📝 Abstract

There are two fair ways to distribute particles in boxes. The first way is to divide the particles equally between the boxes. The second way, which is calculated here, is to score fairly the particles between the boxes. The obtained power law distribution function yields an uneven distribution of particles in boxes. It is shown that the obtained distribution fits well to sociological phenomena, such as the distribution of votes in polls and the distribution of wealth and Benford’s law.

💡 Analysis

There are two fair ways to distribute particles in boxes. The first way is to divide the particles equally between the boxes. The second way, which is calculated here, is to score fairly the particles between the boxes. The obtained power law distribution function yields an uneven distribution of particles in boxes. It is shown that the obtained distribution fits well to sociological phenomena, such as the distribution of votes in polls and the distribution of wealth and Benford’s law.

📄 Content

Sociological Inequality and the Second Law

Oded Kafri

Varicom Communications, Tel Aviv 68165 Israel.

Abstract

There are two fair ways to distribute particles in boxes. The first way is to divide the particles equally between the boxes. The second way, which is calculated here, is to score fairly the particles between the boxes. The obtained power law distribution function yields an uneven distribution of particles in boxes. It is shown that the obtained distribution fits well to sociological phenomena, such as the distribution of votes in polls and the distribution of wealth and Benford’s law.

1 It seems that nature dislike equality. In many cases distributions are uneven, a few have a lot and many have to be satisfied with little. This phenomenon was observed in many sociological systems and has many names. In economy it is called Pareto law [1,2], in Sociology it is called Zipf law [3,4] and in statistics it is called Benford law [5-7]. These distributions differ from the canonic (exponential) distribution by a relatively moderate decay (a power-law decay) of the probabilities of the extremes that enables a finite chance to become very rich. Here it is shown that the power law distributions are a result of standard probabilistic arguments that are needed to solve the statistical problem of how to distribute P particles in N boxes.
Intuitively one tends to conclude that P particle will be distributed evenly among N boxes, since the chance of any particle to be in any box is equal, namely, N 1 . However, this is an incorrect conclusion, because the odds that each box will score the same amount of particle are very small. Usually there are some lucky boxes and many more unlucky ones. The distribution function of particles in boxes should maximize the entropy. This is because in nature, fairness does not mean an equal number of particles to all boxes N, but an equal probability to all the microstates (configurations) . The equal probability of all the microstates is the second law of thermodynamics, which, exactly for this reason, causes heat to flow from a hot place to a cold place.
Ω Calculating the distribution of P particles in N boxes with an equal chance to any configuration is not simple, as the number of the configurations ) , ( N P Ω is a function of both P and N namely,
! )! 1 ( )! 1 ( ) , ( P N P N P N − − +

Ω .

(1)

2 The derivation of the distribution function to Eq.(1) is not new. Planck published it in 1901 in his famous paper in which he deduced that the energy in the radiation mode is quantized [8,9]. Here the Planck’s calculation is followed with the modifications needed to fit our, somewhat simpler, problem. Planck first expressed the entropy, namely ( is the Boltzmann constant), as a function of the number of modes N and the number of light quanta Ω

ln B k S B k P in a mode N P n = . Using Stirling formula, he obtained that } ln ) 1 ln( ) 1 {( n n n n N k S B − + +

. Then he used the Clausius inequality in equilibrium [10] to calculate the temperature T, from the expression, T q N T Q S δ δ δ

= , where Q is the energy of all the radiation modes and q is the energy of a single radiation mode. Therefore, the temperature is S q N T ∂ ∂

. Then, Planck made his assumption that ν nh q = , namely S n Nh T ∂ ∂

ν . Therefore, T h N n n N k n S B ν

= ∂ ∂ ) 1 ln( , this is the famous Planck equation, namely, the number of quanta in a radiation mode is, 1 1 −

T k h B e n ν . The calculation of Planck is comprised of three steps. First he expressed the entropy S by the average number of quanta n in a box and the number of boxes (radiation modes) N. Next, he used the Clausius equality to calculate the temperature. The equality sign in Clausius inequality expresses the assumption of equilibrium in which all the configurations have the same probability. Then Planck added a new law that was verified by the data of the blackbody radiation that the energy of the quant is proportional to the frequency. This law is responsible for the observation that in the higher frequencies n is lower.

3 In our problem we do not have energies or frequencies. We just have particles and boxes. Therefore, we will write the dimensionless entropy, namely the Shannon information as a function of and N, and obtain that n } ln ) 1 ln( ) 1 {( n n n n N I − + +

. Parallel to Planck, we calculate the dimensionless temperature Θ according to I n n N I P ∂ ∂

∂ ∂

Θ ) ( φ . Here we replace the total energy Q by P and q by ) (n nφ , where ) (n φ is a distribution function that tells us the number of boxes having n particles. ) (n φ is the analogue of Planck’s ν h . Changing the frequency enabled Planck to change the number of the particles in a mode at a constant temperature. Here we change the probability of a box with n particles at a constant temperature. The sociologic tempe

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