The work integral is something I saw long time ago and in completely understood it. \begin{align} W_{12} & =\int F(x)dx=m\int^{t_2}_{t_1}adx=m\int\left(\frac{dv}{dt}\right)dx=m\int\left(\frac{dv}{dx}\right)\left(\frac{dx}{dt}\right)dx\\ &=m\int\left(\frac{dx}{dt}\right)dv=\frac12\left(mv_2^2-mv_1...

This is in reference to page 326, 327 of introduction to solid state physics, 8th edition by Charles Kittel The mean field theory does not give a good description of the variation of $M$ at low temperature. For $T<<T_{c}$ the argument of $\tanh$ in (9) is large and $\tanh(\xi) = 1 - 2e^{-2\xi}$ ...

On the one hand, black holes are said to be able to grow by acquiring mass. In order to see the event horizon’s radius increase, you first have to see mass fall into it. But on the other hand, I’m led to believe that you can never see any mass cross the event horizon; it will instead look like it...

If you expand a gas adiabatically using a piston, the process is isoentropic. However, if you simply remove the piston and let the gas expand freely, the process is now not isoentropic. What makes these processes different? Is it simply because the free expansion is not a reversible process? But ...