aren't you still in school? why don't you ask your physics teacher?

...oh right, it's saturday.

It's not just Saturday, coming Tuesday would be the only option for the foreseeable future and everyone is obsessed with exams so the timing would still be bad. I may discuss with some university students Monday but I want to know what I can before then :-)

The answer could well be "yes":

http://phys.org/news/2012-11-smallest-logic-circuit-fabricated-single-electron.html

http://www.worldscientific.com/worldscibooks/10.1142/p650

Thanks for the links? If anyone else can point me in a direction or answer my questions, it's appreciated!

Thanks for the links!*

You can use a potential difference to send an electron down a conducting path. It won't look like a traditional circuit but it can be done. 1 e won't power much though. What is it you're actually wondering about?

I'd recommend looking up the double slot experiment.

https://en.m.wikipedia.org/wiki/Double-slit_experiment

I'm familiar with that experiment.

I'm drawing a line between an economic theory of simultaneous games (think prisoner's dilemma) and electrical circuits. Say you have two paths, path 1 with 1 lamp and path 2 with 2 lamps. All lamps are identical. If there's only one electron, I would assume it takes path 2 for an optimal division.
If there are 2 electrons, you get an interesting situation: for each individual electron, the best 'option' is to divide your charge over two lamps, but if they both take path 2, you get a combination which is inferior to one electron picking each path.

I want to find out what happens in practice. Either both electrons take the same path (presumably path 2) or one goes with each path (optimal division for two electrons in this case). Alternatively, you get an interesting 'quantum wave' with each electron heading in both directions until the wave collapses..

I don't know the English terms, but I hope you get what I mean with my thought-experiment.

and I'd eagerly recommend "Quantum Physics. A first encounter" by V. Scarani, it's a short nontechnical book, and I found it very elightning

Key is that there aren't enough electrons to give each lamp an equal charge (in my example, a multitude of 3 electrons for 1/3 of the total charge for each lamp). What happens if that isn't possible?

I've had my first encounter with Quantum Physics a while ago, but I don't know whether it applies here.
Since the electron could move in two opposite directions but nowhere else inbetween, it seems to me that you can't apply a standard quantum wave because there are only two options.

It's an interesting thought experiment, but I think you're confusing it with a variety of unrelated issues.

If you bifurcate a conducting path, I suspect you'd see current flow through both inversely proportional to the resistance of each path. Because of the wave-particle nature of electrons, I don't think it matters that there is only one.

With that being said, if the two paths were separated by a significant distance, I suppose you could trap the electron down one path. That would involve statistics though. I don't think it would always flow down the same one.

Also, for clarity, it's important to remember how electrons actually flow through a circuit. You don't have millions of electrons flying from your outlet to a lamp. Electrons actually travel very short distances at a time because they constantly crash into things (this is known as Mean Free Path). However, the electrons transfer their energy through the circuit very quickly. So, when you have 1 electron, you'd be relying completely on a potential difference attracting an electron to a location. The only way this would work is if you shot it through a vacuum.

Would it be twice as likely to take the path with double resistance though?
I understand the statistics-approach, but it seems to me like something worth researching if possible, because especially if you take two electrons, you statistically expect a significant enough likelihood that both take path 1, while that makes no sense in any way if you assume the electrons go for an optimal division of charge, which is the pattern you observe on a large scale, but could be explained by statistics rather than other forces. If you reduce the scale to single electrons, you can rule out possibilities.

Trapping electrons down one path is something I considered as well. In theory, if you could indirectly observe the electricity by virtue of a hypothetical lamp running on one electron, the electron should be trapped as soon as it meets its first lamp. Three options of identical lamps at varying distances could create interesting questions as well:
Once enough time passes for the first lamp to turn on if the electron was headed in that direction (I know that's not a great description), does the wave collapse even if the lamp doesn't turn on, meaning there's still two options left?

@abge: I did not know that last bit on that level, but it makes sense. I see why that would make it trickier..

In circuits, what you're referring to is a current divider. The formula for the percentage of current that flows down each path is here https://en.m.wikipedia.org/wiki/Current_divider

That really won't translate to the nano scale though. For that you'd need some sort of electron transport model. Unfortunately, that two is not relevant if you're shooting an electron down a vacuum.

This is a hard thing to think about because what you're asking can't really be made in reality, so it's hard to answer what would happen.

Do you mean it can't be made because of what you explained about how electrons flow through a circuit? I'm not sure why that makes it impossible. In the end, it should be a matter of time until an electron reaches the 'crossroads', right? I get that it'll be a different one, but does that matter?

That's not a prisoner's dilemma. I'm pretty sure an electron isn't subject to rational choice theory. Also, the two paths here aren't identical, and the electron is making its "choice" not based on expected outcome or self-interested behavior but due to some other electrical or conductive factor.

I understand there's a difference, I just describe the analogy which helped me formulate the thought-experiment: the analogical 'dominant' situation does not equal the 'optimal' one. Then I want to see what electrons do in such a situation to improve our understanding..

I guess I didn't mean it "can't be made" but the forces that affect the thought experiment are quite complex when you take everything into account that you'd need to make such a device.

@steephie, I love your question and I love the connection to game theory. As it happens, I'm an electrical engineering PhD student studying game theory. :) So maybe I can fumble around with this.

I think you're getting yourself into trouble by talking about electrical resistances and individual electrons in the same sentence. Individual electrons are difficult to predict, control, and even describe. They move around randomly and can't be individually tracked (because of that damn Heisenberg principle). This is what abgemacht was talking about when he said "electrons actually travel very short distances...."

But physicists came up with a nice work-around: since there are billions of electrons in a wire, we can describe their *average* motion very precisely with the notion of electrical currents (measured in "Amperes"). An electrical current is defined (in very broad terms) as a net motion of electrons. Suppose you have two currents: A) 1 electron moving at 1 m/s, and B) 2 electrons moving at 0.5 m/s. Which has more amperes of current? It's a trick question: both have the same number of amperes! (footnote to the pedantic: my units are wonky, and require an uninformative correction factor. Sorry. The concepts are still correct.)

So here's where the answer to your question starts to emerge: Because current is just a measure of average electron motion, we can do something kind of funky: we can describe the above current as 0.5 electrons moving at 2 m/s. Since a current is just an average, fractional electrons are fine. We're not literally saying that the electron is cut in half; we're saying that for all intents and purposes, we might as well describe the electron as being cut in half.

Now let's think about your single-electron experiment in an actual circuit. The quick, dirty answer is that there's no such thing as a single-electron current in a wire, because a current is defined (roughly) as the average motion of *all* electrons in a wire. If you took a single electron and somehow injected it into the circuit and tried to track it, before long it would collide or interact with other stuff in the wire and its motion would average out to be a very very very small electrical current, which would split nicely between your two circuit branches according to the rules of a current divider.

So what's the point of all that? I think I'm trying to get across the idea that your original question is not well-posed: you're asking a question about the behavior of individual electrons, but you're asking it in terms of average electron motion (by referring to resistances and circuits). Thus, the answers are going to sound weird and un-intuitive. Let me know which parts of my answer are the most confusing. :)

@2WL: There are some very nice connections between Nash Equilibria and electrical circuits, particularly in games with an infinite number of players. See Tim Roughgarden's "Selfish Routing and the Price of Anarchy." Steephie's intuition about the connection is correct, even if he's not likely to find satisfying answers.

Just to see if I got this:
You have a (hypothetical) power source which 'emits' (by lack of a better word) one electron at a time. This electron almost instantly bumps into another electron, activating that electron by partially passing over the energy in the original electron to the new one?

I deduct this from your statement that one electron could create start an electrical current which can split two ways. This means that you'd have to split the 'injected' energy over several electrons.

Did I get this right? One 'activated' electron and one 'inactive' electron can bounce and create two 'activated' electrons with half the extra energy?

In that case, I see the problem. Otherwise, I don't see it yet :-)

I'm mostly just bummed that my idea isn't as original as I hoped.

That was supposed to have a smiley, so don't worry :-)

@biophil: Care to give a valid example of where that connection does become apparent?

That's exactly what I was getting at.

Also, if you want to be very literal, you can't talk about "inactive" electrons - all electrons are in motion all the time. In metal, very many of these electrons are moving around the chunk of metal freely - they're not even stuck "orbiting" any particular atom.

Also, I'm not a physicist and have only the weakest grasp of quantum stuff. So it's possible that someone else will come along and say "no no no that's not even close to how it works literally" and then I'll say "but it's exactly how it works on average" and we'll both be correct.

biophil, that's an awesome writeup. Thanks for adding.

I like the article you cited and I can definitely see the applications of game theory when it comes to network traffic. My problem (misunderstanding?) though is how choice applies to the classic prisoner's dilemma, your network traffic example, and an electron. In the PD and network traffic, there is a choice made by self-interested users (network users and prisoners). My understanding of steephie's scenario, as he wrote it up, is that the electron itself is somehow making a choice between two paths. I majored in economics and currently teach it at a high school but have no background in electrical engineering, but my understanding is that an electron "choosing" a path doesn't quite parallel with a self-interested human making a decision. The electron (or current) is simply "choosing" the path of least resistance (or whatever the term is). There's no conscious decision made to pick option A or B.

Actually Steephie, that appears 100% correct. If the initial electron has a lot of kinetic energy then it could represent a large current... And many electrons would light up all the lamps (or burn out the circuit in the extremely large energy scale, as heat prodced by all these collisions will melt you circuit)

How and ever, this seems to ignore the root of your question.

You want to explore what one electron does. Well in practice, you can create a vacumn and an electric field. The electron moves towards the positive field edge (gaining energy along the way) and has some random probability of interacting with some lamps (losing energy to the lamps at that point).

Now if you allow a second electron join the fray - it's probability will be changed - because the first electron has an electric field at one of the lamps, so the attraction towards that single lamp is lowered. For one electron at a time this may not be signifigant, but i'm presuming that for billions of electrons you get this emergent property which you describe as 'optimal division' - i think...

For one electron at a time, the electron fields don't interact, so you shouldn't see any optimal division. At least that is my best guess at an answer.

@steephie: the example I'm most familiar with is not quite what you're looking for, but it's a good one. The game setting is called a non-atomic routing game, described pretty nicely in section 1.1 of http://theory.stanford.edu/~tim/papers/optima.pdf.

The connection to circuits is that you can model traffic flow as an electrical current, a network link with a cost function c(x)=x as a resistor, and a link with cost function c(x)=1 as a diode. Currents flowing in a circuit with resistors and diodes are exactly analogous to Nash Equilibria for the corresponding routing games.

@normal circuits: you have a positive and negative pole.

If you break the circuit at any point then you get an electric field between the two sides of the break - this is electrons building up on one side and being depleted on the other side. It perfecty balances/cancels out the electric field of the power source.

It is the relative build up and depletion of electrons which keeps an electric current going (which is whu you get flow along a circuit and why it seems different in the single electron case - where the electron goes along the electric field lines, rather than towards to lowest concentration of electrons - which is bound to the wire)

In the extreme case, you have enough energy in the electric field at that circuit gap to ionise the air, which then becomes a conductor, and electrons flow through the air plasma (as lightning). Also ions of air flowing the opposite direction can carry current - though their much larger mass means they will usually be flowing much more slowly and thus be negligible...

@2WL, one connection is in the theory of potential games; these are so-called for their nice physical analogies to stuff happening in response to a potential gradient. You're right, the electrons aren't literally making rational choices, but they are responding to an electric potential, which can be modeled as a cost. Likewise, players in a PD aren't literally responding to a potential function, but a potential function turns out to be a convenient way to model their cost-reducing deviations.

I'm not sure how valuable the analogy is, but I find it quite cute.

Cute is what I was going for!

On a more serious note, I'm going to check out the link provided once I'm done with a report about a slightly more boring experiment..

couldn't you think of it as pool balls? you hit one (electron) and it hits two electrons (the split in the wire) and sends them both out with half the initial speed. does that work?

That kind of works, except electrons don't sit still next to eachother and I'm not even sure it's possible to hit two electrons at once.

A better way to describe it would be one electron hitting one other electron at such an angle that both the original as well as the hit electron are sent out at half speed. Just for the idea.

Speed may also be nothing more than a metaphor, but I'm not sure about that since I haven't had this in years and only shortly so I simply forgot what exactly was transferred in a collision.