Simon Newcomb (1881) noticed the wear on logarithmic tables was not uniform. Suggested that the a priori assumption of the most significant digit distribution was not uniform. Frank Benford (1938), physicist, tested the hypothesis over many datasets.
Benford's Law (in base 10)Benford's Law (in other bases)
Benford's original paper took data from many disparate sources
Benford's law applies not only to scale-invariant data, but also to numbers chosen from a variety of different sources.
As the number of variables increases, the density function approaches that of the above logarithmic distribution.
It was rigorously demonstrated that the "distribution of distributions" given by random samples taken from a variety of different distributions is, in fact, Benford's law.
Benford's Law is now considered admissible evidence
for fraudulent claims in forensic accounting.
Some numerical distributions follow Benford's law exactly,
such as , and the Fibonacci numbers.
The 54 million real constants in Plouffe's
Inverse Symbolic Calculator database follow Benford's law.
One expects that the average internal energy , of a system in contact with a thermal bath should increase with , the bath temperature.
Why might this be so? Consider
Astronomers have experience with such systems. If the system obeys the Virial theorem with potential energy that scales as with external edge pressure and volume
Isolated gravitational fields give , and
Assign a simple (Boltzmann) transition rate, but couple states and to different thermal baths, say and . For simplicity, disallow .
Using the Master equation, the average energy is
There is a range of values where
The intuitive explanation is the creation of the barrier
of the forbidden .
Suppose a particle moves stochastically but is driven (say by an electric field, with driving energy ) down two lanes. The particle has the following movement probabilities, where
In the uniform case, this gives a current density that is always positive
Now introduce a barrier
The conductivity for sufficiently large !
Biological membranes, different heat
coupling in the membrane and outside
Brownian noise with a driven diffusion
Negative mobility and sorting of colloidal particles
How can be computable if it has an infinite amount of "information"? Consider the following program
How do you compare sets?
We get that . How can we do that if we can't count them? We say that two sets have the same cardinality, , if we can find a bijection (i.e. a mapping from one to another)
This works for infinite sets as well (see Irr. Topics 3).
Think about vs .
The smallest infinity is , but
While there are just as many fractions as integers, there are "more" real numbers than fractions! (details: if the continuum hypothesis holds ).
Think of all possible programs one could write (say in python), let this set be . Some of these programs run and spit out an answer in a finite amount of time and halt, some of them run forever (some programs have syntax errors and never run, give them a run time of zero).
There is a real number, called Chitin's constant, that represents the average halting probability for all programs.
is not computable. This is the Church-Turing thesis, and is directly related to Gödel's incompleteness theorems.
Since the programs in can be enumerated,
we must have the case that
This means that all computable real numbers
are equal in size to the whole numbers.
Which means that a random point on the
real number line is not-computable.
This implies that (almost) every number we use in physics amounts
to an infinitesimal fraction of the real number line!
The question of physics and the real number line is intimately tied to the discreteness of Nature.
If one built a machine (more powerful than Turing) that could use all of the real numbers, one could solve all of the "hard" problems easily. It would imply that .