Monday 21 March 2016

Review of The Theoretical Minimum - Chapter 1

This post is really the first post where I will use my blog for what I intent, and that is to review the things I have learned, or should I say relearned.

I have worked through Chapter 1 of Leonard Susskind first book in his "Theoretical Minimum" series. The aim of these books, along with his lecture videos, is to give the reader the minimum amount of physics knowledge to understand cutting edge physics. The first book covers classical physics and in the first chapter he sets out a number of topics. These include a description of what is meant by classical physics, a description of simple dynamical systems and their space-of-states,  dynamical laws which are allowable along with those that are not, how cycles within a systems space-of-states lead to conservations laws and finally how the initial conditions of system can never be know with infinite accuracy and therefore measurements carried out upon a system will always be subject to uncertainty.

What is Classical Physics?

Susskind sets out that classical physics/mechanics is used to describe physical phenomena for which a quantum uncertainties need not be accounted for. These include Newton's Laws of motion, Maxwell's and Faradays laws of electromagnetism, and Einstein's Special and General theory's of Relativity.

He also alludes that the job of classical mechanics is to predict the future. However, to predict the future every thing about a system at a given point in time needs to be known and how the system will change with time. He also sets out that the laws by which systems change with time must be deterministic and reversible, that is not only must they be able to predict the future, they must also be able to be used to determine what happened in the past.

Simple Dynamical Systems and Space of States

Two types of system are set out, where is system is simply a collection of objects:
  • Closed system - either the entire universe or a collection of objects that are some remote that they behave as if nothing else exists
  • Open system - A collection of objects which are open to external influence.
What is meant by deterministic and reversible laws are set out using some very elementary examples of systems. A coin which is affixed to a table, which obviously can't change state, as its space of state is very narrow, containing just the one state.  The law which describes how this system changes with time is both perfectly deterministic and reversible. The coin remains in the same state for all time, it has no choice.

A coin that is free to flip has a slightly wider space of states, that is, it has two states which is either heads or tails. There are many laws which could be set out to define how this system evolves with time however, two of the simplest are used. The first is if the coin starts at heads, it flips to tails, and if it is tails it flips to heads. The second is if the coin starts at tails, it flips to heads and if it is heads it flips to tails. (These two laws are in fact the same)

The laws which describe the time evolution of both of the above systems are perfectly deterministic and perfectly reversible, if the state of the coin is known at any point in time then the law(s) can be used to determine what will happen in the future or what has happened in the past with absolute certainty.

The concepts from the two 'coin' systems above are then expanded to include six sided dice, that is a system which has six states. Numerous dynamical laws could be use to describe the evolution of this system, for example a simple cyclic law where the dice cycles though sides 1 to 6 in order and then back to 1.

Or a cycle for example 1 to 3 to 2 to 6 to 4 to 5 and back to 1. This second cycle is in fact the same as the first as each in the state in the second cycle could simply be relabelled  to arrive at the first. However, here multiple cycles within a system is introduced. For example, two cycles - 1 to 2 to 3 and back to 1, and 4 to 5 to 6 and back to 4 - or even three cycles - 1 to 2 and back to 1, 3 to 4 and back to 3, and 5 to 6 and back to 6. The point is that even with multiple cycles within the space of states, the law which describes the whole system is perfectly deterministic and reversible - if the system is in any one of the states, then the next or previous state can be determined from the law.

Laws not Allowed

Examples of the types of laws that are not allowed are also given using simple systems similar to those used above. For example a three state system where the system cycles from 1 to 2 to 3 and back to 2. This system at first glance appears to be deterministic, however, if this cycle is reversed it leads to a problem, in that it is not clear which state is next if the system is in state 2, should it be 1 or 3? Also if the system is in state 1, which state is next? Therefore this law is deterministic, however it is not reversible. Laws of this type violate one of the central cannons of physics and that is the conservation of information. The information about which state the system starts in is lost.

Infinite Space of States and Conservation Laws

More realistic systems obviously have an infinite space of states, however, cycles within such systems can and do exist. In fact, such cycles are equivalent to quantities which are conserved, and directly lead to conservation laws


Finally it is pointed out that it is not possible to measure the initial conditions of the system with infinite accuracy, therefore for any set of measurements carried out on a system there will always be uncertainly in the prediction of the outcome of the system even if the dynamical law that describes the system of interest is known.

That's it for chapter 1. The next I shall review from this book is interlude 1, a section on Spaces, Trigonometry and Vectors.

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