Solution of the fractional Liouville equation by using Riemann–Liouville and Caputo derivatives in statistical mechanics

We solve the fractional Liouville equation by using Riemann–Liouville and Caputo derivatives for systems exhibiting noninteger power laws in their Hamiltonians. Based on the fractional Liouville equation, we calculate the density function (DF) of a classical ideal gas. If the Riemann–Liouville derivative is used, the DF is a function depending on both the momentum \(p\) and the coordinate \(q\), but if the derivative in the Caputo sense is used, the DF is a constant independent of \(p\) and \(q\). We also study a gas consisting of \(N\) fractional oscillators in one-dimensional space and obtain that the DF of the system depends on the type of the derivative.