# Random Walk of Thermal Electrons

The drift velocity of electrons in a typical electronic circuit might be measured in mm per second. In contrast, the thermal velocity of electrons is in the vicinity of km per second.

Because of the slow drift velocity, beginners in electronics sometimes get the mistaken impression that an electron leaving a battery's negative electrode and travelling through a circuit will only arrive at the battery's positive electrode after a relatively lengthy period of time.

Rather, I expect that the electron at one end of a wire, even with no applied electric field, will very rapidly move a short distance, (it's mean free path), have it's velocity and direction randomized, and then continue on in a random walk. At some point in it's random walk it will reach the other end of the wire. I am guessing that for relatively short lengths of wire, this could take less time than for an electron travelling directly at drift velocity.

My question is this.

On average, how long would it take for an electron to travel, via random walk, from one end of a wire of say 1 meter length to the other end? For your answer you may assume that the wire is copper (or anything else you choose), that the electrons are moving under thermal motion, with no drift velocity, that the temperature is any value you choose near 20 degrees Celsius. You may make any other assumptions you wish if you make them explicitly.

• Electrons aren't localized in a conductor to begin with. Are we supposed to ignore the fact that they are fermions? What are the suppositions we have to make here? Commented Oct 17, 2021 at 8:36
• Make the suppositions that you believe are reasonable. However, in the classical view of electricity, an individual electron might be found at one end of the wire at one moment, and then after the elapse of some time that same electron might be found at the other end of the wire. In this classical view, if the time elapsed is small, the probability of finding the electron at the far end is small. Say, after time t, the cumulative probability that the electron could have been found at the far end is 50%. That is, classically 50% probability it traveled to the far end, even if no longer there. Commented Oct 17, 2021 at 13:39
• I have two comments in reply to your question (a) A scenario with no drift velocity would imply that there is no potential difference in the wire and hence effectively the total electron flow (sum of incoming and outgoing) electrons would be zero (otherwise, a wire kept at a constant temperature would start generating electricity). (b) You should realise that thermally agitated random walk is symmetric in space, therefore the mean displacement will be zero, but the root mean squared displacement (RMSD) will follow the rule $\sqrt{2Dt}$; hence if you know D (diffusivity),can calculate the RMSD. Commented Oct 28, 2021 at 8:28