On Achieving the Entropy Maximum in Mechanical Systems

Consideration has been given to the well-known argument that in a conservative mechanical system existing in an equilibrium state, entropy reaches a maximum. First, using the example of a system of material points modeling a rarefied gas, it has been shown that a maximum of entropy defined through the number of microstates is achieved on a solution that cannot be approximated by an exponential distribution corresponding to the Maxwell-Boltzmann distribution. There are many such solutions and each of them satisfies both the condition of invariability of the number of points and the condition of constancy of the system ' s total energy. Next, consideration has been given to an alternative definition of entropy through the density of distribution of the probability of a mechanical system being in an assigned volume of phase space. It has been shown that a conservative mechanical system of general type assigned by its Hamiltonian, in selecting various sets in the phase space for seeking an extremum, has an entropy maximum on various distributions in the case of one and the same energy. A conclusion has been drawn on the occurrence of a Maxwell-Boltzmann distribution in conservative systems of classical mechanics for reasons unrelated to entropy maximization.