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Agenda: Information, Probability, Statistical Entropy
• Information and probability
simple combinatorics, random variables.
Probability distributions, joint probabilities,
correlations
Statistical entropy
Monte Carlo simulations
• Partition of probability
• Phase space evolution (Eta Theorem)
• Partition functions for different degrees of freedom
• Gibbs stability criteria, equilibrium
Reading
Assignments
Weeks 3
&4
LN III:
Kondepudi Ch. 20
Additional Material
McQuarrie & Simon
Ch. 3 & 4
Math Chapters
MC B, C, D,
W. Udo Schröder 2021 Information&Probability 1
Information&Probability 2 W. Udo Schröder 2021
Statistics and Probability
Particles in a multi-particle systems move
often in random directions and with
quasi-random velocities, a result of
multiple scattering, see example of
molecular scattering off a crystal lattice.
Therefore, one can use statistical-
probabilistic arguments in discussing the
average properties of such systems and
perhaps fluctuations about that average.
Assumption of molecular chaos: All microscopic states that are energetically
available to a system are occupied by the system with equal probability (Ergodic
System) → same macro-state.
Aspects of Information Theory
In the absence of information, probability replaces certainty.
Information theory provides probability as an objective link be-tween randomness and certainty.
Important theorist Claude Shannon, “Father of Information Theory”
W. Udo Schröder 2021 Information&Probability 3
Illustration:
Two possible 2-dim micro-states for
a system of 100 particles distributed
differently over the available phase
space.
Random positions = first guess if nothing known about particles
“If a situation (event) is very likely to occur (high probability) →
information provided with its occurrence is low.” And vice versa.
Particles cluster ina corner → deducemutual attraction→ significant info
Claude Shannon 1948
Information&Probability 4 W. Udo Schröder 2021
Simple Probability Concepts
Probability: understood as being relative to set of many uncertain events, experiments, measurements, outcomes, …If instances of a type of event xn occur in random selection, without hidden preference (bias), one can estimate
Probability → combinatoricsConduct experiment/measurement →
probability for event type x
( )k kN frequency o nx f eve ts x=
( )
( )k
N
k
n
nP
xx = Lim
N( )
N x→
Probability a posteriori
Example: experiment (unbiased) rolling 1 dice 1000x →
Expect (a priori) face with “6” P(6) = 1/6 = 0.1667 → N(6) = 166 (167)
observe (a posteriori) face with “6” : N(6) = 173 → P(6) = 0.173 ≈ 1/6
Information&Probability 5 W. Udo Schröder 2021
Probability Distributions
Conduct many (M=5) measurements of 1000 rolls of one die → M-5 different
probabilities for event type x =“6”→ a posteriori probability
( )k kN frequency o nx f eve ts x=
( )
( )1 6
; 1,..6
6 ,i
iN
ik
NP( ) = Lim i M
kN→
=
Mean/average probability
{N(6)}= {175, 160, 155, 167, 182}
0.1
0.2
0
6i
P ( )
Measurement # i
( ) ( ) ( )1 1
16 : 6 6
M M
i i iii i
P w P PM= =
= =
( )
1
6 0.168
i
i
Measurements of equal quality
Equal weights w M
P
→ =
→ =
Information&Probability 6 W. Udo Schröder 2021
Probability Distributions
Conduct many (M=5) measurements of 1000 rolls of one die → M-5 different
probabilities for event type x =“6”→ a posteriori probability
( )k kN frequency o nx f eve ts x=
( )
( )1 6
; 1,..6
6 ,i
iN
ik
NP( ) = Lim i M
kN→
=
Variance/Standard Deviation
{N(6)}= {175, 160, 155, 167, 182}
0.1
0.2
0
6i
P ( )
Measurement # i
( ) ( )( )
( ) ( )
22
1
2 5
16 : 6
( 1)
6 2.4 10 6 0.005
M
P ii
P P
P PM M
=
−
= −−
= → =
( ) ( )6 0.168 0.005i
P =
( ): 6 1 6 0.167a prio Pri = =
Information&Probability 7 W. Udo Schröder 2021
Probability Combinations
Additive, inclusive or exclusive, cumulative,
multiplicative, conditional probabilities.
Sum & product rules for disjoint (independent)
probabilities
Outcome of one trial (→Event E1) has no effect
on the result of the next trial (→Event E2). The
corresponding probabilities are independent of
one another and add
A priori probability to get a face “6” in either of two trials, the first or the
second throw of a dice, equals the sum of both
1 2 1 2
1 1 1P (6) P (6) P (6)
6 6 3 = + = + =
Simultaneous (joint) a priori probability to get a face “6” in both trials,
the first and the second throw of a dice, equals the product of both
1 2 1 2
1 1 1P (6 | 6) P (6) P (6)
6 6 36 = = =
OR inclusive
AND
Properties of a priori Probabilities
• The probability for an event is 0 ≤ P ≤ 1
• The probability for any of the possible outcomes to occur is
• The probability of an impossible event is zero, P = 0.
• If two events (E1 and E2) are independent (disjoint, mutually
exclusive), the probability of the sum (“or”) event is the sum of the
probabilities,
• If two events (E1 and E2) are independent (disjoint, mutually exclusive),
the probability for the simultaneous event is the product of the
probabilities,
• If two events are not mutually exclusive,
• Additional considerations for conditional (marginal) probabilities,
example
W. Udo Schröder 2021 Information&Probability 8
1 2 1 2P P P = +
1 2 1 2P P P =
1 2 1 2 1 2P P P P = + −
1 2 1 2P E | E : Probability E , given E=
1ii
P P= =
Conditional Probabilities
Constraints on a set of events → conditional probability P{E|condition}.
Example drawing from 3 balls (R, G, G) hidden in a box. What are a priori probabilities for a sequence 1. draw, 2. draw,…. e.g. G2,R1…
Depends on 1. draw returned or not! If returned, uncorrelated draws →
W. Udo Schröder 2021 Information&Probability 9
( ) ( ) ( )2 1 2 1
2 21P G R P G P R =
3 3 9= =
If ball is not returned, correlated=conditional draws →
( ) ( )122 111
1
3
1|
3P G R RG PP R→ = =
Given that Red was drawn first
3. draw ( ) ( )1 13 2 2 13 2|
1 1P G G R P G R =
3 3G G 1RP= =
Statistical Event Domains
Possible relations are illustrated between hypothetical domains of probable events
W. Udo Schröder 2021 Information&Probability 10
and 1 2 1 2P E , P E , P E E
Independent events E1 and E2 or if E1 E2 → E2 has no influence on the
probability for E1 → Conditional
( ) 121 2E giveP n E TE | P E= =
If two events are mutually exclusive,
1 2 1 2P E E 0 P E | E = =
( ) ( )2 1
1 2 12
P E | EP E | E P E
P E=
( )1Prior probability P E suspicion, guess=
( )2 1 2
Fractional probability for
E T(if E T ) / Total P E= =
2E
1 2P E | E 0
( )
( )
1 2
1 2 1 2 2
2 1 1
If E ,E not disjo int :
P E E P E | E P E
P E | E P E
=
=
Discrete Multivariate Probability Distributions
Experiment: Draw random dies out of bag with colored dies (i=1-5). Roll each dies a large number of times. Record frequencies of faces (j=1,….,6) Normalize to total number of rolls. → 5x6 dimensional total probability domain {pij}. Independent constraints:
W. Udo Schröder 2021 Information&Probability 11
After Dill & Bromberg
6 5 5 6
1 1 1 1
1 1Facei ij i j ij jj i j j
C o u pol r u p = = = =
= → = = → = Check!
Colori
Face j
24p
Discrete Multivariate Probabilities
Compare a priori with posteriori probabilities to find bias. Example: blue die with face “4” from overlap (simultaneous) probability domains.
W. Udo Schröder 2021 Information&Probability 12
Data after Dill & Bromberg
2 4
: ( , #) 1 5; ( ,4) 1 6 ( ) (4) 1 30 0.033
( ) (4 9: ) 0.3 0.162 0.04
A P blue any P ai n
l
prior
A p
y P blue P
P b ue Ps uo teriori
= = → = =
= = =
20.049 0.033 1.5 0.2But u =
Continuous Probability Distributions
( ),P x y
xy
( ), ,... , ,...JDegrees of freedom x y Probabit lity P x yoin→
( ),y,...
)
1
(
dx dy P x
dx
N
Partia
l
l or c
b
onditional probability
P
ormalized Pro abi ity
x
=
( )
( ) ( )
,y,... 1
( ) ,y,... 1
dy P
P dx dy x
x
x a xP a
= −=
( )
0
22
( , ,. .....)P
xP
Average x P x y dx
and
x dy
x x
=
= −
( )2
22
1( ) exp
22 xx
Px
xx
− = −
Define normalized
Gaussian probability
W. Udo Schröder 2021 Information&Probability 13
Continuous Probability Distributions
( ),P x y
xy
( )
( )
( ) ( )
( ) , ,... 1
( ) , ,...
1
1
,y,...
Partial or conditional p
d
N b
y
ormalized Prob
P
a
y
il
robability
P dx P x
P dx y P x
dx dy x
y
ity
b
y
y b
=
=
=
−
( )
0
22
( , ,...) ...P
yP
Average y P x y d
y
x
an
y
yd
=
= −
( )2
22
1( ) exp
22 yy
Py
yy
− = −
Define normalized
Gaussian probability
W. Udo Schröder 2021 Information&Probability 14
( ), ,... , ,...JDegrees of freedom x y Probabit lity P x yoin→
W. Udo Schröder 2021 Information&Probability 15
Correlations in Joint Distributions
For independent events (d.o.f) probabilities multiply. Correlations within distributions. Example: y increases with increasing x
( ) ( )2 2
2 22 2
( , ) ( ) ( )
1exp
2 24
= =
− − = − +
unc
x yx
P x y P x P y
x x y y
( ) ( ) ( )( )( )
2 2 2
2 2 2
2 2 2
1( , )
2
2exp
2
= −
− + − − − − −
−
corr
x y xy
x y xy
x y xy
P x y
x x y y x x y y
( ) ( )
( )
22 2 2 2 21
cot 42
; 1 1
x y x y xy
xy
xy xy x y xycorrelation coeffi rcie rnt
= − + − +
= −
( ) ( ): ( , ) −= − xy x x x dxdyCovarian yce Py y
x
y
Uncorrelated Punc(x,y)
x
yCorrelated Pcorr(x,y)
Probability Generating Functions
( )( )
P
Characteristic functions of P x
sam
Laplace tran
in
sforma
e
tio
o
n of
f
( ) ( )
( ) ( )
0
0
( ) :
:n n
s x
n n
n n s x
D
s
d ds dx e P
P
erivatives of
xds d
d e
s
x x x
−
−
=
= −
W. Udo Schröder 2021 Information&Probability 16
( ),P x y
xy
( ): 1Example dimensional system P x−
( ) ( ): s xs dx e P x− =
( ) ( )0
1n
nn
n
s
dx s
ds=
= −
( ) ( )0
:
s
ds dx
l
x P
simi ar
x xds =
= =
→
-(Set s=0)
Properties of a priori Probabilities
• The probability for an event is 0 ≤ P ≤ 1
• The probability for any of the possible outcomes to occur is
• The probability of an impossible event is zero, P=0.
• If two events (E) 1 and 2 are independent (disjoint, mutually
exclusive), the probabilities of the sum (“or”) event is the sum of the
probabilities,
• If two events 1 and 2 are independent (disjoint, mutually exclusive), the
probability for the simultaneous event is the product of the
probabilities,
• If two events are not mutually exclusive,
• Additional considerations for conditional (marginal) probabilities,
example
W. Udo Schröder 2021 Information&Probability 17
1 2 1 2P P P = +
1 2 1 2P P P =
1 2 1 2 1 2P P P P = + −
1 2 1 2P E | E : Probability E , given E=
1ii
P P= =