general considerations monte carlo methods (i). averages x = (x 1, x 2, …, x n ) – vector of...
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General considerations Monte Carlo methods (I)
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11
1
N
N
xx
xx
dVf
dVfAA
X
X
X
XX
Averages
X = (x1, x2, …, xN) – vector of variablesP – probability density function
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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
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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
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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
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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
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Some examples of central moments
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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
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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
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Ergodicity
AA
AN
A
dAt
A
N
ii
N
t
t
theoremergodic
ρ toacc.sampled
1
0
1lim
1lim
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Trajectories in the phase space. For ergodic sampling, sub-trajectories should be part of a trajectory that passes through all points.
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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.
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Kinds of Monte Carlo methods
The von Neumann (rejection) sampling
The Metropolis (in general: Markov chain) sampling. Also known as „importance sampling”
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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)
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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
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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
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Application of the rejection sampling to compute the number
1
41
21 2
lim
totN
N
nS
S
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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.
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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
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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)
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E0
E1
Accept with probability exp[-(E2-E1)/kBT]
E1
Accept
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Space representation in MC simulations
• Lattice (discrete). The particles are on lattice nodes
• Continuous. The particles move in the 3D space.
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On- and off-lattice representations
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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.
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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.
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Detailed balance (Einstein’s theorem)
)()()()( onnnoo NN
old
new
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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”
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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..
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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
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Translational perturbations (straightforward)
5.0():
5.0():
5.0():
1()int:
ranfozz
ranfoyy
ranfoxx
ranfNo
new
new
new
x[o]
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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.
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Rotational perturbations: rigid linear molecules
Orientational perturbation
uu’
Genarate a random unit vector v
Compute t:=u+v
Compute u’:= t/||t||
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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
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Rotation matrix
rr
qqqqqqqqqqqq
qqqqqqqqqqqq
qqqqqqqqqqqq
R
R
'
23
22
21
2010322031
103223
22
21
203021
2031302123
22
21
20
22
22
22
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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.
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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.
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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.