ITU Termodynamics Lab Report

1. PHYSICS ENGINEERING DEPARTMENT
FIZ341E - Statistical Physics and Thermodynamics
Laboratory
Name of Exp. :Binomial Distribution, Probability and
Entropy
Date of Exp. : 26.11.2014
BARIŞ ÇAKIR
090100235

2. Introduction
Theory of experiment can be explained under three main topic.
Binomial Distribution: It investigates systems whose elements can be
present in two states only and mainly it states that if the trial numbers increased,
system will approach to the equilibrium state. If trial numbers goes infinite both
probabilities of each state will be equal each other.
Micro –State; The state which contains microscopic parameters. For
example; velocities of particles in a gas molecule.
Macro –State; The state which contains macroscopic parameters. For
example; pressure of a gas molecule.
If we search relationship between macro and micro states, we can explain it by
an sample, assume we flip 5 coins and they came respectively; H,T,T,H,T in this
situation macro state is; 3 tails and 2 heads, but micro state also includes the
positions of heads and tails. So, we can write so many combinations whose
macro states equal but micro states are different (like, H,T,H,T,T or H,H,T,T,T).
Experimental Procedure
Tools and devices: Coins
Procedure: First, we flip 10 coins for 140 trials and we calculate number
of tails for each trial. Second, we make histograms (Fit table) for 10th
, 20th
,40th
,80th
,140th
trials. Finally we calculate standart deviations for each step.
In this experiment the likelihood of occurrence of various distributions in
a series of coin flips were observed and probability distribution and standard
deviation were calculated. Also we calculated standart deviations and
probability distributions for some steps with following formulas;
< n > = nP(n)
< n > = n P(n)
= (< > −< > )
Also relation between standart deviations should be like;
> > > >

3. Data Analysis
Calculated head and tail numbers for each step is given in table1 and
table2.
Number of
try Number of tail
Number
of try
Number
of tail
Number
of try
Number
of tail
Number
of try
Number
of tail
1 6 21 4 41 5 61 3
2 7 22 4 42 8 62 6
3 6 23 6 43 2 63 5
4 3 24 8 44 6 64 4
5 3 25 6 45 5 65 6
6 5 26 6 46 6 66 4
7 6 27 5 47 3 67 5
8 7 28 2 48 5 68 8
9 5 29 6 49 6 69 4
10 7 30 6 50 5 70 5
11 7 31 4 51 6 71 5
12 5 32 4 52 5 72 5
13 8 33 4 53 5 73 4
14 3 34 6 54 7 74 5
15 5 35 5 55 5 75 4
16 5 36 6 56 4 76 5
17 2 37 5 57 6 77 7
18 4 38 6 58 4 78 6
19 5 39 2 59 5 79 5
20 6 40 5 60 5 80 6
Table1. Measurements
Number of try Number of tail
Number of
try
Number of
tail
Number of
try
Number of
tail
81 4 101 6 121 3
82 5 102 7 122 3
83 2 103 7 123 4
84 6 104 8 124 4
85 6 105 5 125 3
86 4 106 7 126 3
87 5 107 3 127 7
88 6 108 4 128 5
89 6 109 4 129 6
90 6 110 6 130 6
91 7 111 4 131 4
92 5 112 2 132 4
93 4 113 6 133 6
94 6 114 3 134 4

4. 95 6 115 5 135 7
96 6 116 6 136 5
97 8 117 4 137 3
98 6 118 3 138 3
99 5 119 4 139 5
100 3 120 6 140 8
Table2. Measurements
Graphics of the histograms whose made after 10th
,20th
,40th
,80th
and
140th
steps.
Graphic1. Histograms
Probability distributions and standart deviations for each step;
For 10th
step;
< > =
6
10
+
10
10
+
18
10
+
21
10
= 5.5
< > =
18
10
+
50
10
+
108
10
+
147
10
= 32.3
= 1.431
For 20th
step;

5. < > =
9
20
+
4
20
+
30
20
+
24
20
+
35
20
= 5.1
< > =
27
20
+
4
20
+
150
20
+
144
20
+
245
20
= 28.5
= 1.428
For 40th
step;
< > =
4
40
+
9
40
+
24
40
+
50
40
+
72
40
+
35
40
+
8
40
= 5.05
< > =
8
40
+
27
40
+
96
40
+
250
40
+
432
40
+
245
40
+
64
40
= 28.05
= 1.596
For 80th
step;
< > =
6
80
+
15
80
+
52
80
+
135
80
+
126
80
+
49
80
+
24
80
= 5.0875
< > =
12
80
+
45
80
+
208
80
+
675
80
+
756
80
+
343
80
+
192
80
= 27.8875
= 1.416
For 140th
step;
< > =
10
140
+
36
140
+
104
140
+
180
140
+
228
140
+
91
140
+
48
140
= 4.9786
< > =
20
140
+
72
140
+
416
140
+
900
140
+
1368
140
+
637
140
+
384
140
= 27.12
= 1.527
Conclusion
Experiments main purpose was proved by expected n values, our
experimental n values were approached to 5 when number of trials increased.
Also, decrease at standart deviations when number of trials were increased,
however, this fact could not come true.
< < < <

6. Resources
- http://en.wikipedia.org/wiki/Microstate_%28statistical_mechanics%29
- Thermodynamics LAB FÖY
- http://en.wikipedia.org/wiki/Binomial_distribution
Answers
1- 10 coined system’s expected possibility density is 5, the closest entropy
value can be produced is 5.5294Kb and it can be calculated by 252 micro
states.
= ∗ ln( )
2- This system got 6 microstates, the probability of each microstate is
%16,66 and the entropy of the system can be calculated with the equation
above.
= ∗ 6 = 1.79