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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.

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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.

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Now suppose we have a law that, given the current state of the system, predicts the next state.Preliminaries:

Evolutionary Law:

State Machines (cont)

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a function

n an integer

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Reassurance

Don’t Panic™.

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State Law of Motion

The law of motion:(something) is a function, a mathematical machine. It has• a domain – all of the things we can

legally feed into its parentheses, and• a range – all of the things it can

possibly return to us. 6

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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.

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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. 8

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Back to Physics and States

Consider once again

Note that only knows about Then can only depend upon .

That is, if we know the current state of the system, we know everything there is to know about its future. That’s determinism.

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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 itsfuture

- Laplace10

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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.12

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Types of Infinity

Infinity comes in a variety of flavors:

• The set of natural numbers,, is countably infinite or denumerably infinite.

• Georg Cantor defined the cardinality (size) of the set of integers as , aleph-null, a transfinite number.

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Types of Infinity (cont.)

• Another type of infinity is that of the real numbers.

• Cantor called the cardinality of this set and found the remarkable result

A set like this is sometimes called a nondenumerable infinity. 14

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Nondenumerable Infinities

Rational numbers are numbers of the

form where both n and m are positive integers and Rational numbers include integers.• Decimal expansions of either

terminate or repeat.• Decimal expansions that terminate or

repeat represent rational numbers. 15

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Real Numbers

Real numbers are all the numbers on the real axis:

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0 152 𝜋

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Nondenumerable Infinities (cont.)

• Between any two members of , i.e. any two rational numbers, there are an infinite number of real numbers.

• All non-rational real numbers are irrational numbers. Almost all reals are irrational.

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Hotel Tricks with

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?

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Back to Work

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Infinite Numbers of States

• Map states onto countably infinite () integers:

• Note that every state is an integer and every integer is a state.

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Remember This?

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nowthe future

the evolutionary function

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Rules for Infinite Numbers of States is a function encapsulating the history of a sequence of states. (Same as .)

For example:

leads to

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Λ (𝑘 )=𝑘+1• lead to

which satisfies determinism.

• We can trivially show

which satisfies reversibility.

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Some More Rules

• (next slide)

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𝑁 (𝑛+1 )=(−1 )𝑁 (𝑛 )𝑁 (𝑛)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

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Here are Some Cycles of States

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871

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7We’ll see later that cyclesare associated with theconservation of something,usually energy.

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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. 27

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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.

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