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From: Jarek Duda on 14 Jul 2010 08:26
While considering random walk on a graph, we usually assume that for
given vertex, each outgoing edge is equally probable - maximizing
uncertainty/entropy locally.
It has recently occured that we can also do it globally: for given
graph choose stochastic process (probabilities for choosing edges),
which really maximizes entropy:
http://link.aps.org/doi/10.1103/PhysRevLett.102.160602
http://demonstrations.wolfram.com/GenericRandomWalkAndMaximalEntropyRandomWalk/
We can imagine this condition also as that:
- for each pair of vertices, each path of given length between them is
equally probable, or
- that we take uniform distribution in the space of possible
(infinite) paths.

These conditions allow to calculate probabilities of edges
combinatorially: we consider the space of all paths with some fixed
part and in each step elongate them on both sides - in the limit we
get probability distribution of this fixed part (2nd section of
http://arxiv.org/abs/0710.3861 ):
while being in i vertex, probability of using edge to j is (1/
lambda)* (psi_j /psi_i)
where psi is the dominant eigenvector of the adjacency matrix to
lambda eigenvalue.
Its stationary probability of being in i vertex is proportional to
(psi_i)^2.
We get 'the squares' because we consider situation inside paths -
while considering situation on ends of paths, probability distribution
would be just proportional to psi_j.

The adjacency matrix occurs to correspond to physical Hamiltonian and
this natural statistical ensemble leads precisely to the lowest
discrete quantum state probability density.
We can generalize this model to use Boltzmann distribution instead of
uniform distribution (optimizing free energy instead of entropy) and
take infinitesimal limit of such lattice covering the space to get
thermodynamical going to the real quantum local lowest energy
eigenstate in continuous limit.
'The squares' make that qunatum mechanics doesn't fulfill Bell
inequalities - this model naturally explains it as the result of that
we are living in spacetime - particle's trajectories doesn't end in
this moment, but goes further into the future ( http://arxiv.org/abs/0910.2724
).

We always believed that natural 'locally maximizing entropy random
walk' was the fundamental one - it leads to Brownian motion in
continuous limit: 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) - when we cannot trace the evolution, we use maximal
uncertainty principle: maximize entropy - 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.

What do you think about it?
From: kunzmilan on 15 Jul 2010 05:10
On Jul 14, 2:26 pm, Jarek Duda <duda...(a)gmail.com> wrote:
> While considering random walk on a graph, we usually assume that for
> given vertex, each outgoing edge is equally probable - maximizing
> uncertainty/entropy locally.
> It has recently occured that we can also do it globally: for given
> graph choose stochastic process (probabilities for choosing edges),
> which really maximizes entropy:http://link.aps.org/doi/10.1103/PhysRevLett.102.160602http://demonstrations.wolfram.com/GenericRandomWalkAndMaximalEntropyR...
> We can imagine this condition also as that:
> - for each pair of vertices, each path of given length between them is
> equally probable, or
> - that we take uniform distribution in the space of possible
> (infinite) paths.
>
> These conditions allow to calculate probabilities of edges
> combinatorially: we consider the space of all paths with some fixed
> part and in each step elongate them on both sides - in the limit we
> get probability distribution of this fixed part (2nd section ofhttp://arxiv.org/abs/0710.3861):
> while being in i vertex, probability of using edge to j is    (1/
> lambda)* (psi_j /psi_i)
> where psi is the dominant eigenvector of the adjacency matrix to
> lambda eigenvalue.
> Its stationary probability of being in i vertex is proportional to
> (psi_i)^2.
> We get 'the squares' because we consider situation inside paths -
> while considering situation on ends of paths, probability distribution
> would be just proportional to psi_j.
>
> The adjacency matrix occurs to correspond to physical Hamiltonian and
> this natural statistical ensemble leads precisely to the lowest
> discrete quantum state probability density.
> We can generalize this model to use Boltzmann distribution instead of
> uniform distribution (optimizing free energy instead of entropy) and
> take infinitesimal limit of such lattice covering the space to get
> thermodynamical going to the real quantum local lowest energy
> eigenstate in continuous limit.
> 'The squares' make that qunatum mechanics doesn't fulfill Bell
> inequalities - this model naturally explains it as the result of that
> we are living in spacetime - particle's trajectories doesn't end in
> this moment, but goes further into the future (http://arxiv.org/abs/0910.2724
> ).
>
> We always believed that natural 'locally maximizing entropy random
> walk' was the fundamental one - it leads to Brownian motion in
> continuous limit: 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) - when we cannot trace the evolution, we use maximal
> uncertainty principle: maximize entropy - 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.
>
> What do you think about it?

Entropies and information indices of star forests: Coll. Czech. Chem.
Commun. 51, 1856-1863, (1986).
kunzmilan
From: Jarek Duda on 15 Jul 2010 14:11
Please expand

If you want to some discussion about it, see for example
http://groups.google.com/group/sci.physics/browse_thread/thread/cad35cab9548a59f#
http://physicsworld.com/cws/article/news/43203
From: Jarek Duda on 25 Jul 2010 02:38
I've just made demonstration about using MERW to model electron
conductance - with stationary probability density exactly as in QM:
http://demonstrations.wolfram.com/preview.html?draft/93373/000008/ElectronConductanceModelUsingMaximalEntropyRandomWalk
In classical model for even the smallest potential applied, we
immediately get almost uniform current flow through the whole sample,
while in this new models we usually require some nonzero minimal
potential gradient to 'soak' out of entropy wells through complicated
entropic landscape.
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