The spherical $p$-spin model is a fundamental model in statistical mechanics of a disordered system with a random first-order transition. The dynamics of this model is interesting both for the physics of glasses and for its implications on hard optimization problems. Here, we revisit the out-of-equilibrium dynamics of the spherical mixed $p$-spin model, which differs from the pure $p$-spin model by the fact that the Hamiltonian is not a homogeneous function of its variables. We consider quenches (gradient descent dynamics) starting from initial conditions thermalized in the high-temperature ergodic phase. Unexpectedly, we find that, differently from the pure $p$-spin case, the asymptotic states of the dynamics keep memory of the initial condition. The final energy is a decreasing function of the initial temperature, and the system remains correlated with the initial state. This dependence disproves the idea of a unique “threshold” energy level attracting dynamics starting from high-temperature initial conditions. Thermalization, which could be achieved, e.g., by an algorithm like simulated annealing, provides an advantage in gradient descent dynamics and, last but not least, brings mean-field models closer to real glass phenomenology, where such a dependence is observed in numerical simulations. We investigate the nature of the asymptotic dynamics, finding an aging state that relaxes towards deep, marginally stable minima. However, careful analysis rules out simple generalizations of the aging solution of the pure model. We compute the constrained complexity with the aim of connecting the asymptotic solution to the energy landscape.

中文翻译：

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对平均场玻态动力学及其与能量景观的关系的重新思考：球形mixed-Spin模型的令人惊讶的情况

球形 $p$自旋模型是具有随机一阶跃迁的无序系统的统计力学中的基本模型。该模型的动力学特性对于眼镜的物理性质及其对硬优化问题的影响都很有趣。在这里，我们回顾了球形混合的不平衡动力学$p$自旋模型，不同于纯 $p$哈密​​顿量不是其变量的齐次函数这一事实。我们考虑从在高温遍历阶段热化的初始条件开始的猝灭（梯度下降动力学）。出乎意料的是，我们发现与纯$p$在自旋情况下，动力学的渐近状态保留了初始条件的记忆。最终能量是初始温度的递减函数，并且系统仍与初始状态相关。这种依赖性反驳了从高温初始条件开始吸引动态的独特“阈值”能级的想法。可以通过例如模拟退火之类的算法实现的热化在梯度下降动力学方面具有优势，最后但并非最不重要的一点是，其平均场模型更接近于真实的玻璃现象学，在数值模拟中可以观察到这种依赖性。我们研究了渐近动力学的性质，发现了朝着深度稳定的极小值松弛的衰老状态。然而，仔细的分析排除了纯模型老化解决方案的简单概括。我们以将渐近解连接到能源格局为目标，计算约束复杂度。