Mathematical Modeling to Study the Dynamics of A Diatomic Molecule N2 in Water

In the present work an attempt has been made to study the dynamics of a diatomic molecule N2 in water. The proposed model consists of Langevin stochastic differential equation whose solution is obtained through Euler’s method. The proposed work has been concluded by studying the behavior of statistical parameters like variance in position, variance in velocity and covariance between position and velocity. This model incorporates the important parameters like acceleration, intermolecular force, frictional force and random force.

💡 Research Summary

The paper presents a computational study of the stochastic dynamics of a nitrogen (N₂) diatomic molecule immersed in liquid water. The authors formulate the problem using a one‑dimensional Langevin equation that incorporates four essential forces: (i) the inertial term (mass m), (ii) a deterministic intermolecular restoring force modeled as a harmonic spring with constant k, (iii) a viscous drag proportional to the velocity with coefficient γ, and (iv) a random force R(t) representing thermal fluctuations of the surrounding water molecules. The random force is assumed to be Gaussian white noise with zero mean and autocorrelation ⟨R(t)R(t′)⟩ = 2γk_BT δ(t‑t′), which guarantees consistency with the fluctuation‑dissipation theorem and yields the Einstein diffusion coefficient D = k_BT/γ.

To obtain numerical solutions, the authors discretize the Langevin equation using the explicit Euler scheme. With a time step Δt, the update rules are
v_{n+1} = v_n + (Δt/m)