Created: August 28, 2022
Modified: August 28, 2022

rate equation

This page is from my personal notes, and has not been specifically reviewed for public consumption. It might be incomplete, wrong, outdated, or stupid. Caveat lector.

The rate equation or master equation for a continuous-time Markov stochastic process describes how the probability density of the process evolves over time.

As illustration, consider a discrete random walk on an infinite 1D lattice: if the system at time $t$ is at lattice point $n$ , then at time $t + 1$ it moves to the adjacent points $n + 1$ and $n-1$ each with probability $k$ , and stays at $n$ with the remaining probability $1 - 2k$ . Since the system cannot 'skip' a lattice point, the change in probability mass $\Delta P_n$ at point $n$ therefore consists of probability- $k$ inflows from the two adjacent points and a probability- $2k$ outflow of mass previously at this point moving to those adjacent points:

\begin{align*} \Delta P_n &\coloneqq P(x_t=n) - P(x_{t-1}=n)\\ &= k(P(x_{t-1}={n-1}) + P(x_{t-1} = {n+1}) - 2P(x_{t-1}=n)) \end{align*}

If we squint at this, we see

rate equation

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