Life with Physics

I reviewed some basics of thermodynamics as well as the Boltzmann entropy in this post . Here, I continue to review the Gibbs entropy, the information entropy as well as the cross-entropy that is widely used in machine learning. In the modern view, entropy is a measure of the ignorance of an observer on the system. Boltzmann Entropy Physics students start to learn the concept of entropy from the Carnot engine and Boltzmann entropy. The Boltzmann entropy applies to a single system with the fixed energy: \begin{equation}S=k_B\log \Omega\,,\tag{1}\end{equation}where $k_B$ is the  Boltzmann constant  and $\Omega$ is the number of microstates corresponding to a system's macrostate. Boltzmann also derived another form of entropy that is very close to the modern form in mathematics. Consider a single system consisting of $N$ particles, we partition $N$ particles into $n$ groups. Let $N_i$ be the number of particles in the group $i$, then the number of microstates is \begin{equation}\Omeg

This post is for knowledge sharing only. It is not intended to be tax or investment advice. 1. Contribute to pre-tax 401K and HSA As well known, pre-tax 401K and health saving account (HSA) are the only a few tax shelters for W2 workers.  2. Tax-loss harvesting If your total capital losses exceed your capital gains in a tax year, you can deduct up to \$3,000 (married filing jointly) from your ordinary income. If your losses are more than $3,000, you can carry over the remaining amount to future years, continuing to offset your income until all the losses are used up. A popular method to generate these deductible losses is  tax-loss harvesting . This involves selling a security at a loss to offset gains or reduce ordinary income, and optionally purchasing a similar, but not "substantially identical", security to avoid missing out on the market. However, be aware of the  wash-sale rule , which disallows claiming a loss if you buy the same or a substantially identical security