渐近完全的自由能耗散：任意正温度下成立的粗粒度MLSI
Asymptotically complete free-energy dissipation: a coarse MLSI holds at any positive temperature
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中文总结 AI 辅助
本研究针对经典伊辛Glauber动力学，通过引入任意正温度下成立的粗粒度修正对数索伯列夫不等式，严格证明了其自由能渐近完全耗散的问题。
中文摘要 AI 辅助
在学校中人人都知道，耦合到固定温度热浴的非平衡系统会随时间演化至热力学平衡，过程中自由能只会减小。至少自Holley 1971年的工作起，数学家在经典伊辛Glauber动力学的有限范围内也知晓这一点。但自由能是否会渐近减小到平衡态的自由能？典型的学生会说“当然是的”，但由于自由能仅为下半连续，这个问题比初看起来更复杂。据我们所知，除了可利用经典函数不等式的唯一性区域外，该问题此前尚未得到严格解决。我们通过引入一种粗粒度修正对数索伯列夫不等式（coarse modified log-Sobolev inequality），证明了经典伊辛Glauber动力学的自由能渐近完全耗散，该不等式在任意正温度下均成立，尤其适用于相共存区域。
英文摘要
Everybody learns in school that an out-of-equilibrium system coupled to a heat bath at a fixed temperature evolves to thermodynamic equilibrium as time goes on, and the free energy will only decrease on its way there. At least since Holley's 1971 work, mathematicians know this too, in the modest context of classical Ising Glauber dynamics. But does the free energy also asymptotically decrease to the free energy of an equilibrium state? A typical school kid would say "yes, of course", but since the free energy is only lower semicontinuous, this question is less straightforward than it initially seems. To the best of our knowledge, apart from the uniqueness regime, where one can make use of classical functional inequalities, this question has not previously been resolved rigorously. We prove the asymptotically complete dissipation of the free energy for the classical Ising Glauber dynamics by introducing a coarse modified log-Sobolev inequality, which holds at every positive temperature, in particular in the phase-coexistence regime.
