Ok i thought i had this under control, but now i come to think about it, i get more confused. I know d sigma/d U for any system must be positive because the temperature must be positive (sigma is the entropy and U is the energy). But what about this? Consider the model of the paramagnetic system where theres a constant N particles in a constant volume. The energy is proportional to how many spin down - spin up particles there is. So lets say for system 1, N= 30. 29 are spin down, 1 is spin up (Lets just say energy is 28). Lets say this system comes in thermal contact with another system (system 2). (the systems are allowed to exchange energy). So initially the entropy of the 1st system is 30 choose 1 = 30. Now lets say system 1 exchanges its spin up for spin down (Change in energy is +2). But now entropy is 30 choose 0 = 1. So change is entropy is -29. so change is entropy wrt to change in energy is negative? And temperature which is 1 over that is negative?