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So especially in the previous section, you've seen that we can use fitting procedures to describe very

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complicated physical problems and to understand then what is going on and to do this, we have formulated

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a lot of the function.

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We have then formulated an oral function between the data and our model function.

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And then we used the so-called gradient descent methods to minimize this error until we had the perfect

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fit.

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However, it turns out that when you have many objects, for example, in the previous section, many

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of these oscillators, then it takes an enormous amount of time to really do this gradient descent and

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to minimize the error in this way.

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So instead, a different approach would be to not use gradient descent, so not just update the parameters

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along the gradient direction, but instead to use randomness.

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So it means we randomly change one of the parameters.

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And if this change reduces the error function, then it's considered a good change and it is accepted.

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However, when the change increases the energy, it's considered a bad change and then it will just

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be disregarded.

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So it means that we use randomness to gain knowledge, and this is really the idea of Monte Carlo simulations.

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So really exploiting randomness and luck, and we will start with a very simple and mathematical example.

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We will approximate the value of PI by basically placing random points in a square.

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And then we're counting how many points are inside of a certain region, which is a circle and how many

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points are outside.

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And then we will turn to a physical problem that we will simulate the collective behavior of a magnet.

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So you see, a magnet consists of individual tiny magnets.

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And all of these magnets can point along all three directions in space, and they interact with each

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other.

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So it means when you try to minimize here or error, then it really takes an enormous amount of time.

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So instead, you minimize the error by just randomly reorienting individual magnets and checking if

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the error has reduced.

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And if this is the case, then you accept the chain to change.

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And if not, then you disregard the change.

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And we will really try this for this magnet and we will see that it brings about a lot of advantages.