MTH3007b Lecture 8

Me, in the lecture

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This session continues from lecture 7 by applying both threads - the diffusion equation and stochastic processes - to more interesting setups. We look at a singular initial condition for the diffusion equation and introduce the first-passage time problem for the OU process.

Diffusion with a Dirac Delta Initial Condition

Instead of a smooth initial profile, suppose all the “heat” (or probability mass) starts concentrated at a single point $x=x_M$ at $t=0$. The initial condition is then a Dirac delta function:

$$u(x,0)=\delta (x-x_M)$$

The Dirac Delta Function

The Dirac delta function $\delta (x-x_M)$ is defined by two properties:

• It is zero everywhere except at $x=x_M$, where it is a spike.
• Its integral over any interval containing $x_M$ equals 1:

$${\int }_{-\infty }^{\infty }\delta (x-x_M)dx=1$$

The sifting property states that for any smooth function $f$:

$${\int }_{-\infty }^{\infty }f(x)\delta (x-x_M)dx=f(x_M)$$

Discrete Approximation

On a grid with spacing $\Delta x$, the Dirac delta is approximated by setting the grid value at $x_M$ to $1/\Delta x$ and all other values to zero:

$$u[x_M]=\frac{1}{\Delta x},u[x_i]=0\text{ for }{x}_i\ne x_M$$

This ensures the discrete sum approximates the integral: ${\sum }_{i}u[x_i]\Delta x=1$.

Analytical Solution

The analytical solution to the diffusion equation with a delta function initial condition is a Gaussian:

$$\rho (x,t)=\frac{1}{\sqrt{4\pi \alpha t}}\exp \left(-\frac{x^2}{4\alpha t}\right)$$

Running FTCS with the discrete delta IC recovers this spreading Gaussian numerically, making it a good verification test for the code.

Stochastic Processes: Continued

The following topics continue from lecture 7. Recall that a a random variable takes values drawn from a probability distribution, and that the Wiener and OU processes are built from Gaussian increments $Z\sim \mathcal{N}(0,1)$.

Recap: Gaussian Random Numbers

If $X\sim \mathcal{N}(0,1)$, then $kX\sim \mathcal{N}(0,k^2)$. In Python: np.random.normal(mu, sigma, N).

Recap: Wiener and OU

• Wiener process: $W(t+\Delta t)=W(t)+\sqrt{\Delta t}Z$
• Ornstein-Uhlenbeck process: $V(t+\Delta t)=(1-k\Delta t)V(t)+\sqrt{\Delta t}Z$

First-Passage Time

The First-passage time is the time at which a stochastic process first reaches or exceeds a threshold $b$. For the OU process $V(t)$:

1. Run the OU simulation with the standard update: $V(t+\Delta t)=(1-k\Delta t)V(t)+\sqrt{\Delta t}Z$.
2. At each step, check whether $V(t)\ge b$.
3. When the condition is met, record the elapsed time $T$ and stop that simulation.
4. Repeat for many independent realisations (walkers).
5. Compute the average first-passage time $⟨T⟩$ from the ensemble.

This is purely a simulation exercise - run the process, apply the stopping criterion, and average the results.

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Pre-Lecture Notes from University Notes

• Dirac delta IC: spike at $x_M$ with unit area; discrete approximation $u[x_M]=1/\Delta x$, rest zero.
• Sifting property: $\int f(x)\delta (x-x_M)dx=f(x_M)$.
• Analytical solution with delta IC is a Gaussian: $\rho =(4\pi \alpha t{)}^{-1/2}\exp (-x^2/(4\alpha t))$.
• Running FTCS with the delta IC numerically reproduces this Gaussian spread.
• First-passage time: run OU, stop when $V\ge b$, record time; average over many walkers.
• Next session: FTCS stability analysis and the implicit BTCS scheme.