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Lecture 2• Molecular dynamics simulates a system by
numerically following the path of all particles in phase space as a function of time
• the time T must be long enough to allow the system to explore all accessible regions of phase space
• the time average of a quantity A is calculated from0
0
1lim ( )
t T
T
t
A A t dtT
3 3
1 2 1 2... ( , ,..., , ,...) N NA r r p p dr dp
Monte Carlo
• no dynamics but random motion in configuration space due to random but uncorrelated forces
• generate configurations or states with a weight proportional to the canonical or grand canonical probability density
• actual steps of calculation depend on the model
Review ofProbability and
Statistics
Introduction• Probability and statistics are the foundations of
both statistical mechanics and the kinetic theory of gases
• what does the notion of probability mean?
• Classical notion: we assign, a priori, equal probabilities to all possible outcomes of an event
• Statistical notion: we measure the relative frequency of an event and call this the probability
Classical Probability
• Count the number W of outcomes and assign them equal probabilities pi = 1/W
• for example: a coin toss
• each “throw” is a trial with W=2 outcomes
• pH = pT = 1/2
• for N consecutive trials, a particular sequence of heads and tails constitutes an event HTTHHHTT...
• there are 2N possible outcomes and the probability of each “event” is pi = 1/2N
Classical Probability• We cannot predict which sequence (event) will
occur in a given trial
• hence we need a statistical description of the system => a description in terms of probabilities
• instead of focusing on a particular system or sequence, we can think of an assembly of systems called an ensemble
• repeat the N coin flips a large number (M) of times
• if event ‘i’ occurs mi times in these M members of the ensemble, then the fraction mi/M is the probability of the event ‘i’
Probability of a head (H) isthe number of coins nH with a H divided by the total number M in the ensemble
lim HH M
np
M
Classical Probability• in statistical mechanics we use this idea by assuming
that all accessible quantum states of a system are equally likely
• basic idea is that if we wait long enough, the system will eventually flow through all of the microscopic states consistent with any constraints imposed on the system
• measurements must be treated statistically• the microcanonical ensemble corresponds to an
isolated system with fixed total energy E• however this is not the most convenient approach
Statistical Probability
• Experimental method of assigning probabilities to events by measuring the relative frequency of occurrence
• if event ‘i’ occurs mi times in M trials, then
pm
Mi Mi F
HGIKJ lim
Independent Events• If events are independent, then the probability that
both occur pi,j = pi pj
• e.g coin toss with 2 trials => 4 outcomes
• pH,H=pT,T=pH,T=pT,H= (1/2)(1/2)=1/4
• but probability of getting one head and one tail in 2 trials = 1/4 + 1/4 = 1/2 (order unimportant!)
• probability of 2 heads and 2 tails (independent of order) in 4 tosses is
prob p pH T4
2 22 2!
! !
Random Walks
• Consider a walker confined to one dimension starting at the point x=0
• the probability of making a step to the right is p and to the left is q=1-p ( p+q=1)
• each step is independent of the preceding step
• let the displacement at step i be denoted as si where si= ±a
• each step is of the same magnitude
• where is the walker after N steps?
a
( ) ( ) ( )is a a p ap q q
Random Walk
x N sii
N
( )
1
Net displacement
x N s s s
s s s
ii
N
i jj
N
i
N
ii
N
ii j
N
j
2
1
2
11
2
1
( ) FHGIKJ
Averages
x N s N pa q a
Na p q
ii
N
( ) ( ( ))
( )
1
x N sii
N
( )
1
The average of a sum of independent random variables is equal to the sum of the averages
Averages
x N s s s
Na N N a p q
x N pqa N
ii
N
i ji j
N2 2
1
2 2
2 2
1
4
( )
( ) ( )
( )
The average of the product of two statisticallyindependent random variables is equal to the product of the averages
x N s s sii
N
ii j
N
j2 2
1
( )
x N x N x N
x x pqa N
2 2
2 2 2 24
( ) ( ) ( )
c h
Notex N
x N
x N
a pqN
p q aN N
:( )
( )
( )
/
2 1 2
2 1
b g
Dispersion or Variance
Walker does not get very far from its mean value if N>>1 !
Probability Distribution• What is the probability P(x,N) that walker
ends up at point x in N steps?
• Total number of steps N= nR + nL
• probability of nR steps to right is pnR
• probability of nL steps to left is qnL
• number of ways = N!/nR!nL!
• but x = (nR - nL)a
• hence nR = (N+ x/a)/2
• nL= (N - x/a)/2 set a=1
p x NN
N x N xp q
N x N x
( , )!
! !
( ) ( )
FHG
IKJ
FHG
IKJ
2 2
2 2
Set a=1
N=20
N=40
p x NN
N x N xp q
N x N x
( , )!
! !
( ) ( )
FHG
IKJ
FHG
IKJ
2 2
2 2
p r NN
N r N rp q
N r N r
( , )!
( )!
( )!
(( )
) (( )
)
FHG
IKJ
FHG
IKJ
12
12
1
2
1
2
-N<x<N
Define r=x/N where -1<r<1
( , )
(0, )
p r N
p N
p x NN
N x N xp q
N x N x
( , )!
! !
( ) ( )
FHG
IKJ
FHG
IKJ
2 2
2 2
x
x p x N
p x N
n
n
x N
N
x N
N
( , )
( , )
p x Nx N
N
( , ) 1
Show
x N p q
x x
pqN
( )
2 2 2
4
For large N, p(x,N) approaches a continuous distribution
lim ( , )
( )
N
x x
p x N
p xe
x
b g22
2
2
2
x p q N
pqN
( )
2 4