# Why is $\left(P_{j+1}-P_{j}\right)$ proportional to the transmission rate from $(j+1)$th state to $j$th state

In the book of Zwanzig, Nonequilibrium statistical physics, at page 63, while giving an example of how to use Master equation, he states that

A common application of master equations is in the treatment of random walks on a lattice. For simplicity, consider an infinite one-dimensional lattice, with sites labeled by $$j .$$ In this application, by "state" we mean the location of the walker. By "transition," we mean the movement of the walker from $$j$$ to $$j+1$$ or $$j-1 .$$ The probability that the walker is in state $$j$$ at time $$t$$ is $$P_{j}(t) .$$ This satisfies the master equation $$\frac{d}{d t} P_{j}=w\left(P_{j+1}-P_{j}\right)+w\left(P_{j-1}-P_{j}\right)$$ The rate of change of $$P$$, is the rate $$w$$ of arrival from $$j+1$$ or $$j-1,$$ less the rate of departure from $$j$$

However, I cannot understand why, for example, $$\left(P_{j+1}-P_{j}\right)$$ is proportional to the transmission rate from $$(j+1)$$th state to $$j$$th state.

• This "master equation" looks strange. A 1d random walk is typically characterised by three values: Probability to walk in +-direction $w_+$, probability to walk in --direction $w_-$ and probability to not move $w_0$. Which of these is $w$ supposed to be here? Is there perhaps context for this from an earlier discussion of random walk in the book? Apr 11, 2020 at 12:46
• @ACuriousMind No there is not; this is the starting paragraph of the section called Random Walk. Probably the author assumes that $P_k$ is a continuous function of time, so that it takes care of "not move" case and since the second term in RHS is written with so that the negative sign is incorporated into the paranthesis.
– Our
Apr 11, 2020 at 13:38

The master equation is for the populations $$P_j$$ only. I.e., it assumes that all coherences decay fast and are uncoupled to the populations. The equation seems to assume equal probability to jump forward or backward, thus $$w_+ = w_- = w$$ (and there is no remain probability).
To your question: just add up the jump probabilities. You can jump from $$j+1$$ to $$j$$, thus the term $$w P_{j+1}$$, and you can jump from $$j-1$$ to $$j$$, thus the term $$w P_{j-1}$$. When you are on site $$P_j$$, you can jump to $$j+1$$, which makes one term $$- w P_{j}$$, and you can jump to $$P_{j-1}$$, which makes the other term $$- w P_j$$.
Now group the terms to get your equation, which is quite intuitive. Because all the jump probabilities are equal, and there is no remain probability, the change on site $$j$$ is proportional to the imbalance between site $$j$$ and $$j+1$$, and between site $$j$$ and $$j-1$$.