Quantum effects in very small scale numerical modelling

In summary, the conversation discusses the potential impact of increased computational power on numerical modelling, specifically in relation to quantum scale discretisation and the use of classical equations. There is discussion about the challenges of modelling quantum systems and the need for extrapolation to an infinite system. The possibility of solving Schrodinger's equation for fluid dynamical systems is also mentioned.
  • #1
Tyro
105
0
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 Schrodinger'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.
 
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  • #2
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 2N, 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??
 
  • #3
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.
 

1. What are quantum effects in numerical modelling?

Quantum effects in numerical modelling refer to the incorporation of quantum mechanics principles into simulations of very small scale systems, such as atoms and molecules. This allows for a more accurate representation of the behavior of these systems and their interactions.

2. Why is it important to consider quantum effects in numerical modelling?

Quantum effects play a crucial role in the behavior of very small scale systems and can have a significant impact on their properties and interactions. Neglecting these effects can lead to inaccurate results and hinder our understanding of these systems.

3. How are quantum effects incorporated into numerical modelling?

Quantum effects can be incorporated into numerical modelling through various methods, such as density functional theory, quantum Monte Carlo simulations, and quantum molecular dynamics. Each method has its own strengths and limitations and is chosen based on the specific system being studied.

4. What types of systems can benefit from including quantum effects in numerical modelling?

Any system that operates at a very small scale, such as atoms, molecules, and nanoparticles, can benefit from including quantum effects in numerical modelling. This includes systems in materials science, chemistry, and biology.

5. What are some challenges of incorporating quantum effects into numerical modelling?

Incorporating quantum effects into numerical modelling can be computationally demanding and require high-performance computing resources. It also requires a deep understanding of quantum mechanics principles and the ability to accurately translate them into numerical simulations. Additionally, different methods may be more suitable for certain types of systems, making it important to carefully select the appropriate approach for each study.

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