Question

The idea in this problem is to derive the solution to the one-dimensional diffusion equation for a point source, given by Equation \(13.32 .\) The tools we invoke in this problem may seem heavy-handed on the first try, but illustrate a bevy of important ideas from mathematical physics. (a) Take the Fourier transform of the diffusion equation by transforming in the spatial variables to obtain a new differential equation for \(\tilde{c}(k, t).\) (b) Solve the resulting differential equation for \(\tilde{c}(k, t) .\) Then compute the inverse Fourier transform to arrive at the solution in real space, \(c(x, t).\) (c) Show that the solution for an arbitrary initial concentration distribution \(c(x, t=0)\) can be written as an integral over the solution for a point source. In particular, consider an initial concentration profile of the form \(c(x, 0)=c_{0}\) for \(x<0\) and \(c(x, 0)=0\) for \(x>0\) and find the resulting diffusive profile. (d) Formally derive the relation \(\left\langle x^{2}\right\rangle=2 D t.\)

Short answer

The solution to the one-dimensional diffusion equation is obtained through a series of steps involving Fourier transform, subsequent solution of the transformed differential equation, Inverse Fourier transform of the obtained solution, derivation of solution for an arbitrary initial concentration profile and a formal proof of the relation \(\left\langle x^{2}\right\rangle=2 D t.\)

Step-by-step solution

Perform Fourier transform

Applying Fourier transform to the diffusion equation, we transform in the spatial variable to obtain a new differential equation. This transformation gives us the frequency domain representation of the diffusion equation (frequencies are denoted by k).

Solve differential equations

Now solve this new differential equation for \(\tilde{c}(k, t)\), which is the Fourier transform of the concentration c.

Perform Inverse Fourier Transform

After obtaining \(\tilde{c}(k, t)\), now calculate the inverse Fourier transform to return to the original spatial domain and get the solution in real space, \(c(x, t)\).

Solve for an arbitrary initial concentration distribution

The next step is to show that the solution for an arbitrary initial concentration distribution \(c(x, 0)\) can be written as an integral representation over the solution for a point source. To do this, consider an initial given concentration profile and find the resulting diffusive profile. Note that integrating based on initial particle concentration will give us the probability distribution function at a later time instant.

Derive Diffusive Property

Next, derive the relation \(\left\langle x^{2}\right\rangle=2 D t\). This relation is key to understanding the fundamental property of diffusion, which governs the mean squared displacement of particles.

Key concepts

Fourier Transform

The Fourier transform is a powerful mathematical tool used to analyze and solve differential equations, particularly in the field of physics and engineering. It transforms a function of time or space into a function of frequency, enabling us to work with signals or systems in the frequency domain.

When applied to the diffusion equation, the Fourier transform simplifies the problem by converting the spatial variable, that is the position, into a frequency variable often represented by the symbol 'k'. This step effectively 'diagonalizes' the differential operator, making the equation easier to solve because the new equation is now in terms of \(k\) and not the original space variable. After solving the new equation in the frequency domain, the inverse Fourier transform is taken to convert the solution back into the original space domain, providing us with a comprehensive insight into how a substance diffuses over time.

Differential Equation

A differential equation is an equation that involves the derivatives of a function as well as the function itself. These derivatives represent rates of change, which are crucial for describing various physical phenomena, including motion, growth, decay, and diffusion.

In the context of the diffusion equation, the differential equation describes how the concentration of a substance changes in space and time. To solve this type of equation, we employ the Fourier transform to convert the differential equation into an algebraic one, which is often much simpler to handle. Once the transformed equation is solved, the inverse Fourier transform helps in retrieving the original concentration profile, \(c(x,t)\), showing the spatial distribution of the substance at any given time.

Initial Concentration Distribution

The initial concentration distribution refers to the starting spatial arrangement of a substance before it begins to diffuse. Understanding and describing this distribution is essential since it serves as the initial condition for solving the diffusion equation.

The solution to the diffusion equation often depends on these initial conditions, which might vary from one scenario to another. For instance, in the textbook problem, we are asked to find the diffusive profile that results from an initial concentration of \(c_0\) for \(x < 0\) and \(0\) for \(x \geq 0\). This kind of step-function initial condition can be challenging, but with Fourier transforms, we can express the final concentration distribution at any later time as an integral over the point source solution — a technique that proves to be very effective for more complex initial conditions.

Mean Squared Displacement

Mean squared displacement (MSD) is a statistical measure that represents the average square distance that a particle travels from its original position over time in a diffusion process. It is a central concept in understanding how substances spread out in space due to random motions.

The relationship \(\langle x^2 \rangle = 2Dt\) is a cornerstone in the study of diffusion, encapsulating that the mean squared displacement is directly proportional to time for a diffusive process. Constant \(D\) is the diffusion coefficient, which quantifies how quickly or slowly the diffusion occurs. This linear relationship with time is a signature characteristic of Brownian motion and diffusion processes, and it provides us with a quantifiable measure of the spread of the diffusing substance over time.