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