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Adiabatic perturbations -- why the name?

  1. Jun 6, 2015 #1
    In cosmological perturbation theory one can make an argument that very early on the relation between dark-matter perturbations and photon-temperature perturbations satisfies
    $$\delta = 3\Theta_0 + \text{constant}.$$
    If ##\text{constant} = 0## we call these primordial perturbations adiabatic and one can show that this leads to an everywhere constant ratio ##n_{dm}/n_\gamma##.

    Now an adiabatic process is a process that happens without the change of heat.

    Question: Is there any connection between the above definition of adiabatic perturbations and the usual meaning of an adiabatic process? Why is this name chosen?
  2. jcsd
  3. Jun 6, 2015 #2


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    I'm not sure any of what I'm going to offer here is relevant. I THINK OF ADIABATIC IN THE USUAL SENSE OF NO TRANSFER OF HEAT and it seems to me obvious that with quantum fluctuations in dark matter you wouldn't have any heat transfer. Overdense patches would contract, develop momentum, expand, oscillate, without any heating and cooling, without radiative transfer.

    But when I looked up adiabatic perturbation theory (especially in a quantum mechanics context) I got some confusing details. I want to mention this as a caveat. Something to keep in mind.
    I checked http://en.wikipedia.org/wiki/Adiabatic_process#Divergent_usages_of_the_word_adiabatic
    I do not know definitely that this is relevant to your question but it suggests that one should be on the lookout for divergent usages of the word:
    Quantum mechanics and quantum statistical mechanics, however, use the word adiabatic in very a different sense, one that can at times seem almost opposite to the classical thermodynamic sense. In quantum theory, the word adiabatic can mean something perhaps near isentropic, or perhaps near quasi-static, but the usage of the word is very different between the two disciplines.

    On one hand in quantum theory, if a perturbative element of compressive work is done almost infinitely slowly (that is to say quasi-statically), it is said to have been done adiabatically. The idea is that the shapes of the eigenfunctions change slowly and continuously, so that no quantum jump is triggered, and the change is virtually reversible. While the occupation numbers are unchanged, nevertheless there is change in the energy levels of one-to-one corresponding, pre-and post-compression, eigenstates. Thus a perturbative element of work has been done without heat transfer and without introduction of random change within the system. For example, Max Born writes "Actually, it is usually the 'adiabatic' case with which we have to do: i.e. the limiting case where the external force (or the reaction of the parts of the system on each other) acts very slowly. In this case, to a very high approximation

    that is, there is no probability for a transition, and the system is in the initial state after cessation of the perturbation. Such a slow perturbation is therefore reversible, as it is classically."[11]
    I also checked:
    ==quote from "Perturbation theory (quantum mechanics)"==
    and we consider the question if a dual Dyson series exists that applies in the limit of a perturbation increasingly large. This question can be answered in an affirmative way [7] and the series is the well-known adiabatic series.[8] This approach is quite general and can be shown in the following way.
    references [7] and [8] are:
    Frasca, M. (1998). "Duality in Perturbation Theory and the Quantum Adiabatic Approximation". Physical Review A 58 (5): 3439. arXiv:hep-th/9801069.
    Mostafazadeh, A. (1997). "Quantum adiabatic approximation and the geometric phase,". Physics Review A 55 (3): 1653. arXiv:hep-th/9606053.

    Again, I'm not sure any of that is relevant. To repeat: I THINK OF ADIABATIC IN THE USUAL SENSE OF NO TRANSFER OF HEAT and it seems to me obvious that with quantum fluctuations in dark matter you wouldn't have any heat transfer. Overdense patches would contract inward, develop momentum, pass through themselves, expand outward, oscillate, without any heating and cooling, without radiative transfer.
    Last edited: Jun 6, 2015
  4. Jun 6, 2015 #3


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    In this case I'm pretty sure the meaning they use is "constant entropy per comoving volume". The basic argument is that entropy generation by irreversible processes (e.g. gravitational collapse) is essentially a rounding error on the entropy of the photons at early times. It isn't until long after the emission of the CMB, when stars and galaxies start to form, that it no longer makes sense to call the expansion adiabatic. Usually the adiabatic expansion is only used for describing the universe prior to the emission of the CMB, long before any bound states begin to form.
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