Why don't marbles naturally arrange themselves like a crystal? Most solids are crystalline in nature because the energy released during the formation of ordered structure is more than that released during the formation of disordered structure such that the crystalline state is the lower energy state. So if we take different marbles in a box and shake it then shouldn't they arrange themselves in order to get to a low energy state? But we see they arrange in a disorderly way. Why do different phenomenon occur in these two cases?
 A: As HolgerFielder and Pieter said in their answers, marbles do form crystalline arrangements.  Notice, though, in Holger Fielder's illustration that the arrangement is much less ordered near a boundary.  
If marbles were confined in a way that did not impose hard boundary conditions, then they would almost always form perfectly crystalline arrangements.  A jar imposes boundary conditions that are geometrically incompatible with a perfectly crystalline arrangement.
A: This seems to be more a question for chemistry. The reason lays on the atomic bonds.
First at all, for high enough temperatures all solids will go to a liquid or and gaseous-like state and behave like marbles. And for temperatures near 0 Kelvin the marbles will behave more or less like a crystal.

... if we take different marbles in a box and shake it then shouldn't they arrange themselves in order to get to a low energy state.

It does. Nearly all marbles will lay in the closed-packing of equal spheres.
And why the balls do not stick together an the crystalline level? Because it is needed some activation energy. But that’s all about chemistry.
A: Interaction between marbles is very similar to the hard sphere (HS) interaction model i.e. a pair-wise potential energy which is zero if spheres do not overlap and $+\infty$ elsewhere.
Hard spheres are one of the first systems studied via computer simulation and one of the first big surprise was that by increasing pressure, they are able to crystallize from a disordered fluid to an fcc crystal, in 3D, or to a triangular lattice in 2D. After the first pioneering studies the scenario has been confirmed many times and fully understood. Moreover,  in the nineties, the experiments by the Pusey's group in UK have shown that the theoretical scenario is closely followed by colloidal systems designed to mimic as closely as possible a real system of HS  (Pusey, P. N., & Van Megen, W. (1986). Phase behaviour of concentrated suspensions of nearly hard colloidal spheres. Nature, 320(6060), 340.)
It is interesting to notice that the HS crystal is stable on the base of  entropic reasons. Neither attraction nor quantum mechanics are needed and the density of the coexistent solid at freezing is about 30% smaller than the close packing density (which means that in the HS crystal at the freezing point spheres collides frequently but do not touch all the time). Probably one of the most interesting things about the HS solid is that is a very nice illustration why the naïf equation entropy="spacial disorder" is wrong: the HS crystal has a higher entropy per particle than the coexisting liquid.
What can be said about marbles, taking into account HS? Although their interaction is a very good representation of the HS potential, usually they lack the dynamics underlying the behavior of a true thermodynamic system. Dissipative effects are quite strong and in a short time, without an external  continuous feed of energy, the kinetic energy of marbles gets dissipated. In the very old times of the study of liquids, somebody performed experiments with a 2D system of marbles in a tray put on top of a hi fi speaker as a tool to feed kinetic energy randomly. However, without such a flux of energy, what can be observed by shaking a 2D or 3D container almost filled with marbles is that, if the system is highly disordered at the beginning, after some shaking part of the "defects" are eliminated and, at least locally, the system looks like a crystalline solid at the close packing. But this is a situation not directly related with the thermodynamic transition. It has more to do with the stability with respect to perturbations of purely mechanical equilibrium configurations. As a last comment, I would add that the dynamic behavior of marble-like particles has been and still is an active research topic in the physics of granular media.
A: You are confusing two systems. The quantum mechanical of crystalline structure of solids, and the classical marbles , even if they are supposed to be perfect spheres. There is no quantization in the classical state to "lock" a marble in a position, it is free to assume any rotational position and translational on the horizontal, so it becomes a classical statistics problem. If they are packed tight they will organize themselves into a regular structure. If there is space, a single marble on a horizontal level of marbles, (gravity organizes them at levels because of potential energy)  cannot "stick" to any position without the slightest impulse sending it sliding over the lower level. There are no bound states.
A: They do. It is easiest to do show this in two dimensions. I used to demonstrate this on an overhead projector, with lead shot in a transparent CD-case. It is probably better to use smaller spheres (more spheres) than marbles. The other classic demo is with bubble rafts, which can also demonstrate the movement of dislocations.
In three dimensions, it is difficult to see this in a jar. One only observes the regions close to the glass. But I made this video, where I had prepared a regular surface of spheres as a seed:
https://play.lnu.se/media/t/0_bmg6kye7 (after 1:00 minute in the Swedish video)
And very small spheres of glass or plastic can form colloidal close-packed crystals. In nature, this has created gems, opal, when the lattice constant is of the order of the wavelength of visible light.
A: "So if we take different marbles in a box and shake it then shouldn't they arrange themselves in order to get to a low energy state?"
They certainly do - they will adopt a hexagonal (2D) or close packed (3D) configuration. In a real life scenario we may not immediately see that. That is so because of friction between marbles at their point (or rather area) of contact. However, if you remove that constraint, i.e. assume absence of friction, the entire collection of marbles will adopt the ordered configurations (hexagonal or close packed) after each time you shake the container.
A: You should study annealing.  Vastly oversimplifying: 


*

*If you cool the sample slowly enough, it retains enough energy long enough to explore its state space and find very low entropy, crystalline, states.

*If you cool a sample rapidly, it loses energy too rapidly to explore more than a tiny neighborhood of its state space, and produces non-crystalline states.


In your example with shaking marbles, you cool very rapidly.  To simulate slow cooling, you would shake for a very long time, gradually tapering the amplitude of the shaking.  Given the energy barrier to dislocation with reasonably sized marbles, you would have to taper very slowly.
Note that shaking may not be the best way to provide energy to the system to quickly find deep valleys in the energy landscape.  "Dice Become Ordered When Stirred, Not Shaken"  (The article also shows that too rapid stirring prevents settling to an ordered state -- the system continues exploring nearby less ordered states.)
A: The van der waals forces between marbles are very small when compared to other forces acting on the marbles such as gravity and friction. This is due to their relatively large size when compared with atoms.
However, when you go down to the size of individual atoms in a crystal lattice, the van der waals forces between them begin to become more comparable in size to other forces, which means that in some cases they can overcome these forces in order to arrange themselves in the lowest energy state, dependant on the material, temperature and pressure.
Only the atoms on the very edge of the marbles can contribute to van der waals forces between the marbles. The amount of atoms on the edge in comparison to the amount of atoms inside is extremely small. For individual atoms in a crystal lattice, the whole atom takes part in van der waals forces.
