Regulation of entropic one over x force by effective mass and stochastic mass interpretation

Abstract

This preprint studies a toy model for entropic force regularization. We consider an entropic force F = C / x where C = k_B T, and an effective mass that depends on acceleration and velocity as m_eff = m - h * a / v^3, with h = Planck constant. The equation of motion becomes quadratic in acceleration. It admits two exact branches: a classical branch that reduces to C / (m x) for small h, and an exotic branch proportional to m v^3 / h. A critical length appears: x_c = 4 h C / (m^2 v^3). Below this length no real solution for acceleration exists. This provides a natural cutoff that regularizes the 1 / x divergence. Numerical values at T = 300 K:- electron with v = 1000 m/s: x_c = 1.32 cm, critical acceleration 6.87e11 m/s2, mass correction -0.66 percent at 0.5 m- electron with v = 100000 m/s: x_c = 13.2 nm- macroscopic 1 kg with v = 1 m/s: x_c = 1.09e-53 m, correction 1e-54 (pure classical) We then promote the effective mass to a stochastic variable. Acceleration fluctuations induce mass fluctuations with diffusion D_m = h^2 D_a / v^6. We write Langevin equation dm_eff = -gamma (m_eff - mean) dt + noise, derive Fokker-Planck equation and stationary distribution. Mass fluctuations renormalize the entropic coupling as C_eff = C times (1 + variance / mean^2). Near x_c variance becomes order one. The model separates quantum regime (cm scale cutoff for slow electrons) and classical regime (cutoff far below Planck length for macroscopic masses) without ad hoc potentials. It is testable with slow electron experiments at 50 cm distance. Files: main.pdf contains full derivation and figures. README.md contains reproducibility code. Keywords: entropic force, effective mass, regularization, stochastic mass, critical lengthLicense: CC BY 4.0