A Brownian particle has mean squared displacement ⟨x²⟩ = 2Dt. If you observe the particle for 4 seconds instead of 1 second, the typical displacement (root-mean-square displacement) will:
AQuadruple, because displacement is proportional to time in any motion
BDouble, because displacement scales as √t — a factor of 4 in time gives a factor of √4 = 2 in displacement
CStay the same, because Brownian motion is random and unpredictable
DIncrease by a factor of 16, because the variance grows as t²
The √t scaling is the defining signature of a random walk: each step is independent, so after N steps the displacement grows as √N, not N. Since time is proportional to N (steps per unit time is constant), displacement grows as √t. This is slower than directed motion (which would give displacement ∝ t). Quadrupling time doubles the typical distance, not quadruples it. Option D confuses variance (which grows as t) with displacement (which grows as √t). The practical consequence: diffusion is effective over short distances but very slow over long ones.
Question 2 Multiple Choice
Einstein's 1905 theoretical treatment of Brownian motion was scientifically decisive because:
AIt proved that pollen grains have a nervous system that drives their motion
BIt established that fluid viscosity decreases with temperature
CIt provided a quantitative relation between macroscopic observables (diffusion, viscosity, temperature) and molecular properties, allowing Perrin to deduce Avogadro's number and empirically confirm the atomic theory
DIt introduced the concept of entropy into classical mechanics
Einstein derived D = kT/γ (where γ = 6πηr for a sphere), linking the diffusion coefficient to temperature, viscosity, and particle size — all macroscopically measurable quantities. Jean Perrin then measured Brownian displacements under a microscope, plugged them into Einstein's formula, and extracted Boltzmann's constant k — and therefore Avogadro's number N_A = R/k. This was decisive evidence for the reality of atoms at a time when some physicists (notably Mach) still denied their existence. Brownian motion provided the atomic theory with a precise, quantitative, experimental anchor.
Question 3 True / False
Brownian motion appears random, but this is an artifact of limited measurement precision — the particle actually follows a deterministic path if you track it finely enough.
TTrue
FFalse
Answer: False
Brownian motion is genuinely stochastic, not deterministically chaotic. The particle's trajectory is continuous but nowhere differentiable — it has no well-defined velocity at any instant, and it changes direction on every timescale. This is not measurement noise; it reflects the fundamental randomness of thermal fluctuations at the molecular level. Einstein's analysis doesn't seek the trajectory but the statistical distribution of displacements, precisely because the individual trajectory is irreducibly random. The √t scaling of displacement emerges from the statistics of this randomness, not from an underlying hidden trajectory.
Question 4 True / False
According to the fluctuation-dissipation theorem, a Brownian particle in a higher-viscosity fluid will experience less random diffusion (smaller D), because the same molecular collisions that cause drag also cause random kicks — and more drag means the collisions are more damped.
TTrue
FFalse
Answer: True
The Einstein relation D = kT/γ makes this explicit: D is inversely proportional to the drag coefficient γ. Higher viscosity means higher γ (more drag), which means lower D (less diffusion). This is not a coincidence — it is the fluctuation-dissipation theorem: drag and diffusion are two manifestations of the same molecular collisions. The collisions that slow a moving particle (drag) are exactly the collisions that kick a stationary particle randomly (diffusion). You cannot have one without the other, and they are quantitatively linked by the temperature.
Question 5 Short Answer
Why does the mean squared displacement of a Brownian particle grow as t rather than t², and what does this reveal about the qualitative difference between random-walk motion and directed motion?
Think about your answer, then reveal below.
Model answer: In directed motion (constant velocity), displacement grows as t, so mean squared displacement grows as t². In a random walk, each step is independent and equally likely to go in any direction. After N steps, the displacements add as vectors — the squared magnitude of the sum of N independent random vectors grows as N (not N²), because the cross terms average to zero. Since N ∝ t, mean squared displacement grows as t. The physical consequence is that diffusion is much less efficient than directed transport over long distances: to diffuse ten times farther requires one hundred times longer. Cells exploit this by using active molecular motors for long-distance transport while relying on diffusion only for short-range delivery.
This √t vs t distinction is not just a formula — it reflects a deep difference in the correlations between successive displacements. In directed motion, each step adds coherently to the previous ones (positive correlation). In a random walk, steps are uncorrelated — each step forgets all previous steps. This is the mathematical signature of a Markov process, which Brownian motion exemplifies.