  The model of physical Brownian motion describes the dynamics of a Brownian
particle experiencing both friction and external magnetic field. It was
investigated as a physically meaningful approach to realising the standard
''mathematical'' Brownian motion, via sending the mass $m \to 0^+$ and
performing natural scaling. In this paper, we are concerned with the singular
limit analysis of a generalised stochastic differential equation (SDE) model,
motivated by and encompassing the physical Brownian motion. We show that the
expected signature of the solution $\left\{P_t\right\}_{t \geq 0}$ for the
generalised SDE converges in the limit $m \to 0^+$ to a nontrivial tensor, at
each degree of tensors and on each compact time interval $[0,T]$. The solution
$\left\{P_t\right\}_{t \geq 0}$, viewed as a rough path, generalises the
momentum of particle in classical physical Brownian motion. This singular limit
is identified through a delicate convergence analysis based on the graded PDE
system for the expected signature of It\^{o} diffusion processes. Explicit
closed-form solutions exhibiting intriguing combinatorial patterns are obtained
when the coefficient matrix $\mathbf{M}$ in our SDE, which generalises the
stress tensor in physical Brownian motion, is diagonalisable. Our work appears
among the very first endeavours to study the singular limit of expected
signature of diffusion processes, in addition to the literature on the analysis
of the expected signature of stochastic processes.
