Computational Physics/Chapter 1

The Spreadsheet Program
This book will describe how specifically to use Microsoft Excel, as this is a widely used program. If you have access to this and are farmiliar with it is suggested that you use it. If you have another preferred spreadsheet program please feel free to use that, however be aware that you will have to refer to the help section where your spreadsheet program differs from this book.

If you would like a free spreadsheet program that works with nearly any operating system then 'Calc' from OpenOffice.org is highly recommended. Hopefully in the future this book will describe how to use this as well as Microsoft Excel.

Why use a spreadsheet? This chapter aims to show you that you can do things quickly and simply without having to learn and type excessive amounts code. We will see towards the end that using a spreadsheet has its limitations, but hopefully also the power of spreadsheets other than for just plotting graphs of your experimental results!

Using Spreedsheets to calculate the Numerical Solutions of Ordinary Differential Equations
If x is a function of a real variable t and f is a function of both x and t, then the equation

x’(t) = f(x(t), t) 		(1.1)

is called a first order differential equation. Solving such an equation involves more than algebraic manipulation; indeed, although the equation itself involves three quantities, x, f, and t, to find a solution we must identify a function x, defined solely in terms of the independent variable t, which satisfies the relationship of (1.1) for all t in some open interval. For many equations, exact solution is not possible and we have to rely on approximations. We will discuss one technique for finding both approximate and, where possible, exact solutions to differential equations. We have already seen many examples of differential equations we discussed finding the position of an object moving in a straight line given its velocity function and its initial position, when we discussed the motion of a projectile, when we considered the two-body problem. Indeed, in many ways the study of differential equations is at the heart of calculus. To study the interaction of physical bodies in the world is to study the ramifications of physical laws such as the law of gravitation and Newton’s second law of motion, laws which frequently lead questions involving the solution of differential equations. Newton was the first to realize the power of calculus for solving a vast array of physical problems. The mathematicians that followed him enlarged and refined his techniques until they began to believe that the entire future of the universe, as well as its past, could be discerned from the knowledge of the current positions and velocities of all physical bodies and the forces at work between them. Today we know more about the limits to our knowledge, but, nevertheless, the study of differential equations remains a key component to our understanding of the universe. To begin our study, consider the equation

x’(t) = f(x(t), t)		(1.2)

with the initial condition x(t0) = x0. To simplify notation, we will frequently omit the independent variable when referring to x(t) and write simply

x’ = f(x, t)		 (1.3)

Now if f(x, t) depends only on the value of t, that is, if f(x, t) = g(t) for all values of x, where g is a function of t alone, then we may solve (1.3) by integration. That is, integrating both sides (1.3) gives us

x(t) - x(t0) =   integral of [g(t), over [t,to]]	(1.4)

which gives

x(t) = x(t0) + integral of [g(t), over [t,to]]	(1.5) Assuming we can compute the definite integral, which can at least always be done numerically, we have solved the differential equation.

For Example

Consider the equation x’ = 4 sin(3t)

with initial condition x(0) = 5. Then

x(t) = 5 - 4/3 cos3t + 4/3 = 1/3 (19 - 4 cos(3t)).

Knowing the value of t we can calculate the exact value of x at t.

Indeed, knowing the difficulty of evaluating ordinary definite integrals, it is not hard to believe that many, if not most; differential equations may be solved only through numerical approximation. Although the function x in equation (1.3) is unknown, we do have enough information to find its best affine approximation at t0. Namely, the best affine approximation to x at t0 is obtained by the Taylor theorem using subinterval [t0,t1] of an entire interval [t0,t].

x(t1) = x0 + f(x0, t0)(t - t0) + 0((t - t0)2 +...). where t1 = t0+h 	(1.6) Equivalently,

x(t0 + h) = x0 + hf(x0, t0)		(1.7)

Thus for a small value of h,

x1 = x0 + hf(x0, t0)			(1.8)

Where h = t1 – t0/n [in the above case n = 1]

Will provide a good approximation to x(t0 + h). However, we want to do more than this; since x is a function, we want to be able to approximate its values over an entire interval, say [t0, t]. To do so, we choose a small value for h and iterate the process that gave us x1. Specifically, we let t = t0 + nh, n = 0, 1, 2. 3,…, where n is chosen large enough and the generalized form of the Euler method is

xn+1 = xn + hf(xn, tn) 			(1.9)

That is, we compute an approximation for x(tn+1) by applying the best affine approximation to our approximation for x(tn), repeating the process enough times until we have approximated x over the entire interval [t0, t]. This method of approximation is known as Euler’s method.

Euler’s method
To approximate a solution to the equation

x’ = f(x, t)

with initial condition x(t0) = x0 on an interval [t0, t], choose a small value for h > 0 and an integer n such that t0 + nh = t. Letting t = t0 + nh, compute x1, x2,. . ., xn+1 using the difference equation xn+1 = xn + hf(xn, tn). (1.10). Then xn+1 is an approximation for x(t0 + nh).

Consider the differential equation x’ = 0.04x with initial condition x(0) = 50. We know that the solution to this equation is

x(t) = 50e 0.04t.

In particular, x(50) = 50e 2 = 369.45, rounding our answer to the second decimal place. To approximate x on the interval [0, 50] using Euler’s method, we will first take h = 1 and, starting with x0 = 50, compute x1, x2. . . x50, where in this case xn+1 = xt will approximate x(t). Using (1.9) with f(x, t) = 0.04x and rounding to two decimal places, we have

n = 1:x1 = x0 + h(0.04x0) = 50 + (1)(0.04)(50) = 50 + 2 + 52.00

n = 2:x2 = x1 + h(0.04x1) = 52 + (1)(0.04)(52) = 52 + 2.08 = 54.08

n = 3: 	x3 = x2 + h(0.04x2) = 54.08 + (1)(0.04)(54.08) = 54.08 + 2.16 = 56.24

and so on, ending with x50 = 355.33.

Notice that the error in our approximation, that is, the difference between xn and x(n), increases as n increases. For example,

x(5) - x5 = 0.24, whereas

x(50) - x50 = 14.12.

This should be expected since the error of the approximation on each step is compounded by the errors made in each of the preceding steps. The only way we can control the amount of error in our approximations is to decrease the step size. For example, if we reduce the step size to h = 0.1, we obtain

x1 = x0 + h(0.04x0) = 50 + (0.1)(0.04)(50) = 50 + 0.2 = 50.2000, x2 = x1 + h(0.04x1) = 50.2 + (0.1)(0.04)(50.2) = 50.4008, x3 = x2 + h(0.04x2) = 50.4008 + (0.1)(0.04)(50.4008) = 50.6024, ………………………………………………………………….	……………………………………………………………….. and ending with

x500 = 367.98 (This can be easily calculated using computers)

We have to calculate the value of x500 and take it equivalent to x50 because from the definition of the parameter h

h = (t – t0)/n

putting h=0.1 clearly the value of n, which actually is the number of iterations to be made, becomes 500 for t0 = 0 and t = 50. [Alay]

Using Spreadsheets to calculate the Numerical Solutions of Integrals
In calculus we come across very complicated and difficult integrals. They are difficult to solve using analytical methods for finding their numerical solutions. This problem can be solved by using numerical integrations e.g. Rectangular rule, limit of Sum definition, Trapezoidal Rule and Simpson’s Rule.

The Trapezoidal Rule
Consider y = f[x] bounded by x=a to x=b. Than by definition integration of the function of x i.e. y w.r.t x is the area under the curve AB shown in figure

Area = F[x] = integral [f(x), over[a,b]dx

now dividing the trapezoid into a Rectangle and an approximate triangle i.e. number of rectangles n = 1 we can write F[x] = h f[a] + ½ h {f[b] – f [a]}

= h {f[a] +1/2f[b] – 1/2f[a]}

= h/2 {f[a] + f[b]} = h/2 {f[x0] + f[x1]}

for n = 2 we have two rectangles and two triangle such that the approximate area under the curve now become

F[x] = h/2 {f[a] +f[{a+b}/2]}+ h/2 {f[{a+b}/2] + f[b]}

= h/2 {f[a] + 2f[{a+b}/2] + f[b]}

= h/2 {f[x0] + 2f[x1] + f[x2]}

So generalizing for n divisions of the trapezoid we have

F[x] = h/2 {f[x0] + 2f[x1] + 2f[x2] + … + 2f[xn-2] + 2f[xn-a] + f[xn]}

Is an approximation of the integral of y = f[x].

Where h = b – a/n

n can be even or odd. And approximation can be improved using large value of n. [Alay]

The Random Walk


This is a book for physicists, so let's get right in and describe the physics we want to investigate. The random walk is a problem that occurs in various physics areas such as Brownian motion, which describes how molecules move about in a fluid (a fluid can either be a liquid or a gas). It also plays a role in quantum field theory, but this is much more complicated. Outside of physics the random walk can be applied to share prices. It is interesting that by investigating one problem we find solutions to entirely different ones, that have no other relationship to each other. This is something that occurs throughout physics, such as the way in which waves can be used to model traffic flows.

So what is a random walk? Imagine a drunk guy (yourself after a good night out if you like) who has just left their local bar. He is standing outside the bar and, being rather drunk, he randomly takes one step North up the street or one step South down the street. The probability of him going North or South is equal and all his steps are of the same length. If he does this for 100 steps how far on average has he got away from the bar? You may think on average he ends up very close to the bar, but actually this is not so! (Remember, distance means how many meters is he from the bar, and doesn't take account of where he is. If we found his average position treating North as positive, and South as negative, then by the symmetry of the situation we know that he would indeed end up outside the bar on average.)

We will investigate this problem and find how far he actually ends up away. This is a known as a one-dimensional random walk, as the drunkard walks only North or South. The same ideas can be applied in two dimensions, say a drunk in a field who staggers North, South, East or West. Three dimensions can similarly be investigated, and even four or more, although this is a bit of an abstract idea!

Can you see now how this links to Brownian motion (where molecules move randomly) or to share prices which have approximately equal chances of rising or falling? [John]

Making a one dimensional random walk
On a blank worksheet in your spreadsheet double click on cell B4 and type =RAND and press enter. In cell B4 you should now see a decimal number between 0 and 1. Try hitting the F9 key a few times, which recalculates the entire spreadsheet. You should see numbers being randomly generated in cell B4. By the way cell B4 is specified so there can be some consistency when referring you to specific cells. It will also allow the spreadsheet to be neatly formatted later if you're so inclined! On the subject of formatting the whole random number may not be showing currently, if you click on cell B4 so that it is highlighted and then go to the format menu, select column and then autofit selection the entire contents of the cell is shown. This can be done for the entire selecting the whole sheet, you can do this by clicking the grey box at the top left of the sheet,

So, why did we type what we typed? You may well know if you're familiar with spreadsheets, = specifies that we're typing a mathematical function and not just text. RAND is a function that the spreadsheet recognises as a request to generate a random number. If this isn't clear right now it will become clear as we work through the rest of the chapter.

How do we transform this into a random walk? We need to transform this random number into an event with a 50:50 probability. What we need to do is generate a step South if the number is less than 0.5 and a step North if the number is more than 0.5. This is where we see how a simple spreadsheet can be transformed into a program.

Double click on cell C4 and type: =IF(B4<0.5,-1,1). This was a bit more of a complicated function than the previous! What the IF function does is to set the value of a cell to one value if the condition specified is true, or another value if the condition is false. The structure of this function goes like this: IF(Condition, Value if True, Value if False). B4<0.5 is the condition, this means 'if cell B4 is less than 0.5'.

You should now be able to work out what values of the random number will make it take a step South (generating a -1) or a step North (generating a 1). If not try it and find out! It should work the same way as it was said it would a bit earlier.

Taking one step isn't particularly exciting or impressive (or if you think it is you're going to absolutely love the rest of the book!) so we'll see how we can expand it to 100 steps. It's fairly easy actually, simply click on cell B4 so it is selected and the move the mouse over the little square in the bottom right corner. The mouse cursor will change and you can then drag down to fill the cells below with the same formulae as in this cell. Drag down to cell B103 so that we will be taking 100 steps. It is even easier to fill the cells in column C, all you need do now is double click on the little black square in the bottom corner of cell C4, this will fill it down to the same row as column B is filled down to. Notice that in cell C4 we have a function that relies on the value of the function in B4. But when we fill the other cells they automatically change so that the function depends on the value of the cell in the left. This is usually very handy, but we'll see in a bit that sometimes we need to override it.

Now we can hit F9 and each time generate a random walk! But it's still not very interesting to look at, or a very informative, we can't see where the drunkard is unless we add up all the cells at the moment, nor can we see where he is it any particular position. This can be easily corrected, enter into D4 the function SUM(C$4:C4). Hopefully you can guess what this does, it adds up (sums) all the cells between C4 and, well, um C4. Not too useful as it is you might think, and why is that $ sign there? Well, if you fill down into the cells below you'll see that each cell adds up the total of all the steps taken, and gives the current position of the drunkard after this step. If you look at cell D20 you should get an idea of what's happening, this cell sums between C4 and C20. The $ sign acts to fix a value when you fill cells below.

Now you can use the spreadsheet to make a graph of the random walk, if you don't know how to do this check the help section of your spreadsheet. As a couple of hints to make the graph look good put a '0' in cell D3 so that the drunkard can start outside the bar and select a line graph. Press F9 to generate new graphs.

Do the graphs you've created look like what you would expect, do they actually look random? You'll probably see some strange patterns in some graphs, but they are honestly random! This will hopefully make you think about what a random pattern truly looks like. [John]

How far does the drunk get?
Try and work this out while this section is written!