I know that for some elementary reaction $m\text{A}+n\text{B}\rightarrow\text{C}$, the rate of reaction is given by $-\frac{\text{dA}}{\text{dt}}=k[\text{A}]^m[\text{B}]^n$.
Now if we consider the case $m=1$ and $n=1$ and where A and B are gas molecules, then, from collision theory we know that the rate of reaction can also be written as collision frequency $\cdot$ fraction of molecules with sufficient energy $\cdot$ steric coefficient: $$\sigma\left(\frac{8\text{RT}}{\pi\mu}\right)^{1/2}\text{N}_\text{A}^2[\text{A}][\text{B}]\cdot e^{\frac{-E_{min}}{\text{RT}}}\cdot P$$
Where $\mu=\frac{\text{M}_A\text{M}_{B}}{\text{M}_A+\text{M}_B}$ ($\text{M}_A$ and $\text{M}_B$ are molar masses of A and B, respectively), $N_\text{A}$ is Avogadro's number, and $P$ is the steric coefficient. With this equation, if we let $$k=\sigma\left(\frac{8\text{RT}}{\pi\mu}\right)^{1/2}\text{N}_\text{A}^2\cdot e^{\frac{-E_{min}}{\text{RT}}}\cdot P$$ then we get the general equation for the rate of reaction, rate = $k[A][B]$, and we also understand the relationship between the rate constant, temperature, activation energy, and the pre-exponential factor.
I am wondering if there is any way to explain how the equation can be modified to explain the general form for rates of elementary reactions ($\text{rate}=k[\text{A}]^m[\text{B}]^n$), and why it would be the case that concentrations of A and B are raised to the their stoichiometric coefficients.
Equation from: Atkins, P., & Jones, L. (2009). Collision Theory. In Chemical Principles: The Quest For Insight (5th ed., pp. 591–594). essay, W. H. Freeman and Company.
