Szilard Engine and Classical Versus Statistical Single Particle?

The Szilard engine example seems to be based on a single particle whose location may be measured like a classical particle, but then seems to behave in a statistical/thermodynamic manner. To quantify this statement, the ideal case law PV=nRT is used to compute work in (1) using: Integral nRT dV/V. The idea seems to be that a reservoir supplies heat to keep T constant as the single particle, measured by a Maxwell demon to be on the left hand side of a container when a partition is added to the middle, as an isothermal expansion occurs until the container is back to size V. In such a case, entropy decreases in the reservoir and universe and heat is converted completely into work, violating the second law of thermodynamics.

   We try to argue that there is a fundamental problem with this example and that is the mixing of  the notion of a classical particle with a statistical one. We first point out that the ideal gas law is often linked with the usual maximization of entropy ln (N!/ Product over i n(ei) !)  subject to the constraint Sum over i  ei p(ei), with Stirling’s approximation being used. Stirling’s approximation cannot be used unless there are a large number of particles, not one. If one considers this problem with a large number of ideal gas particles in a volume V at T, then the equivalent situation of finding a single particle on the left hand side when a partition is added to the middle is the process of pushing the gas back to the left half of the container while keeping T constant. This involves work -W and heat gain of Q which exactly balances the work W and heat loss -Q from the reservoir as the system expands toward the right hand wall.. Thus, no net work is gained and there is no net loss of entropy, thus nullifying the Szilard engine argument.

   An issue now arises with respect to a single particle. A main idea of statistical mechanics/thermodynamics is that a particle, even in a single one, may be found anywhere in V. Thus, having a demon (1) measure the particle to be in the left hand side breaks the notion of  a statistical particle and replaces it with classical one. Even in the case of a dilute gas which thermalizes only through collisions with the wall, there is still equal probability for a gas particle to be anywhere in V (no potential) and so localizing a single particle which is said to represent the entire gas breaks the entire notion of statistical mechanics/thermodynamics. Thus, if the single particle is measured by the demon to be on the left hand side of the container, we argue that this means that it must have been pushed isothermally by a partition on the right hand side which moves to the center doing -W work and obtaining Q heat. These two values must be used together with results from subsequent processes, i.e. the process of the single particle pushing the partition back to the right hand wall.