Chapter 4, Stowe

8/13/2019 Chapter 4, Stowe

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=21p

Npqnn = 21

2

pNn=

CHAPTER 4 : STATISTICS FOR SYSTEMS OF

MANY ELEMENTS

A. Fluctuations For a large number of identically prepared systems (an ensemble), theaverage number of elements per system that satisfy the criterion is

p = probability that a single element satisfies it

N = # of elements per system

Example : If a large number of coins were flipped many times, how many wouldland heads up, on the average?

probability of 1 coin to land heads 2

N

pNn =Example : If a large number of dice are rolled many times,how many would land with 1 dot up, on the average? 6

1=p6

Nn=

Standard Deviation square root of the average fluctuations squared

Relative Fluctuation Npq

NpNpq

n=

Half-width at half maximum

Jitter aboutthe mean


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For largeN, the distribution of n about the mean is given by a GAUSSIANdistribution. The probability that n is within 1 standard deviation of is 0.68.

Example : 100 flipped coins and 10,000 flipped coins ?and,, =n

n

100 flipped coins 5010021100 =pNn 525

2

1

2

1100100 =

Npq

%1000,550

000,10

000,10 =n 000,5000,102

110,000 =pNn

5025002

1

2

1000,1010,000 =

Npq

%1050

5

100

100 =n

10,000 flipped coins

For 100 flipped coins, the number of heads will usually be (i.e., 68% of the time)

between 40 and 60% of the total, while for the 10,000 coins system, the number ofheads will usually be between 49 and 51% of the total.

The larger the system, its behavior will be more predictable.

n

forheads


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B. The Gaussian Distribution

For systems with largeN, the probability that n elements satisfy the criterion and theremaining (N-n) do NOT is

] 22 if2

1 22


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Example : How many different combinations of the air molecules are there, suchthat 1,100 are in the front third of the room?

85810!1900!1100

!3000!!

! =nNn N

Binomial coefficient

(by Stirlings formula)

Binomial distribution is correct for all

systems but is cumbersome for large N

C. Random Walk

study of the net motion of objects where the motion occurs in individual discretesteps with random direction and is independent of the direction of the preceding orsucceeding steps

Examples of random walk problemsDiffusion of an ammonia molecule through airTravel of electrons through a metalDiffusion of a hole through a semiconductor


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For the random walk problem in any 1-dimension, if P(s)ds is the probability that asingle step falls in the range between s and s+ds, then after N steps

sNS= 22 sN

dsssPs dssPss 22 22222 ssdssPsssss

Example : migration of a conduction electron in a metal

1012 collisions per second with atoms and other electrons (N=1012). 1 root-mean-square distance between collisions

angstromsEwith 410,0 = 22 1angstroms =

metersNS 2412 10angstrom1010 =

meter

angstromsN

4

21222

10

110

=

Standard deviation forN steps

Average distancecovered after 1012 steps

The average position of the objects,

The square of the standard deviation of their positions,

Where :


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ASSIGNMENT (yellow pad, due Jan 19)

4-4 (Stowe, 1984) For air at room temperature, the probability of one molecule

being in an excited electronic state is about 10-10

(i.e.,p=1010

, q=1). In atypical room there are about 1028 air molecules. Using these numbers,calculate the following.

(a) The mean number of electronically excited air molecules in a room at any time.(b) The standard deviation about this value.

(c) The relative fluctuation about this value.

4-5 (Stowe, 1984) Consider identically prepared systems, each having 600 rolleddice. Suppose you are interested in the number of dice per system, n, with six

dots showing upward.(a) What is the average value of n?(b) What is the standard deviation, ?(c) In the distribution of probabilities over the various possible states,

2600 nnBAenP =What are the values of the coefficients A and B?

(d) If you rolled 600 dice, what is the probability of exactly 100 landing with 6 dots up?(e) What is the probability of exactly 96 landing with 6 dots up?


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4-9 (Stowe, 1984) An ammonia bottle is opened very briefly in the center of a largeroom releasing many ammonia molecules into the air. These ammoniamolecules go an average of so = 10 5 m in any direction between collisions with

air (and other ammonia) molecules, and collide on the average of 107

times persecond. After each collision they are equally likely to go in any direction.(a) What is the average displacement in one dimension (say the z-dimension) for any

one step? (Hint. sz=socos

in spherical coordinates. Taking the average over all

solid angles gives

(b) What is the square of the standard deviation for any one step? (Hint.

is obtained by squaring the answer to part (a), and is obtained using

(c) What is the average displacement in the z-direction of the escaped NH3molecules after 2 s?(d) What is the standard deviation about the value obtained in part (c)?(e) If you were standing 6 m from the bottle, how long would it take before more than

32% of the ammonia molecules were farther from the bottle than you?

ddss oz sincos41

ddss oz sincos41 22 222 zzz sss

2zs

Chapter 4, Stowe

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