A Brownian particle is confined in a harmonic potential V = ½kx² and coupled to a thermal reservoir. After a long time, what does the Fokker-Planck equation predict for the stationary probability distribution P(x)?
AA uniform distribution across all positions, since noise can push the particle anywhere
BA delta function at x = 0, since the potential minimum attracts all trajectories
CThe Boltzmann distribution P_eq ∝ exp(−V/k_BT), a Gaussian centered at x = 0
DNo stationary distribution exists — thermal noise prevents the system from settling
The Fokker-Planck equation with harmonic drift A(x) = −kx/γ and thermal diffusion has a stationary solution where drift and diffusion balance: P_eq ∝ exp(−kx²/2k_BT) = exp(−V/k_BT). This is the Boltzmann distribution, confirming that stochastic dynamics with thermal noise correctly recovers equilibrium statistical mechanics.
Question 2 Multiple Choice
What physical effect does the drift term −∂/∂x[A(x) P] represent in the Fokker-Planck equation?
ARandom spreading of the probability distribution due to thermal fluctuations
BDeterministic flow of the probability density under a systematic external force
CDecay of total probability over time as particles escape the system
DCoupling between the velocity and position degrees of freedom
The drift term is a continuity equation: −∂/∂x[A(x) P] = −∇·(AP), describing how probability flows with the deterministic velocity field A(x). If all particles are pushed right by a force, the probability density shifts right. The diffusion term ∂²/∂x²[D P] separately handles the stochastic spreading.
Question 3 True / False
The Fokker-Planck equation is a deterministic partial differential equation, even though it describes an inherently stochastic process.
TTrue
FFalse
Answer: True
This is the core advantage of the Fokker-Planck approach. Although the underlying dynamics are random, the probability density P(x,t) evolves deterministically according to a PDE. This lets you apply PDE machinery (Green's functions, eigenfunction expansions, numerical solvers) to a problem that started as a stochastic one.
Question 4 True / False
The Fokker-Planck equation and the Langevin equation describe the same stochastic system at the same level of description — they are just different notations for the same mathematical object.
TTrue
FFalse
Answer: False
They describe the same physics but at different levels. The Langevin equation describes individual stochastic trajectories x(t), requiring noise terms and statistical averaging. The Fokker-Planck equation describes the probability density P(x,t) directly — a deterministic PDE. Deriving the Fokker-Planck from the Langevin requires computing moments of the displacement and taking a continuum limit.
Question 5 Short Answer
A free Brownian particle starts at x = 0 at t = 0 (so P(x,0) = δ(x)) with no external force. Using the structure of the Fokker-Planck equation, describe qualitatively how P(x,t) evolves over time.
Think about your answer, then reveal below.
Model answer: With no external force, A = 0 and the drift term vanishes. The Fokker-Planck equation reduces to the diffusion equation ∂P/∂t = D ∂²P/∂x². Starting from a delta function at the origin, P(x,t) spreads into a Gaussian with mean 0 (no drift) and variance 2Dt growing linearly in time. The probability density stays symmetric and centered at x = 0 but broadens indefinitely.
The diffusion term spreads the distribution; without drift, the mean stays fixed at 0. The solution P(x,t) = (4πDt)^{−1/2} exp(−x²/4Dt) is the fundamental solution of the diffusion equation — a broadening Gaussian. This matches the physical picture of random-walk diffusion.