Renyi相对熵Lyapunov泛函在耗散随机哈密顿系统中收敛到不变测度的应用
Renyi's Relative Entropy Lyapunov Functional for Convergence to Invariant Measure in Dissipative Stochastic Hamiltonian Systems
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- School of Engineering, Australian National University（澳大利亚国立大学工程学院）
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中文总结 AI 辅助
本文针对耗散随机哈密顿系统，证明χ²散度（Renyi二阶相对熵）作为Lyapunov泛函，通过位置-动量条件化确保其严格递减，从而保证系统收敛到Boltzmann不变测度。
中文摘要 AI 辅助
本文关注多变量随机哈密顿系统，该系统由位置变量的常微分方程和动量变量的Ito随机微分方程共同控制。动量动力学包含保守力和非保守力，包括Langevin粘性阻尼和作为随机力的Wiener过程。该设置允许旋转自由度、非二次势能函数以及依赖于位置的扩散和阻尼矩阵。除了对受随机力作用的非线性物理动力学进行建模外，该系统还从随机优化的角度出发，其中通过收敛到有利于该函数较小值的Boltzmann平衡测度来寻找势能的全局最小值（例如，使用梯度下降）。在一定的正则性条件下，我们证明了相对于不变测度的χ²散度（或等价地，Renyi二阶相对熵）具有非正的时间导数，从而为这种收敛提供了一个Lyapunov泛函候选。然而，该导数在某些时刻消失，这与给定位置条件下动量的概率分布行为有关，并与由底层经典系统位置空间参数化的一族辅助量子谐振子哈密顿量有联系。这种位置-动量条件化被用来证明熵耗散中的这种预平衡中断时间是孤立的，从而使χ²散度成为控制此类系统中联合位置-动量分布的Fokker-Planck-Kolmogorov方程的严格递减Lyapunov泛函。
英文摘要
This paper is concerned with multivariable stochastic Hamiltonian systems governed by an ordinary differential equation for the position and an Ito stochastic differential equation for the momentum. The momentum dynamics involve both conservative and nonconservative forcing including Langevin viscous damping and the Wiener process as a random force. The setting allows for rotational degrees of freedom, a non-quadratic potential energy function and position-dependent mass, diffusion and damping matrices. In addition to modelling nonlinear physical dynamics subject to random forcing, the system is motivated by a stochastic optimisation viewpoint, where the search for global minima of the potential energy (for example, using the gradient descent) is carried out through convergence to a Boltzmann equilibrium measure which favours smaller values of this function. Under certain regularity conditions, we show that the $χ^2$-divergence or, equivalently, Renyi's second-order relative entropy with respect to the invariant measure has a nonpositive time derivative and thus provides a Lyapunov functional candidate for this convergence. However, this derivative vanishes at some moments of time, which is related to the behaviour of the probability distribution of the momentum conditioned on the position and has a link to a family of auxiliary quantum harmonic oscillator Hamiltonians parameterised by the position space of the underlying classical system. This position-momentum conditioning is used in order to prove that such pre-equilibrium break times in the entropy dissipation are isolated, thus making the $χ^2$-divergence a strictly decreasing Lyapunov functional for the Fokker-Planck-Kolmogorov equation which governs the joint position-momentum distribution in such systems.
