Numerical Work
Art Credit: BLEACH Volume 48
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Bootstrap Method
Bootstrap Method
The bootstrap method is a numerical technique that can find bounds on observables using a positivity constraint, which simply states that positive semi-definite matrices must have non-negative determinants/eigenvalues. Both single particle and matrix models can be bootstrapped, although their implementations are vastly different. In the former case, one finds a recursion relation based on the potential in order to solve for the energy eigenvalues of the system. In the latter scenario, one constructs a matrix of observables to obtain a lower bound on the energy of the system with respect to certain observables. A detailed description of both can be found in the UC Davis REU Summary in the Publications/Documents section.
Some examples (the system and the graph found through bootstrapping) can be found below. The work done (including the variables present in the graph) to find these graphs and more examples can be found in the aforementioned link.
Bootstrapped region as a function of E. K is roughly the strength of the positivity constraint. Note that the higher value of K correctly produces more energy eigenvalues (with ω = 1).
Harmonic Oscillator
Pöschl-Teller Potential
Bootstrapped region as a function of sech^2(x) and E. The peaks indicate the energy eigenvalues of this system, in which λ = 3 (λ is a constant in the potential).
Lower Bound of E/N^2 as a function of g, the coupling constant. L here is the number of observables considered in the matrix that the positivity constraint is applied to.
Anharmonic Oscillator Matrix Model
D0-Brane Matrix Model
Bosonic and Fermionic Contributions to the lower bound of λ^(-4/3)<tr(X̃^4)> with respect to E/(N^2λ^(1/3)). Here, N is the dimension of the matrices in the model and λ is a function of the coupling constant (of the bosons and fermions) and N.
Monte Carlo on Path Integrals
Monte Carlo on Path Integrals
Path integrals in (Euclidean time signature) quantum field theories are not generally convergent. As such, determining the ratio of such integrals - which is convergent - is a difficult task. Traditional numerical techniques, which would solve for each integral individually, cannot be relied on to evaluate correlation functions in field theoretic systems. This is especially true in higher dimensions.
The Monte Carlo technique can be used to evaluate this ratio without finding each integral. The elementary Monte Carlo method can do this as well, but the variance produced by this simpler technique is much higher and requires more runs to get close to the exact value. The Markov Chain Monte Carlo (MCMC) method instead uses importance sampling and Markov Chains (found through algorithms such as the Metropolis-Hastings) to minimize variance.
I am in the process of learning this method.
Simulations
Simulations
The following simulations were obtained using the Leapfrog numerical technique*, which updates the position of an object given its acceleration. Most examples below are from John R. Taylor's textbook Classical Mechanics, but the work done is my own. More information on the code can be found on my GitHub.
**Note: The GIFs, unless stated otherwise, are the first four minutes of the program running. The top plot is a schematic of the system, while the bottom plot is of the respective quantities' x values. To see plots of the systems when the program ends, please go to my GitHub**
* The Driven Damped Pendulum simulation can be improved by using the Runge Kutta Nystrom method (or similar methods), which properly accounts for the first derivative term in the equation of motion of phi.
Helpful Link(s):
Double Pendulum
Pendulum on a Block Attached to a Spring
Three Pendulums with Two Springs
Two Blocks with Three Springs
RLC Circuit
Simulation of an underdamped RLC circuit.
Simulation of an overdamped RLC circuit.
Simulation of a critcially damped RLC circuit.
Chaotic Driven Damped Pendulum
2D N-Body Simulation (with Collisions)
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