基于矩阵Riccati方程的非平衡动力学大偏差前因子的路径积分推导及数值计算
Path integral derivation and numerical computation of large deviation prefactors for non-equilibrium dynamics through matrix Riccati equations
摘要：
对于许多由小噪声驱动的非平衡动力学，在物理学、化学、生物学或经济学中，罕见事件确实很重要。大偏差理论则解释了主要统计量的首阶项具有指数行为。指数速率通常作为作用量的下确界获得，它沿着瞬子最小化。在本文中，我们考虑的次阶次指数前因子的计算，这是至关重要的大量的应用。遵循路径积分的方法，我们推导出周围的瞬子的高斯波动的动力学和计算从它的次指数前因子。正如所料，形式主义导致计算的功能决定因素和矩阵Riccati方程。通过与具有细致平衡或广义细致平衡的平衡动力学的情况相比，我们强调Riccati方程解的特定非定域性：前因子依赖于沿瞬子的涨落，而不仅仅是在其起点和终点。我们解释如何数值计算前因子。统计平稳量的情况下，需要考虑非平凡的初始条件的矩阵Riccati方程。
英文摘要：
For many non-equilibrium dynamics driven by small noise, in physics, chemistry, biology, or economy, rare events do matter. Large deviation theory then explains that the leading order term of the main statistical quantities have an exponential behavior. The exponential rate is often obtained as the infimum of an action, which is minimized along an instanton. In this paper, we consider the computation of the next order sub-exponential prefactors, which are crucial for a large number of applications. Following a path integral approach, we derive the dynamics of the Gaussian fluctuations around the instanton and compute from it the sub-exponential prefactors. As might be expected, the formalism leads to the computation of functional determinants and matrix Riccati equations. By contrast with the cases of equilibrium dynamics with detailed balance or generalized detailed balance, we stress the specific non locality of the solutions of the Riccati equation: the prefactors depend on fluctuations all along the instanton and not just at its starting and ending points. We explain how to numerically compute the prefactors. The case of statistically stationary quantities requires considerations of non trivial initial conditions for the matrix Riccati equation.
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