General considerations Monte Carlo methods (I)

11

1

N

N

xx

xx

dVf

dVfAA

X

X

X

XX

Averages

X = (x1, x2, …, xN) – vector of variablesP – probability density function

Random variables and their distributions

A random variable: a number assigned to an event

Distribuand:

Probability density:

Any function of a random variable is a random variable.

XYPXF

dX

XdPXf

Moments of a distribution

n

iii

n

iii

xxPxHH(x)E

xxPxE({x})

1

1

dxxfxHxHE

dxxxfx{x}E ˆ

Continuous variables:

If H(x)=(x-xc)n then E{H(X)} is called the nth moment of x with respect to c; if c= then E is the nth central moment, n({x}).

x̂

Some useful central moments

Variance

dxxfx̂xxx 22

2

Skewness

dxxfx̂xx

1

x

xx 3

32/32

3

Kurtosis

3dxxfx̂x

x

13

x

xx 4

422

4

For a set of points:

3

)1n(

x̂x

)1n(

x̂x

xxn1n

1x̂x

1n

1

xn

1x̂

4

n

1i

4i

3

n

1i

3i

2n

1ii

n

1i

2i

n

1i

2i

2

n

1ii

Some examples of central moments

Boltzmann distribution

E BB

NN

B

NN

B

NN

B

i

iB

ii

dETk

EEQdE

Tk

EE

QdEEP

ddTk

EQdd

Tk

E

QddP

Tk

EQ

Tk

E

QP

expexp1

,exp

,exp

1,

expexp1

pqpq

pqpq

pqpqq p

kB (Boltzmann constant) = 1.3810-23 J/K

NAkB = R (universal gas constant) = 8.3143 J/(molK)

q – positions; p – momenta; (E) – density of states with energy E

Thermodynamic quantities

VN

TTBV

B

BVN

Bi

iiB

TNB

i B

iiT

VNB

i B

iiT

T

UEETkC

QTkF

QkT

QTkPPTkS

V

QTk

TkVQpp

T

QTk

TkQEU

,

222

,

,

,

2

ln

lnln

ln

lnexp

1

lnexp

1

Ergodicity

AA

AN

A

dAt

A

N

ii

N

t

t

theoremergodic

ρ toacc.sampled

1

0

1lim

1lim

Trajectories in the phase space. For ergodic sampling, sub-trajectories should be part of a trajectory that passes through all points.

Monte Carlo methods: use of random-number generators to follows the evolution of a system according to a given probability-distribution function.

Beginnings: Antiquity (?); estimation of the results of dice game.

First documented use: G. Comte de Buffon (1777): estimation of the number p by throwing a needle on a sheet of paper with parallel lines and counting the number of hits.

First large-scale (wartime) application: J. von Neumann, S. Ulam, N. Metropolis, R.P. Feynman i in. (1940’s; the Manhattan project) calculations of neutron creation and scattering. For security, the calculations were disguised as ,,Monte Carlo” calculations.

Kinds of Monte Carlo methods

The von Neumann (rejection) sampling

The Metropolis (in general: Markov chain) sampling. Also known as „importance sampling”

Simple Monte Carlo averaging

N

iii xfxA

NA

1

1

x

f(x)

Sample a point on x from a uniform distribution

Compute f(xi)

Rejection or hit-and-miss sampling

x

f(x)

Sample a point on x from a uniform distrubution

Sample a point on f

accept

reject

acceptedi

iaccepted

xAN

A1

Algorithm

Generate a random point X in the

configurational space

Generate a random point y in [0,1]

P(X)>yAccept X

A:=A+A(X)Reject X

Application of the rejection sampling to compute the number

1

41

21 2

lim

totN

N

nS

S

Applications

• For one-dimensional integrals classical quadratures (Newton-Cotes, Gauss) are better than that.

• For multi-dimensional integrals sampling the integrations space is somewhat better; however most points have zero contributions.

• We cannot compute ensemble averages of molecular systems that way. The positions of atoms would have to generated at random, this usually leading to HUGE energies.

Illustration of the difference between the direct- and importance-sampling methods to measure the depth of the river of Nile

Von Neumann: all points are visited

Metropolis: The walker stays in the river

Perturb Xo: X1 = Xo + X

Compute the new energy (E1)

Configuration Xo, energy Eo

E1<Eo ?

Draw Y from U(0,1)

Compute W=exp[-(E1-Eo)/kT]

W>Y?

Xo=X1, Eo=E1

N

Y

Y

N

A:=A+A(Xo)

E0

E1

Accept with probability exp[-(E2-E1)/kBT]

E1

Accept

Space representation in MC simulations

• Lattice (discrete). The particles are on lattice nodes

• Continuous. The particles move in the 3D space.

On- and off-lattice representations

Application of Metropolis Monte Carlo

• Determination of mechanical and thermodynamic properties(density, average energy, heat capacity, conductivity, virial coefficients).

• Simulations of phase transitions. • Simulations of polymer properties.• Simulations of biopolymers.• Simulations of ligand-receptor binding.• Simulations of chemical reactions.

Computing averages with Metropolis Monte Carlo

N

iiAN

A1

1

It should be noted that all MC steps are considered, including those which resulted in the rejection of a new configuration. Therefore, if a configuration has a very low energy, it will be counted multiple times.

Detailed balance (Einstein’s theorem)

)()()()( onnnoo NN

old

new

In real life the detailed balance condition is rarely satisfied…

These gates at a Seoul subway station do not satisfy the detailed-balance condition: you can go through but you cannot go back…

A famous Russian proverb states: „A ruble to get in, ten rubles to get out”

I thought there was a way out….

I was sssoooo busy working for my Queen and Community….

Nature teaches us the hard way that detailed balance is not something to meet in the macro-world..

It is only too easy to violate the

detailed-balance conditions

Monte Carlo with minimization: energy is minimized after each move. Transition probability is proportional to basin size.

Little chance to get from C to B

A B C

pertubation

minim

ization

pertubation

Minim

ization

brings back to C

Translational perturbations (straightforward)

5.0():

5.0():

5.0():

1()int:

ranfozz

ranfoyy

ranfoxx

ranfNo

new

new

new

x[o]

Orientational perturbations

Euler angles. We rotate the system first about the z axis by so that the new x axis is axis w, then about the w axis by so that the new z axis is z’ and finally about the z’ axis by so that the new x axis is x’ and the new y axis is y’.

Uniform sampling the Euler angles would result in a serious bias.

Rotational perturbations: rigid linear molecules

Orientational perturbation

uu’

Genarate a random unit vector v

Compute t:=u+v

Compute u’:= t/||t||

Rigid non-linear molecules

Sample a unit vector on a 4D sphere (quaternion) (E. Veseley, J. Comp. Phys., 47:291-296, 1982), then compute the Euler angles from the following formulas:

2sin

2cos

2sin

2sin

2cos

2sin

2cos

2cos

1;,,,

3

2

1

0

23

22

21

203210

q

q

q

q

qqqqqqqqQ

Rotation matrix

rr

qqqqqqqqqqqq

qqqqqqqqqqqq

qqqqqqqqqqqq

R

R

'

23

22

21

2010322031

103223

22

21

203021

2031302123

22

21

20

22

22

22

Choosing the initial configuration and step size

• Generally: avoid clashes.

• For rigid liquids molecules start from a configuration on a lattice.

• For flexible molecules: build the chain to avoid overlap with the atoms already present.

MC step size

• Too small: high acceptance rate but poor ergodicity (can’t get out of a local minimum).

• Too large: low acceptance rate.

• Avoid accepting „good” advices that the acceptance rate should be 10/20/50%, etc. Do pre-production simulations to select optimal step size

• This can help:– Configurational-bias Monte Carlo,– Parallel tempering.

Reference algorithms for MC/MD simulations (Fortran 77)

M.P. Allen, D.J. Tildesley, „Computer Simulations of Liquids” , Oxford Science Publications, Clardenon Press, Oxford, 1987

http://www.ccp5.ac.uk/software/allen_tildersley.shtml

F11: Monte Carlo simulations of Lennard-Jones fluid.