Entropy variation rate divided by temperature always decreases

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For an isolated assembly that comprises a system and its surrounding reservoirs, the total entropy (Sₐ) always monotonically increases as time elapses. This phenomenon is known as the second law of thermodynamics (Sₐ≥0). Here we analytically prove that, unlike the entropy itself, the entropy variation rate (B=dSₐ/dt) defies the monotonicity for multiple reservoirs (n≥2). In other words, there always exist minima. For example, when a system is heated by two reservoirs from T=300K initially to T=400K at the final steady state, B decreases steadily first. Then suddenly it turns around and starts to increases at 387K 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.