Statistical Sociology
by Dr. Willy H. Gerber



Abstract

Based on the mathematical methods used in the description of multiple particle system in Physics, I develop a model that descript how a society will distribute resources (Lorenz Curve, Gini factor), create unemployment and low level of resources can trigger and sustained mass movements.

Because that particular method is called in Physics "Statistical Mechanics" I name this approach "Statistical Sociology".



1. Working hypothesis

For building the model, I state two basic hypothesis on which the model structure is based.

The first one has to do with the parameters we need to describe the state of one individual in particular and society as a whole.

Hypothesis 1: The state of a person can be describe by a scalar measurable quantity "ω" we will call "well-being".

Each person in our society has a particular level of well-being. To this quantity contributes not only material things like the earning and savings of the person, also the social acceptance and other factors necessary to succeed in society.

One of the key results of this model is to estimate how this individual "well-being" is distribute among the different people in this society. The quantity could be concentrated only in few or equally around the system. Each situation is a possible state of our society. The second hypothesis will state that we are not defining a "a priory" distribution:

Hypothesis 2: Each possible state of the society is equally alike.

Starting with this basic hypothesis and few simple assumptions we hope to describe the behaviour of a society.


2. Counting States

Suppose that our society has a limited amount of well-being W to be distribute among there members. In case of the material needs, it could be in a first approximation the Gross Domestic Product (GDP). The condition of a fix maximal Well-being for the society implies that the sum of all individual well-being wi of each person i must be:

(2.1)

The question now is how probable it is, that a single person has a well-being level between ω and ω+dω. By definition of probability, this will be the fraction of cases where the person has the level w to the total number of situations we can find our society. If Ω( ω) is the number of states of the society where the person has the well-being level w and Ω is the total number of states the society can be in, the probability will read:

(2.2)

To calculate the number of states Ω, lets assume that the total amount of well-being W can be split in

(2.3)

little segments of length Δ. We have now to distribute the N persons in this different cells assuring that at all time the condition (2.1) is fulfil. Lets start with the simple case we have only one person. In this case there are only 1 alternative because the person has a well-being level equal to the total amount of well-being in the system.

(2.4)

In case we have two persons one can be in any of the M cells. The second can only be in the cell with the level of well-being that satisfies the condition (2.2) and there fore the number of states is

(2.5)

Lets now consider the case with three persons. Lets suppose that the third person has a well-being of ω. The other two persons can be in

(2.6)

possible states with a total well-being of the remaining W-ω. Since the third person can occupy any of the M cells, the total number of states will be

(2.7)

In a similar form we can calculate the case with four persons getting the number of states:

(2.8)

If we repeat this exercise N times we will get the number of possible states for N persons to be

(2.9)

In this case is M >> N and also N >> 1, so that we can rewrite the coefficient

(2.10)

and get in a very good approximation for the number of states of N person with a total well-being of W is

(2.11)

The number of states for the case one person has a well-being level of w is equal to the number of states for only N-1 persons with a total well-being of W-w:

(2.12)

Inserting (2.11) and (2.12) into (2.2) we get for the probability:

(2.13)

If we introduce the average well-being as the total well-being divide by the number of people in the society:

(2.14)

we get in case of a large number of persons (N >>1) the expression:

(2.15)

This is the idealized case where we have assumed that each of the N person has free access to well-being. Before we model a more realistic case lets analyse the implications for (2.15).


3. The Lorenz curve and Gini factor for the idealized case

In this case we can sketch a Lorenz curve of the cumulative distribution function of well-being in function of associated cumulate population. We can define a cumulated population function integrating (7) over well-being up to a fraction µ of the total amount W:

(3.1)

In a similar form we can introduce a cumulated well-being integrating population density (2.13) multiplying by the level of well-being:

(3.2)

With a partial integration we can express the cumulated well-being in terms of the cumulated population getting an expression for the Lorenz curve:

(3.3)

The graphic representation of this equation shows the classical curve:


The Gini factor can be easily calculate sustracting from the surface under the curve γ=µ the integral of (3.2):

(3.4)

Before we proceed to study a more realistic case we should setup the general frame of the method we are applying.


4. Partition Function

One of the central concepts of statistical mechanics I will borrow for this study is the concept of partition function or as it is call in german "Zustandssumme". In analogy to (7) we define the partition function for a systems of person as

(4.1)

Together with the border condition

(4.2)

the partition function can be used to calculate the mean value

(4.3)

the quadratic mean

(4.4)

and the variance of the system

(4.5)

This schema delivers the same result we already develop in point 1; if we suppose that the possible well-being states is any positive value we have

(4.6)

and we can calculate β

(4.7)

delivering a distribution identical to (2.15)

Now we can generalize the partition function for the case the total number of member is not necessary fixed. If the number of persons with a well-being level ωr is nr the partition function will read

(4.8)

with the condition

(4.9)

Equation (4.9) can be add on the number of persons per state simplifying the Partition function to

(4.10)

or in the logarithmic form

(4.11)

The partition function can be used to calculate the mean value

(4.12)

the quadratic mean

(4.13)

and the variance of the system

(4.14)

Now we can study the more realistic case implied by the partition function (4.10).


5. Well-being distribute in a society

Before we can estimate the Lorenz curve for this case, we must calculate the factor α to establish that the number of persons witch have access to well-being is less-equal the total number of persons N:

(5.1)

If we also assume that the mayor contribution to the well-being level, is due to the wages the persons are receiving, we could introduce a minimal wages ω0. In this case we can calculate the number of persons with access to well-being applying (4.12) to the partition function (4.13) in the continuous approximation (4.6):

(5.2)

After integration we get an equation for the factor α:

(5.3)