “extended system”: - the real system - anexternal (virtual)system, acting as a heatreservoir Theenergyisallowedtoflowdynamicallyfromthereservoirtothesystem and back.Theheatreservoircontrolsthetemperature of thegivensystem, i. e., thetemperaturefluctuatesaround a target value.

22. Constant temperature molecular dynamics: variables of the Nose-Hoover thermostat Theheatbathisconsidered as an integral part of thesystembyaddition of - anextra degree of freedom, - anextra mometum, - a “mass”, , [dimensions: (energy)  (times)2] . This variable controlstherate of temperaturefluctuations SYSTEM T Reservoir

23. Constant temperature molecular dynamics: variables of the Nose-Hoover thermostat: the artificial variable SYSTEM T Reservoir The extra degree of freedom (artificial parameter) s, playsthe role of a time scalingparameter. The time scale in the extended system, , isscaledbythe factor s Virtual time in the extended system Real time

24. Constant temperature molecular dynamics: variables of the Nose-Hoover thermostat: the artificial momentum SYSTEM T Reservoir The atomic coordinates are identical in both systems Atomic coordinates in the extended system Atomic coordinates in the real system Momentum in the extended system Momentum in the real system

25. Constant temperature molecular dynamics: variables of the Nose-Hoover thermostat: the extended Lagrangian The Lagrangian of the extended system is Kinetic energy of the thermostat Extra potential Kinetic energy minus the potential energy of the real system The extra potential is chosen to ensure that the algorithm produces a canonical ensemble Numbers of degree of freedom: 3N 3N-3 ifthe total momentumisfixed +1 because of the extra degree of freedom of thethermostat

26. Constant temperature molecular dynamics: the Lagrangian equations for the Nose-Hoover thermostat The Lagrangian equations of motions are

27. Constant temperature molecular dynamics: the Lagrangian equations for the Nose-Hoover thermostat The Lagrangian equations of motions are

28. Constant temperature molecular dynamics: the Lagrangian equations for the Nose-Hoover thermostat The Lagrangian equations of motions are Theequations of motion are solvedusingthestandard predictor-corrector method These equations sample a microcanonical ensemble in the extended system. The extended system Hamiltonian is conserved However, the energy of the real system is not constant. Accompanying the fluctuations of s, heat transfers occur between the system and a heat bath, which regulate the system temperature.

29. Constant temperature molecular dynamics: Choice of Q in the Nose-Hoover thermostat regain conventional Molecular Dynamics Too high Too low Long lived weakly damping oscillations of energy occur, resulting in poor equilibration Slow energy flow between the system and the reservoir