Mean squared displacement

Key takeaways

Mean squared displacement belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Mean squared displacement to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Mean squared displacement from memory before moving on to harder problems.

Reference excerpt

In statistical mechanics, the mean squared displacement (MSD), also called 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 MSD is measured over time to determine if a particle is spreading slowly due to diffusion, or if an advective force is also contributing. Another relevant concept, the variance-related diameter (VRD), defined as twice the square root of MSD, is also used in studying the transportation and mixing phenomena in 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 at time t {\displaystyle t} is defined as an ensemble average:

MSD ≡ ⟨ | x ( t ) − x 0 | 2 ⟩ = 1 N ∑ i = 1 N | x ( i ) ( t ) − x ( i ) ( 0 ) | 2 {\displaystyle {\text{MSD}}\equiv \left\langle \left|\mathbf {x} (t)-\mathbf {x_{0}} \right|^{2}\right\rangle ={\frac {1}{N}}\sum _{i=1}^{N}\left|\mathbf {x^{(i)}} (t)-\mathbf {x^{(i)}} (0)\right|^{2}}

where N is the number of particles to be averaged, vector x ( i ) ( 0 ) = x 0 ( i ) {\displaystyle \mathbf {x^{(i)}} (0)=\mathbf {x_{0}^{(i)}} } is the reference position of the i {\displaystyle i} -th particle, and vector x ( i ) ( t ) {\displaystyle \mathbf {x^{(i)}} (t)} is the position of the i {\displaystyle i} -th particle at time t.

Derivation of the MSD for a Brownian particle in 1D 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.)

∂ p ( x , t ∣ x 0 ) ∂ t = D ∂ 2 p ( x , t ∣ x 0 ) ∂ x 2 , {\displaystyle {\frac {\partial p(x,t\mid x_{0})}{\partial t}}=D{\frac {\partial ^{2}p(x,t\mid x_{0})}{\partial x^{2}}},}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Mean squared displacement

Start with the simplest possible case. Write down what Mean squared displacement claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Mean squared displacement before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Mean squared displacement ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Mean squared displacement

In research: Mean squared displacement appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Mean squared displacement in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Mean squared displacement is common in secondary-school and first-year university syllabi. It links to neighbouring topics Motion (physics), Statistical deviation and dispersion, Statistical mechanics, so understanding it makes those chapters shorter.
In everyday life: Look for Mean squared displacement outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Mean squared displacement in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Mean squared displacement means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Mean squared displacement out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Mean squared displacement in simple terms?

In statistical mechanics, the mean squared displacement (MSD), also called 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…

Why does Mean squared displacement matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Mean squared displacement?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Mean squared displacement.