Quantum effects in very small scale numerical modelling

[Tyro]

For those of you familiar with numerical modelling of various phenomena, you will know about work like the various discretisation schemes, stability/gradient limiters for high order schemes and so on. The most broad sweeping improvement to the field of numerical modelling would ultimately be computational power, which would allow a either a more refined model/mesh calculated in the same time or the same one calculated in less time.

Going forward a several years to decades (depending if Moore's Law holds), when the computational power available is such that the finest mesh spacings -- at a quantum scale -- can be modeled, how will this be done?

Take a simple backward-biased first order discretisation of the linear advection equation for a fluid in a channel. Setting up the mesh is pretty straightforward, and the process is entirely deterministic. If the mesh were to be so fine that quantum effects became involved, the numerical model is now probabilistic. On a large scale, however, it has to be deterministic (talk about a computational analogue of Schrödinger's Cat ). Any ideas on how one would go about setting up the computation? Would the control volume approach be completely inadequate because the material being modeled is no longer continuous? Or will probabilities be sufficient to suitably render the CV properties as 'continuous'?

Some may point out that the current level of discretisation for the above problem is more than adequate, with no quantum scale discretisation necessary. But the problem above is just a simple example. Materials modelling at a quantum scale or CFD of rarefied gases come to mind.

 

[heumpje]

I'm not sure wether this is going to answer your question, but I have some experience with modelling quantum systems. The way they do it know is to take a grid and use a Hamiltonian to build up Hilbert space (via exact diagonalization or using a Monte Carlo method). Given the limited computer time you can only simulate systems of say 4x4 sites depending on the complexity of the system. To eliminate finite size effects, you should take a number of systems say 4x4, 6x6 and 8x8. The next step is the tricky one because you then have to extrapolate to an infinite system. The only improvement gained by more computer power is that you can go up to, say, 32x32 grids. The problem with quantum systems is that you have to know the entire Hilbert space ( remember: the groundstate of the system is in principle a linear combination of ALL basis states). For instance the dimension of Hilbert space for a spin system grows as 2^N, N the number of spins. So for a system of 10x10 you already have an extremely large Hilbert space. Are you familiar with the Renormalization Group theory??

 

[christench]

When you are trying to solve fluid dynamical problems numerically then you use classical equations. The numerical accuracy of the result will improve with a higher mesh density. However, too high a mesh density will introduce numerical error and eventually lead to the wrong answer. In the end, if you try to solve a classical equation numerically the best you can do is accurately model the classical picture.

If, on the other hand, you think you can solve shrodingers equation given the boundary conditions of your fluid dynamical system, then you will end up with a time dependent wave function. This you just interpret probabilistically.

By the way, if the basis states are orthogonal, then the ground state cannot be constructed from the other basis states.