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# Probability theory. Probability Distributions Statistical Entropy

## 1.

MEPhI General PhysicsMolecular Physics. Statistics

Lecture 08

Probability theory.

Probability Distributions

Statistical Entropy

## 2.

Thermodynamics and Physical StatisticsThermodynamic Approach

- defines correlations between the

observed physical quantities

(macroscopic),

- relies mostly on the

experimentally established

dependencies

- allows generally to consider the

physical essence of the problem - does not require precise

information about the microscopic

structure of matter.

## 3.

Thermodynamics and Physical StatisticsStatistical Approach

Thermodynamic Approach

- defines correlations between the observed physical quantities

(macroscopic),

- relies mostly on the

experimentally established

dependencies

- allows generally to consider the

physical essence of the problem - does not require precise

information about the microscopic

structure of matter.

Based on certain models of the

micro-structure of matter

defines correlations between the

observed physical quantities,

starting from the laws of motion of

molecules using the methods of

probability theory and mathematical

statistics

allows to understand the random

fluctuations of macroscopic

parameters.

## 4.

Thermodynamics and Physical StatisticsStatistical Approach

Thermodynamic Approach

- defines correlations between the observed physical quantities

(macroscopic),

- relies mostly on the

experimentally established

dependencies

- allows generally to consider the

physical essence of the problem - does not require precise

information about the microscopic

structure of matter.

Based on certain models of the

micro-structure of matter

defines correlations between the

observed physical quantities,

starting from the laws of motion of

molecules using the methods of

probability theory and mathematical

statistics

allows to understand the random

fluctuations of macroscopic

parameters.

Statistical approach strongly complements thermodynamics. BUT! To

implement it effectively we need proper mathematical tools – first of all – the

probability theory.

This will be the focus of today’s lecture!…

## 5.

ProbabilitiesProbability = the quantitative measure of possibility for certain event to occur.

EXAMPLE 1. Eagle and Tails Game:

Test = throw of a coin;

Results of a test (events); eagle.

or tail

If the game is honest – possibilities to obtain each result must be equal.

Quantitative measure of possibility = probability: Pe=0,5 – eagle; Pt=0,5 - tail.

P = 1 is the probability of a “valid” event, which will occur for sure.

Pe + Pt=1 – either eagle or tail will fall out for sure (if rib is excluded ).

The rule of adding probabilities P (1 OR 2) = P(1) + P(2)

The normalization rule: the sum of probabilities for all possible results of a test

is evidently valid and thus is equal to 1.

## 6.

ProbabilitiesProbability = he quantitative measure of possibility for certain event to occur.

EXAMPLE 2. Dice game. Test = throw of a dice;

Results of a test (events); 1, 2, 3, 4, 5, 6

If the game is “honest”: P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6.

Example 2.1: Probability to obtain the odd number as the result of a test:

P(1) + P(3) + P(5) = 1/6 + 1/6 + 1/6 = 1/2

Example 2.2: Probability to obtain “double 6”: from the first throw P(6) = 1/6,

and then – out of this probability only 1/6 of chances that 6 will occur again.

Finally P( 6 AND 6) = P(6)P(6) = 1/6*1/6 = 1/36

The rule of multiplying probabilities P (1 AND 2) = P(1)P(2)

## 7.

Distribution of ProbabilitiesEXAMPLE 3. Rotating of a top toy.

If initially the red mark was exactly to the right from the

top toy’s axis (in direction X), after rotation stops - it

may stop at any angle .

The question: what is the probability that the toy wil

stop EXACTLY turned by the angle φ from the initial

orientation? The answer: P(φ) = 0 (!) – as absolutely

precise experiments are impossible in principle. There

always will be some error, or deviation, or uncertainty.

φ

X

The CORRECT question: what is the probability for the top toy to stop within

certain range of angles: between φ and (φ + dφ)?

Evidently, this probability MAY depend on φ and MUST be proportional to dφ:

dP(φ to φ + dφ) = f(φ)dφ

Here f(φ) is the probability distribution function.

## 8.

Distribution of ProbabilitiesSO: the PROBABILITY for the top toy to stop between φ

and (φ + dφ) equals: dP(φ to φ + dφ) = f(φ)dφ

f(φ) is the probability distribution function.

φ

The Normalization Rule: the integration of the

probability distribution function over the range of

all possible results MUST give 1

In our example: 2π∫0 f(φ)dφ = 1

The Even Distribution: if all possible results are

equally probable – the distribution function equals

constant.

In our example: f(φ) = Const = 1/ 2π

X

## 9.

Even Distribution on the PlaneEXAMPLE 4: a point is dropped (randomly) on the table

with the area S. What is the probability that it will fall

within the little square within dS?

f(x,y) = C: the even probability distribution function.

The Normalization Rule: ∫S Cdxdy = CS = 1 => C

= 1/S

The result: dP(dS) = f(x,y)dS = dS/S

S

dS

## 10.

Even Distribution in PhysicsEXAMPLE 5: Molecules in gas – at the state of

thermal equilibrium, when the concentration of

molecules is the same all throw the volume are

evenly distributed over all the coordinates:

f(x,y,z) = Const = 1/V =

dP(x,y,z) = f(x,y,z)dV = dxdydz/V

V

dV

## 11.

Probability Distribution and Average ValuesKnowing the distribution of a random x we may calculate its average value <x>:

x xdPx xf ( x)dx,

Moreover, we may calculate the average for any function ψ(x):

( x) ( x)dPx ( x) f ( x)dx

THE PROPERTIES OF AVERAGES.

• Average of the sum of two values equals to the sum of their averages

<x + y> = <x> + <y>

• Average of the product of two values equals to the product of their averages ONLY

in case if those two values DO NOT depend on each other

<xy> = <x><y> only if x and y are independent variables

## 12.

Probability Distribution and Average ValuesExamples: in case of even distribution of molecules

over certain spherical volume V (balloon with radius R):

Y

X

x xdPx xf ( x)dx, = 0

<x +y> = 0; <x y> = 0

<x2> = R2/5 >0

<r2> = < x2 +y2 +z2> = 3R2/5 > <r>2

<r> = <(x2 +y2 +z2)1/2> = 3R/4

Calculation – ‘live”…

V

Z

## 13.

Different Kinds of Averages<r> = <(x2 +y2 +z2)1/2> = 3R/4 - average

(<r2>)1/2 = 0,61/2R > <r> - squared average

<x> = 0; <x2>1/2 = R/51/2 >0

Y

Median average rmed ; the quantity of molecules

with r<rmed equals to the quantity of molecules Z

with r>rmed

rmed=R/21/3=0,7937R >

(<r2>)1/2 =0,7756R >

<r>

=0,75R

V

X

This all is about even distribution of molecules over space in spherical balloon.

What about the distribution of molecules over velocities and energies?

It can be spherically symmetric, but it can not be even as formally there is no

upper limit of velocity…

## 14.

Normal DistributionEagle and Tails game

## 15.

Eagles and Tails GameEXAMPLE 1: One throw of a coin = one Test (N = 1)

Test Results: type 1 – “Tail”;

If we do very many tests (throws)

Ni

Pi lim

N N

T

= 0,5

type 0 – “Eagle”

both for types 1 and 0.

NT – number of tests (throws),

Ni – number of tests with the result of the type i

Рi – probability to obtain the result of the type i

Ni

Pi lim

N N

## 16.

Eagles and Tails GameEXAMPLE 2: 1 Test (or one series of tests) = 2 throws of a coin (N = 2)

Test results (types):

0 –2 Eagles

1 –1 Eagle and 1 Tail

or

2 –2 Tails

Ni

Pi lim

N N

T

= 0,25 probability of types 0 and 2

= 0,5 – probability of type 1

N – the length of a test series (in this case N = 2) ,

NT – total number of tests

Ni – number of tests with result of the type i

Рi – probability to obtain the result of the type i

## 17.

SOME MATHEMATICSThe product of probabilities:

The first test – the probability of result “1” - P1

The second test – the probability of result “2” = P2.

The probability to obtain the result ‘1’ at first attempt and then to obtain result

‘2’ at second attempt equals to the product of probabilities

P(1 AND 2) = P(1 ∩ 2) = P1P2

EXAMPLE: Eagle AND Tail = ½ * ½ = ¼

The SUM of probabilities:

One test – the probability of result “1” - P1; the probability of result “2” = P2.

The probability to obtain the result ‘1’ OR ‘2’ equals to the SUM of

probabilities: P(1TOR 2) = P(1 U 2) = P1 + P2

EXAMPLE: Eagle OR Tail = ½ + ½ = 1

EXAMPLE 2: (Eagle AND Tail) OR (Tail AND Eagle) = ¼ + ¼ = ½

## 18.

Eagles and Tails GameGeneral Case: The test length = N (N throws = 1 test)

All the types of test results can be numbered i from 0 to N by the number

of the tailspins in each test. Each type of results will have the probability,

proportional to the number of variants, that produce that result

Type

Number of variants producing this type of result

i=0

1 (all the throws = eaglespins)

i=1

N (only one there was a tailspin (either at first throw, or

at the 2nd, or at the 3rd, … or at the last)

i=2

N(N -1)/2

i=3

N(N-1)(N-2)/6

…..

i=k

N(N-1)(N-2)….(N-k+1)/k! = N!/k!(N-k)!

….

i = N-1

N

i=N

1 (all the throws = tailspins)

## 19.

Eagles and Tails GameThe test series has the length N (N throws = attempts)

Number of variants to obtain the result of the type k (with k tailspins):

Ωk= N!/k!(N-k)!

ΣΩk= 2N

k

Probability to obtain the result of the type k:

250

Pk= N!/k!(N-k)!2N

200

If N>>1 we may apply the Sterling's approximation:

ln(n!) ~= n ln(n/e)

Pk= (2/πN)1/2exp(-2(k-N/2)2/N)

Для N = 10

150

100

here n = (k - N/2) – is the deviation of the 50

obtained result from the average N/2. Probabilities

are noticeable when

n<~N1/2 << N

k

0 2 4 6 8 10

## 20.

SOME MATHEMATICS: the probability distributionSmooth CHART – distribution function

f(x)

BAR CHART

Nx

Na

Area =dPx

ΔP

area =

x

0

X

x x+a

dPx f ( x)dx

N

P

i

i

N

i

1,

0

x x+dx

X

dN x

Px dPx

f ( x) lim

lim

N Ndx

x x

dx

dP f ( x)dx 1

x

## 21.

SOME MATHEMATICSN – number pf tests,

Ni – number of results of the type i

Рi – probability of the result of the type i

Ni

Pi lim

N N

For continuously distributed quantities X:

differential probability to find the random value X within the range from X to X

+ dX

dP(x) = f(x)dx

f(x) – is the probability distribution function

Probability to find the random value X within

x2 the range from x1 to x2 :

P( x1 x x2 )

f ( x)dx

For continuously distributed quantities thex1probability to have exactly some value

x0, is equal to zero P(x = x0)=0.

## 22.

Normal distributionPk=(2/πN)1/2exp(-2(k-N/2)2/N) =>

The normal (or Gauss) distribution –

is the smooth approximation of the

Newton’s binomial formula

k

250

Parameters of Gauss distribution:

x – some random value

μ — the most probable (or

expected) value of the random (the

maximum of the distribution function)

σ — dispersion of the random value.

In case of the Eagles and Tails game: μ

200

150

100

50

k

0 2 4 6 8 10

= N/2, σ = (N/2)1/2 << N

## 23.

Gauss DistributionThe normal distribution is very often

found in nature.

Examples:

• Eagle and Tails game

• Target striking

• the deviations of experimental

results from the average (the

dispersion of results = the

experimental error)

## 24.

MEPhI General PhysicsNormal (Gauss) Distribution

and Entropy

## 25.

Normal Distribution.EXAMPLE: Eagles and Tails Game

Number of variants leasing to the result k (if N =10):

Ωk= N!/k!(N-k)!

Type

k=0

k=1

ΣΩk= 2N

Realizations (variants)

0000000000

1000000000 0100000000

0010000000 … 0000000001

…

k=5

0100110011 1101000101 … - total 252 variants.

If N>>>1 – figures may be exponentially high

k

250

200

N = 10

150

100

1. The more realizations are possible – the

higher is the probability of the result.

2. The more realizations are possible – the

less looks the “degree of order” in this result

50

k

0 2 4 6 8 10

## 26.

The Entropy of ProbabilityThe more realizations are possible – the more is the probability of it and the

less is the “degree of order” in this result. For long series of tests (N>>>1)

numbers of variants of realizations may be exponentially high and it may be

reasonable to apply logarithmic functions)

The definition of the entropy of probability: S(k)

= ln(Ωk) -

If N = 10

S(0) = S(10) = ln(1) = 0

S(5) = ln(252) = ~5,6

N = 10

9

8

7

6

S(k) = Aln(Pk) = Aln(Ωk) - ANln2

The higher is the “order” – the lower is entropy

The higher is probability – the higher is entropy

P(k и i) = PkPi => S(k и i) = S(k)+S(i)

Entropy is the additive function!

5

4

3

2

1

0

## 27.

Entropy in InformaticsType of result

k=0

k=1

k=5

Realizations

0000000000

1000000000 0100000000 ...

0100110011 1101000101 …

Any information or communication can be coded as a string of zeroes and

units: 0110010101110010010101111110001010111…. (binary code)

To any binary code with length N (consisting of N digits, k of which are

units and (N-k) – are zeroes) we may assign a value of entropy:

S(N,k) = ln(ΩN,k), where ΩN,k – is the number of variants how we may

compose a string out of k units and (N-k) zeroes

• Communications, looking like 00000000.. 111111111… S = 0

• Communications with equal number of units and zeroes have maximal

entropy .

• Entropy of 2 communications equals to the sum of their entropies.

## 28.

Entropy in InformaticsAny information or communication can be coded as a string of zeroes and

units: 0110010101110010010101111110001010111…. (binary code)

The entropy of information or communication, containing k units and (N-k)

zeroes is defined like: S(N,k) = ln(ΩN,k), where ΩN,k – is the number of

variants how we may compose a string out of k units and (N-k) zeroes

• Communications, looking like 00000000..

111111111… S = 0 … the

informational value of such ‘communication’ is also close to zero

• Communications with equal number of units and zeroes have maximal

entropy . Most probably they are the sets of randomly distributed symbols,

thus also having practically no informational value

• Entropy of 2 communications equals to the sum of their entropies.

Real informative communications as a rule

• do include fragments with noticeable predominance of either zeroes or

units, and

• Have the entropy noticeably different from both maximum and minimum

## 29.

Entropy in InformaticsThe deffinition of entropy as the measure of

disorder (or the measure of informational

value) in informatics was first introduced by

Claude Shannon in his article «A

Mathematical Theory of Communication»,

published in Bell System Technical Journal

in 1948

Those ideas still serve as the base for the

theory of connunications, tmethods on

encoding and decoding, are used in

linguistics etc…

Claude Elwood Shannon

1916 - 2001

## 30.

MEPhI General PhysicsStatistical Entropy in Physics

## 31.

Distribution of Molecules over possible“micro-states”

Imagine that we have К possible “states” and we have N “molecules” that

can be somehow distributed over those “states”

The row of numbers (n1, n2, …nK) = n(i) forms what is called the «macrostate» of the system. Each macro-state can be realized by the tremendously

great number of variants Ω(n(i)). The number of variants of realization is

called the “statistic weight” of the macro-state.

The more is the “statistic weight” – the greater is the probability to realize

this state.

Even distribution of molecules over space has the biggest statistical weight

and thus id usually associated with thermo-dynamical equilibrium.

## 32.

Statistical Entropy in Physics.Statistical Entropy in Molecular Physics: the logarithm of the number of

possible micro-realizations of a state with certain macro-parameters,

multiplied by the Boltzmann constant.

S k ln J/K

1.

2.

3.

4.

5.

In the state of thermodynamic equilibrium, the entropy of a closed

system has the maximum possible value (for a given energy).

If the system (with the help of external influence)) is derived from the

equilibrium state - its entropy can become smaller. BUT…

If a nonequilibrium system is left to itself - it relaxes into an equilibrium

state and its entropy increases

The entropy of an isolated system for any processes does not

decrease, i.e. ΔS > 0 , as the spontaneous transitions from more

probable (less ordered) to less probable (more ordered) states in

molecular systems have negligibly low probability

The entropy is the measure of disorder in molecular systems.

## 33.

Statistical Entropy in Physics.For the state of the molecular system with certain macroscopic parameters

we may introduce the definition of Statistical Entropy as the logarithm of the

number of possible micro-realizations (the statistical weight of a state Ώ) of a this state, multiplied by the Boltzmann constant.

S k ln J/К

Entropy is the additive quantity.

2

1

i

3

p p1 p2 ... pN

pi

i

1 2 ... N

N

S k ln k ln 1 ln 2 ... ln N

S Si

i 1

## 34.

Statistical Entropy and the Entropy of Ideal GasNot a strict proof, but plausible considerations.

• the number of variants of realization of a state (the statistical weight of a

state) shall be higher, if the so called phase volume, available for each

molecule (atom), is higher: Phase volume Ω1 ~Vp3~VE3/2 ~VT3/2

• the phase volume for N molecules shall be raised to the power N:

Ω ~ VNT3N/2.

For multy-atomic molecules, taking into account the possibilities of rotations

and oscillations, we shall substitute 3 by i : Ω

~ VNT iN/2

• As molecules are completely identical, their permutations do not change

neither the macro-state, nor the micro-states of the system. Thus we have

to reduce the statistical weight of the state by the factor ~ N! (the number

of permutations for N molecules)

Ω~ VNTiN/2 /N!; S=k ln Ω =kNln(VTi/2/NC)=

= v(Rln(V/v) + cVlnT +s0)

## 35.

Statistical Entropy and the Entropy of Ideal GasΩ~ VNTiN/2 /N!; S=k ln Ω =kNln(VTi/2/NC)=

= v(Rln(V/v) + cVlnT +s0)

The statistical entropy proves to be the same physical quantity, as was

earlier defined in thermodynamics without even referring to the molecular

structure of matter and heat!

## 36.

2nd Law of Thermodynamics and the ‘Time arrow’“The increase of disorder, or the

increase of the Entropy, of the

Universe over time is one of the possibilities to define the

so-called “Time arrow”, that is the

direction of time, or the ability to

distinguish the past from the future,

Stephen Hawking

(1942-2018)

## 37.

MEPhI General PhysicsThe Distributions of Molecules

over Velocities and Energies

Maxwell and Boltzmann Distributions

That will be the Focus of the next lecture!