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Boltzmann entropy, the holographic bound and Newtonian cosmology
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1  Universidad Politécnica de Valencia


The holographic principle sets an upper bound on the total (Boltzmann) entropy content of the Universe at around $10^{123}k_B$ ($k_B$ being Boltzmann's constant). In this work we point out the existence of a remarkable duality between nonrelativistic quantum mechanics on the one hand, and Newtonian cosmology on the other. Specifically, nonrelativistic quantum mechanics has a quantum probability fluid that exactly mimics the behaviour of the cosmological fluid, the latter considered in the Newtonian approximation. One proves that the equations governing the cosmological fluid (the Euler equation and the continuity equation) become the very equations that govern the quantum probability fluid after applying the Madelung transformation to the Schroedinger wavefunction. Under the assumption that gravitational equipotential surfaces can be identified with isoentropic surfaces, this model allows for a simple computation of the gravitational entropy of the Universe.

In a first approximation we model the cosmological fluid as the quantum probability fluid of free Schroedinger waves. We find that this model Universe saturates the holographic bound. As a second approximation we include the Hubble expansion of the galaxies. The corresponding Schroedinger waves lead to a value of the entropy lying three orders of magnitude below the holographic bound. Although a considerable improvement, this still lies above existing phenomenological estimates of the entropy of the Universe. Current work on a fully relativistic extension of our present model can be expected to yield results in even better agreement with empirical estimates of the entropy of the Universe.

Keywords: Newtonian cosmology, holographic principle, quantum mechanics
Comments on this paper
Asmaa Shalaby
Repulsive harmonic potential
Dear Dr. Jose'

What is the Repulsive harmonic potential, I want to know the physical meaning of ;Repulsive; harmonic potential her . Forces may be repulsive and attractive but potential how?.

Philip Broadbridge
My understanding is that this is simply the potential energy for a repulsive force that is proportional to displacement. The potential for a harmonic oscillator is 0.5*k*r^2 where r is distance from the origin and k is the positive spring constant. The potential for the repulsive analogue is -0.5*k*r^2 . Sinusoidal 1D solutions x(t) in the attractive case are replaced by exponential or sinh functions in the repulsive case. These repulsive modes come up at very long wavelengths when we decompose a field that is coupled to an exponentially expanding universe, e.g. Broadbridge and Zulkowski Rep Math Phys 2006. Of course, Dr Isidro may think differently.

J.M. Isidro
Dear Colleagues,

I agree with Dr. Broadbridge's reply. Forces are gradients of a potential (I mean conservative forces of course). Hence potentials can be attractive or repulsive, just as forces. Changing the sign before the force vector amounts to changing the sign before the potential. The potential we use is the sign-opposite of the standard harmonic oscillator (which is a binding potential: it has bound states but no scattering states). Our potential admits no bound states, just scattering states, as it is supposed to model the expansion of the Universe.

Incidentally, we use Verlinde's idea (see the ref [8] in the slides) that forces are ENTROPY gradients. This has deep physical implications (see Verlinde's paper in the refs, and also Padmanabhan as quoted in Cabrera et al, ref [1]). In our case the entropy potential function and the standard harmonic potential function are simply proportional to each other.

Best regards
Asmaa Shalaby
Many Thanks, Prof. Broadbridge and Dr Jose'

got it.