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A Can KE be reformulated using |v| instead of v^2?

  1. Nov 13, 2017 #1
    While doing some calculations on v_rms using the Maxwell-Boltzmann distribution, I noticed that v_rms and v_avg are pretty similar (https://casper.berkeley.edu/astrobaki/index.php/File:MaxwellSpeedDist.png).
    In fact, really it's just the choice of using the 1-norm (|v|_avg) vs. 2-norm sqrt(v^2 avg). When deriving v_rms or <KE> from the Maxwell-Boltzmann distribution , we get <KE> = 0.5m<v^2> = 3kT/2; however, using v_avg from the Maxwell-Boltzmann distribution we get v_avg = sqrt(8kT/(pi*m)) and then 0.5m<v>^2 = 4kT/pi.

    In computer science/statistics, people often choose the distance metric <|x|> vs. sqrt<(x^2)> based on their needs, and generally <|x|> is thought to be better and more robust to outliers (yet has far worse properties, e.g., no convergence). I'm familiar with the work-energy theorem and the derivation of KE = 0.5mv^2 from Newton's 2nd law, but am trying to figure out if there are alternative formulations using |v|. I'm also familiar with the equipartition theorem with energies of form Ax^2 being assigned 0.5kT energy and am not sure how this would be reformulated using |v|.

    So two questions:
    1. Can KE/classical laws be reformulated using |v| instead of |v^2|, or has anything been published using the 1-norm (absolute value) or any other norm besides the 2-norm (square)? I realize the units of energy won't make sense with just changing how KE is computed. But, for example, speed could be defined as (sum |v|) instead of (sum v^2), though I'm not sure if people do this.

    2. Are there cases where v_avg is useful to use instead of v_rms?

    Thanks in advance for the help/dicussion! This has been gnawing at me since in CS/stats we usually have the choice of using |v| or v^2 to compare distributions, but I'm not sure if we're "forced" to use v^2 in physics because of fundamental laws (and if so which ones?).

    Image is from: https://casper.berkeley.edu/astrobaki/index.php/Maxwellian_velocity_distribution
     
    Last edited: Nov 13, 2017
  2. jcsd
  3. Nov 14, 2017 #2

    russ_watters

    User Avatar

    Staff: Mentor

    Are you familiar with momentum?
     
  4. Nov 14, 2017 #3
    Yep! I know that momentum = mv. I've only seen momentum used as a property of individual molecules as opposed to ensembles in stat mech (and also never seen a scalar formulation of momentum analogous to speed for velocity) . Is the average momentum of a system ever considered in a way similar to KE? Maybe (q1) is better posed as -- is there a macroscopic energy-like description of a system using |v| instead of v^2? For example, this could be useful in non-equillibrium settings where few particles have very high velocity and are outliers, and where v^2 could be misleading.

    (q2) still remains -- is there a case where v_avg is useful to describe systems instead of v_rms? Or, is it really only used to describe the momentum of particles.
     
    Last edited: Nov 14, 2017
  5. Nov 14, 2017 #4

    Dale

    Staff: Mentor

    The units don’t work out as written. However, since v is a vector over the reals then |v^2|=|v|^2
     
  6. Nov 14, 2017 #5
    Poor notation on my part. I meant reformulated with <|v|> instead of <v^2>. But also curious about if <|v|>^2 is ever useful to use.
     
  7. Nov 15, 2017 #6
    Both total energy and total momentum are conserved. But total speed is not conserved, so it is less physically relevant. Of course, there are some uses, but not nearly as many.

    Energy, momentum, and speed will all change depending on your frame of reference. But energy and momentum form a 4-vector, so laws involving them are covariant. Speed isn't a component of a tensor, so any laws involving speed will have to change depending on your frame of reference.

    Also, the absolute value function is hard to work with since it's nonholomorphic.
     
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