Maximal Entropy of the Boltzmann Distribution: A Quantum Perspective

Specifically, in the classical limit (when βEn << 1), the quantum energy levels become very close to each other, and the partition function can be approximated by an integral over continuous energy states rather than a sum over discrete states. In this limit, the quantum Boltzmann distribution approaches the classical result...

Is the Transition from Univariate to Multivariate Gaussian Distribution Linear?

Now that we have the foundation in place, let’s shift gears and consider the generalization of the univariate Gaussian to higher dimensions. In the multivariate case, we are no longer dealing with a single random variable, but rather a vector of random variables...

Problem Ⅲ: Group Theory

Problem Ⅱ: Linear Transformations