 ### Wrote up to the simple example of MC integration

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 ... @@ -5,16 +5,26 @@ ... @@ -5,16 +5,26 @@ ## Introduction ## Introduction The Monte Carlo method is a very powerful tool of statistical physics. Monte Carlo methods are as useful as they are widespread. For example, one can also compute [molecular dynamics using Monte Carlo methods](https://en.wikipedia.org/wiki/Monte_Carlo_molecular_modeling). There's a reason it's named after Monaco's famous casino; it utilises probability and randomness. In most cases, a system is evolved to a new state which is chosen from a randomly generated ensemble of possible future states. Then, using some criteria, this new state is accepted or rejected with a certain probability. This can be used in many different areas of statistics, with end goals ranging from: reaching a Bose-Einstein ground state, minimizing an investment portfolio risk or [optimizing the boarding process of an airplane](https://arxiv.org/abs/0802.0733?). Considering the breadth of applications, we choose to center this second project on Monte Carlo methods. Lecture notes will be published soon! ## Monte Carlo integration ## Monte Carlo integration While there are multiple categories of Monte Carlo Methods, we will focus on Monte Carlo integration. To see the advantage of this technique, consider a system that is described by a Hamiltonian \$H(R)\$ depending on \$R\$. Let's call \$R\$ the set of all system degrees of freedom. This might include terms like a magnetic field \$B\$, potential \$V\$, etc. We're interested in a specific observable of this system called \$A(R)\$. Specifically, we'd like to know it's expectation value \$\langle A\rangle\$. From statistical physics, all system state likelihoods can be summarized in the partition function: \$\$Z=\int e^{=\beta H(R)}dR\$\$ Where \$\beta=1/(k_BT\$) and the Hamiltonian \$H(R)\$ is integrated over all system degrees of freedom. Here, the Boltzmann factor weighs the probability of each state. The expression for the expectation value can then be expressed as: \$\$\langle A\rangle = 1/Z \int e^{=\beta H(R)} A(R)dR\$\$ ### A simple example For most systems, \$R\$ is a collection of many parameters. Hence, this is a high-dimensional integral. This means an analytic solution is often impossible. A numerical solution is therefore required to compute the expectation value. In the next section, we will demonstrate the purpose of sampling an integral and convert it into a sum, which is easier to solve for computers. Then, a bit later, you will see why Monte Carlo integration becomes beneficial quite quickly. ### A simple example Take a general, one-dimensional integral \$I=\int_a^bf(x)dx\$. We can rewrite this integral into a summation as follows: \$\$\int_a^bf(x)dx = (b-a)\int_a^b \frac{1}{b-a} f(x)dx= \lim_{N \rightarrow \infty} \frac{b-a}{N} \sum_i^N f(x_i)\$\$ Where, \$x_i = a + i\ \frac{b-a}{N}\$ One could say, the \$\{x_i\}\$ are distributed uniformly in the interval \$[a,b]\$ The limit is only needed for the integral and summation to be exacftly the same. From probability theory, we learn that: \$\$\int p(x)f(x)dx \approx \frac{1}{N}\sum_i f(x_i)\$\$ Now, the \$x_i\$ are randomly drawn from \$p(x)\$. In other words: we are sampling the function \$f(x)\$ with values from \$p(x)\$. This way, the result of the integral can be constructed from the finite summation. In the previous example, the \$x_i\$ weren't random but rather evenly distributed. ### Why Monte Carlo integration becomes beneficial for high-dimensional integrals ### Why Monte Carlo integration becomes beneficial for high-dimensional integrals \$\$\int_a^bf(x)dx = (b-a)\int_a^b \frac{1}{b-a} f(x)dx=\frac{b-a}{N} \sum_i^N f(x_i)\$\$ ## Importance sampling ## Importance sampling ... ...
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