microscopic states (microstates)or microscopic configurationsunder external

constraints (N or , V or P, T or E,

etc.) Ensemble (micro-canonical,

canonical, grand canonical, etc.)

Average over a collection

of microstates

Macroscopic quantities (properties, observables)• thermodynamic – or N, E or T, P or V, Cv, Cp, H, S,

G, etc.• structural – pair correlation function g(r), etc.• dynamical – diffusion, etc.

These are what are measured in true experiments.

they’re generated naturally from thermal fluctuation

In a real-space experiment

In a virtual-space simulation

How do we mimic the Mother Nature in a virtual space to realize lots of microstates, all of which correspond to

a given macroscopic state? By MC or MD method!

it is us who needs to generate them by MC or MD methods.

t1 t2 t3~1023

particles

Beyond 1D integrals: A system of N particles in a container

of a volume V in contact with a thermostat T (constant NVT)

(=1/kT) or for discrete microstates

for discrete microstates

• Particles interact with each other through a potential energy U(rN) (~ pair potential).

• U(rN) is the potential energy of a microstate {rN} = {x1, y1, z1, …, xN, yN, zN}.

• (rN) is the probability to find a microstate {rN} under the constant-NVT constraint.• Partition function Z (required for normalization) = the weighted sum of

all the microstates compatible with the constant-NVT condition

• Average of an observable O, <O>, over all the microstates compatible with constant NVT

or for discrete microstates

ensemble average

“canonical ensemble”

external constraint

3N-dimension integration

3N-dimension

• 1885 – Johann Balmer – Line spectrum of hydrogen

• 1886 – Heinrich Hertz – Photoelectric effect experiment

• 1897 – J. J. Thomson – Discovery of electrons from cathode rays experiment

• 1900 – Max Planck – Quantum theory of blackbody radiation

• 1905 – Albert Einstein– Quantum theory of photoelectric effect

• 1910 – Ernest Rutherford – Scattering experiment with -particles

• 1913 – Niels Bohr – Quantum theory of hydrogen spectra

• 1923 – Arthur Compton – Scattering experiment of photons off electrons

• 1924 – Wolfgang Pauli – Exclusion principle – Ch. 10

• 1924 – Louis de Broglie – Matter waves

• 1925 – Davisson and Germer – Diffraction experiment on wave properties of

electrons

• 1926 – Erwin Schrodinger – Wave equation – Ch. 2

• 1927 – Werner Heisenberg – Uncertainty principle – Ch. 6

• 1927 – Max Born – Interpretation of wave function – Ch. 3

Ludwig Boltzmann in the Maxwell-Boltzmann distribution

Boltzmann

a pioneer in atomic theory

was used in the derivation of

Can we use Monte Carlo to compute Z? (even for a very simple, minimum-size, discrete case)

• Consider a model of spins on a 2D lattice (idealized magnetic model). ⇒ a discrete system

• Each of N spins can take 2 states: up (↑) and down (↓). ⇒ 2N microstates

• Each spin interacts only with its nearest neighbors (nn). ⇒ 4 neighbors for a 2D square lattice

• Suppose that it takes 10-6 (or 10-15) s to compute the interaction of a spin with its neighbors. ⇒ Time to calculate the energy Ei of a microstate i = N x 10-6 (or N x 10-15) s

• For N = 100 spins:

- 2100 ~ 1030 microstates - 10-3 (or 10-12) s to calculate the energy of a microstate

⇒ 1027 (or 1018) s to estimate Z! (~ age of the universe ~ 13.8 billion years ~ 4 x 1017 s)

• The situation gets worse with a real (continuous, larger, with beyond-nn interaction) system!

a microstate

⇒ We cannot compute Z (and the absolute free energy kT ln Z) for real systems! (However, we can compute a relative free energy between two states.)

• What contributes to <O> the most? Instantaneous values {Oi} of microstates with high probability

(= with large Boltzmann weight = with low energy)

• We just showed that we cannot compute all the microstates.• Let’s take a subset, for example, 106 states for the spin system in the previous example.• How do we choose these states?

a) Brute force Monte Carlo (hit & miss, sample mean by uniform sampling)

Randomly pick a microstate i (i.e. the orientation of each spin). → Too high probability that the subset doesn’t contribute to the average → The situation is the worst for a continuous dense system!

b) Importance samplingPick a microstate i with high probability (i.e. large i). Sample according to i.

→ We need to use a normalized distribution {i}. → The normalization requires Z, and we showed that we cannot compute Z!

c) Metropolis importance samplingPick a microstate i with large i without calculating the normalization.

→ biased random walk in the phase space

Can we use Monte Carlo to compute <O>?

Hard sphere1/10260

Importance sampling for ensemble averages- Biased random walk

Frenkel and Smit, Understanding Molecular Simulations

quickdepth info: yesarea info: no

depth info: yesarea info: yesslow~impossible

Finished?

Yes

No

Give the particle a randomdisplacement. Calculate thenew energy. ')'( UrU

Accept the move with

'exp

1min)'(

UUrr

Select a particle at random.Calculate the energy. UrU )(

1

)(

tottot

tottot rAAA

Calculate the ensemble Average.

tot

totAA

Initialize the positions 0 ;0 tottotA

Biased random walk in configuration space:Metropolic Monte Carlo method

Kristen A. Fichthorn, Penn. State U.

Stochastic process is a movement through a series of well-defined microstates (or configurations or states in short) in a way that involves some element of randomness.

Markov process is a stochastic process that has no memory.Selection of next state depends only on the current state, not on prior states.A process is fully defined by a set of transition probabilities ij.

Transition probability (ij) or ij is the probability to go from the state i to j.- Non-negative, not greater than unity (0 1)- Probability of staying in present state may be non-zero.

Transition probability matrix collects all ij.

- sum of each row = 1- (example) system with three states

Markov process & Transition probability

11 12 13

21 22 23

31 32 33

0.1 0.5 0.4

0.9 0.1 0.0

0.3 0.3 0.4

If in state 1, it will move to state 3 with probability 0.4

If in state 1, it will stay in state 1 with probability 0.1.

It never go from state 2 to state 3.

),( trP

= Probability to be at state at time tr

= Transition probability per unit time from to'r

r

r

'r

The master equation for Markov processes

)'( rr

''

)'(),'()'(),(),(

rr

rrtrPrrtrPdt

trdP

)(),( rrtrP

)'(),( rrtrP

Detailed balance (or Microscopic reversibility) criterion at equilibrium: key of Metropolis

algorithm

)'(),'( );(),( rrPrrP

After a long time, the system reaches equilibrium. The transition probabilities should satisfy the condition that they do not destroy the equilibrium distribution once it is reached. Thus, at equilibrium, we have a stationary Markov provess:

)'()'()'()( rrrrrr

One way to satisfy this would be a (more stringent) detailed balance condition.

)(

)'(

)'(

)'(

r

r

rr

rr

''

)'(),'()'(),(0),(

rr

rrrPrrrPdt

rdP

''

)'()'()'()(rr

rrrrrr

StationaryMarkov:Average population does not change with time (i.e., MC steps).

Choices of Metropolis

)'()'()'( rraccrrrr

1. The transition probability can be divided into two parts:(1)Choosing a new configuration with a probability .(2)Accepting or rejecting this new configuration with an acceptance probability acc.

where = probability to choose a particular move, acc = probability to accept the move

N. Metropolis et al. J. Chem. Phys. 21, 1087 (1953)

Then, the detailed balance condition becomes:

)'()'()'()'()'()( rraccrrrrraccrrr

)(

)(

)(

)'(

)'(

)'(

rr

rr

r

r

rracc

rracc

Choices of Metropolis for canonical ensemblesN. Metropolis et al. J. Chem. Phys. 21, 1087 (1953)

1. Detailed balance condition

Z

errrP

Z

errrP

rU

NVT

rU

NVT

)()(

)'()(),'( ;)()(),(

)(

)(

)(

)'(

)'(

)'(

rr

rr

r

r

rracc

rracc

2.Symmetric :

3.Limiting probability distribution for canonical ensemble = Boltzmann

Transition probability & acceptance probability acc should satisfy:

4. Define the acceptance probability as:

Nrrrr /1)'()'(

UrUrU

rU

rU

eeZe

Ze

)()(

)(

)'(

/

/

)()'( i.e., ,0 if , 1

)()'( i.e., ,0 if , )()(

rUrUU

rUrUUee UrUrU

)(

)'(

)(

)'(

)(

)'(

r

r

rracc

rracc

rr

rr

NVT

NVT

)'( rracc

Accept all

the downhill moves.

Accept uphill moves only whennot too uphill.

naturally satisfies this condition.

Choice of Metropolis: Metropolis MC algorithmN. Metropolis et al. J. Chem. Phys. 21, 1087 (1953)

)'(exp1

1

min

if , )'(exp1

if , 1

)'(

UUN

N

UUUUN

rUUrUUNrr

Finished?

Yes

No

Give the particle a randomdisplacement. Calculate thenew energy. ')'( UrU

Accept the move with

'exp

1min)'(

UUrr

Select a particle at random.Calculate the energy. UrU )(

1

)(

tottot

tottot rAAA

Calculate the ensemble Average.

tot

totAA

Initialize the positions 0 ;0 tottotA

Biased random walk in configuration space:Metropolic Monte Carlo method

Kristen A. Fichthorn, Penn. State U.

Almost always involves a Markov processMove to a new configuration from an existing one according to a well-defined transition probability

Simulation procedure

1. Generate a new “trial” configuration by making a perturbation to the present configuration

2. Accept the new configuration based on the ratio of the probabilities for the new and old configurations, according to the Metropolis algorithm

3. If the trial is rejected, the present configuration is taken as the next one in the Markov chain

4. Repeat this many times, accumulating sums for averages

new

old

U

U

e

e

state k

state k+1

Metropolis Monte Carlo Molecular Simulation

David A. Kofke, SUNY Buffalo

For a new configuration of the same volume V and number of molecules N,displace a randomly selected atom to a point chosen with uniform probability inside a cubic volume of edge 2 centered on the current position of the atom.

Examine underlying transition

probability to formulate the acceptance

criterion

Displacement trial move. 1. Specification

?

Select an atom at random.

Consider a region centered

at it.

Move atom to a point chosen

uniformly in region.

Consider acceptance of

new configuration.

2

Step 1 Step 2 Step 3 Step 4

general hitherto

David A. Kofke, SUNY Buffalo

Detailed specification of trial move and transition probability

Event[reverse event]

Probability[reverse probability]

Select molecule k[select molecule k]

1/N[1/N]

Move to rnew

[move back to rold]1/v

[1/v]

Accept move[accept move]

min(1,)[min(1,1/)]

Forward-step transition probability

(2)d

1 1min(1, )

N v

Reverse-step transition probability

11 1min(1, )

N v

is formulated to satisfy the detailed balance condition.

Displacement trial move. 2. Analysis of transition probabilities

David A. Kofke, SUNY Buffalo

Detailed balance

Forward-step transition probability

1 1min(1, )

N v

Reverse-step transition probability

11 1min(1, )

N v

i ij j ji=

Limiting distributionof canonical ensemble

( )1( )

NN N U N

Nd e d

Z rr r r

Displacement trial move. 3. Analysis of detailed balance

David A. Kofke, SUNY Buffalo

Detailed balance

Forward-step transition probability

1 1min(1, )

N v

Reverse-step transition probability

11 1min(1, )

N v

i ij j ji

old newU Ue e

=

( )new oldU Ue Acceptance probability for canonical ensemble

11 1 1 1min(1, ) min(1, )

old newU N U N

N N

e d e d

Z N v Z N v

r r

Displacement trial move. 3. Analysis of detailed balance

David A. Kofke, SUNY Buffalo

public void thisTrial(Phase phase) { double uOld, uNew; if(phase.atomCount==0) {return;} //no atoms to move

int i = (int)(rand.nextDouble()*phase.atomCount); //pick a random number from 1 to N Atom a = phase.firstAtom(); for(int j=i; --j>=0; ) {a = a.nextAtom();} //get ith atom in list

uOld = phase.potentialEnergy.currentValue(a); //calculate its contribution to the energy a.displaceWithin(stepSize); //move it within a local volume phase.boundary().centralImage(a.coordinate.position()); //apply PBC uNew = phase.potentialEnergy.currentValue(a); //calculate its new contribution to energy

if(uNew < uOld) { //accept if energy decreased nAccept++; return; } if(uNew >= Double.MAX_VALUE || //reject if energy is huge or doesn’t pass test Math.exp(-(uNew-uOld)/parentIntegrator.temperature) < rand.nextDouble()) { a.replace(); //...put it back in its original position return; } nAccept++; //if reached here, move is accepted }

Displacement trial move. 4. Code example (Java)

David A. Kofke, SUNY Buffalo

Displacement trial move. 4. Pseudo Code (Louis)

Initialization

Reset block sums

Compute block average

Compute final results

“cycle” or “sweep”

“block”

Move each atom once (on average) 100’s or 1000’s

of cycles

Independent “measurement”

moves per cycle

cycles per block

Add to block sum

blocks per simulation

New configuration

New configuration

Entire SimulationMonte Carlo Move

Select type of trial moveeach type of move has fixed probability of being selected

Perform selected trial move

Decide to accept trial configuration, or keep original

David A. Kofke, SUNY Buffalo

Displacement trial move. 5. Implementation

Rule of thumb: Size of the step is adjusted to reach a target acceptance rate of displacement trials, which is typically 50%.

Large step leads to less acceptance but bigger moves.

Small step leads to less movement but more acceptance.

Displacement trial move. 6. Step size tuning