Determinism: Difference between revisions

Revision as of 14:21, 17 July 2019

By Areeba Abid

Determinism is the idea that if the physical state of a system is known, including the momentums, positions, and forces of each object in the system, the behavior of the system can be calculated using iterative prediction to determine its behavior at any given time. This idea is based on Newton's second Law: the Momentum Principle, developed by Isaac Newton.

Main Idea

Concepts of Physical and Philosophical Determinism

Newton's laws allow us to predict the motion of an object if we know the initial position, initial momentum, and forces acting on the object. If this idea is extrapolated to all objects in the universe, it would seem that we could predict the future of the entire universe if we only knew the initial conditions (the starting positions, momenta, and interaction forces) of each object. In philosophy and metaphysics, this principle has been used to argue that humans do not have free will, because the atoms that make up our bodies simply continue to move in the paths we would expect based on their initial conditions.

=Deterministic Chaos

Based on an understanding of Newton's Laws, is may seem certain that determinism is correct. However, the story is more complicated, and in one manner or another those complications are the basis of most modern physics.

The first practical limitation is the fact that the initial conditions of a system may only be measured as accurately as our instruments allow. In most systems, uncertainties grow over time, which means the even small errors at the beginning may lead to drastic differences between predictions and reality after a given time period. This is why iterative prediction is generally only reliable for small time scales, although the exact nature of the system may cause the relevant time scales to vary drastically. In an idealized projectile motion problem, errors in the initial condition will generally have a linear effect upon the error of the prediction, which is manageable. If one introduces more complex forces, such as air resistance, these errors may be amplified much more substantially.

For an example of a chaotic system, consider a game of pool. The blue and yellow paths are the result of very similar forces on the cue ball, but the outcomes are not similar. Slight changes in the force applied to the ball result in a wildly different outcome seconds later.

A related limitation is our use of phenomenology. We treat a billiard ball, for example, as a single object, when in fact it is a collection of atoms. Modelling the ball correctly would require modelling each of these atoms and their interactions, but this is well beyond our capabilities, and so we use approximate forces instead of considering the truly fundamental forces. Even if we did have a unified theory of everything, it would be functionally impossible for us to apply it to the universe, and so the dream of fully predicting a system is beyond our reach as humans.

Chaos

It is to some extent a misnomer to discuss chaos in a section on determinism, since there are in fact many entirely deterministic forms of chaos^[1]. However, the concepts are related, and a discussion of chaos fits well with the broader discussion of iterative prediction of motion, since only through computational methods are we able to generate and evolve systems which are chaotic, and yet still simple enough to understand. Chaos is one of the great frontiers of present physics, and what we do know of it is extremely complicated. One may note, for example, that there is a million dollar prize for anyone who can prove that solutions exist for the Navier-Stokes equations, which govern the behavior of fluids^[2]. As such, this section will focus n use computational modeling to observe the behavior of chaos, rather than trying to explain it. That being said, chaos theory is a specialty of the Georgia Tech physics department, and if you are interested in learning more there are a wealth of resources to explore.

Stochastic Systems

A stochastic system is one with an inherent amount of randomness. This may be introduced by the limitations above, but it may also have more fundamental roots. When one moves past Newton's theory, the idea of determinism is called into question. Relativity behaves nicely from this respect, being (to the best of our knowledge) a purely deterministic theory. Quantum mechanics, however, seems by its very nature to defy determinism: quantum states always have a degree of randomness, and so it is impossible to fully determine the future of a quantum system. None of this is certain, of course, since we still lack a unified field theory, and there is an argument to be made, such as is made by Gerard t'Hooft^[3], that this theory will be deterministic, but we have no way to know at this time.

Mathematical Model

As per the above, in so far as we consider any modeling, we will focus on the interesting case of deterministic chaos, rather than on the stochastic (that is, non-deterministic) systems which occur as a result of quantum mechanics, or systems with noise in general.

Computational Modeling

Examples

All of these examples should be performed with the use of a numerical solution program, such as this

Simple

Use the chaotic spring system from Predicting Change in multiple dimensions, set [math]\displaystyle{ k = 4.9 }[/math], [math]\displaystyle{ \theta = \pi/6 }[/math], equilibrium length [math]\displaystyle{ L = 4 \; m }[/math], and initial extension [math]\displaystyle{ r_0 = 5 \; m }[/math].

Middling

Now, put the following system into a numerical solution program:

[math]\displaystyle{ F_x = (\frac{1}{10})(x^2-y^2-x) }[/math]

[math]\displaystyle{ F_y = (\frac{1}{19})(2xy + y) }[/math]

Difficult

Connectedness

Connection to Personal Interest: Philosophical Determinism

How is this topic connected to something that you are interested in?
Determinism has fascinating implications for the concept of free will. Even if it is impossible for us to calculate the future behavior of a system accurately, it is still without a doubt the direct result of the initial conditions, as directed by Newton's laws. Consider the human body as a physical system. If the body is simply a collection of particles and thoughts in the brain are just neurons firing, is every feeling experienced and every decision made by a person simply the result of whatever initial conditions they started with? Is it possible to differentiate between the laws that regulate matter and the laws that regulate our own bodies? What happens when we try to apply the laws of physics to the workings of our minds? It's difficult to think of your body as a collection of chemicals that behave like a very complicated deterministic system, but the implications are mind blowing and have been the subject of controversy since ancient times.

Connection to Major: Engineering Applications

How is it connected to your major (biomedical engineering)?
Predicting the behavior of physical systems is important for any engineer, and determinism makes this possible to some extent.

Connection to Industry: Ethernet Data Transfer

Is there an interesting industrial application?
Data transmission takes time, and knowing the time that it takes for information to travel from one location to another is important is some industrial applications, such as sending control variables to a high-speed CNC mill. Engineers need to be able to know how long it will take for a machine to receive data, and if a system is deterministic this can be calculated. However, Ethernet systems are not deterministic and the rate of data transfer is unpredictable, which poses problems for engineers.^[4]

History

The concept of determinism has been around for centuries and was explored by thinkers around the world. Physical determinism was bolstered by Newton's Laws, and the first publication discussing it in a scientific context, "LaPlace's demon", was written by Pierre Laplace in 1814. He wrote that a hypothetical omniscient observer could potentially predict the entire future of the universe. Even though this is practically impossible for any human or computer to ever achieve, LaPlace argued that it was still hypothetically possible if physical determinism were true. However, this was before 1927. when the Heisenberg Uncertainty Principle asserted that initial conditions cannot be known with complete accuracy, even hypothetically.

References

↑ http://chaosbook.org/
↑ https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness
↑ https://arxiv.org/pdf/quant-ph/0212095.pdf
↑ http://www.automationworld.com/networking-amp-connectivity/understanding-ethernet-speed-and-determinism Understanding Ethernet Speed and Determinism

[1] ttp://chaosbook.org/

[2] ttps://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness

[3] ttps://arxiv.org/pdf/quant-ph/0212095.pdf

[4] ttp://www.automationworld.com/networking-amp-connectivity/understanding-ethernet-speed-and-determinism Understanding Ethernet Speed and Determinism

[1]

[2]

[3]

[4]

@@ Line 8: / Line 8: @@
 Newton's laws allow us to predict the motion of an object if we know the initial position, initial momentum, and forces acting on the object. If this idea is extrapolated to all objects in the universe, it would seem that we could predict the future of the entire universe if we only knew the initial conditions (the starting positions, momenta, and interaction forces) of each object. In philosophy and metaphysics, this principle has been used to argue that humans do not have free will, because the atoms that make up our bodies simply continue to move in the paths we would expect based on their initial conditions.
-====Limitations====
+====Deterministic Chaos===
+Based on an understanding of Newton's Laws, is may seem certain that determinism is correct. However, the story is more complicated, and in one manner or another those complications are the basis of most modern physics.
-While the idea of determinism seems intuitively true based on our everyday experience, the true story is more complicated, though it remains unresolved.
+The first practical limitation is the fact that the initial conditions of a system may only be measured as accurately as our instruments allow. In most systems, uncertainties grow over time, which means the even small errors at the beginning may lead to drastic differences between predictions and reality after a given time period. This is why iterative prediction is generally only reliable for small time scales, although the exact nature of the system may cause the relevant time scales to vary drastically. In an idealized projectile motion problem, errors in the initial condition will generally have a linear effect upon the error of the prediction, which is manageable. If one introduces more complex forces, such as air resistance, these errors may be amplified much more substantially.
-One practical limitation is the fact that initial conditions can only be measured as accurately as our instruments allow, and after a few time steps, small inaccuracies build on each other. This means that over time, our model of the system and our predictions of its future state stray farther and farther away from the actual future state of the system. This is why iterative prediction is generally only reliable for small time scales, although the exact nature of the system may cause the relevant time scales to vary drastically. In an idealized projectile motion problem, errors in the initial condition will generally have a linear effect upon the error of the prediction, which is manageable. If one introduces more complex forces, such as air resistance, these errors may be amplified much more substantially.
 [[File:determinismpooltable.png|frame|left|For an example of a chaotic system, consider a game of pool. The blue and yellow paths are the result of very similar forces on the cue ball, but the outcomes are not similar. Slight changes in the force applied to the ball result in a wildly different outcome seconds later.]]
@@ Line 18: / Line 17: @@
 A related limitation is our use of ''phenomenology''. We treat a billiard ball, for example, as a single object, when in fact it is a collection of atoms. Modelling the ball correctly would require modelling each of these atoms and their interactions, but this is well beyond our capabilities, and so we use approximate forces instead of considering the truly fundamental forces. Even if we did have a unified theory of everything, it would be functionally impossible for us to apply it to the universe, and so the dream of fully predicting a system is beyond our reach as humans.
-These limitations prevent us from ever exactly determining the evolution of a system, but they don't necessarily say anything about the nature of the universe itself. When one moves past Newton's theory, however, the entire idea of determinism is called into question. Relativity behaves nicely from this respect, being (to the best of our knowledge) a purely deterministic theory. Quantum mechanics, however, seems by its very nature to defy determinism: quantum states ''always'' have a degree of randomness, and so it is impossible to fully determine the future of a quantum system. None of this is certain, of course, since we still lack a unified field theory, and there is an argument to be made, such as is made by Gerard t'Hooft<ref> https://arxiv.org/pdf/quant-ph/0212095.pdf </ref>, that this theory will be deterministic, but we have no way to know at this time.
+====Chaos====
+It is to some extent a misnomer to discuss chaos in a section on determinism, since there are in fact many entirely deterministic forms of chaos<ref> http://chaosbook.org/ </ref>. However, the concepts are related, and a discussion of chaos fits well with the broader discussion of iterative prediction of motion, since only through computational methods are we able to generate and evolve systems which are chaotic, and yet still simple enough to understand. Chaos is one of the great frontiers of present physics, and what we do know of it is extremely complicated. One may note, for example, that there is a million dollar prize for anyone who can prove that solutions ''exist'' for the Navier-Stokes equations, which govern the behavior  of fluids<ref>https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness</ref>. As such, this section will focus n use computational modeling to observe the behavior of chaos, rather than trying to explain it. That being said, chaos theory is a specialty of the Georgia Tech physics department, and if you are interested in learning more there are a wealth of resources to explore.
+====Stochastic Systems====
+A stochastic system is one with an inherent amount of randomness. This may be introduced by the limitations above, but it may also have more fundamental roots. When one moves past Newton's theory,  the idea of determinism is called into question. Relativity behaves nicely from this respect, being (to the best of our knowledge) a purely deterministic theory. Quantum mechanics, however, seems by its very nature to defy determinism: quantum states ''always'' have a degree of randomness, and so it is impossible to fully determine the future of a quantum system. None of this is certain, of course, since we still lack a unified field theory, and there is an argument to be made, such as is made by Gerard t'Hooft<ref> https://arxiv.org/pdf/quant-ph/0212095.pdf </ref>, that this theory will be deterministic, but we have no way to know at this time.
-====Chaos====
-It is to some extent a misnomer to discuss chaos in a section on determinism, since there are in fact many entirely deterministic forms of chaos<ref> http://chaosbook.org/ </ref>. However, the concepts are related, and a discussion of chaos fits well with the broader discussion of iterative prediction of motion, since only through computational methods are we able to generate and evolve systems which are chaotic, and yet still simple enough to understand. Chaos is one of the great frontiers of present physics, and what we do know of it is extremely complicated. One may note, for example, that there is a million dollar prize for anyone who can prove that solutions ''exist'' for the Navier-Stokes equations, which govern the behavior  of fluids<ref>https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness</ref>. As such, this section will focus n use computational modeling to observe the behavior of chaos, rather than trying to explain it. That being said, chaos theory is a specialty of the Georgia Tech physics department, and if you are interested in learning more there are a wealth of resources to explore.
 ===Mathematical Model===