Method of averaging velocities of electron when deriving drift velocity

Question

Here is derivation of drift velocity.

Assume there is a field $\vec{E}$ inside the conductor(wire). Using equations of motion we can say that for every charge inside the conductor, $$\vec{v_1}=\vec{u_1}+\frac{\vec{E}e}{m}t_1$$ $$\vec{v_2}=\vec{u_2}+\frac{\vec{E}e}{m}t_2$$ $$.$$ $$.$$ $$.$$ $$\vec{v_n}=\vec{u_n}+\frac{\vec{E}e}{m}t_n$$ where $t_1,t_2,...t_n$ are the relaxation times. Summing them and dividing by the number of charge particles($N$) we get, $$\sum_{i=1}^n\frac{\vec{v_i}}{N}=\sum_{i=1}^n\frac{\vec{u_i}}{N}+\frac{\vec{E}e}{m}\sum_{i=1}^n \frac {t_i}{N}$$

Since initial velocities are random $$\sum_{i=1}^n\vec{u_i}=0.$$ Thus we get, $$\vec{v_d} = \frac{\vec{E}e}{m}\tau.$$

My question is why do we use relaxation time instead of using $\Delta t$ and then averaging the velocity as time interval for all electron's velocity in the wire?

As Wikipedia states "drift velocity is the average velocity attained by charged particles, such as electrons, in a material due to an electric field". So shouldn't we use change in very small time instead of using relaxation time?

Here's the link https://en.m.wikipedia.org/wiki/Drift_velocity#:~:text=In%20physics%2C%20a%20drift%20velocity,an%20average%20velocity%20of%20zero.

Answer 1

Average velocity

As Wikipedia states "drift velocity is the average velocity attained by charged particles, such as electrons, in a material due to an electric field". So shouldn't we use change in very small time instead of using relaxation time?

Averaging can be done only over a finite time interval - taking a very small interval $\Delta t$ and then taking it to zero would produce an instantaneous velocity - we will have learned nothing new from it. In practice, one often averages quantities over physically small time (or physically small volume, as in the case of elasticity or continuous electrodynamics). This time should be sufficiently short on the experimental time scale, but long on the microscopic time scale. In other words, e.g., if we are looking at the ac current, the averaging time should be much smaller than the period of the ac oscillations, but this would likely still be much longer than the time between electron collisions with lattice or other electrons.

How averaging is done in practice
There are different ways how averaging can be done within Drude-like models of conductivity:

• Ad-hoc scattering time One assumes that the times between collision events are all equal to the mean scattering time $\tau$. That is, the velocity just before the scattering even is $$\mathbf{v}=\mathbf{u}+\frac{e\mathbf{E}\tau}{m}.$$
• Poisson process One can asume that the probability of an interval $t$ between collisions is given by $$P(t)=\frac{1}{\tau}e^{-\frac{t}{\tau}},$$ calculate the electron velocity assuming scattering at times $t_1,t_2...$, and then average over the above probability distribution. This produces the same result as the previous approach, but allows also for calculating more complicated quantities (such as distribution of velocities around the drift velocity, i.e., the noise). The approach discussed in the OP is somewhere between these two, with $$\tau = \frac{\sum_{i=1}^N t_i}{N}$$
• Hydrodynamic approach/diffusion One could use hydrodynamic approach, where the drift velocity is described by the Newton's equation $$ m\dot{\mathbf{v}}=-\frac{m\mathbf{v}}{\tau}+e\mathbf{E}. $$ Here $1/\tau$ plays the role of the effective friction coefficient due to electron being scattered multiple times. This produces the same result as the above approaches. One could further augment this approach by adding a random scattering force in the Newton's equation, converting it into a Langevin equation.

Note finally, that besides averaging over the scattering times, there is also averaging over Maxwell-Boltzmann distribution of the initial electron velocities (although it is not mentioned in simple derivations).

See here and here for related discussions.

Answer 2

Indeed, the terminology is confusing. Here “average velocity” is the average instantaneous velocity with the average taken over all of the free electrons for a single instant. But in introductory physics class the change in position divided by the change in time is also called “average velocity” with the average taken over time for a single object. Unfortunately, we don’t have separate terms for the average taken over a bunch of particles at a single time versus the average taken over an interval of time for a single particle. They are both called “average velocity”.

We do not know the velocities of the individual electrons, so here we are interested in the average over particles. Furthermore, we usually assume a steady state, so that the average over time is not changing. So the most interesting one in this context is the average over particles. But the meaning of “average velocity” in general must be determined from context, it is inherently ambiguous.

Answer 3

I would like to bring up some counter questions and points which I think would help you understand what is going on here.

Firstly, suppose we use $\Delta t$, then how large do we make it to be? $.01$ seconds, $.02$..? All the electrons collide at different times.. so how to choose a good value of time interval such that we get the drift velocity?

Secondly, what is the idea behind this derivation? If we really think about it, the set up is we take the first look at the system when the battery is just switched on, and, average the velocity of each and every electron over the time it takes for those electrons to hit another one.

The reason that you can say that the average of initial velocity is zero is because initially, before battery started pushing electron, the total current is zero.

Suppose we consider some time after the current is switched on, and try to compute change in average velocity. We would find that it is zero. So, this derivation wouldn't conceptually work then but the final result would hold true.