我在想(在最简单的模型中)，地球每单位时间发射$N$光子，一些比例$p$击中温室气体粒子，并将以$0.5$的概率重新发射回地球。因此，温室气体粒子越多，$p$就越大，地球的温度也就越高。对于一个改进的模型，我认为大气更像一个连续的介质，光子在粒子之间反弹并加热它们。在这种情况下，是否有一个简单的类比，或者可能是一个1维微分方程模型(比如通过介质的热流)?热在物质中的扩散行为是否与辐射在大气中的传播方式相似?

我对理解温室气体的简单易懂和大致准确的模型感兴趣

我的回答集中在一个“简单、容易理解和大致准确的理解温室气体的模型”。这与你的问题的评论中使用的模型是一样的。理解温室效应最基本的模型是基于辐射平衡的。入射的太阳辐射与射出的(长波)辐射相平衡。从能量的角度来说:在大的时间尺度上，输出的能量必须等于输入的能量。

一个在大气顶部面向太阳的单位正方形接收$S_0 \约1360$ $\text{W}\text{m}^{-2}$，其中$S_0$为< A href="https://en.wikipedia.org/wiki/Solar_constant" rel="nofollow noreferrer">太阳常数。被太阳辐射击中的区域是一个地球半径$r$。因此，地球接收到的能量/时间为$E_I = S_0\pi r^2$。一部分辐射被反射(见< A href="https://en.wikipedia.org/wiki/Albedo" rel="nofollow noreferrer">albedo)，得到$E_I = S_0(1-\alpha)\pi r^2$，其中$\alpha$为反照率。地球反照率在$\alpha = 0.3$。

If we think about earth being a black body with no atmosphere we can use the Stefan-Boltzmann law to estimate the outgoing radiative energy to be $E_O = \sigma T_s^4 4 \pi r^2$, with Stefan-Boltzmann constant $\sigma = 5.67 \times10^{-8}$ $\text{W}\text{m}^{-2}\text{K}^{-4}$ and earths surface Temperature $T_s$. The factor 4 occurs because for the outgoing radiation we need to consider earths whole surface (I gave an explanation on why this is a reasonable assumption here).

Since energy needs to be balanced we can equate $E_I = E_O$ and solve for $T_s$ which will yield us insight on how severe the greenhouse effect actually is.

\begin{equation} T_s = \left(\frac{S_0(1-\alpha)}{4\sigma} \right)^{\frac{1}{4}} \approx 255 \, \text{K} \approx -18°\text{C} \end{equation}

We observe that our estimated temperature is a lot colder than earths average Temperature, which is around $15°\text{C}$ or $288.15 \, \text{K}$. This corresponds to a greenhouse effect of approximately $\sigma(288^4 \text{K}^4 - 255^4 \text{K}^4) \approx 150 \, \text{Wm}^{-2}$. This is the amount of energy/time/unit square earth receives because of the atmosphere.

We could refine the model by adding atmospheric layers to the problem, however I think this captures the essence.