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I Virial Theorem and the Ergodic hypothesis

  1. Sep 7, 2017 #1
    I'm currently reading "The Virial Theorem in Astrophysics" by G.W. Collins (the book is available as a free web edition at http://ads.harvard.edu/books/1978vtsa.book/) in which the author claims the importance of the ergodic hypothesis when applying the virial theorem to astrophysical system.
    After explaining what an ergodic system is, he says:

    "At this point the reader is probably wondering what all this has to do with the virial theorem. Specifically, the virial theorem is obtained by taking the time average of Lagrange's identity. Thus​

    $$\lim_{T\to\infty} \frac{1}{2}\int_{t_0}^{t_0+T} \frac{d^2 I}{dt^2}\,dt=<2T>_t + <U>_t \quad (1.5.2)$$​

    and for systems which are stable the left hand side is zero. The first problem arises with the fact that the time average is over infinite time and thus operationally difficult to carryout. Farquhar points out that the time interval must at least be long compared to the relaxation time for the system and in the event that the system crossing time is longer than the relaxation time, the integration in equation(1.5.2) must exceed that time if any statistical validity is to be maintained in the analysis of the system. It is clear that for stars and star-like objects these conditions are met. However, in stellar dynamics and the analysis of stellar systems they generally are not. Indeed, in this case, the astronomer is in the envious position of being in the reverse position from the thermodynamicists. For all intents and purposes he can perform an 'instantaneous' ensemble average which he wishes to equate to a 'theoretically determined' time average. This interpretation will only be correct if the system is ergodic in the sense of satisfying the 'quasi-ergodic hypothesis'. Pragmatically if the system exhibits a large number of degrees of freedom then persuasive arguments can be made that the equating of time and phase averages is justified. However, if isolating integrals of the motion exist for the system, then it is not justified, as these integrals remove large regions of phase space from the allowable space of the system trajectory. Lewis' theorem allows for ergodicity in a sub-space but then the phase averages must be calculated differently and this correspondence to the observed ensemble average is not clear. Thus, the application of the virial theorem to a system with only a few members and hence a few degrees of freedom is invalid unless care is taken to interpret the observed ensemble averages in light of phase averages altered by the isolating integrals of the motion. Furthermore, one should be most circumspect about applying the virial theorem to large systems like the galaxy which appear to exhibit quasi-isolating integrals of the motion. That is, integrals which appear to restrict the system motion in phase space over several relaxation times. However, for stars and star-like objects exhibiting 1050 or more particles undergoing rapid collisions and having short relaxation times, these concerns do not apply and we may confidently interchange time and phase averages as they appear in the virial theorem. At least we may do it with the same confidence of the thermodynamicist."​

    I understand his reasoning, but I have one question related to the sentence in bold: what does he mean when he says that astronomers are well capable of doing "instantaneous" ensemble averages?
    Let's suppose that I'm looking at a cluster of galaxies: I understand why one could call the measurement "instantaneous" since the relaxation time of the system is very big, but it's still just one system.. Where's the ensemble?

    In Statistical Mechanics, a statistical ensemble is a collection of systems that share the same macrostate but have different microstates. So, in my view, looking at a cluster gives me a picture of only one microstate but from there I'm lost... How can you average anything from that?

    I'm not familiar with observational methods in Astrophysics, so I might simply be missing something. Any ideas?

  2. jcsd
  3. Sep 7, 2017 #2
    If you open a bottle of perfume the room will stink a few minutes later. I believe this will happen because of unfortunate experience and because of thermodynamics theory. In any of those experiences I never knew the exact path taken by any one of the molecules of stink that invaded the sensor in my nostril.

    Astronomers have the reversed information. They know where stars are now and they can measure things like momentum. They cannot observe galaxies for a few billion years to see how the stars mix up. Technically they can watch for a few billion years. Even a million or few hundred thousand years might be enough for a lot of stars to interact. Most astronomers are looking for funding and/or publication within human lifespans. So nearly instantaneous.

    For an "ensemble" you would be stating something about galaxies if you look at clusters. Or making a statement about stars if you are looking at a galaxy.
  4. Sep 8, 2017 #3
    Thank you for your answer!
    I think I understand now what the author meant. It simply didn't exactly resonate with the concept of statistical ensemble that I have from thermal physics.

    As you said, astronomers are capable of taking a "snapshot" at the galaxy and determine the position and momentum of every star, as one might be theoretically be able to do with a gas of particles. However this snapshot represents a single microstate of the system "galaxy", and there seems to be no ensemble on which one should average.

    Let's say that I want to calculate the mean kinetic energy in gas at a certain temperature. I would have to consider all the possible ##N## "snapshots" (microstates) consistent with the macroscopic observables, compute the kinetic energy in each of this configurations and then average over all of them.
    $$<T>_ {ensemble} = \frac{1}{N} \sum_{k=1}^{N} T(\vec{x}_k) $$
    where ##\vec{x}_k## is a vector in phase space that contains the position and momentum of each particle of the gas in the ##k## microstate.

    However, it seemed to me that astronomers can determine ##T(\vec{x}_k)## for only one ##k## so no average was possible. I would have expected to need an ensemble of galaxies.

    Taking the stars as the elements of the ensemble is probably what the author was referencing in that sentence. It's still not a perfect analogy to what I'm used to in thermal physics, but I'll think about it in more depth.
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