Immortal Unbounded Universe
    By Johannes Koelman | April 22nd 2014 11:21 AM | 13 comments | Print | E-mail | Track Comments
    About Johannes

    I am a Dutchman, currently living in India. Following a PhD in theoretical physics (spin-polarized quantum systems*) I entered a Global Fortune


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    Our universe is eternal, expanding indefinitely, and not threatened by any form of 'heath death'. All evidence points into this direction. In fact, at the end of the nineteenth century Ludwig Boltzmann had collected all the evidence to draw such conclusions. His consistent reasoning on how to render the reversible laws of physics compatible with the second law of thermodynamics brought him close, but he failed to make the final decisive steps.

    Ok, I realize that at Boltzmann's time our knowledge of the universe didn't stretch beyond the Milky Way, the island of stars that we inhabit. And yes, I admit reasoning here with the benefit of a heap of hindsight. But even now, in the twenty-first century, this hindsight is apparently not distributed uniformly.

    What is flabbergasting is that the line of argument "the second law of thermodynamics naturally emerges in a universe that is expanding indefinitely under the influence of reversible laws" is still not widely accepted amongst physicists. If you Google "past hypothesis" or "Weyl curvature hypothesis" you discover that lots of physicists lure themselves into thoughts of an unnatural (divinely tuned?) low-entropy initial condition at the start of the universe being required for the second law of thermodynamics and the 'arrow of time' to emerge.

    Finite Systems Fallacy

    Today I watched a newly issued FQXi video featuring Alan Guth entitled "Entropy and the Arrow of Time". Guth tackles the question "Can an arrow of time develop in a system with reversible laws and no special initial conditions?". He summarizes his conclusion with the statement "in a universe with an 'infinite developable phase space' the maximum possible entropy is infinite, and therefore any entropy you start with is a low entropy".

    That's a concise summary of what I have been trying to sell here to all of you since 2009 (see this September 2009 post and this August 2011 post), although I prefer the wording 'an infinitely expandable universe' or simply 'an open universe' over the more cryptic 'a universe with an infinitely developable phase space'. But in any case, it is reassuring to witness a 'physics superstar' like Guth, who soon will have amassed more physics prize money than anyone else in the world,* to support the claim that unbounded reversibility equates to persistent entropy growth. 

    Alan Guth, soon the world's most successful bounty hunter in physics? 

    The key take away is that physicists who perceive an incompatibility between the reversible fundamental laws of physics and the second law of thermodynamics are invariably mislead by an intuition that is based on 'finite systems thinking'. Any reversible system with unbounded dynamics is capable of developing an arrow of time in the form of unstoppable entropy increase. Even a single particle model suffices.

    Harmonic Accelerator

    The toy model to consider consists of nothing more than a harmonic oscillator 'turned upside-down'. This means that the harmonic restoring force is replaced by a harmonic acceleration force (akin to dark energy). To coin the term 'harmonic accelerator' for this model seems quite appropriate. You can visualize this model as a ball moving without friction over an infinite 'harmonic hill' under the sole influence of a uniform gravitational acceleration. You might start the ball in upward motion, but inevitably after some time it will start sliding down and it will forever persist in this downward motion. This despite the fact that the ball follows a fully reversible dynamics: every trajectory has a corresponding time-reversed trajectory that is obtained simply by reversing the ball's velocity.

    Mankind's attempts at building harmonic accelerators invariably fall short of reaching unbounded displacements

    For this 'harmonic hill model' we can formulate a law that expresses its irreversible behavior with the simple words "in the end everything will go downhill". A succinct summary of the second law of thermodynamics!

    Is it all that simple? Yes it is. In an infinitely expandable reversible universe observing stones falling in ponds and causing their surfaces to ripple, but never witnessing ripples on a pond throwing out a stone, is in essence no weirder than observing only downward motion in a harmonic accelerator.

    But still... a ball sliding down an infinite hill seems way too trivial an illustration for irreversible thermodynamic behavior. Such a simple system features no more than a single degree of freedom, and can therefore hardly be seen as truly thermodynamic in nature. That might be true in the classical sense of the term 'thermodynamical', but since Boltzmann we know that thermodynamic properties like entropy can be assigned to any dynamic system that at each time can be assigned a specific state. The instantaneous entropy of such a system is nothing more than the length of the description required to fully specify the state. The state of our infinite hill model at any moment in time is fully described by providing two numbers: the ball displacement and the ball velocity. Doing the math, it follows that ignoring short-time transient behaviors, the entropy grows proportional with time. As there is no limit to this entropy growth, the entropy never saturates and no equilibrium is ever reached. A harmonic accelerator will never meet its heath death.

    Is this all at odds with the reversible dynamics defining the model? On the contrary. Thanks to the fact that the displacements follow a time-symmetric dynamics, entropy increases both for growing positive times as well as for growing negative times. If we reverse the particle velocity at any time, the reverse motion would be followed, but again after a transient period the same downhill motion would start. Bottom line is that in a time-symmetric (or rather CPT-invariant) universe an indefinite expansion can only be realized in a 'bow tie pattern' that ensures indefinite entropy increase in either direction. Needless to say that time in a 'bow tie universe' will always march in the direction of expansion.

    A universe filled exclusively with dark energy produces a perfect bow-tie in the form of a De Sitter spacetime


    * Both Alan Guth and Andrei Linde have pocketed the inaugural Fundamental Physics Prize, the Gruber Cosmology Prize, and the ICTP Dirac Prize, and if the BICEP2 results will hold up it's just a matter of time for both to receive a Physics Nobel. That will bring the physics prize money for each of them close to an unprecedented USD 4 million. However, Guth has also pocketed the GBP 1,000 Isaac Newton Medal, thereby narrowly beating Linde who hasn't.


    OK, I'm not sure I understand but trying to put it in simplest terms...

    If we assume an infinite space then the gas from an explosion could expand indefinitely and so never reach an equilibrium state.

    Well that may be true but the energy density would decline to the point that the universe isn't very interesting. For example if the expansion of the universe continues to accelerate then the universe will have an infinity developable phase space. But...

    Johannes Koelman
    If we assume an infinite space then the gas from an explosion could expand indefinitely and so never reach an equilibrium state. Well that may be true but the energy density would decline to the point that the universe isn't very interesting.

    I am tempted to agree, but hesitate because we humans are very poor at predicting emergent phenomena. The point is: one could apply the same pessimistic reasoning during the epoch of recombination. The prospect were dull: temperatures will continue to drop to eV-level and further, electrons and protons will cluster into uninteresting electrically neutral clumps, and all the fascinating magneto-hydrodynamic turbulence will disappear. Who would have foreseen intelligent life to emerge at temperatures of a few tens of meV?
    I am not sure I understand it correctly. The visible universe will empty (dilute) over time.

    In the end, the visible entropy will be the "uncertainty" whether or not the volume within the horizon contains a photon or not?

    Does this mean that the entropy of the visible universe (within the horizon) shows a decreasing entropy, while the (unobservable) universe as a whole has increasing entropy?

    This should be possible because the visible universe is not a closed system.

    Johannes Koelman
    The observable universe will dilute (but never empty out), with an entropy that will increase indefinitely.
    Reading this brought to mind the following, which I posted as a comment on an earlier article of yours.

    It is a poem by James Clerk Maxwell, as brought to us on Science 2.0 by Monte Davis in In The Field With James Clerk Maxwell:

    Till in the twilight of the Gods,
    When earth and sun are frozen clods,
    When, all its energy degraded,
    Matter to ether shall have faded,
    We, that is, all the work we've done
    As waves in ether shall forever run
    In ever widening spheres through heavens beyond the sun.

    I hope you don’t mind the repetition!

    Robert H. Olley / Quondam Physics Department / University of Reading / England
    Johannes Koelman
    Thanks for the link to this most charming article on JCM!
    Entropy may be increasing eternally but it could still be approaching an asymptote, couldn't it?

    Johannes Koelman
    Do you mean "could it be that although the entropy keeps increasing, each moment in time the universe is in equilibrium and the entropy maximal"? The answer is "no". You can't assign a finite equilibrium entropy to unbounded systems like harmonic accelerators.
    Quentin Rowe
    A universe filled exclusively with dark energy produces a perfect bow-tie in the form of a De Sitter spacetime
    Can you elaborate on the axis represented... Does your bow-tie graphic represent two arrows of time (one positive, one negative)?
    Johannes Koelman
    Yes. The "arrow of time" is an emergent property always pointing in the direction of entropy increase. So in a 'bow tie universe' there will be two such opposing arrows.
    Is that equivalent to saying that the 'bow tie universe' consists of one half 'filled' with matter and another with anti-matter?

    Johannes Koelman
    A perfectly symmetric bow-tie universe would have each half being the CPT transform of the other. So, Both halves would be each other's mirror image with every particle replaced by its anti-particle. Because the laws of physics are CPT invariant, both halves would basically represent the same universe.

    At this stage, this is all speculation. CPT-symmetric bow-ties are possible, but (as far as we know) non-symmetric bow-ties are not forbidden, and both halves could represent a distinct universe. But it certainly is a very interesting thought.
    I don't understand how the entropy of the harmonic accelerator could increase if the the number of degrees of freedom is fixed at two.