In an answer to the post about Microscopic Definition of Heat and Work, Ronan says,
$$<dE> = \sum \epsilon_idp_i + p_id\epsilon_i$$
We can see that the change in average energy is partly due to a change in the distribution of probability of occurrence of microscopic state $\epsilon_i$ and partly due to a change in the eigen values $\epsilon_i$ of the N-particles microscopic eigen states.
and the first term corresponds to heat: $$TdS = \sum \epsilon_idp_i.$$ I am having a hard time imagining what it means. If the following is true:
- probabilities $p_i$ can not change the shape of Gibbs distribution if we assume that the new state is in equilibrium
- $\sum p_i = 1$
then the temperature must have risen (without any work done)! Is that correct, or can anything else happen?