Entropy and internal energy + molecule Internal energy can be defined theoretically for one molecule (U = 1/2 Kb T) for example but entropy is defined for a system thus for many molecules. Then we define temperature equal to $ \delta U $ / $ \delta S $ but here U can be defined for one molecule, so S can also be defined for one molecule? How?
 A: The problem is that entropy is not defined for a system, but for an ensemble of systems (if your system is sufficiently large and in equilibrium you can obviously cheat and consider your system an ensemble of smaller systems, that only interact weakly ...).
So the answer depends on the thermodynamic ensemble you are in.
If you consider the molecule in the microcanoncial ensemble (totally separated from the outside and at a given energy), it will have a well defined entropy of $\ln \nu$, where $\nu$ is the amount of states the molecule can have at the given energy. The temperature is defined as $\frac 1 T = \partial_U S$ in the microcanonical ensemble (this is just a restatement of the formula you gave), but for a single molecule you will not get a well-defined temperature. As the possible states will be discrete due to the quantum mechanics of a particle in a box, and the steps not small compared to the total energy, you cannot define the derivative above, which needs approximately continuous energy values.
If you couple your single molecule to a heat bath, however, it will have a well defined entropy (because then your total ensemble of the particle and the heat bath will be large), by the gradient relation
$$ S = -\partial_T F, $$
with the free energy $F$. But in this ensemble the temperature is not defined in the way above, but is a property of the heat bath.
Note, that you already confuse the setting in your question, by assuming the single molecule has a temperature a priori (by defining its internal energy to be $U = \frac 1 2 kT$), this is the setting of the canonical ensemble this internal energy is actually the average value of the internal energy of the system, where the average is taken over the ensemble of systems. If you just had a single molecule in a tight box the internal energy would, in contrast, be a specific value, which is, as described above, not connected with a well defined temperature.
