The Nature of Classical Physics A Rehash Followed by Some New Stuff 1 Nature of Classical Physics TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A

Determinism and Reversibility We may regard the present state of the universe as the effect of its past and the cause of its future. —Pierre-Simon Laplace, A Philosophical Essay on Probabilities Given the present there can be only one future and there can have been only one past. Nature of Classical Physics 2

State Machines We used the notion of state machines to provide a simple analogy to examine the structure of physical laws. The analogy is imperfect and we should be careful not to push it too far. Nature of Classical Physics 3

State Machines (cont) Nature of Classical Physics 4 a function n an integer

Reassurance Don’t Panic™. Nature of Classical Physics 5

State Law of Motion Nature of Classical Physics 6

A Function Is A Rule We stuff an object from the function’s domain, such as a number, into its parentheses, and it returns an object from its range, such as a different number. Nature of Classical Physics 7

State Functions In our case, the input to  () is one of the system’s states, so  ()’s domain is the set of system states. When we give  () a state as input, it gives us back a state as output so  ()’s range is (also) the set of system states. Nature of Classical Physics 8

Back to Physics and States Nature of Classical Physics 9

Determinism and Reversibility Say There is exactly one arrow entering each state box and exactly one arrow leaving. the present state of the universe is the effect of its past and the cause of its future - Laplace Nature of Classical Physics

Now The New Stuff Nature of Classical Physics 11

A Short Digression on Infinities It’s good to be a tourist: we get to watch the slide shows and skip the final. Infinities make for fascinating pillow talk. I think it’s interesting. Nature of Classical Physics 12

Types of Infinity Nature of Classical Physics 13

Types of Infinity (cont.) Nature of Classical Physics 14

Nondenumerable Infinities Nature of Classical Physics 15

Real Numbers Real numbers are all the numbers on the real axis: Nature of Classical Physics 16 01

Nondenumerable Infinities (cont.) Nature of Classical Physics 17

Suppose we own a hotel with the countably infinite rooms numbered 1, 2, …. Suppose our hotel is full. 1.How would you add one guest? 2.How would you add a countably infinite number of guests? Nature of Classical Physics 18

Back to Work Nature of Classical Physics 19

Infinite Numbers of States Nature of Classical Physics 20

Remember This? Nature of Classical Physics 21 now the future the evolutionary function

Rules for Infinite Numbers of States Nature of Classical Physics 22

Nature of Classical Physics 23

Some More Rules Nature of Classical Physics 24

Easiest to work out specific cases. Then we find 0 a static state +1, -1 a cycle 2 a static state -2 a static state +3, -3 a cycle et cetera et cetera et cetera Nature of Classical Physics 25

Here are Some Cycles of States Nature of Classical Physics We’ll see later that cycles are associated with the conservation of something, usually energy.

Reversibility and Precision States of “real” systems will be associated with continuous, real values. Capturing the complete state of the system at some instant will require specifying a set of real numbers with unlimited precision. The effects of finite precision will be to introduce error in the backward computation. Nature of Classical Physics 27

Reversibility and Precision (cont.) The cumulative effect of error when we run a system backwards leads to chaotic behavior, in both the technical and practical senses. We’re going to leave this topic for now. Nature of Classical Physics 28

Here Ends This Part Nature of Classical Physics 29