Is there a way to describe the Internet in terms of a physics theory, like how the atom is described by quantum mechanic? If there is, how is it described by this theory.
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First of all, you need to consider that the Internet is at one level a physical thing. It is made of a collection of servers, clients, routers, switches, and the like, all of which are typically metal and plastic boxes full of electronic components. Within these boxes of course you have the physics of semiconductors, power supplies, etc. These boxes are connected to each other by physical things also: metallic wires, glass and plastic fiber-optics, radio waves, microwaves. There you have the physics of electromagnetics, optics, etc. Let's not forget satellites, they're in there too! So we have the physics of orbits and relativity. There are trillions of physical components to the Internet, and they are each made of high-tech materials using modern science.
So the literal answer to your question is that nearly every subfield of modern physics, except perhaps the most theoretical, are involved in making the Internet work.