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Statistical Ensembles. Classical phase space is 6N variables ( p i , q i ) and a Hamiltonian function H( q , p ,t). We may know a few constants of motion such as energy, number of particles, volume... - PowerPoint PPT Presentation
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May 2001 D. Ceperley Simulation Methods 1
Statistical Ensembles• Classical phase space is 6N variables (pi, qi) and a Hamiltonian
function H(q,p,t).• We may know a few constants of motion such as energy, number
of particles, volume... • Ergodic hypothesis: each state consistent with our knowledge is
equally “likely”; the microcanonical ensemble.• Implies the average value does not depend on initial conditions.• A system in contact with a heat bath at temperature T will be
distributed according to the canonical ensemble: exp(-H(q,p)/kBT )/Z
• The momentum integrals can be performed. • Are systems in nature really ergodic? Not always!
May 2001 D. Ceperley Simulation Methods 2
Ergodicity• Fermi- Pasta- Ulam “experiment” (1954) • 1-D anharmonic chain: V= [(q i+1-q i)2+ (q i+1-q i)3]
• The system was started out with energy with the lowest energy mode. Equipartition would imply that the energy would flow into the other modes.
• Systems at low temperatures never come into equilibrium. The energy sloshes back and forth between various modes forever.
• At higher temperature many-dimensional systems become ergodic. • Area of non-linear dynamics devoted to these questions.
May 2001 D. Ceperley Simulation Methods 3
Let us say here that the results of our computations were, from the beginning, surprising us. Instead of a continuous flow of energy from the first mode to the higher modes, all of the problems show an entirely different behavior. … Instead of a gradual increase of all the higher modes, the energy is exchanged, essentially, among only a certain few. It is, therefore, very hard to observe the rate of “thermalization” or mixing in our problem, and this wa s the initial purpose of the calculation.
Fermi, Pasta, Ulam (1954)
May 2001 D. Ceperley Simulation Methods 4
• Equivalent to exponential divergence of trajectories, or sensitivity to initial conditions. (This is a blessing for numerical work. Why?)
• What we mean by ergodic is that after some interval of time the system is any state of the system is possible.
• Example: shuffle a card deck 10 times. Any of the 52! arrangements could occur with equal frequency.
• Aside from these mathematical questions, there is always a practical question of convergence. How do you judge if your results converged? There is no sure way. Why? Only “experimental” tests.– Occasionally do very long runs.– Use different starting conditions.– Shake up the system.– Compare to experiment.
May 2001 D. Ceperley Simulation Methods 5
Statistical ensembles• (E, V, N) microcanonical, constant volume• (T, V, N) canonical, constant volume• (T, P N) constant pressure• (T, V , ) grand canonical
• Which is best? It depends on:– the question you are asking– the simulation method: MC or MD (MC better for phase
transitions)– your code.
• Lots of work in last 2 decades on various ensembles.
May 2001 D. Ceperley Simulation Methods 6
Definition of Simulation
• What is a simulation?An internal state “S”A rule for changing the state Sn+1 = T (Sn)
We repeat the iteration many time.
• Simulations can be– Deterministic (e.g. Newton’s equations=MD)– Stochastic (Monte Carlo, Brownian motion,…)
• Typically they are ergodic: there is a correlation time T. for times much longer than that, all non-conserved properties are close to their average value. Used for:– Warm up period– To get independent samples for computing errors.
May 2001 D. Ceperley Simulation Methods 7
Problems with estimating errors
• Any good simulation quotes systematic and statistical errors for anything important.
• Central limit theorem: after “enough” averaging, any statistical quantity approaches a normal distribution.
• One standard deviation means 2/3 of the time the correct answer is within of the estimate.
• Problem in simulations is that data is correlated in time. It takes a “correlation” time to be “ergodic”
• We must throw away the initial transient and block successive parts to estimate the mean value and error.
• The error and mean are simultaneously determined from the data. We need at least 20 independent data points.
May 2001 D. Ceperley Simulation Methods 8
Estimating Errors Trace of A(t): Equilibration time. Histogram of values of A ( P(A) ). Mean of A (a). Variance of A ( v ). estimate of the mean: A(t)/N estimate of the variance, Autocorrelation of A (C(t)). Correlation time ( ). The (estimated) error of the (estimated) mean ( ). Efficiency [= 1/(CPU time * error 2)]
May 2001 D. Ceperley Simulation Methods 9
Statistical thinking is slippery
• “Shouldn’t the energy settle down to a constant” – NO. It fluctuates forever. It is the overall mean which converges.
• “My procedure is too complicated to compute errors”– NO. Run your whole code 10 times and compute the mean and
variance from the different runs
• “The cumulative energy has converged”.– BEWARE. Even pathological cases have smooth cumulative
energy curves.
• “Data set A differs from B by 2 error bars. Therefore it must be different”. – This is normal in 1 out of 10 cases.
May 2001 D. Ceperley Simulation Methods 10
Characteristics of simulations.
• Potentials are highly non-linear with discontinuous higher derivatives either at the origin or at the box edge.
• Small changes in accuracy lead to totally different trajectories. (the mixing or ergodic property)
• We need low accuracy because the potentials are not very realistic. Universality saves us: a badly integrated system is probably similar to our original system. This is not true in the field of non-linear dynamics or, in studying the solar system
• CPU time is totally dominated by the calculation of forces. • Memory limits are not too important.• Energy conservation is important; roughly equivalent to time-reversal
invariance.: allow 0.01kT fluctuation in the total energy.
May 2001 D. Ceperley Simulation Methods 11
The Verlet Algorithm
The nearly universal choice for an integrator is the Verlet (leapfrog) algorithm
r(t+h) = r(t) + v(t) h + 1/2 a(t) h2 + b(t) h3 + O(h4) Taylor expandr(t-h) = r(t) - v(t) h + 1/2 a(t) h2 - b(t) h3 + O(h4) Reverse timer(t+h) = 2 r(t) - r(t-h) + a(t) h2 + O(h4) Addv(t) = (r(t+h) - r(t-h))/(2h) + O(h2) estimate velocities
Time reversal invariance is built in the energy does not drift.
2 3 4 51 6 7 9 10 11 12 138
May 2001 D. Ceperley Simulation Methods 12
How to set the time step• Adjust to get energy conservation to 1% of kinetic energy.• Even if errors are large, you are close to the exact
trajectory of a nearby physical system with a different potential.
• Since we don’t really know the potential surface that accurately, this is satisfactory.
• Leapfrog algorithm has a problem with round-off error.• Use the equivalent velocity Verlet instead:
r(t+h) = r(t) +h [ v(t) +(h/2) a(t)]v(t+h/2) = v(t)+(h/2) a(t)v(t+h)=v(t+h/2) + (h/2) a(t+h)
May 2001 D. Ceperley Simulation Methods 13
Spatial Boundary Conditions
Important because spatial scales are limited. What can we choose?
• No boundaries; e.g. droplet, protein in vacuum. If droplet has 1 million atoms and surface layer is 5 atoms thick 25% of atoms are on the surface.
• Periodic Boundaries: most popular choice because there are no surfaces (see next slide) but there can still be problems.
• Simulations on a sphere• External potentials• Mixed boundaries (e.g. infinite in z, periodic in x and y)
May 2001 D. Ceperley Simulation Methods 14
Minimum Image Convention: take the closest distance:|r|M = min ( r+nL)
• Potential is cutoff so that V(r)=0 for r>L/2 since force needs to be continuous. How about the derivative?
• Image potentialVI = v(ri-rj+nL)
• For long range potential this leads to the Ewald image potential. You need a back ground and convergence method (more later)
Periodic distances
x
-L -L/2 0 L/2 L
May 2001 D. Ceperley Simulation Methods 15
Complexity of Force Calculations
• Complexity is defined as how a computer algorithm scales with the number of degrees of freedom (particles)
• Number of terms in pair potential is N(N-1)/2 O(N2)• For short range potential you can use neighbor tables to
reduce it to O(N)– (Verlet) neighbor list for systems that move slowly– bin sort list (map system onto a mesh and find neighbors from the
mesh table)
• Long range potentials with Ewald sums are O(N3/2) but Fast Multipole Algorithms are O(N) for very large N.
May 2001 D. Ceperley Simulation Methods 16
Constant Temperature MD• Problem in MD is how to control the temperature. (BC in
time.)• How to start the system? (sample velocities from a
Gaussian distribution) If we start from a perfect lattice as the system becomes disordered it will suck up the kinetic energy and cool down. (v.v for starting from a gas)
• QUENCH method. Run for a while, compute kinetic energy, then rescale the momentum to correct temperature, repeat as needed.
• Nose-Hoover Thermostat controls the temperature automatically by coupling the microcanonical system to a heat bath
• Methods have non-physical dynamics since they do not respect locality of interactions. Such effects are O(1/N)
May 2001 D. Ceperley Simulation Methods 17
Quench method• Run for a while, compute kinetic energy, then rescale the
momentum to correct temperature, repeat as needed.• Control is at best O(1/N)
2
'
3( 1)
i iI
i iI
i
Tv vT
m vTN
May 2001 D. Ceperley Simulation Methods 18
Nose-Hoover thermostat• MD in canonical distribution (TVN)• Introduce a friction force (t)
T Reservoir
SYSTEM
p(t))(t)F(q,dtdp t
Dynamics of friction coefficient to get canonical ensemble.
Tk2
3Nmvdt
Qdb
221 Feedback restores
makes kinetic energy=temperature
Q= “heat bath mass”. Large Q is weak coupling
May 2001 D. Ceperley Simulation Methods 19
Effect of thermostat
System temperature fluctuates but how quickly?
Q=1
Q=100DIMENSION 3TYPE argon 256 48. POTENTIAL argon argon 1 1. 1. 2.5DENSITY 1.05TEMPERATURE 1.15TABLE_LENGTH 10000LATTICE 4 4 4 4SEED 10WRITE_SCALARS 25NOSE 100.RUN MD 2200 .05
May 2001 D. Ceperley Simulation Methods 20
• Thermostats are needed in non-equilibrium situations where there might be a flux of energy in or out of the system.
• It is time reversable, deterministic and goes to the canonical distribution but:
• How natural is the thermostat?– Interactions are non-local. They propagate instantaneously– Interaction with a single heat bath variable-dynamics can be
strange. Be careful to adjust the “mass”
REFERENCES1. S. Nose, J. Chem. Phys. 81, 511 (1984); Mol. Phys. 52, 255 (1984).2. W. Hoover, Phys. Rev. A31, 1695 (1985).
May 2001 D. Ceperley Simulation Methods 21
Constant Pressure• To generalize MD, follow similar
procedure as for the thermostat for constant pressure. The size of the box is coupled to the internal pressure
TPN•Volume is coupled to virial pressure
•Unit cell shape can also change.
,2P 13
di j dr
i j
K r
May 2001 D. Ceperley Simulation Methods 22
Parrinello-Rahman simulation•500 KCl ions at 300K
•First P=0
•Then P=44kB
•System spontaneously changes from rocksalt to CsCl structure
May 2001 D. Ceperley Simulation Methods 23
• Can “automatically” find new crystal structures• Nice feature is that the boundaries are flexible• But one is not guaranteed to get out of local minimum• One can get the wrong answer. Careful free energy
calculations are needed to establish stable structure.
• All such methods have non-physical dynamics since they do not respect locality of interactions.
• Non-physical effects are O(1/N)REFERENCES1. H. C. Andersen, J. Chem. Phys. 72, 2384 (1980).2. M. Parrinello and A. Rahman, J. Appl. Phys. 52, 7158 (1981).
May 2001 D. Ceperley Simulation Methods 24
Brownian dynamics
• Put a system in contact with a heat bath• Will discuss how to do this later.• Leads to discontinuous velocities.• Not necessarily a bad thing, but requires some physical
insight into how the bath interacts with the system in question.
• For example, this is appropriate for a large molecule (protein or colloid) in contact with a solvent. Other heat baths in nature are given by phonons and photons,…
May 2001 D. Ceperley Simulation Methods 25
Monitoring the simulation
• Static properties: pressure, specific heat etc.• Density• Pair correlation in real space and fourier space.• Order parameters: How to tell a liquid from a solid
May 2001 D. Ceperley Simulation Methods 26
Thermodynamic properties
• Internal energy=kinetic energy + potential energy• Kinetic energy is kT/2 per momentum• Specific heat = mean squared fluctuation in energy• pressure can be computed from the virial theorem.• compressibility, bulk modulus, sound speed• But we have problems for the basic quantities of entropy
and free energy since they are not ratios with respect to the Boltzmann distribution. We will discuss this later.
May 2001 D. Ceperley Simulation Methods 27
May 2001 D. Ceperley Simulation Methods 28
Microscopic Density(r) = < i (r-r i) >
Or you can take its Fourier Transform:
k = < i exp(ikri) >(This is a good way to smooth the density.)
• A solid has broken symmetry (order parameter). The density is not constant.• At a liquid-gas transition the density is also inhomgeneous.• In periodic boundary conditions the k-vectors are on a grid: k=2/L
(nx,ny,nz) Long wave length modes are absent.• In a solid Lindemann’s ratio gives a rough idea of melting:
u2= <(ri-zi)2>/d2
May 2001 D. Ceperley Simulation Methods 29
Order parameters
• A system has certain symmetries: translation invariance.• At high temperatures one expect the system to have those
same symmetries at the microscopic scale. (e.g. a gas)• BUT as the system cools those symmetries are broken. (a
gas condenses).• At a liquid gas-transition the density is no longer fixed:
droplets form. The density is the order parameter.• At a liquid-solid transition, both rotational symmetry and
translational symmetry are broken• The best way to monitor the transition is to look for the
dynamics of the order parameter.
May 2001 D. Ceperley Simulation Methods 30
May 2001 D. Ceperley Simulation Methods 31
Electron Density during exchange2d Wigner crystal (quantum)
May 2001 D. Ceperley Simulation Methods 32
Snapshots of densities
Liquid or crystal or glass? Blue spots are defects
May 2001 D. Ceperley Simulation Methods 33
Density distribution within a helium dropletDuring addition of molecule, it travels from the surface to the interior.
Red is high density, blue low density of helium
May 2001 D. Ceperley Simulation Methods 34
Pair Correlation Function, g(r)Primary quantity in a liquid is the probability distribution of pairs of
particles. Given a particle at the origin what is the density of surrounding particles
g(r) = < i<j (ri-rj-r)> (2 /N2)
Density-density correlation
Related to thermodynamic properties.
3NV ( ) d rv(r)g(r)2ij
i j
v r
May 2001 D. Ceperley Simulation Methods 35
g(r) in liquid and solid helium• First peak is at inter-particle
spacing. (shell around the particle)
• goes out to r<L/2 in periodic boundary conditions.
May 2001 D. Ceperley Simulation Methods 36
(The static) Structure Factor S(K)• The Fourier transform of the pair correlation function is
the structure factor S(k) = <|k|2>/N (1)
S(k) = 1 + dr exp(ikr) (g(r)-1) (2)
• problem with (2) is to extend g(r) to infinity• This is what is measured in neutron and X-Ray scattering
experiments. • Can provide a direct test of the assumed potential.• Used to see the state of a system:
liquid, solid, glass, gas? (much better than g(r) )• Order parameter in solid is G
May 2001 D. Ceperley Simulation Methods 37
• In a perfect lattice S(k) will be non-zero only on the reciprocal lattice vectors G: S(G) = N
• At non-zero temperature (or for a quantum system) this structure factor is reduced by the Debye-Waller factor
S(G) = 1+ (N-1)exp(-G2u2/3)• To tell a liquid from a crystal we see how S(G) scales as the
system is enlarged. In a solid, S(k) will have peaks that scale with the number of atoms.
• The compressibility is given by:• We can use this is detect the liquid-gas transition since the
compressibility should diverge as k approaches 0. (order parameter is density)
) TS(0)/(kBT
May 2001 D. Ceperley Simulation Methods 38Crystal liquid
May 2001 D. Ceperley Simulation Methods 39
Here is a snapshot of a binary mixture. What correlation function would be important?
May 2001 D. Ceperley Simulation Methods 40
• In a perfect lattice S(k) will be non-zero only on the reciprocal lattice vectors: S(G) = N
• At non-zero temperature (or for a quantum system) this structure factor is reduced by the Debye-Waller factor
S(G) = 1+ (N-1)exp(-G2u2/3)• To tell a liquid from a crystal we see how S(G) scales as the
system is enlarged. In a solid, S(k) will have peaks that scale with the number of atoms.
• The compressibility is given by:
We can use this is detect the liquid gas transition since the compressibility should diverge. (order parameter is density)
) TS(0)/(kBT
May 2001 D. Ceperley Simulation Methods 41
Dynamical Properties• Fluctuation-Dissipation theorem:
Here A e-iwt is a perturbation and (w) e-iwt is the response of B. We calculate the average on the lhs in equilibrium (no external perturbation).
• Fluctuations we see in equilibrium are equivalent to how a non-equilibrium system approaches equilibrium. (Onsager regression hypothesis)
• Density-Density response function is S(k,w) can be measured by scattering and is sensitive to collective motions.
dtdA(0)B(t)dte)(
0
it
May 2001 D. Ceperley Simulation Methods 42
•Diffusion constant is defined by Fick’s law and controls how systems mix
•Microscopically we calculate:
•Alder-Wainwright discovered long-time tails on the velocity autocorrelation function so that the diffusion constant does not exist in 2D
)t,r(Ddtd 2
(t)(0) dt
)t6/((t))-(0)(D
031
2
vvrr
Diffusion Coefficient
May 2001 D. Ceperley Simulation Methods 43
Mixture simulation with CLAMPS
Initial condition Later
May 2001 D. Ceperley Simulation Methods 44
Transport Coefficients
• Diffusion: Particle flux• Viscosity: Stress tensor• Heat transport: energy current• Electrical Conductivity: electrical current
These can also be evaluated with non-equilibrium simulations use thermostats to control.
• Impose a shear flow• Impose a heat flow