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6.055J / 2.038J The Art of Approximation in Science and Engineering
Spring 2008
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View Full Document Chapter 9
Discretization
9.1 Diaper usage
9.2 Pendulum period
9.3 Random walks
Random walks are everywhere. Do you remember the card game War? How long does
it last, on average? A molecule of neurotransmitter is released from a vesicle. Eventually
it binds to the synapse, and your leg twitches. How long does it take to get there? On a
winter day, you stand outside wearing only a thin layer of clothing. Why do you feel cold?
These physical situations are examples of random walks. In a physical random walk, for
example a gas molecule moving and colliding, the walker moves a variable distance and
can move in any direction. This general situation is complicated. Fortunately, the essential
features of the random walk do not depend on these complicated details.
Simplify by discarding the generality. The generality arises from the continuous degrees of
freedom: the direction is continuous and the distance between collisions is continuous. So,
discretize the direction and the distance: Assume that the particle travels a ﬁxed distance
between collisions and that it can move only along the coordinate axes. Furthermore, ana
lyze the special case of onedimensional motion before going to the more general cases of
two and threedimensional motion.
In this discretized, onedimensional model, a particle starts at the origin and moves along
a line. At each tick it moves left or right with probability
1
/
2
in each direction. Let the
position after
n
steps be
x
n
, and the expected position after
n
steps be
h
x
n
i
. Because the
random walk is unbiased – because moving in each direction is equally likely – the expected
position remains constant:
h
x
n
i
=
h
x
n
−
1
i
.
So
h
x
i
, the socalled ﬁrst moment of the position, is an invariant. However, it is not a
fascinating invariant because it does not tell us much that we do not already understand
intuitively.
92
6.055 / Art of approximation
Given that the ﬁrst moment is not interesting, try the nextmostcomplicated moment: the
second moment
h
x
2
i
. This analysis is easiest in special cases. Suppose that after a while
wandering, the particle has arrived at
7
, i.e.
x
=
7
. At the next tick it will be at either
x
=
6
or
x
=
8
. Its expected squared position – not its squared expected position! – has become
h
x
2
i
=
2
1
±
6
2
+
8
2
²
=
50
.
The expected squared position increased by 1.
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This note was uploaded on 02/24/2012 for the course MECHANICAL 6.055J taught by Professor Sanjoymahajan during the Spring '08 term at MIT.
 Spring '08
 SanjoyMahajan

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