Why the first digit of the values of
(Jeffreys) physical quantities
tends to be 1 or 2 ?
Albert Tarantola
Preamble
Needs@"Histograms`"D
SeedRandom@0D
First run with 64 points (to checks if everything is OK)
Let us select 64 as the number of values, to start with.
n = 64;
We generate 64 values with constant probability density in the range (-100,+100) :
x = Table@Random@Real, 8-100, +100<D, 8n<D
8-93.6293, 28.0416, 44.8605, 16.1768, 86.2303, 27.3468, 85.6849, -90.212,
40.4607, -14.9532, 38.7222, -96.2569, 48.3508, 60.7061, -37.4265, 94.712,
93.1128, 93.9266, 59.0126, 86.9209, 35.4998, 10.3493, -94.8879, 61.389,
29.1291, 82.3077, -39.7484, -54.7878, 42.8988, -45.0391, -25.4333, -64.5758,
-97.5619, 69.914, 35.8446, -68.3189, -45.9128, -90.7921, -26.729, -63.0309,
-39.0256, -84.7187, 14.2584, -49.9518, 25.4746, 4.93198, 9.14623, -11.3408,
96.3455, 22.6243, -51.1054, -56.5529, -46.5533, -32.3366, 74.3279, -91.9771,
-48.9913, -2.25059, -61.5167, 76.3418, 96.9215, -11.4585, 65.2123, 39.3727<
2
BenfordLaw.nb
The histogram of the 64 values is totally unremarkable:
Histogram@x, HistogramCategories Ø 4D
15
10
5
-50
0
50
100
We now take the exponential of these values (remark the first digit is more frequently
an "one" or a "two" than a "eight" or a "nine"):
X = Exp@xD
92.17432 µ 10-41 , 1.50775 µ 1012 , 3.03857 µ 1019 , 1.0605 µ 107 , 2.8141 µ 1037 , 7.52621 µ 1011 ,
1.63116 µ 1037 , 6.62843 µ 10-40 , 3.73141 µ 1017 , 3.20563 µ 10-7 , 6.5587 µ 1016 ,
1.57096 µ 10-42 , 9.96534 µ 1020 , 2.31391 µ 1026 , 5.57056 µ 10-17 , 1.35796 µ 1041 ,
2.74402 µ 1040 , 6.19154 µ 1040 , 4.25461 µ 1025 , 5.61376 µ 1037 , 2.61438 µ 1015 ,
31 235.8, 6.1763 µ 10-42 , 4.5804 µ 1026 , 4.47298 µ 1012 , 5.56892 µ 1035 , 5.46396 µ 10-18 ,
1.60674 µ 10-24 , 4.27272 µ 1018 , 2.75262 µ 10-20 , 9.00452 µ 10-12 , 9.0175 µ 10-29 ,
4.25974 µ 10-43 , 2.30824 µ 1030 , 3.69055 µ 1015 , 2.13536 µ 10-30 , 1.14905 µ 10-20 ,
3.71096 µ 10-40 , 2.46459 µ 10-12 , 4.22701 µ 10-28 , 1.12564 µ 10-17 , 1.61115 µ 10-37 ,
1.55716 µ 106 , 2.02408 µ 10-22 , 1.15739 µ 1011 , 138.654, 9378.99, 0.0000118788,
6.95556 µ 1041 , 6.69281 µ 109 , 6.38557 µ 10-23 , 2.75029 µ 10-25 , 6.05591 µ 10-21 ,
9.04512 µ 10-15 , 1.90628 µ 1032 , 1.1346 µ 10-40 , 5.28868 µ 10-22 , 0.105337, 1.92153 µ 10-27 ,
1.4283 µ 1033 , 1.23725 µ 1042 , 0.0000105596, 2.09581 µ 1028 , 1.25701 µ 1017 =
BenfordLaw.nb
To isolate the first digit, we evaluate the mantissa of the numbers
XX = MantissaExponent@XD
Y = Table@XX@@i, 1DD, 8i, 1, n<D
880.217432, -40<, 80.150775, 13<, 80.303857, 20<, 80.10605, 8<, 80.28141, 38<,
80.752621, 12<, 80.163116, 38<, 80.662843, -39<, 80.373141, 18<, 80.320563, -6<,
80.65587, 17<, 80.157096, -41<, 80.996534, 21<, 80.231391, 27<, 80.557056, -16<,
80.135796, 42<, 80.274402, 41<, 80.619154, 41<, 80.425461, 26<, 80.561376, 38<,
80.261438, 16<, 80.312358, 5<, 80.61763, -41<, 80.45804, 27<, 80.447298, 13<,
80.556892, 36<, 80.546396, -17<, 80.160674, -23<, 80.427272, 19<, 80.275262, -19<,
80.900452, -11<, 80.90175, -28<, 80.425974, -42<, 80.230824, 31<, 80.369055, 16<,
80.213536, -29<, 80.114905, -19<, 80.371096, -39<, 80.246459, -11<,
80.422701, -27<, 80.112564, -16<, 80.161115, -36<, 80.155716, 7<, 80.202408, -21<,
80.115739, 12<, 80.138654, 3<, 80.937899, 4<, 80.118788, -4<, 80.695556, 42<,
80.669281, 10<, 80.638557, -22<, 80.275029, -24<, 80.605591, -20<, 80.904512, -14<,
80.190628, 33<, 80.11346, -39<, 80.528868, -21<, 80.105337, 0<, 80.192153, -26<,
80.14283, 34<, 80.123725, 43<, 80.105596, -4<, 80.209581, 29<, 80.125701, 18<<
80.217432,
0.373141,
0.274402,
0.447298,
0.425974,
0.112564,
0.695556,
0.528868,
0.150775,
0.320563,
0.619154,
0.556892,
0.230824,
0.161115,
0.669281,
0.105337,
0.303857, 0.10605, 0.28141, 0.752621, 0.163116, 0.662843,
0.65587, 0.157096, 0.996534, 0.231391, 0.557056, 0.135796,
0.425461, 0.561376, 0.261438, 0.312358, 0.61763, 0.45804,
0.546396, 0.160674, 0.427272, 0.275262, 0.900452, 0.90175,
0.369055, 0.213536, 0.114905, 0.371096, 0.246459, 0.422701,
0.155716, 0.202408, 0.115739, 0.138654, 0.937899, 0.118788,
0.638557, 0.275029, 0.605591, 0.904512, 0.190628, 0.11346,
0.192153, 0.14283, 0.123725, 0.105596, 0.209581, 0.125701<
We multiply ther mantissa by 10 and take the integer part:
YY = 10 Y
Z = IntegerPart@YYD
82.17432, 1.50775, 3.03857, 1.0605, 2.8141, 7.52621, 1.63116, 6.62843, 3.73141, 3.20563,
6.5587, 1.57096, 9.96534, 2.31391, 5.57056, 1.35796, 2.74402, 6.19154, 4.25461,
5.61376, 2.61438, 3.12358, 6.1763, 4.5804, 4.47298, 5.56892, 5.46396, 1.60674,
4.27272, 2.75262, 9.00452, 9.0175, 4.25974, 2.30824, 3.69055, 2.13536, 1.14905,
3.71096, 2.46459, 4.22701, 1.12564, 1.61115, 1.55716, 2.02408, 1.15739, 1.38654,
9.37899, 1.18788, 6.95556, 6.69281, 6.38557, 2.75029, 6.05591, 9.04512, 1.90628,
1.1346, 5.28868, 1.05337, 1.92153, 1.4283, 1.23725, 1.05596, 2.09581, 1.25701<
82, 1, 3, 1, 2, 7, 1, 6, 3, 3, 6, 1, 9, 2, 5, 1, 2, 6, 4, 5, 2, 3, 6, 4, 4, 5, 5, 1, 4, 2, 9, 9,
4, 2, 3, 2, 1, 3, 2, 4, 1, 1, 1, 2, 1, 1, 9, 1, 6, 6, 6, 2, 6, 9, 1, 1, 5, 1, 1, 1, 1, 1, 2, 1<
3
4
BenfordLaw.nb
And we can now make the histogram of the first digit of the quantities X = Exp[x] :
Histogram@Z, HistogramCategories Ø 80.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, 7.5, 8.5, 9.5<D
20
15
10
5
2
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Second run with 24000 points
n = 24 000;
x = Table@Random@Real, 8-100, +100<D, 8n<D;
Histogram@x, HistogramCategories Ø 20D
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1000
800
600
400
200
-50
0
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100
BenfordLaw.nb
X = Exp@xD;
XX = MantissaExponent@XD;
Y = Table@XX@@i, 1DD, 8i, 1, n<D;
YY = 10 Y;
Z = IntegerPart@YYD;
H1 =
Histogram@Z, HistogramCategories Ø 80.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, 7.5, 8.5, 9.5<D
7000
6000
5000
4000
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1000
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8
This is the Benford law! Here we have built it mathematically. But we find it everywhere
in Nature -> Nature works somehow this way.
In fact, let us compare this experimental histogram with the Benford law. The Benford
law is:
Benford@k_D = Log@10, Hk + 1L ê kD;
Its plot is:
H2 = ListPlot@n Table@Benford@kD, 8k, 1, 9<D, PlotRange Ø 80, 7300<D
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BenfordLaw.nb
We can plot together the experimental histogram and the Benford law (an almost
perfect agreement):
Show@8H1, H2<D
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