Root mean square fluctuation

In statistical mechanics, the mean squared displacement (MSD, also mean square displacement, average squared displacement, or mean square fluctuation) is a measure of the deviation of the position of a particle with respect to a reference position over time. It is the most common measure of the spatial extent of random motion, and can be thought of as measuring the portion of the system 'explored' by the random walker. In the realm of biophysics and environmental engineering, the Mean Squared Displacement is measured over time to determine if a particle is spreading solely due to diffusion, or if an advective force is also contributing. Another relevant concept, the Variance-Related Diameter (VRD, which is twice the square root of MSD), is also used in studying the transportation and mixing phenomena in the realm of environmental engineering. It prominently appears in the Debye–Waller factor (describing vibrations within the solid state) and in the Langevin equation (describing diffusion of a Brownian particle). The MSD is defined as In statistical mechanics, the mean squared displacement (MSD, also mean square displacement, average squared displacement, or mean square fluctuation) is a measure of the deviation of the position of a particle with respect to a reference position over time. It is the most common measure of the spatial extent of random motion, and can be thought of as measuring the portion of the system 'explored' by the random walker. In the realm of biophysics and environmental engineering, the Mean Squared Displacement is measured over time to determine if a particle is spreading solely due to diffusion, or if an advective force is also contributing. Another relevant concept, the Variance-Related Diameter (VRD, which is twice the square root of MSD), is also used in studying the transportation and mixing phenomena in the realm of environmental engineering. It prominently appears in the Debye–Waller factor (describing vibrations within the solid state) and in the Langevin equation (describing diffusion of a Brownian particle). The MSD is defined as where N is the number of particles to be averaged, x n ( 0 ) = x 0 {displaystyle x_{n}(0)=x_{0}} is the reference position of each particle, x n ( t ) {displaystyle x_{n}(t)} is the position of each particle in determined time t. The probability density function (PDF) for a particle in one dimension is found by solving the one-dimensional diffusion equation. (This equation states that the position probability density diffuses out over time - this is the method used by Einstein to describe a Brownian particle. Another method to describe the motion of a Brownian particle was described by Langevin, now known for its namesake as the Langevin equation.) given the initial condition p ( x 0 , t = 0 ∣ x 0 ) = δ ( x − x 0 ) {displaystyle p(x_{0},t={0}mid x_{0})=delta (x-x_{0})} ; where x ( t ) {displaystyle x(t)} is the position of the particle at some given time, x 0 {displaystyle x_{0}} is the tagged particle's initial position, and D {displaystyle D} is the diffusion constant with the S.I. units m 2 s − 1 {displaystyle m^{2}s^{-1}} (an indirect measure of the particle's speed). The bar in the argument of the instantaneous probability refers to the conditional probability. The diffusion equation states that the speed at which the probability for finding the particle at x ( t ) {displaystyle x(t)} is position dependent.