Macquarie University The Heat Equation and Diffusion PHYS by Lesa Moore DEPARTMENT OF PHYSICS

Macquarie University Diffusion of Heat The diffusion of heat through a material such as solid metal is governed by the heat equation. We will not try to derive this equation. We will compare results from the heat equation with our studies of the random walk.

Macquarie University Initial Temperature Distribution Consider diffusion in 1D (let a thin copper wire represent a one- dimensional lattice). Let u(t,x) be the heat at point x at time t, with x and t integers, u(t=0,x=0)=1 and u(t=0,x)=0 if x is not zero.

Macquarie University The Partial Differential Equation The heat equation is a partial differential equation (PDE): k is the diffusion coefficient. Assume the initial distribution is a spike at x=0 and is zero elsewhere.

Macquarie University Partial Derivatives For functions of more than one variable, the partial derivative is the rate of change with respect to one variable with the other variable(s) fixed. :

Macquarie University The PDE in full :

Macquarie University Converting to a Difference Equation Don’t take the limits as intervals approach zero. Take finite time steps (  t=1) and finite positions steps (  x=1).

Macquarie University Simplifying … 1 1 1

Macquarie University Rearranging … Want all t+1 terms on l.h.s. and everything else on r.h.s.

Macquarie University Modelling in Excel Columns are x-values. Rows are t-values. The difference equation relates each cell to three cells in the row above.

Macquarie University The Excel Spreadsheet The first row (t=0) is all zeros except for the initial spike: u(t=0,x=0) = 1. The same formula is entered in every cell from row 2 down: A1 holds the value of k (k = 0.1) AA3=$A$1*(Z2-2*AA2+AB2)+AA2

Macquarie University Filling the Spreadsheet In Excel, it is easiest to insert the formula in the top left cell of the range, select the range and use Ctrl+R, Ctrl+D to fill the range: -20 ≤ x ≤ 20; 0 ≤ t ≤ 60.

Macquarie University Boundary Conditions What happens at the boundaries? Setting columns at x=±21 equal to zero stops the spatial evolution of the model – is this a problem? Provided that values in neighbouring columns (x=±20) are still small at the end of the simulation, the choice of boundary conditions is not so important. u=0 is equivalent to an absorbing boundary.

Macquarie University Snapshots

Macquarie University Plotting the Heat Spread

Macquarie University Spreadsheet Results Conservation of heat can be demonstrated by adding the values in a row (a row is a time step). Values in a row should add to 1. Checking the sum in a row is good test of numerical accuracy. Heat diffusion looks like a Gaussian distribution.

Macquarie University The Distribution The simulation satisfies conservation of energy (total heat along a row = 1). Does the Gaussian distribution satisfy this condition too (area under curve = 1)? The initial spike can be thought of as a very sharp, very narrow Gaussian. For t>0, need to integrate the Gaussian. “Normalised” if integral yields unity.

Macquarie University Normalisation of the Gaussian Formula for Gaussian with  = 0. Use a trick for the integral:

Macquarie University  The integral becomes

Macquarie University  But using and

Macquarie University  Cancelling

Macquarie University  Then use the substitution:

Macquarie University  And finally:

Macquarie University The integral proves that the Gaussian is normalised to unity – the area under the curve is one. But the heat equation is a function of x and t, and uses a constant k. k and t must be included in the  term of the Gaussian if we are to say our model satisfies this distribution.

Macquarie University What is  ? From the Random Walk, we learned that   √t. Try a guess: The Gaussian becomes:

Macquarie University Derivatives of the Gaussian Space derivatives: The time derivative is left as an exercise …

Macquarie University The Gaussian satisfies the Heat Equation It can be shown that the heat equation is satisfied by our guess. The distribution integrates to unity (conservation of energy). The spread of heat is given by  of the Gaussian (normal) distribution.

Macquarie University Diffusion and the Random Walk The initial temperature spike grows into a Gaussian distribution according to the 1D heat equation. The width  grows in proportion to the square root of elapsed time. Heat and diffusion can be understood in terms of the “random walk”.

Macquarie University Other Conditions The initial condition may not be a spike, but could be some initial distribution: u(x,0)=g(x). The boundary conditions may not be absorbing, but could be continuous. The thermal diffusivity constant k may not be constant, but may vary with x or t.

Macquarie University Summary The heat equation is a PDE. By separating space and time variables, we see that a Gaussian that spreads as √t is a solution. We can model the differential equation as a difference equation in Excel and see the same effect. The spread of heat is a physical example of a random walk.

Macquarie University Acknowledgements This presentation was based on lecture material for PHYS220 presented by Prof. Barry Sanders, Additional Reference: Folland, Fourier Analysis and its Applications, 1992.