论文标题
牛顿动力学的随机调节和引起潜在的 - 对螺旋星系的应用和黑暗电势
Stochastic modication of Newtonian dynamics and Induced potential -application to spiral galaxies and the dark potential
论文作者
论文摘要
使用[J. Cresson，D。Darses，J。Math。物理。 48，072703（2007）]，我们研究了在该方程可以将随机过程允许作为溶液中的随机过程允许的假设下变形的经典牛顿方程的动力学如何变形。我们专注于随机牛顿方程式的两个定义，称为dierential and差异。我们首先证明了一种随机病毒定理，这是经典案例的自然概括。随机性通过添加一个称为诱发电位的潜在术语来改变病毒关系，该势力在量子力学中对应于BOHM电位。此外，dierential随机牛顿方程自然提供了一个动作函数，该功能是随机的汉密尔顿 - 雅各比方程。该方程的实际部分对应于经典的汉密尔顿 - 雅各布方程，其额外的潜在项对应于随机病毒定理中已经观察到的诱导电位。诱导电势具有明确的形式，具体取决于随机牛顿方程的随机过程解决方案的密度。事实证明，这种密度符合非线性schr {Ö} dinger方程。将这种形式主义用于开普勒电位，人们证明，诱导的电势与用于在螺旋星系旋转曲线恢复的临时“黑暗电位”相吻合。然后，我们讨论了以前的形式主义在螺旋星系中的应用，按照[D. Da Rocha和L. Nottale，混乱，孤子和分形，16（4）：565-595，2003]，其中“黑暗电位”的出现被视为在规模相对理论背景下空间分形的结果。
Using the formalism of stochastic embedding developed by [J. Cresson, D. Darses, J. Math. Phys. 48, 072703 (2007)], we study how the dynamics of the classical Newton equation for a force deriving from a potential is deformed under the assumption that this equation can admit stochastic processes as solutions. We focus on two denitions of a stochastic Newton's equation called dierential and variational. We rst prove a stochastic virial theorem which is a natural generalization of the classical case. The stochasticity modies the virial relation by adding a potential term called the induced potential which corresponds in quantum mechanics to the Bohm potential. Moreover, the dierential stochastic Newton equation naturally provides an action functional which sat-ises a stochastic Hamilton-Jacobi equation. The real part of this equation corresponds to the classical Hamilton-Jacobi equation with an extra potential term corresponding to the induced potential already observed in the stochastic virial theorem. The induced potential has an explicit form depending on the density of the stochastic processes solutions of the stochastic Newton equation. It is proved that this density satises a nonlinear Schr{ö}dinger equation. Applying this formalism for the Kepler potential, one proves that the induced potential coincides with the ad-hoc ''dark potential'' used to recover a at rotation curve of spiral galaxies. We then discuss the application of the previous formalism in the context of spiral galaxies following the proposal and computations given by [D. Da Rocha and L. Nottale, Chaos, Solitons and Fractals, 16(4):565-595, 2003] where the emergence of the ''dark potential'' is seen as a consequence of the fractality of space in the context of the Scale relativity theory.
