Entropy variation rate divided by temperature always decreases

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.