Can thermodynamical model be fundamental: reason not result?
Jarek Duda
theory – statistical result of some fundamental physics below, but
recently there became popular theories starting from ‘entropic force’
as fundamental (basing on holographic scenarios, like in
http://arxiv.org/abs/1001.0785 ).
For a simple mathematician like me it sounds like a nonsense – in
fundamental theory describing evolution of everything there should be
one concrete history of our universe – there is no place for direct
probabilities of scenarios required to define e.g. entropy.
So I wanted to ask if someone could explain why we can even think
about fundamental ‘entropic’ theories?
To start the discussion I would like to briefly remind/discuss looking
clear for me distinction between deterministic and stochastic/
thermodynamical models:
DETERMINISTIC models – the future is completely determined
- evolution of gas in a tank is full dynamics of all its particles -
for given valve opening there escaped concrete number of particles,
- it's usually Lagrangian mechanics of some field – there is some
scalar/vector/tensor/’behavior of functional'(QFT) in each point of
our spacetime, such that ‘the action is optimized’ – each point is in
equilibrum with its four-dimensional neighborhood (spacetime is kind
of ‘4D jello’),
- evolution equations (Euler-Lagrange) are HYPERBOLIC PDE - linearized
behavior of coordinates in the eigenbase of the differential operator
is
d_tt x = - lambda x
(0 < lambda = omega^2 )
so in linear approximation we have superposition of rotation of
coordinates – ‘unitary’ evolution – and so such PDE are called
wavelike – the basic excitations on water surface, in EM, GR, Klein-
Gordon are just waves,
- the model has FULL INFORMATION – there is no place for direct
probability/entropy in electromagnetism, general relativity, K-G etc.
– the model has some TIME (CPT) SYMMETRY INVARIANCE (no 2nd law of
thermodynamics – there is still unitary evolution in thermalized gas
or a black hole)
THERMODYNAMICAL/STOCHASTIC models – there is some probability
distribution among possible futures
- gas in a tank is usually seen as thermalized, what allows to
describe it by a few statistical parameters like entropy (like sum of –
p*lg(p) ) or temperature (average energy per degree of freedom) - for
a specific valve opening, the number of escaped particles is given by
a probability distribution only,
- it is used when we don’t have full information or want to simplify
the picture – so we assume some mathematically universal STASTICAL
ENSEMBLE among POSSIBLE SCENARIONS (like particle arrangements) –
optimizing entropy (uniform distribution) or free energy (Boltzmann
distribution),
- thermodynamical/stochastic evolution is usually described by
difussion-like: PARABOLIC PDE – linearized behavior of coordinates in
the eigenbase of the
differential operator is
d_t x = - tau x
(tau - ‘mean lifetime’ )
so in linear approximation we have exponential decay (forgetting) of
coordinates – evolution is called thermalization: in the limit there
survive only ones with the smallest tau – we call it thermodynamical
equilibrium and usually can be describe using just a few parameters,
- these models don’t have time symmetry – we cannot fully trace the
(unitary?) behavior so we have INFORMATION LOST – entropy growth – 2nd
law of thermodynamics.
Where I’m wrong in this distinction?
I agree that ‘entropic force’ is extremely powerful, but still
statistical result – for example if while random walk instead of
maximizing entropy locally what leads to Brownian motion, we do it
right: globally, we thermodynamically get going to the lowest quantum
state – single defects create macroscopic entropic barriers/wells/
interactions:
http://demonstrations.wolfram.com/GenericRandomWalkAndMaximalEntropyRandomWalk/
For me the problem with quantum mechanics is that it’s between these
pictures – we usually have unitary evolution, but sometimes entropy
grows while wavefunction collapses – there is no mystical
interpretation needed to understand it: entropy maximizing from
mathematically universal uncertainty principle is just enough (
http://arxiv.org/abs/0910.2724 ).
What do you think about this distinction?
Can thermodynamical models be not only effective (result), but
fundamental (reason)?
Can quantum mechanics alone be fundamental?
Jarek Duda
simplified effective picture in which we assume statistically typical
behavior, like that when we completely don't know which scenario is
happening, we should assume maximizing entropy uniform distribution
http://en.wikipedia.org/wiki/Microcanonical_ensemble
Unfortunately in world of quantum mechanics which is generally
believed to be impossible to understand but still fundamental - the
logic of reason-result distinction is no longer binding...
The belief that QM is fundamental leads to many worlds interpretation
- that our spacetime is infinitely quickly branching tree of parallel
universes ...
... while field theories we use on all scales (GR, EM, Klein-Gordon,
QFT) are deterministic and clearly say what our spacetime is - in
these theories we live in static 4D action optimizing solution - each
point is in equilibrium with its 4D neighborhood - spacetime is kind
of '4D jello'.
They are deterministic and like QM mechanics have 'wavelike/unitary'
evolution.
So what's happening when we cannot fully trace the evolution? ... for
example the behavior of a single particle ...
In such situations we have to use some thermodynamical model - assume
some statistical ensemble among possible scenarios for example to
maximize entropy - assume that the particle makes some random walk ...
Maximizing entropy locally leads to Brownian motion in continuous
limit - but when we do it right: assume global entropy maximum (like
in models I advocate) - we get thermodynamical going to squares of
coordinates of the dominant eigenvector of discrete Hamiltonian (and
finally the real Hamiltonian while assuming Boltzmann distribution
among trajectories).
http://link.aps.org/doi/10.1103/PhysRevLett.102.160602
These new but fundamental stochastic models finally show what was
missing - that in field theories on thermodynamical level: when we
cannot fully trace the evolution, we should assume collapse to some
local lowest quantum state.
Living in specetime ('4D jello') leads to many nonintuitive 'quantum'
consequences - like (confirmed) Wheeler's delayed choice experiment,
that in models with limited information to translate what we are
working on (amplitude) into the real probabilities - we should
'square' it against Bell's intuition, or allows for 'quantum'
computations:
http://www.thescienceforum.com/Four-dimensional-understanding-of-quantum-computers-24936t.php
Sam Wormley
A Scientist Takes On Gravity
by DENNIS OVERBYE
Published: July 12, 2010
http://www.nytimes.com/2010/07/13/science/13gravity.html
"It’s hard to imagine a more fundamental and ubiquitous aspect of life
on the Earth than gravity, from the moment you first took a step and
fell on your diapered bottom to the slow terminal sagging of flesh and
dreams".
"But what if it’s all an illusion, a sort of cosmic frill, or a side
effect of something else going on at deeper levels of reality"?
"So says Erik Verlinde, 48, a respected string theorist and professor of
physics at the University of Amsterdam, whose contention that gravity is
indeed an illusion has caused a continuing ruckus among physicists, or
at least among those who profess to understand it. Reversing the logic
of 300 years of science, he argued in a recent paper, titled “On the
Origin of Gravity and the Laws of Newton,” that gravity is a consequence
of the venerable laws of thermodynamics, which describe the behavior of
heat and gases".
Darwin123
> For me the problem with quantum mechanics is that it’s between these
> pictures – we usually have unitary evolution, but sometimes entropy
> grows while wavefunction collapses – there is no mystical
> interpretation needed to understand it: entropy maximizing from
> mathematically universal uncertainty principle is just enough (http://arxiv.org/abs/0910.2724).
>
> What do you think about this distinction?
> Can thermodynamical models be not only effective (result), but
> fundamental (reason)?
> Can quantum mechanics alone be fundamental?
I think there are already theories (or interpretations?) accepted by
the mainstream (or a tributary thereof) that treat quantum mechanics
this way. These are sometimes called coherence theories of quantum
mechanics. Coherence theory doesn't start out with string theory. I
believe that parts of coherence theory are being incorporated into
string theory.
I don't have any references on me. However, I remember reading
books on it. Coherence theory is basically an "all wave"
interpretation of quantum mechanics. The measuring apparatus is
basically a very complex waveform, with a lot of degrees of freedom.
That is why it acts "classical."
When a sample s being examined with the measuring apparatus, it
causes the system to change two ways. First, the system decoheres. The
phases of the sample are locked together, so that the sample ends up
turning into a wave pulse. Then, the entropy of the sample increases.
There really isn't a fundamental change in "waviness" of the
sample. However, the result is a type of wave-form collapse. The
sample wave form becomes like a particle.
The loss of information is hidden by the changes in the measuring
apparatus. There are so many degrees of freedom in the measuring
apparatus that it is impossible to measure the phase of each
component.
Needless to say, there are still some problems that come with
this interpretation. However, it seems more logically self consistent
than the Copenhagen interpretation. Not every scientist is comfortable
with the Copenhagen interpretation of quantum mechanics.
Darwin123
of cosmology and high energy physics. Even if people are disappointed
in string theory as a fundamental theory, there are complex systems
that map neatly onto string theory.
Okay, this is sort of a cop out. We can't use string theory the
way we originally hoped. However, we do use it for something else
which may be technologically important.
The mathematics of string theory turns out to have interesting
applications in condensed matter physics. String theory mathematics is
being used to describe "perfect gases." Experiments are being
performed with "perfect gases." The result match experiments.
"Perfect gases" are fluids that have negligible viscosity and yet
act like they are made of Newtonian particles. Perfect gases are
formed in the collision of nucleii with high atomic number, and in
some very cold gases.
This is different from superfluids. Superfluids have negligible
viscosity but have quantum mechanical properties. These include helium
III, superconductors and Bose-Einstein gases.
It is ironic that string theory ultimately explains why some
systems act classical rather than quantum mechanical. This is not
fundamental in terms of fundamental particles. Too bad. However, it is
tied to reality. So string theory may become very important.
Rock Brentwood
> I always thought that thermodynamics/statistical physics is effective
> theory – statistical result of some fundamental physics below, but
> recently there became popular theories starting from ‘entropic force’
> as fundamental (basing on holographic scenarios, like inhttp://arxiv.org/abs/1001.0785).
Newspapers need to learn how to properly frame stories. This is not
some "new and sensational discovery announced here today!", but just
an extension of what's already a running thread in the literature and
has been for nearly 20 years (and more).
If I didn't see Jacobson's name in the reference list, I would have
called it plagarism. Look up his article in the reference list. That's
what it's all about and this paper is just a "I'm going to add my 2
cents so I can have an excuse to publish the same damn thing from
'another point of view'" type thing. The literature is replete with
these.
Jacobson didn't say that thermo is fundamental. He said -- to distill
it in a way that best fits how you think of thermo -- that gravitation
can be derived from the laws of thermo, when the following specific
interpretations are taken:
0th law: the Hawking Unruh temperature is taken as T in the second law
1st law: the continuity equation for the stress tensor is taken as the
first law
2nd law: dQ = T dS, where Q is the energy flux across a causal horizon
3rd law: the entropy is taken as proportional to the area over a
"causal horizon", this gives you S
Combined, this gives you Einstein's equations. In addition (a point
Jacobson failed to make, but implied in his paper): the actual
constant of proportionality is determined by the condition that
everything be made to fit Newton's inverse r^2 formula with Newton's
constant G as the proportionality factor.
The way Jacobson went on to explain matters is that the "graviton" is
no more fundamental than the phonon is in solid state physics. That
means: (1) the "graviton" is for all intents and purpose a vacuum
phonon and (2) there is a ultraviolet breakdown, entirely analogous to
that for phonons in solid state physics.
Sam Wormley
> String theory has turned out to have testable predictions outside
> of cosmology and high energy physics. Even if people are disappointed
> in string theory as a fundamental theory, there are complex systems
> that map neatly onto string theory.
What is the testable prediction?
Sam Wormley
> Newspapers need to learn how to properly frame stories. This is not
> some "new and sensational discovery announced here today!", but just
> an extension of what's already a running thread in the literature and
> has been for nearly 20 years (and more).
>
> If I didn't see Jacobson's name in the reference list, I would have
> called it plagarism. Look up his article in the reference list. That's
> what it's all about and this paper is just a "I'm going to add my 2
> cents so I can have an excuse to publish the same damn thing from
> 'another point of view'" type thing. The literature is replete with
> these.
>
Dr. Verlinde said he had read Dr. Jacobson’s paper many times over the
years but that nobody seemed to have gotten the message. People were
still talking about gravity as a fundamental force. “Clearly we have to
take these analogies seriously, but somehow no one does,” he complained.
His paper, posted to the physics archive in January, resembles Dr.
Jacobson’s in many ways, but Dr. Verlinde bristles when people say he
has added nothing new to Dr. Jacobson’s analysis. What is new, he said,
is the idea that differences in entropy can be the driving mechanism
behind gravity, that gravity is, as he puts it an “entropic force.”
Jarek Duda
gravity - it only says that mathematics is universal ...
Please answer to logical/philosophical(?) question from the subject.
Having some concrete situation in our spacetime, like solution from
deterministic physics, we can introduce thermodynamical picture OVER
THIS SOLUTION: in each point of spacetime we take a ball and average
over it to get local effective statistical parameter field like
entropy or temperature - it allows to focus on SIMPLIFIED picture in
which we focus on statistically typical behavior.
The 2nd law of thermodynamics says that four-dimensional gradient of
such introduced scalar field of entropy agrees with our time arrow -
there has to be entropy minimum in Big Bang, so it probably created
the gradient...
But how such effective picture over some concrete solution can be
fundamental - The Reason?
Darwin, I couldn't find this 'coherence theories of quantum
mechanics', but what you say sounds somehow similar.
We always believed that natural 'locally maximizing entropy random
walk' was the fundamental one, it leads to Brownian motion - it's good
enough approximation for diffusion in fluids, but has nothing to do
with QM.
Now we finally have the real Maximum Entropy Random Walk and it says
exactly what was needed: that on thermodynamical level (of e.g. field
theories) we should assume 'wavefunction collapse' to the local lowest
energy state precisely like in QM - so the only nonmistical: Born's
ensemble interpretation is finally enough to understand QM.
Darwin123
> Darwin, I couldn't find this 'coherence theories of quantum
> mechanics', but what you say sounds somehow similar.
I think I used the wrong buzzwords. I think the model is more
generally called "quantum decoherence." It involves the environment
which is a complex wavefunction "continuously measuring" the sample
wave function. This eventually causes the sample wave function to
collapse into a wave packet, which resembles a particle.
The topic has gained an importance because of advances in
technology. Quantum decoherence is very important for developing
quantum computers and quantum communications. Quantum communications
are starting to be used for sending encoded messages. Look up quantum
cryptology.
Reexamination of the Copenhagen interpretation has left the field
of philosophy because of these new technological applications of
quantum mechanics. The Copenhagen interpretation is no longer the
final say in quantum mechanics. Since general relativity and quantum
mechanics still have fundamental contradictions, quantum decoherence
still isn't the final say in everything.
Quantum decoherence is more fundamental than string theory, at
least to me. String theory is a model within quantum mechanics. So I
am not surprised that a theorist wants to bring together string theory
with quantum decoherence. Quantum decoherence answers the question of
whether the electron is a particle or a wave (a wave!). Still, you one
wants to know if gravitational energy is a particle or a wave. So
efforts to unify general relativity and quantum mechanics will at some
point have to collide with quantum decoherence.
I originally went through articles and books on the subject. This
was the bad old days before "google". However, I did a short google
search for you. I came up with the following set of links.
Specific overviews of quantum decoherence
http://en.wikipedia.org/wiki/Quantum_decoherence
http://arxiv.org/abs/quant-ph/0312059
General overviews of quantum mechanical interpretations, which contain
a description of quantum decoherence.
http://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics
http://www.bibliotecapleyades.net/archivos_pdf/quantum_mechanics_crossroads.pdf
I didn't list every hit that came up. However, I hope these
links are a good beginning. If you seriously research it yourself, I
am sure you will come up with better links. Maybe even a few good
books, like what I used to have!
Jarek Duda
is thermodynamical result of interactions with environment.
The essence of thermodynamics is using mathematical theorems like
maximum uncertainty principle.
Standard random walk was successfully pretending to already do it -
its continuous limit is enough to model diffusion in fluids, but from
QM or for example recent STM pictures of electron stationary
probability density on a surface of semiconductor, we clearly see that
fixed structure of defects in condensed matter makes this approximated
thermodynamical model inappropriate.
But when we do it right - use the real Maximal Entropy Random Walk and
generalized models, we get exactly what's needed - that we should get
going to the square of coordinates of the dominant eigenvector of
(discrete) Hamiltonian - that when we cannot trace unitary evolution,
we should assume 'wavefunction collapse' - explaining this decorence
interpretation.
Here is new discussion about it:
http://physicsworld.com/cws/article/news/43203
Rock Brentwood
> His paper, posted to the physics archive in January, resembles Dr.
> Jacobson’s in many ways, but Dr. Verlinde bristles when people say he
> has added nothing new to Dr. Jacobson’s analysis. What is new, he said,
> is the idea that differences in entropy can be the driving mechanism
> behind gravity, that gravity is, as he puts it an “entropic force.”
I read it fully now. His own account of what's new is off. The only
real things he added to the thread were: (1) a somewhat better
separation of the fundamentals (much along the lines I spelled out in
my last article, in fact) and (2) the identification of *which*
surfaces to take as "cutoff" surfaces. The other stuff was pretty much
an elaboration of what I already described in the previous article,
apart from the discussion of the interpretation of the potential.
But I've been floating a better suggestion in SPR on both fronts
lately -- (1) the Cosmological Horizon already gives you a ready-made
cutoff for the state space, by its very definition. And it grows with
time so you get something like a 2nd law as emergent (in contrast,
Verlinde's analysis took at as a postulate). Plus, you already have
ready-made horizons, over and above all this, by doing the
construction for the Noether theorem the right way.
(2) the analysis is *still* not getting to fundamentals, despite the
fact that he nearly hit on the central idea. It's still "stuck in the
present" in 1990's 2000's string-speak and Holography-speak, when all
of these issues are utterly irrelevant. The REAL issue almost alluded
to in paragraphs 4 and 5 of section 1 is that there is a breakdown of
the Noether Theorem for any symmetry that involves having to move
points.
The Noether theorem only applies to compact regions only. (And, so the
construction alluded to above, is the one made necessary by this fact
-- a local foliation of a saucer-shaped compact region generated by a
vector field that drops off to 0 on a fixed 2-D surface on the rim --
the "Noether Horizon" for lack of a better name).
For local symmetries, points are fixed, so no problem. The Noether
theorem works. For classical theory, moving a region is no problem,
you account for the difference by the boundary terms. It's "no
problem" with a proviso -- the so-called "constant" conserved currents
are not constant. They're functions of the shape of the region over
which the local foliation is done. They're functions of the Noether
horizon. If the region moves, the "conserved" currents change, unless
the symmetry leaves the horizon fixed.
But in any case: for Quantum theory -- BIG problem. When you move a
region, you change its state space and even the very definition of
"quantization". The new state space is not equivalent to the old state
space.
Part of what was in the original region is outside, part of what was
outside is inside. The result is an incomplete Bologiubov transform.
The situation is entirely analogous to what underlies the Hawking-
Unruh effect. You get an introduction of a kind of "anomalous" entropy
associated with the motion of the region -- a cut-off entropy. In
turn, this produces an anomalous contribution to the Noether current
associated with the diffeomorphism symmetry. That contribution goes on
the left-hand side the equation that has the stress tensor (the
Noether current) on the right.
There is nothing here involving issue-of-the-day-speak (i.e. AdS/CFT,
Holography, or whatever other largely fad issues have been in the
journals in the past 10-20 years). It's simply a matter of properly
addressing fundamentals.
In any case, his analysis closely fits the description I already gave.
In fact, you can even line up the equations (and this is something
even he missed).
Under "0th law": (3.8), (4.17)
Under "1st law": (3.11), (4.19)
Under "2nd law": (2.3)
Under "3rd law": (3.6), (3.10), (4.18).
The only really new idea there is that geodesic motion (meaning free
fall + inertia) only requires equations that fall under laws 0+2+3,
while backreaction requires equations that fall under laws 0+1+3. The
physical interpretation of potential is also interesting.
But, it also bears pointing out that he understated the "robustness of
the derivation" point. It's not that h-bar and c are not essentially
involved, as he pointed out, but that they're completely irrelevant.
You could actually revert the whole set of equations under section 3
to 19th century form by taking 1/(mu epsilon) in place of c^2 and
root(mu/epsilon) e^2/(2 alpha) for h. Only alpha is quasi-new. Then
all the equations (and even the explanations) start to look a whole
lot like something Lorentz has already discussed in the late 1800's or
early 1900's. These fundamentals may even be present in his 1904
paper. So, the "quantum explanation" or "string explanation" can just
as easily be recast as a Lorentz-style "electromagnetic" explanation.
This shows that NEITHER issue is at the root of the matter. Instead,
it's the issue of Noether symmetry that is, and the breakdown (as
described above) for the symmetries under the diffeomorphism group.