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