Why use log-normal distribution?

1. SPEX uses log-normal distribution to model the emission measure distribution

The emission measure distribution is often modeled by a log-normal distribution in SPEX. The emission measure distribution is defined as

$$Y(x)=\frac{Y_0}{\sqrt{2\pi }{\sigma }_T}{e}^{-(x-x0{)}^2/2{\sigma }_T^2}$$

where $Y_0$ is the total, integrated emission measure. By default $x\equiv \log T$ and $x_0\equiv \log T_0$ with $T_0$ the average temperature of the plasma (from Cie).

Such a log-normal distribution may be confusing for a beginner.

2. The log-normal distribution originates from multiplicative processes

2.1 What is a multiplicative process?

The density, pressure, temperature, and other physical properties of the plasma are often influenced by various multiplicative processes, such as shock heating, radiative cooling, and turbulent mixing. After a certain physical process, such as shock heating, the temperature of the plasma will be multiplied by a factor, which is a random variable, that is

$$T_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}}=T_{\mathrm{p}\mathrm{r}\mathrm{e}}\times r,$$

where $r$ is a random variable, $T_{\mathrm{p}\mathrm{r}\mathrm{e}}$ and $T_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}}$ are the temperature before and after the physical process, respectively. The post-process temperature is related to the pre-process temperature by a multiplicative factor $r$, which is why we refer to it as a multiplicative process.

2.2 How does the log-normal distribution arise from multiplicative processes?

For a large number of such processes, the temperature of the plasma will be a product of a large number of random variables. The final temperature of the plasma is related to the initial temperature by

$$T_{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}=T_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}\times r_1\times r_2\times r_3\times \cdots \times r_n,$$

where $r_1$, $r_2$, $r_3$, $\cdots$, $r_n$ are random variables that are independently and identically distributed (regardless of the shape of the distribution), and $n$ is the number of multiplicative processes. $n$ is sufficiently large, and the central limit theorem can be applied, the distribution of the final temperature will be approximately log-normal. Some essential derivations are shown in the following.

$$\log \frac{T_{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}}{T_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}}=\log r_1+\log r_2+\log r_3+\cdots +\log r_n$$

Since $\log r_j$ ($j=1,2,3,\cdots ,n$) are independent and identically distributed, the central limit theorem can be applied, and the distribution of $\log \frac{T_{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}}{T_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}}$ will be approximately normal, that is

$$\log \frac{T_{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}}{T_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}}\sim N(n\mu ,n{\sigma }^2).$$

Here, we suppose that the mean value and variance of $\log r_j$ ($j=1,2,3,\cdots ,n$) are $\mu$ and ${\sigma }^2$, respectively. Let $x\equiv \log T_{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}$ and $x_0\equiv \log T_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}$, then

$$x-x_0\sim N(n\mu ,n{\sigma }^2),$$

namely, the distribution of $x-x_0$ will be

$$f(x-x_0)=\frac{1}{\sqrt{2\pi n{\sigma }^2}}\exp (-\frac{(x-x_0-n\mu {)}^2}{2n{\sigma }^2}).$$

We can clearly see that $\log T_0=x_0+n\mu$.

2.3 Addictive or multiplicative processes?

Likewise, the additive process will lead to a normal distribution, and it is defined as follows.

$$T_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}}=T_{\mathrm{p}\mathrm{r}\mathrm{e}}+\mathrm{\Delta }T,$$

where $\mathrm{\Delta }T$ is a random variable. The post-process temperature of the plasma is related to the pre-process temperature by an additive factor $\mathrm{\Delta }T$, which is why we refer to it as an additive process. Certain shot-noise models could be the additive process. For more details about addictive and multiplicative processes, you can see Uttley et al. (2005).

3. Some first-sight properties of the log-normal distribution

If you find it hard to understand the origin of the log-normal distribution, i.e., you cannot understand the derivation of the log-normal distribution from multiplicative processes, you can also gain some primary insight into the log-normal distribution from its properties.

• (1) The log-normal distribution is more suitable for the physical properties that cannot be negative, such as the temperature, density, and pressure of the plasma. Although you can also use the normal distribution to model the temperature, density, and pressure of the plasma, you must truncate the normal distribution to ensure that the temperature, density, and pressure are non-negative.
• (2) The log-normal distribution is more suitable for the physical properties that range over several orders of magnitude. If you insist on using the normal distribution that centers at ${10}^{5.5}\mathrm{K}$ (whatever the number is) to model the temperature, the contribution from the much lower (like ${10}^4\mathrm{K}$) or higher temperature (like ${10}^6\mathrm{K}$) will be insignificant unless you use a large $\sigma$.