Mean squared displacement
The mean square displacement (often referred to as the mean squared displacement english (MSD ) and denoted by the symbol, see definition below ) is in statistical physics is a measure of the distance that a particle in certain agents (eg, over many trials) in a time travels. This measure is particularly important in the description of Brownian dynamics and other random movements, as there typically exists no excellent direction along which one could measure a distance traveled.
Descriptive description and interpretation
Graphically the mean square displacement is a measure of the volume that roams a particle that performs a random movement in a certain time. As an example we consider a pure Brownian motion in two dimensions (see figure at right). If you let more particles ( in the figure above in various colors) start at the same place, then although individual particles from the starting point to move away, but in different directions. Also, each particle may well return to the starting point. By averaging over all particle positions now after a waiting time τ, so this average will again be near the starting point, the particles have therefore not moving in the middle. Had the particles, in contrast, a preferred direction, so also their average would move at a certain speed in this preferred direction.
However, one now observes that the particles have an even greater area over emphasize the longer you wait (concentric circles in the figure, the outer circles correspond to longer waiting times τ ), that is, the longer you wait, the more likely it is also once a particle in a larger distance from the starting point. For a description of this swept area ( the slow τ with the waiting time grows ) Now you can use the mean square displacement of all particles: your root () designates indeed the radius of this growing counties / area.
One can therefore interpret the curve so that particles that move according to her, were, after a time τ with high probability already at a distance from their starting point to make.
Exact definition
The mean square displacement is defined by the ensemble average over many trajectories:
This is averaged over many particles, which are observed in each of the time period τ. Alternatively ( and especially in theoretical considerations, where these quantities are calculated ), this can also have the probability (see Green's function ) of particles at time τ be written:
Note that the distance to the starting point of the trajectory ( which is placed in the origin ) measures.
Depending on the system, the mean square displacement can be defined in terms of a time mean value over a trajectory of a particle in the space:
This means that only one particle is observed, and then, starting from different points of time t is measured how far the particles up to the time t τ has moved. It is then averaged over the τ shifts during all possible time periods within the duration T of the trajectory.
Both definitions give only the same size when the system under consideration is ergodic. Often also mixed forms of these two ideal definitions are used, especially when in experiments both averages are mixed (for example, fluorescence correlation spectroscopy).
In addition, often the root mean square displacement
Used and then usually referred to as the English root mean squared displacement ( RMSD ).
Importance
Especially with non-directional and random movements, there is often no excellent spatial direction. Averaging over the ( vectorial ) shifts to the starting point is therefore zero, as there is for each movement in a direction, a movement in the opposite direction with the same statistical frequency. For example, the average deflection of a random walk of a single particle for all times be zero. The particles in a given time a certain spatial area, which is characterized by the mean square displacement still covered.
For normal diffusion results in a simple relation for the mean square displacement:
Where D is the diffusion coefficient, and n is the number of spatial dimensions in which the movement takes place.
For normal diffusion in conjunction with a directional movement ( flow speed V ) is obtained further:
In the case of anomalous diffusion, there is often general, the relationship:
Where Γα a general proportionality constant ( generalized diffusion coefficient depends, possibly of α from! ) and α describes the anomaly of the movement. For α <1 is called subdiffusion and for α > 1 of superdiffusion. For the case α = 1 there is again normal diffusion.
In many systems, the anomaly of a movement limited to a specific time range for τ Rouse model).
Measurement and Applications
The mean square displacement is often used for the characterization of random movements in simulations. They can be directly determined from the simulated particle trajectories. With the help of the relation for normal diffusion (*) can be defined as a diffusion coefficient D due to the grid spacing a and the time step At of simulation:
Thus, then the simulation, which often occurs in idealized and normalized coordinates are normalized to real systems.
The mean square displacement is experimentally accessible.
Thus, it can be determined approximately by single-particle tracking techniques directly from the measured trajectories there.
Also with the help of fluorescence correlation spectroscopy is experimentally accessible under certain assumptions.
It can then be determined from the measured curve
Connection to the velocity autocorrelation
The mean square displacement is above the Green- Kubo relation in close connection to the velocity autocorrelation function: