This question has introduced me to the whole "entropic force" area which has several papers during 2010. I see that there are references to "entropic force" explanations for Coulomb's law and other areas too. Here is a link to a simple introduction to these ideas.
The Verlinde paper and others however are deriving Newtons Law, Einstein's GR etc as classical theories. The underlying formulation of course being a stochastic behaviour of unknown microstates. Despite the presence of $\hbar$ and the motivation from the Black Hole area formulae the Verlinde paper does not introduce an explicit link with quantum mechanics. Thus there is no derivation of Schrodinger's equation and no introduction of $\Psi$.
The Kobakhidze paper says "One starts with a "holographic screen” S which contains macroscopically large number of microscopic states which we denote as $\left|i(z)\right\rangle$, $i(z) = 1, 2, ...,N(E(z), z).$ The screen is then described by the mixed state
$$\rho(z)=\sum p_{i(z)}\left|i(z)\rangle \langle i(z)\right|$$
However Verlinde does not explicitly introduce microstates as quantum states, with density matrices etc, although this is a tempting extension.
Now it might be that this is the only sensible quantum development of the stochastic basis of the "entropic idea", but Verlinde has not taken it. So what is disproved is a theory that Verlinde has not written down.
Having said this, there is a resemblance between "entropy" and the idea of introducing "stochastics" into quantum theory. One such attempt is known as "Stochastic Electrodynamics" (link to Wikipedia). As you will see from the summary this has had successes with e.g. the Unruh effect, but problems modelling genuine quantum phenomena.
I dont know whether anyone has considered combining the two areas directly.
Short answer: Yes, I'd buy the Berkeley group's work; their value of $\mu$ is the highest I've seen...
Long answer: Yes. The question arises because of widespread confusion between the terms "adhesion" and "friction".
Crudely, adhesion is a force that resists the separation of in-contact surfaces in the normal direction. Friction is a force that opposes relative tangential motion between two in-contact surfaces. One need not imply the other, their causative mechanisms are distinct and in fact most models exclusively address one or the other.
Adhesion is driven by Van der Waals kinda forces.
"Coulomb" friction (solid-solid) is caused by the presence of asperities (think small bumps or protrusions) on surfaces. Due to the presence of these asperities, the "real" area of contact between two surfaces is much smaller than the "apparent" area of contact.
The Coulomb model is a phenomenological fit to experiments that was shown to be deducible assuming this type of contact. In some sense this sets an upper bound on the resistance to the tangential relative motion between surfaces.
If you somehow ensure these areas are very nearly the same, would $\mu$ then increase?
There are, indeed, "intimate" contacts where the apparent and real areas of contact are very nearly the same and the resistance to sliding large. In such cases, the frictional behavior is intimately linked to the mechanisms of deformation at the small scale (e.g. plasticity in metals). However, even this is not enough to get the largest possible $\mu$.
For instance, the maximum shear stress resisting relative motion in metals is capped to a maximum value, beyond which it cannot increase.
(i.e.) Instead of
Shear stress = $\mu\times$ Normal stress ...(I)
You would've
Shear stress = min ($\tau_{max}$ , $\mu\times$ Normal stress) ...(Ia)
Equation (I) is simply Coulomb's law applied locally.
Q = $\mu N$ ...(II)
Now, if someone devised a contact / material system that produces very high $Q$ for a given $N$ in equation (II) in an experiment, they could claim that they had devised a material with high friction coefficient.
This is essentially what the Berkeley group seems to have done. As I said, this kind of thing is hard to do with metals - even if intimate contact is achieved (say, under a state of severe deformation), something like Eqn. (Ia) kicks in and prevents the shear resistance from rising further. Previously, therefore, people achieved high $\mu$ using compliant, soft materials, but these guys use a microfiber array to engineer a surface with high $\mu$.
Their main advances are
(1) High $\mu$ than reported in soft materials
(2) Allowing control of $\mu$ by controlling the fiber layout etc
(3) Achieving high $\mu$ in combination with low adhesion, which was not the case in softer materials. This is the kind of property combination you'd need for automobile tyre.
Coming to the Gecko paper, it has far more to do with adhesion that friction. The Gecko paper and the Berkeley friction paper have little to do with each other. Also, contrary to popular myth, the Gecko mechanism has nothing to do with "suction". See these papers in Nature - it is largely adhesion driven.
Adhesive force of a single gecko foot-hair
K Autumn, YA Liang, ST Hsieh, W Zesch, WP Chan… - Nature, 2000
Micro-fabricated adhesive mimicking gecko foot-hair
AK Geim, SV Dubonos, IV Grigorieva… - Nature materials, 2003
For physicists interested in these areas - I understand that tribology and solid mechanics are not taught in US physics departments - it might help to refer to standard texts on Tribology by Bowden and Tabor, Kendall, Israelachvili, Persson, Maugis etc.
Or better still, talk to your colleagues who work in tribology (they're usually to be found in mechanical engineering, materials science and chemistry). They will be eager and willing to help, if only for the opportunity to brag at faculty meetings that a physicist asked them for advice :-)
Best Answer
Thermodynamically polymers can be discussed in two different ways:
Statistical mechanics viewpoint
When a single polymer molecule is treated as a thermodynamical system, its entropy comes from the fact that the chain can be folded in many different configurations. The stretched configurations, with the extent of $\sim Nb$ ($N$ - degree of polymerization, $b$ - size of the link) are far less probable than the ball-like configuartions. In fact, the radius of gyration of a polymer is typically given by $$ R_g\approx N^\nu b, $$ where $\nu$ is about $3/5$. Thus, A stretched polymer, when released, returns will behave as an isolated system: evolve towards the state of maximum entropy, i.e., sample different configurations, most of which are not the stretched ones.
Thermodynamics viewpoint
One has to perform work in order to stretch a polymer, i.e., increase its internal energy. When the polymer is releazed, it converts this energy to entropy. Note that supplying system with heat is not the only way to change its entropy, since entropy is a state function, while the heat is not - i.e., the same entropic state can be reach by different paths.
References