A particle of mass \(m\) moves along the positive \(x\)-axis under the influence of a single force given by \(F(x) = \dfrac{\beta}{x}\), where \(\beta\) is a positive constant and \(x > 0\). The particle moves from position \(x = x_0\) to \(x = 2x_0\), and subsequently from \(x = 2x_0\) to \(x = 4x_0\). What is the ratio \(\dfrac{\Delta K_2}{\Delta K_1}\), where \(\Delta K_1\) is the change in the particle's kinetic energy over the first displacement and \(\Delta K_2\) is the change in its kinetic energy over the second displacement?