[The Theoretical Minimum] Can closed systems exist?

[Highwire]

Homework Statement

The first exercise in Susskind and Hrabovsky's The Theoretical Minimum is one that, in the words of Susskind, "is designed to make you think, more than it is designed to test you." The exercise asks:

Since the notion is so important to theoretical physics, think about what a closed system is and speculate on whether closed systems can actually exist. What assumptions are implicit in establishing a closed system? What is an open system?

Homework Equations

In the prior paragraph, the authors define the notion of a system - "A collection of objects" - and a closed system - "A system that is either the entire universe or is so isolated from everything else that it behaves as if nothing else exists".

In a hint, the authors also suggest the student consider the idea of mathematical closure:

Closure can be thought of as a boundary. You could even think of it in terms of mathematical closure—if you do something to a member of a set, closure requires that it remain a member of the same set. Thus adding to natural numbers results in a natural number, the set of natural numbers is closed under addition. Thus, a closed set includes its boundary. Similarly an open set is one that does not include its boundary.

The Attempt at a Solution

Guided by the hint, I first considered what it meant for a set or system to be mathematically closed. From my limited experience with higher mathematics, it only makes sense to say that a set is closed if it's closed under some operation (though a cursory look at some Wikipedia articles makes me think that that definition is naïve). The set of natural numbers is closed under addition, but subtraction can't be defined on the whole set without expanding it out to the integers.

But does it mean anything for a system, especially a physical system, to be closed under something else?

In the physical world, my intuitive assumption is that only the universe itself can be a totally closed system (it contains its own boundary, going by the authors' hint), but certain systems can be treated as closed for the most part. I don't know if this is what's happening when classical physicists make predictions for which quantum uncertainty isn't important, or conversely when quantum physicists make predictions for which classic mechanics don't apply, but I imagine there are situations where treating a system as approximately closed is okay.

Lastly, and open system seems it should be - bluntly - a system that isn't closed, i.e. one that behaves in the context and under the influence of other systems.

(I feel like my brainstorming lacks a whole lot of brevity and rigor, so I just wanted to make sure I get on the right track with this before going on, since the question is so foundational to the rest of the book!)

 

[Stephen Tashi]

From my limited experience with higher mathematics, it only makes sense to say that a set is closed if it's closed under some operation

Another notion of "closed" comes from point set topology. Intuitively, a set of points is "closed" if it contains all points "infinitely close" to it. For example, the set of numbers between 0 and 1 fails to contain 0 and fails to contain 1, so it is not closed. The set of numbers greater than or equal to 0 and equal or less than 1 contains 0 and 1. If you pick any number outside that set, it is a finite distance away.

That idea implements the notion of "closed" as "can be isolated" in the sense of distance. It would be challenging to make a similar definition that applies to physical systems.

 

[mfb]

In the physical world, my intuitive assumption is that only the universe itself can be a totally closed system (it contains its own boundary, going by the authors' hint), but certain systems can be treated as closed for the most part.

Right.
As an example: If you want to describe the orbit of planets in the solar system, it is sufficient to consider the solar system. There are gravitational interactions with other objects (with all objects in the observable universe, even), but they are negligible.

I don't know if this is what's happening when classical physicists make predictions for which quantum uncertainty isn't important

That is a different case - where you use physical laws that are only approximations. Closed vs. open systems are not a statement about the laws used, they are properties of the systems.