Let $x_i,x_i$ be two points. The potential between these two points is

$V(x_i,x_j)$

More prescisely, the potential actually is a function of $x_i-x_j$:

$V(x_i-x_j)=V(x_j-x_i)$

(Only depend on the distance but not direction, even function).

The force applied to $x_i$ is $F_{x_i}=-\nabla_{x_i}V(x_i,x_j)$.

The force applied to $x_j$ is $F_{x_j}=-\nabla_{x_j}V(x_i,x_j)$.

Highlight:

Physical interpretion: easy!  The direction of the force points to the negtive gradient!

The gradient of $V(x_i-x_j)$ with respect to $x_i$ means: if $x_i$ can move ($x_j$ can’t move), moving along the force direction, the potatial decrease the fastest.

Mathematically, it is easy to see $\nabla_{x_i}V(x_i-x_j)=-\nabla_{x_j}V(x_i-x_j)$. Hence $F_{x_i}=-F_{x_j}$.

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EDIT:

The force and potential are only functions of $x_i-x_j$. More specifically, the potential is a function of $\|x_i-x_j\|$. So it can be usually seen that the force and potential are written as

$F(x_{ij})$ and $V(x_{ij})$

where $x_{ij}=x_i-x_j$.

And because $\nabla_{x_{ij}}V(x_i-x_j)=\nabla_{x_{i}}V(x_i-x_j)$ and $\nabla_{x_{ij}}V(x_i-x_j)=-\nabla_{x_{ji}}V(x_i-x_j)$

The force applied to $x_i$ is $F_{x_{ij}}=-\nabla_{x_{ij}}V(x_i,x_j)$.

The force applied to $x_j$ is \$latex F_{x_{ji}}=-F_{x_{ij}}\$.