For example a particle in a gravitational field p2 = 2mE −2m2gq so energy surfaces ... Liouville's theorem (version II): (Fluid) volumes in phase space remain
Mitchell • 4 years ago
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Mitchell • 4 years ago
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Luboš Motl • 4 years ago
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Mitchell • 4 years ago
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Luboš Motl • 4 years ago
As a result of this discussion, I think I have at least identified the crucial proposition of "entropic gravity 2.0", which is that the phase space volume for the fast variables, associated with a uniform gravitational potential surface on which the escape velocity is "v", should be (A/4G)v. When v=c, a horizon forms, and the thermodynamic regime takes over.
Hopefully we can agree that the growth of a horizon is an entropy-maximizing process. So what about horizon formation?
Consider a black hole of a given size. It has a large gravitational entropy. Now consider various ways that it could have formed (gravitational collapse of a gas cloud, infall of gas onto a neutron star, collision of two neutron stars, etc). In each of these situations, before a horizon forms, there is negligible gravitational entropy. The region of phase space containing the black hole is a region of convergence for many dynamical trajectories, which is how a system moving along any such trajectory can experience a big jump in total entropy when the horizon forms. It's entered a bigger region of phase space.
Somehow there is a transition from a situation (no black hole) in which maximization of gravitational entropy plays no dynamical role, to a situation in which it does, because the black hole now exists and its horizon can grow. Erik isn't saying any more that gravity outside of a horizon is maximizing an entropy, but he's saying that its form is determined by conservation of the phase space volume for the fast degrees of freedom, and when slow and fast timescales converge, you get horizons and a transition from a kinetic to a thermodynamic regime.
I still think there must be an insight there which is both not trivial and not wrong. :-)
Consider a black hole of a given size. It has a large gravitational entropy. Now consider various ways that it could have formed (gravitational collapse of a gas cloud, infall of gas onto a neutron star, collision of two neutron stars, etc). In each of these situations, before a horizon forms, there is negligible gravitational entropy. The region of phase space containing the black hole is a region of convergence for many dynamical trajectories, which is how a system moving along any such trajectory can experience a big jump in total entropy when the horizon forms. It's entered a bigger region of phase space.
Somehow there is a transition from a situation (no black hole) in which maximization of gravitational entropy plays no dynamical role, to a situation in which it does, because the black hole now exists and its horizon can grow. Erik isn't saying any more that gravity outside of a horizon is maximizing an entropy, but he's saying that its form is determined by conservation of the phase space volume for the fast degrees of freedom, and when slow and fast timescales converge, you get horizons and a transition from a kinetic to a thermodynamic regime.
I still think there must be an insight there which is both not trivial and not wrong. :-)
Dear Mitchell, you're fooling yourself all the time.
In a unitary quantum theory - and even in a classical theory with a time-dependent "overall" Hamiltonian for all degrees of freedom - the volume of the phase space below some energy (or, quantum mechanically, the number of microstates in it) is *exactly* conserved.
This volume actually doesn't depend on dynamics at all: it is a universal function of the overall energy and the overall energy is exactly conserved. Because this phase space has a constant volume, the "entropy" calculated from this volume in Verlinde's sloppy way (as a logarithm) is also constant which means that the force is zero.
There's really no way - except for introducing errors into physics by hand - how you could get an entropy-based force that would still lead to reversible phenonomena. It's a contradiction. Any dynamics based on entropy differences is irreversible by the first key property of the entropy, the second law of thermodynamics.
Regardless of tiny mutations of the wording, you still haven't gotten rid of the basic error that was present in this Verlinde stuff from the beginning. You still seem to think that the number of microstates - or volume of phase space - corresponding to the Sun and the Earth at different distances depends on the distance (maybe even exponentially with a huge extra coefficient in the exponent).
But this is impossible: it would violate unitarity in a dramatic way. The Sun and the Earth are getting closer and further every year, so whatever microstate for the fast degrees of freedom you have in January when the Sun-Earth distance is minimal must also have their counterparts, in a one-to-one fashion, in July when the distance is maximal. So the phase space volume beneath "E" is obviously independent on how you realize "E" - pretty much by definition. In particular, it must be independent of the Sun-Earth distance, otherwise this distance couldn't change in both ways in a unitary fashion.
Much more strongly, we know from physics - obviously from string theory but we don't really need string theory - that the Sun-Earth system carries no gravitational entropy, surely no entropy that would scale like A/4G with a macroscopic value of A - i.e. an entropy comparable to the black holes of macroscopic dimensions. The Sun and the Earth only carry a very tiny entropy, essentially equal to the number of atoms in them, and this entropy has clearly nothing to do with gravity. In principle, you could consider frozen planets orbiting each other. The gravity wouldn't change but the entropy would be (nearly) zero.
Only event horizons may produce a large entropy of order A/4G. No horizons means no entropy. I can't believe that this elementary point known from the early 1970s is still being misunderstood in 2011.
Cheers
LM
In a unitary quantum theory - and even in a classical theory with a time-dependent "overall" Hamiltonian for all degrees of freedom - the volume of the phase space below some energy (or, quantum mechanically, the number of microstates in it) is *exactly* conserved.
This volume actually doesn't depend on dynamics at all: it is a universal function of the overall energy and the overall energy is exactly conserved. Because this phase space has a constant volume, the "entropy" calculated from this volume in Verlinde's sloppy way (as a logarithm) is also constant which means that the force is zero.
There's really no way - except for introducing errors into physics by hand - how you could get an entropy-based force that would still lead to reversible phenonomena. It's a contradiction. Any dynamics based on entropy differences is irreversible by the first key property of the entropy, the second law of thermodynamics.
Regardless of tiny mutations of the wording, you still haven't gotten rid of the basic error that was present in this Verlinde stuff from the beginning. You still seem to think that the number of microstates - or volume of phase space - corresponding to the Sun and the Earth at different distances depends on the distance (maybe even exponentially with a huge extra coefficient in the exponent).
But this is impossible: it would violate unitarity in a dramatic way. The Sun and the Earth are getting closer and further every year, so whatever microstate for the fast degrees of freedom you have in January when the Sun-Earth distance is minimal must also have their counterparts, in a one-to-one fashion, in July when the distance is maximal. So the phase space volume beneath "E" is obviously independent on how you realize "E" - pretty much by definition. In particular, it must be independent of the Sun-Earth distance, otherwise this distance couldn't change in both ways in a unitary fashion.
Much more strongly, we know from physics - obviously from string theory but we don't really need string theory - that the Sun-Earth system carries no gravitational entropy, surely no entropy that would scale like A/4G with a macroscopic value of A - i.e. an entropy comparable to the black holes of macroscopic dimensions. The Sun and the Earth only carry a very tiny entropy, essentially equal to the number of atoms in them, and this entropy has clearly nothing to do with gravity. In principle, you could consider frozen planets orbiting each other. The gravity wouldn't change but the entropy would be (nearly) zero.
Only event horizons may produce a large entropy of order A/4G. No horizons means no entropy. I can't believe that this elementary point known from the early 1970s is still being misunderstood in 2011.
Cheers
LM
Hi Lubos - I understood Erik to be saying that different degrees of freedom could spontaneously develop different characteristic timescales of evolution, and that when you construct an effective theory for the slow degrees of freedom, it will contain these reaction forces. The phase-space volume beneath a given energy level is very nearly conserved, and the log of this volume is his "entropy".
Dear Mitchell, sorry to say but what you write (and what Verlinde is saying these days) still makes no sense whatsoever. It's just masking the crackpottery he has been previously spreading under a thick layer of fog.
The 2007 paper you referred to is about Berry's phase which is a quantum phase induced by an adiabatic change of parameters. But the adiabatic change of parameters must be pushed by another, external force: the adiabatic change is not the first driver. The external parameters are being slowly changed by an external agent so that the entropy doesn't increase - that's what we mean by "adiabatically" - and we may study the reactions of a particular system to these slow external changes. But the slow external changes are driven by an independent force, e.g. by muscles of a human controlled by the human's "free will": there is no "formula" for "what the adiabatic changes should be". The "spontaneous" force acting on the parameters - without an independent driving force - is zero.
The same is true for any other paper about the "adiabatic reaction force" in this context, e.g. one by Berry himself (with Shukla).
The term "adiabatic reaction force" was just used by Verlinde to replace "entropic force" because everyone has understood by now that gravity isn't an entropic force, also because entropic forces are *irreversible*. But when the entropy actually isn't increasing, the entropic force goes to zero.
So saying that the "adiabatic reaction force" is nonzero is the same childish mistake as talking about "forces induced by a constant gravitational or electric potential". By a simple calculation, d0/dx = 0, we get zero. You can't have it both ways. Forces related to entropy are either irreversible, or zero. Each of these conditions is incompatible with basic properties of gravity.
Cheers
LM
The 2007 paper you referred to is about Berry's phase which is a quantum phase induced by an adiabatic change of parameters. But the adiabatic change of parameters must be pushed by another, external force: the adiabatic change is not the first driver. The external parameters are being slowly changed by an external agent so that the entropy doesn't increase - that's what we mean by "adiabatically" - and we may study the reactions of a particular system to these slow external changes. But the slow external changes are driven by an independent force, e.g. by muscles of a human controlled by the human's "free will": there is no "formula" for "what the adiabatic changes should be". The "spontaneous" force acting on the parameters - without an independent driving force - is zero.
The same is true for any other paper about the "adiabatic reaction force" in this context, e.g. one by Berry himself (with Shukla).
The term "adiabatic reaction force" was just used by Verlinde to replace "entropic force" because everyone has understood by now that gravity isn't an entropic force, also because entropic forces are *irreversible*. But when the entropy actually isn't increasing, the entropic force goes to zero.
So saying that the "adiabatic reaction force" is nonzero is the same childish mistake as talking about "forces induced by a constant gravitational or electric potential". By a simple calculation, d0/dx = 0, we get zero. You can't have it both ways. Forces related to entropy are either irreversible, or zero. Each of these conditions is incompatible with basic properties of gravity.
Cheers
LM
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