Emergence of Newtonian Deterministic Causality from Stochastic Motions in Continuous Space and Time

Since Newton's time, deterministic causality has been considered a crucial prerequisite in any fundamental theory in physics. In contrast, the present work investigates stochastic dynamical models for motion in one spatial dimension, in which Newtonian mechanics becomes an emergent property: We present a coherent theory in which a Hamilton-Jacobi equation (HJE) emerges in a description of the evolution of entropy $-\phi(x,t)=\epsilon \log$(Probability) of a system under observation and in the limit of large information extent $\epsilon^{-1}$ in homogeneous space and time. The variable $\phi$ represents a non-random high-order statistical concept that is distinct from probability itself as $\epsilon=0$; the HJE embodies an emergent law of deterministic causality in continuous space and time with an Imaginary Scale symmetry $(t,x,\phi)\leftrightarrow (it,ix,-i\phi)$. $\phi(x,t)$ exhibits a nonlinear wave phenomenon with a mathematical singularity in finite time, overcoming which we introduce viscosity $\epsilon(\partial^2\phi/\partial x^2)$ and wave $i\epsilon(\partial^2 \phi/\partial x^2)$ perturbations, articulating dissipation and conservation, which break the Imaginary Scale symmetry: They lead to the Brownian motion and Schr\"{o}dinger's equation of motion, respectively. Last but not least, Lagrange's action in classical mechanics acquires an entropic interpretation and Hamilton's principle is established.