Speculation on the Role of Volume in Entropy for an Ideal Gas

   Traditionally, the Maxwell-Boltzmann entropy is given by -k ln(number of states). As a result, it follows immediately that for a fixed T (temperature) and N (number of particles), a larger volume has more states as there are more places to place the particles. In (1), however, we introduced a function: Sum over i p(ei) F(p(ei)) ((1)) with F(p(ei))= -ei/T and p(ei) the unnormalized equilibrium probability. We argued that for the sake of stability, changing p(ei) -> p(ei)+dp(ei) in ((1)) should have the dp(ei) terms  match those of the constraints Sum over i p(ei)=1 and Sum over i ei p(ei) = Eave as these sum to zero. In such a viewpoint, there is no notion of the number of states. We thus ask: How would one think of the second law of thermodynamics in this viewpoint for an ideal gas constrained to the left half of a large volume which has the central barrier removed allowing the gas to fill the entire volume? There is no notion of  the number of states so one cannot state that the entropy ( as state function of T,V,N necessarily increases or decreases because delta E=0  heat=0 and work=0. We argue that in a given volume, particles tend to move away from the center of mass of the volume (due to the direction of their velocity vectors) and so given the chance will do so without any work performed on them, leading to a larger volume with T and V remaining fixed. The reverse process, with delta E = heat + work means that to create a smaller volume, one must do work on the single particles. If delta E remains the same at the beginning and end and work is done, this work must be released as heat. By convention a negative heat means that heat is released. Heat, however, is not linked with an exact differential. Mathematically, the exact differential is TdS, where S(T,V,N). If one considers a quasistatic route with T and N constant (i.e. an isothermal path), then V changes from a large number to small one and dE=0 at each instant. Given that work is done then S must decrease as V changes from a large to small number. This suggests that entropy for fixed T,N is smaller for a smaller volume due to the notion of work going against the motion of a particle with no reference to the number of states in a large versus small volume. This is already known in the literature, but we suggest that the microscopic notion of work being done on particles to push them against their velocity vector direction is sufficient for ideal gas entropy decreasing with V for T,N held constant. One does not need the extra microscopic viewpoint of ln (number of states), we argue.