The Role of Gradient of the Logarithm of Density in the Quantum Mechanics, Classical Mechanics, the Euler Equation, Equilibrium Statistical Mechanics and Brownian Motion

It is interesting to note that  d/dx [ln(d(x))] where d(x) is the spatial density, plays a central role in the time-independent Schrodinger equation, classical mechanics, the Euler equation, equilibrium statistical mechanics and Brownian motion and in all cases, even the quantum one, is related to force as -d/dx V(x), where V(x) is the potential. In this note, we try to examine the role of d/dx [ln(d(x))], and in particular try to see how quantum mechanics is related to classical and fluid mechanics and how it differs.