Nondimensionalisations of the Langevin equation and small-parameter limits

Inspired by the success of dimensional analysis in continuum mechanics and nondimensional numbers such as Péclet, Schmidt, Lewis, and Prandtl, we introduce the study of the Brownian motion nondimensional parameter M=conservativeforce/nonconservativeforce. In particular, we nondimensionalise the Langevin equation for a Brownian particle of mass m moving in a fluid of viscosity γ with diffusivity coefficient σ under the action of a conservative force in the form of a harmonic oscillator of constant k. We identify all the possible time-scales T, and the associated length-scales L, on the basis of the nondimensional parameter M=m k γ−2. Through this nondimensionalisation we study the small-parameter limits with respect to Mα and we identify five different regimes due to α, in opposition to the three identified by the standard analysis on the basis of the time-scales:T1=m/γ, T2=m/k, and T3=γ/k. Because of the ratio-type definition of M the leading force is established by the sign of α. For very short time-scales (α>0), the particle is trapped at its initial condition. For short times (α=0), the particle has the same dynamics of a free Brownian particle. That is, the action of the conservative force is negligible. For intermediate time (−1<α<0), the particle diffuses as a Wiener process, that is as a free Brownian particle at large elapsed times. The over-damped timescale (α=−1) is a critical timescale where the dynamics are given by the Smoluchowski–Kramers approximation. For long times (α<−1), the particle is trapped at the bottom of the potential well.