Physics of uncertainty, the Gibbs paradox and indistinguishable particles

• A probability-theoretic framework is proposed for thermodynamics of gases.
• Entropy is a pure probabilistic notion applied to thermodynamical systems.
• Gas particles are not regarded indistinguishable and the Gibbs paradox is resolved.
• The extensive entropy of thermodynamics is the difference of two probabilistic entropies.
• This framework enables generalization for processes across any type and scale of systems.

The idea that, in the microscopic world, particles are indistinguishable, interchangeable and without identity has been central in quantum physics. The same idea has been enrolled in statistical thermodynamics even in a classical framework of analysis to make theoretical results agree with experience. In thermodynamics of gases, this hypothesis is associated with several problems, logical and technical. For this case, an alternative theoretical framework is provided, replacing the indistinguishability hypothesis with standard probability and statistics. In this framework, entropy is a probabilistic notion applied to thermodynamic systems and is not extensive per se. Rather, the extensive entropy used in thermodynamics is the difference of two probabilistic entropies. According to this simple view, no paradoxical behaviors, such as the Gibbs paradox, appear. Such a simple probabilistic view within a classical physical framework, in which entropy is none other than uncertainty applicable irrespective of particle size, enables generalization of mathematical descriptions of processes across any type and scale of systems ruled by uncertainty.