  For an isolated assembly that comprises a system and its surrounding
reservoirs, the total entropy ($S_{a}$) always monotonically increases as time
elapses. This phenomenon is known as the second law of thermodynamics
($S_{a}\geq0$). Here we analytically prove that, unlike the entropy itself, the
entropy variation rate ($B=dS_{a}/dt$) defies the monotonicity for multiple
reservoirs ($n\geq2$). In other words, there always exist minima. For example,
when a system is heated by two reservoirs from $T=300\,K$ initially to
$T=400\,K$ at the final steady state, $B$ decreases steadily first. Then
suddenly it turns around and starts to increases at $387\,K$ until it reaches
its steady-state value, exhibiting peculiar dipping behaviors. In addition, the
crux of our work is the proof that a newly-defined variable, $B/T$, always
decreases. Our proof involves the Newton's law of cooling, in which the heat
transfer coefficient is assumed to be constant. These theoretical macro-scale
findings are validated by numerical experiments using the Crank-Nicholson
method, and are illustrated with practical examples. They constitute an
alternative to the traditional second-law statement, and may provide useful
references for the future micro-scale entropy-related research.
