``Block what you can and randomize what you cannot'' is the core principle for treatment effect estimation in randomized controlled trials. Although a wealth of allocation strategies has been developed, an explicit trade-off between the covariate balance achieved by blocking and the robustness guaranteed by randomization is seldom quantified. Motivated by the second law of thermodynamics, this work posits a new criterion that lowers the covariate imbalance while maximizing the entropy that quantifies contrast and allocation diversity. The resulting optimal strategy, termed the minimum free energy randomized design, is then derived, thereby formally achieving such a trade-off. To facilitate practical implementation, we further develop a computationally efficient dynamic allocation algorithm with theoretical guarantees. Using a finite-sample variance decomposition, the proposed randomization strategy is shown to control covariate imbalance while preventing unobserved heterogeneity from dominating the mean squared error, thus retaining minimax efficiency under the prescribed design constraints. Extensive numerical simulations demonstrate that our method achieves superior statistical efficiency and greater robustness than existing approaches.

翻译：“区组化能做的，随机化无法做到的”是随机对照试验中处理效应估计的核心原则。尽管已有丰富的分配策略被开发出来，但区组化带来的协变量平衡与随机化保证的稳健性之间的明确权衡却鲜少被量化。受热力学第二定律启发，本研究提出一个新准则，该准则在降低协变量不平衡的同时，最大化量化对比与分配多样性的熵。由此推导出的最优策略，即最小自由能随机化设计，从而形式化地实现了这一权衡。为方便实际应用，我们进一步开发了一种计算高效且具有理论保证的动态分配算法。利用有限样本方差分解，证明所提出的随机化策略能够控制协变量不平衡，同时防止未观测异质性主导均方误差，从而在指定设计约束下保持极小极大效率。广泛的数值模拟表明，我们的方法在统计效率和稳健性方面均优于现有方法。