Consider the "short time" relaxation of a free Brownian particle. The Langevin equation for the velocity is \(m \mathrm{~d} v / \mathrm{d} t=-\gamma v+\xi(t)\). (a) Find the Fokker-Planck equation for the probability, \(P(v, t) \mathrm{d} v\), to find the Brownian particle with velocity, \(v \rightarrow v+\mathrm{d} v\), at time \(t\). (b) Solve the FokkerPlanck equation, assuming that at time, \(t=0\), the velocity is \(v=v_{0}\). (Hint: Use the transform in Section Z.3.3.1 to write the Fokker-Planck in terms of a Hermitian operator. The eigenfunctions of that operator will be Hermite polynomials.)

The Fokker-Planck equation is key for describing the time evolution of the probability distribution of a system's state variables. In this context, it details how the probability density function of the velocity of a Brownian particle changes over time.

Given the Langevin equation, the first step is to determine the drift and diffusion coefficients. These coefficients help translate the stochastic differential equation into the deterministic partial differential equation known as the Fokker-Planck equation.

The drift coefficient represents the systematic part of the motion (e.g., friction), while the diffusion coefficient encapsulates the random forces acting on the particle. For our exercise:

• The drift coefficient is given by \(A_v(v) = -\frac{\gamma}{m} v\).
• The diffusion coefficient derived from the noise term \(\xi(t)\) is \(B_v(v) = \frac{1}{m} \xi(t)\).

Using these coefficients, we form the Fokker-Planck equation:
\[ \frac{\partial P(v,t)}{\partial t} = \frac{\partial}{\partial v} \left( \frac{\gamma}{m} v P(v,t) \right) + \frac{1}{2} \frac{\partial^2}{\partial v^2} \left( \frac{1}{m^2} \xi(t)^2 P(v,t) \right). \] Finally, this equation describes how the probability of finding a particle with velocity v at time t evolves. The initial condition is usually a delta function if the initial velocity is known exactly.

Brownian motion, named after botanist Robert Brown, refers to the random motion of particles suspended in a fluid. This motion results from their collision with fast-moving molecules in the fluid. In physics, it is a fundamental example of a stochastic process, characterized by random and unpredictable changes in a particle's position and velocity over time.

Analyzing a Brownian particle involves considering both deterministic forces (like friction) and random forces (stemming from molecular impacts).

• Deterministic forces often lead to a drift, modeled as a function of velocity.
• Random forces introduce a diffusion term, accounting for the spread in velocity due to stochastic impacts.

The Langevin equation describes the velocity of a Brownian particle, incorporating both these forces together. Switching from the Langevin description to a statistical description using the Fokker-Planck equation allows us to study the overall behavior of many particles over time.
Understanding Brownian motion is crucial across various scientific fields. In finance, it's used to model stock prices; in biology, to track the diffusion of molecules in cells; and in physics, it lays the groundwork for the study of stochastic processes.

Hermite polynomials arise in probability theory, particularly when dealing with Gaussian distributions and solutions to differential equations with Gaussian noise. In the context of the Langevin and Fokker-Planck equations, Hermite polynomials facilitate solving these equations through spectral methods.

Hermite polynomials, \(H_n(x)\), are orthogonal polynomials over the weight function \(e^{-x^2}\), making them suitable for problems involving Gaussian-like processes:

• The zeroth Hermite polynomial is constant: \(H_0(x) = 1 \).
• The first few polynomials include \(H_1(x) = 2x\) and \(H_2(x) = 4x^2 - 2\).
• Orthogonality condition: \[ \int_{-\infty}^{\infty} H_m(x) H_n(x) e^{-x^2} dx = 0 \quad \text{for} \ \ m e n \]

In our specific problem, the Hint suggests transforming the Fokker-Planck equation into a form where these Hermite polynomials are the eigenfunctions of a Hermitian operator. This transformation reduces the original problem into simpler parts. The final probability distribution, \(P(v, t)\), is then often expressed in terms of these polynomials.
Hermite polynomials thus form a powerful toolset for tackling complex differential equations by converting them into solvable algebraic equations.