Gaseous system: Meaning of this integral eq.

[jam_27]

What is the physical or statistical meaning of the following integral

$$\int^{a}_{o} g(\vartheta) d(\vartheta)$$ = $$\int^{\infty}_{a} g(\vartheta) d(\vartheta)$$

where $$g(\vartheta)$$ is a Gaussian in $$\vartheta$$ describing the transition frequency fluctuation in a gaseous system (assume two-level and inhomogeneous) .

$$\vartheta = \omega_{0} -\omega$$, where $$\omega_{0}$$ is the peak frequency and $$\omega$$ the running frequency.

I understand that the integral finds a point $$\vartheta = a$$ for which the area under the curve (the Gaussian) between 0 to a and a to $$\infty$$ are equal.

But is there a statistical meaning to this integral? Does it find something like the most-probable value $$\vartheta = a$$? But the most probable value should be $$\vartheta = 0$$ in my understanding! So what does the point $$\vartheta = a$$ tell us?

I will be grateful if somebody can explain this and/or direct me to a reference.

Cheers

Jamy

 

[azntoon]

angThe integral in question is a type of probability distribution known as a Gaussian or normal distribution. This type of distribution is used to describe the probability of a certain event occurring. In this case, the integral is computing the probability of the transition frequency fluctuation in a gaseous system taking on a particular value (in this case, \vartheta = a). The integral finds a point \vartheta = a for which the area under the curve (the Gaussian) between 0 to a and a to \infty are equal. This is the most probable value of \vartheta - i.e. the value that is most likely to be observed.

 

[Entropia]

The integral in this equation has a physical meaning in the context of a gaseous system. It represents the probability of finding a transition frequency fluctuation, described by a Gaussian function g(\vartheta), within a certain range of values. In this case, the range is from 0 to a and from a to \infty. This integral can also be thought of as the cumulative distribution function (CDF) of the Gaussian function, which gives the probability of a random variable falling within a certain range of values.

The point \vartheta = a represents the boundary between the two regions of the CDF, where the probabilities of finding a transition frequency fluctuation are equal. This means that there is a 50% chance of finding a fluctuation within the range of 0 to a and a 50% chance of finding it within the range of a to \infty.

In terms of statistical meaning, this integral can be used to calculate the standard deviation of the transition frequency fluctuations in the gaseous system. The value of a represents the standard deviation of the Gaussian distribution, which is a measure of the spread of the data points around the mean value (in this case, \vartheta = 0).

Overall, this integral provides insight into the distribution of transition frequency fluctuations in a gaseous system and can be used to calculate important statistical parameters. I would suggest consulting a textbook or scientific paper on statistical mechanics for further information and references.