What is mean square displacement?

Mean square displacement (MSD) is a measure of the average distance that particles or molecules have moved from their initial position over a given period of time. It is often used in the study of particle diffusion in a variety of systems, such as liquids, gases, and biological cells.

Mathematically, MSD is calculated by taking the square of the distance traveled by each particle at each time step, averaging these values over all particles, and then averaging these values over all time steps. The formula for MSD is typically written as:

MSD(t) = 1/N * Σ[|r(t) - r(0)|²]

where N is the total number of particles, r(t) is the position of a particle at time t, and r(0) is the initial position of the particle.

By analyzing the behavior of MSD over time, researchers can gain insights into the dynamics and diffusion characteristics of particles in a system. For example, a linear relationship between MSD and time suggests that particles are undergoing normal diffusion, while deviations from this behavior can indicate anomalous or confined diffusion.

Overall, mean square displacement is a valuable tool for quantifying the movement of particles in a wide range of scientific disciplines, including physics, chemistry, and biology.