In general, interaction of proton magnetization states with neighboring atoms allows states transitions. If we can consider them as small perturbations then in terms of time-dependent perturbation theory the transition probability rate form state i to f is given,
\begin{equation}
W_{fi} = \frac{2\pi}{\hbar} |<f|V|i>|^{2} \delta(E_{f}-E_{i})
\end{equation}
To conserve energy if transition from $m=-1/2$ to $m=+1/2$ happens, it should be followed by another proton from $m=+1/2$ to $m=-1/2$. Then take any two state $a,b$ with b state has higher energy. Denote an upwards energy transition by $a-$ $\rightarrow$ $b+$ and downward transition by $b+$ $\rightarrow$ $a-$.
If $N_{+}$ is the initial number of spins with $m=+1/2$ then,
\begin{equation}
\frac{dN_{+}}{dt} = W_{b+a-}N_{-}n_{a}-W_{a-b+}N_{+}n_{b}
\end{equation}
where $n_{b}$ is the initial number states with higher energy and $n_{a}$ lower energy, related by Boltzmann factor,
\begin{equation}
\frac{n_{a}}{n_{a}} = e^{-\hbar B_{0} \gamma /kT}
\end{equation}
For our case probability transitions are equal, which is the case in $1^{st}$ or perturbation theory,
\begin{equation}
W_{b+a-} = W_{a-b+} = W
\end{equation}
Total number of states is fixed,
\begin{equation}
N = N_{+}+N_{-}
\end{equation}
and writing,
\begin{equation}
N_{\pm} = \frac{N \pm \Delta N}{2}
\end{equation}
Now combining the previous equations,
\begin{equation}
\frac{d \Delta N}{dt} = WN(n_{a}-n_{b})-W\Delta N(n_{a}-n_{b})
\end{equation}
We want in the equilibrium
\begin{equation}
\frac{d \Delta N}{dt} = 0
\end{equation}
But this implies
\begin{equation}
\Delta N_{0} = \frac{n_{a}-n_{b}}{n_{a}+n_{b}}N
\end{equation}
Define $W(n_{a}+n_{b})$ as $\frac{1}{T_{1}}$ longitidunal relaxation time constant, then we get
\begin{equation}
\frac{d \Delta N}{dt} = \frac{\Delta N_{0} -\Delta N}{T_{1}}
\end{equation}
Finally, take the average over the volume and you will get something similar to
\begin{equation}
\frac{d M_{z}}{dt} = \frac{M_{0}-M_{z}}{T_{1}}
\end{equation}
which is the longitudunal component relaxation equation of spin.
