A Guide to Momentum Distributions

In real life, the moments of an object are descriptive rather than quantitative. In mathematics, the moments of an act are descriptive, taking the characteristics of the object into account in determining their values. For instance, when I am walking through a park, I am actively observing the surroundings. The moment that I decide to stop and look around are the moments of my choice, but for the purposes of this article, I will be using the word “moment” to describe the transformation of one object into another.


Moments are of three types: first moments, central moments, and derivatives. First moments are simply the positions and orientations of the points that make up the system of a system with moments. Central moments occur when the system is changing and tends to remain stationary; they are the moments of change for an elliptic curve, for example. A derivative is any change that occurs on a function over time, and derivatives of a constant value are themselves moments of change that modify a function by some quantity.

The second type of moment is any measurable quantity that changes as the system varies. Thus, the third type of moment is a measure of integration or a property of some kind that changes as the system varies. We will use the concept of distribution to describe these moments. In a normal distribution, the mean value of a variable is that value that would be obtained if the value of the variable were equal to the mean value of all other variables. The variance of the distribution, which can also be called the variance of distribution or the deviation of the mean from the normal distribution, can be plotted on a log-normal scale, and graphed as a function of the mean value.

When the distribution of moments is normal, kurtosis occurs. Kurtosis occurs when the lines of perfect correlation (i.e., horizontal tangent) tend to be curved. The curve is called the kurtosis curve, and it can be thought of as the normal curve on a normal distribution. Higher moments, therefore, are indicated by slopes of the kurtosis curve. The higher moments, like the cumulative distribution of moments, occur in random samples rather than as the result of a normal distribution. Distribution of higher moments is mathematically more complicated because of the fact that sampling introduces error into the results.

The distribution of moments is also related to another familiar measure of variance: the variance of mean. The variance of mean is simply the mean of all the sample variances. The formula for computing the variance of means uses one less factor than the formula for computing the variance of events; the formula is:

Momentum, and acceleration are the three forces that combine to create motion. Each of these forces has its own momentum and kinetic energy, which are measured in units of velocity per meter per second. A force may be considered to have a kinetic energy that is conserving, or it may have an equivalent energy that is being converted to heat. In some cases, the conversion may not occur, such as the case where the source of force is an ocean of water, or an air mass with very low molecules. All of these different types of momentum may be derived from an ideal gas or solid and then used to calculate the momentum distribution.