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In the last post, I demonstrated that even if Marty McFly went back in time for just one second, just his gravitational influence would irrevocably change the future. I then claimed that this is actually overkill: that even small changes at interstellar distances will affect things on Earth. Let's see about that.

(Most of this post is a walk-through of a calculation. Jump to the end if you just want to see the conclusion. The conventions in this post are the same as in the last post. All numbers and equations are in mks units. The powers of 10 for scientific notation is indicated following 'e'.)

The distance to the Andromeda Galaxy is 2.5 million light-years. That's 2.4e22 meters.

Consider a single water molecule somewhere in that galaxy. Water is a polar molecule - that is, it has positive and negative parts, separated by some distance. This creates an electric dipole moment. For a single water molecule, this dipole moment has a magnitude of 6.2e-30 coulomb-meters.

Now, dipole moments exert an electric field that depends on their spatial orientation. The full equation for that electric field can be found at any number of places, but we're really only interested in how much this field can change depending on the orientation. It turns out that it can change by about 1/(4 π ε0) * p/r^3. "p" is the magnitude of the dipole moment, "r" is the distance from that dipole, and ε0 is the permittivity of free space; it's a physical constant with a value of 8.85e-12. Plugging in all these numbers, we get a value of 4.0e-87 newtons/coulomb as the electric field.

So, depending on the orientation of a single water molecule in the Andromeda Galaxy, the electric field on Earth can change by about as much as 4.0e-87 newtons/coulomb. That electric field will exert a force of 6.4e-106 newtons on an elementary electric charge here on Earth.

Now, let's take a moment to appreciate how minuscule this number is. Some mathematical tools will simply not compute a number this small, and just display it as "0". It is on the order of a millionth of a googolth of a newton. After all, what can we expect from changing the orientation of a single water molecule in the Andromeda Galaxy? How can that possibly affect us on Earth?

If this force acts on an ionized air molecule - say, a singly ionized nitrogen molecule - that would cause an acceleration of F/m = 1.4e-80 m/s^2. If this acceleration acts for 1.4e-10 seconds (0.14 nanoseconds), it will move this ion by 1.4e-100 meters. This, then, will affect the future collisions of this ion with other air molecules.

So, to review: we are changing the orientation of a single water molecule, somewhere in the Andromeda Galaxy, for 0.14 nanoseconds. We are considering how this would affect the Earth. So far, we've determined that it would move an ionized nitrogen molecule by 1.4e-100 meters (1.4 googolth of a meter), which would then influence the future collisions of that molecule.

We'll model the collisions as if they were classical collisions with hard spheres - basically, as if the molecules were billiard balls. In such collisions, changing the position of the incoming billiard ball by x causes it to bounce off at a different angle. That difference, under some simplifying assumptions, is given by 2 x/d, where d is the diameter of the "billiard balls". In our case these are air molecules, which have a diameter of about 4e-10 meters. So, after bouncing off, our original ionized molecule is now moving at a different angle - one that's 2 x/d = 6.9e-91 radians away from the direction it would have otherwise bounced to. This difference in angle will then cause it to run into the next air molecule from a different position, which will then cause another difference in bouncing angle, and so on and so forth, in an iterative fashion. Eventually, after enough bounces, the deflection will be large enough that our ionized molecule will entirely miss another molecule that it would have otherwise hit. How many collisions does this take? I wrote a small bit of code in Python to find out, using typical values for the relevant conditions for air (such as the time between collisions, or the path length that a molecule travels between collisions):

acceleration = 1.4e-80
path_length = 7e-8
time = 1.4e-10
diameter = 4e-10

n = 0
impact_parameter = 0.5*acceleration*time**2

def one_collision(incoming_d_impact_parameter):
    new_deflection = 2*incoming_d_impact_parameter/diameter
    new_d_impact_parameter = new_deflection*path_length
    return {
        "deflection":new_deflection, 
        "impact_parameter":new_d_impact_parameter}

collision = True
while collision:
    result = one_collision(impact_parameter)
    impact_parameter = result["impact_parameter"]
    n += 1
    out_string = 
        "collisions:" + "{:2d}".format(n) + 
        "  impact_parameter:" + "{:.2e}".format(result["impact_parameter"]) + 
        "  deflection:" + "{:.2e}".format(result["deflection"])
    print out_string
    if result["impact_parameter"] > diameter:
        collision = False
        print "done in " + "{:2d}".format(n) + " collisions"

Running this code gives the following output:

collisions: 1  impact_parameter:4.80e-98  deflection:6.86e-91
collisions: 2  impact_parameter:1.68e-95  deflection:2.40e-88
collisions: 3  impact_parameter:5.88e-93  deflection:8.40e-86
collisions: 4  impact_parameter:2.06e-90  deflection:2.94e-83
collisions: 5  impact_parameter:7.21e-88  deflection:1.03e-80
collisions: 6  impact_parameter:2.52e-85  deflection:3.60e-78
collisions: 7  impact_parameter:8.83e-83  deflection:1.26e-75
collisions: 8  impact_parameter:3.09e-80  deflection:4.41e-73
collisions: 9  impact_parameter:1.08e-77  deflection:1.54e-70
collisions:10  impact_parameter:3.78e-75  deflection:5.41e-68
collisions:11  impact_parameter:1.32e-72  deflection:1.89e-65
collisions:12  impact_parameter:4.64e-70  deflection:6.62e-63
collisions:13  impact_parameter:1.62e-67  deflection:2.32e-60
collisions:14  impact_parameter:5.68e-65  deflection:8.11e-58
collisions:15  impact_parameter:1.99e-62  deflection:2.84e-55
collisions:16  impact_parameter:6.96e-60  deflection:9.94e-53
collisions:17  impact_parameter:2.44e-57  deflection:3.48e-50
collisions:18  impact_parameter:8.52e-55  deflection:1.22e-47
collisions:19  impact_parameter:2.98e-52  deflection:4.26e-45
collisions:20  impact_parameter:1.04e-49  deflection:1.49e-42
collisions:21  impact_parameter:3.65e-47  deflection:5.22e-40
collisions:22  impact_parameter:1.28e-44  deflection:1.83e-37
collisions:23  impact_parameter:4.48e-42  deflection:6.39e-35
collisions:24  impact_parameter:1.57e-39  deflection:2.24e-32
collisions:25  impact_parameter:5.48e-37  deflection:7.83e-30
collisions:26  impact_parameter:1.92e-34  deflection:2.74e-27
collisions:27  impact_parameter:6.72e-32  deflection:9.60e-25
collisions:28  impact_parameter:2.35e-29  deflection:3.36e-22
collisions:29  impact_parameter:8.23e-27  deflection:1.18e-19
collisions:30  impact_parameter:2.88e-24  deflection:4.11e-17
collisions:31  impact_parameter:1.01e-21  deflection:1.44e-14
collisions:32  impact_parameter:3.53e-19  deflection:5.04e-12
collisions:33  impact_parameter:1.23e-16  deflection:1.76e-09
collisions:34  impact_parameter:4.32e-14  deflection:6.17e-07
collisions:35  impact_parameter:1.51e-11  deflection:2.16e-04
collisions:36  impact_parameter:5.29e-09  deflection:7.56e-02
done in 36 collisions

"impact_parameter" is the difference in position as our ion approaches the next molecule. You see that, after the 36th collision, this value exceeds the diameter of a molecule, causing our ion to miss a collision that would have otherwise taken place. Since air molecules make a collision about every 1.4e-10 seconds, these 36 collisions only take 5.0e-9 seconds - that is, 5 nanoseconds.

The rest of the story is similar to how they occurred in the last post. A missed collision means that the states of the two molecules that would have been involved are now completely different, completely changed from what they would otherwise be. Then each of their future collisions also completely changes the states of these new molecules they run into. This means that with each round of collisions, the number of affected air molecules doubles. In far less than a second, all of the molecules around the original ion are affected. Furthermore, this is happening starting from every single ionized molecule in Earth's atmosphere - so in no time at all, the microstate of Earth's entire atmosphere is completely changed. This then leads to macroscopic changes (like Marty McFly never being born) soon enough.

So, that is the magnitude of the future's unpredictability. That is the degree of the universe's inter-connectivity and complexity. A SINGLE WATER MOLECULE, located in ANOTHER GALAXY, changing its orientation for a FRACTION OF A SECOND, will completely alter the microstate of the Earth's atmosphere nearly instantly (after accounting for the speed-of-light delay), and change our macroscopic future in short order. Your life, brain, and very existence is the result of all these effects combining to create you in your current state. "Subtle is the LORD" indeed.

Next week, we'll discuss a different way of thinking about the future.