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 walk
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space.
An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...
er. In the realm of
biophysics
Biophysics is an interdisciplinary science that applies approaches and methods traditionally used in physics to study biological phenomena. Biophysics covers all scales of biological organization, from molecular to organismic and populations. ...
and
environmental engineering
Environmental engineering is a professional engineering discipline that encompasses broad scientific topics like chemistry, biology, ecology, geology, hydraulics, hydrology, microbiology, and mathematics to create solutions that will protect and ...
, the Mean Squared Displacement is measured over time to determine if a particle is spreading slowly due to
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemica ...
, 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
Environmental engineering is a professional engineering discipline that encompasses broad scientific topics like chemistry, biology, ecology, geology, hydraulics, hydrology, microbiology, and mathematics to create solutions that will protect and ...
. 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 at time
is defined as an
ensemble average
In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents ...
:
:
where ''N'' is the number of particles to be averaged, vector
is the reference position of the
-th particle, and vector
is the position of the
-th particle at time ''t''.
Derivation of the MSD for a Brownian particle in 1D
The
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
(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
; where
is the position of the particle at some given time,
is the tagged particle's initial position, and
is the diffusion constant with the S.I. units
(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
is position dependent.
The differential equation above takes the form of 1D
heat equation. The one-dimensional PDF above is the
Green's function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if \operatorname is the linear differenti ...
of heat equation (also known as
Heat kernel in mathematics):
:
This states that the probability of finding the particle at
is Gaussian, and the width of the Gaussian is time dependent. More specifically the
full width at half maximum
In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve mea ...
(FWHM)(technically/pedantically, this is actually the Full ''duration'' at half maximum as the independent variable is time) scales like
:
Using the PDF one is able to derive the average of a given function,
, at time
:
:
where the average is taken over all space (or any applicable variable).
The Mean squared displacement is defined as
:
expanding out the ensemble average
:
dropping the explicit time dependence notation for clarity. To find the MSD, one can take one of two paths: one can explicitly calculate
and
, then plug the result back into the definition of the MSD; or one could find the
moment-generating function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compare ...
, an extremely useful, and general function when dealing with probability densities. The moment-generating function describes the
moment of the PDF. The first moment of the displacement PDF shown above is simply the mean:
. The second moment is given as
.
So then, to find the moment-generating function it is convenient to introduce the
characteristic function:
:
one can expand out the exponential in the above equation to give
:
By taking the natural log of the characteristic function, a new function is produced, the
cumulant generating function,
:
where
is the
cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
of
. The first two cumulants are related to the first two moments,
, via
and
where the second cumulant is the so-called variance,
. With these definitions accounted for one can investigate the moments of the Brownian particle PDF,
:
by completing the square and knowing the total area under a Gaussian one arrives at
:
Taking the natural log, and comparing powers of
to the cumulant generating function, the first cumulant is
:
which is as expected, namely that the mean position is the Gaussian centre. The second cumulant is
:
the factor 2 comes from the factorial factor in the denominator of the cumulant generating function. From this, the second moment is calculated,
:
Plugging the results for the first and second moments back, one finds the MSD,
:
Derivation for ''n'' dimensions
For a Brownian particle in higher-dimension
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, its position is represented by a vector
, where the
Cartesian coordinates are
statistically independent
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of o ...
.
The ''n''-variable probability distribution function is the product of the
fundamental solutions in each variable; i.e.,
:
The Mean squared displacement is defined as
:
Since all the coordinates are independent, their deviation from the reference position is also independent. Therefore,
:
For each coordinate, following the same derivation as in 1D scenario above, one obtains the MSD in that dimension as
. Hence, the final result of mean squared displacement in ''n''-dimensional Brownian motion is:
:
Definition of MSD for time lags
In the measurements of single particle tracking (SPT), displacements can be defined for different time intervals between positions (also called time lags or lag times). SPT yields the trajectory