Furthermore, any probabilistic prediction system should be derived, ideally, from the basic physical principle for predicting probability, that is, the Liouville equation
(Yano and Ouchtar 2017), although its practical use may appear difficult (see "Data assimilation" section).
Equation (1) has a widespread adoption in many physical branches, such as conserved current of Liouville equation
, two-dimensional quantum gravity gauge field, and conformal field theory [7-13].
Saarloos shown that the density function (mass, momentum and energy fields) obeys a Liouville equation
for hydrodynamics ideal fluid.
Tarasov in  based on the Liouville equation
obtained the fractional analogues of the classical kinetic and transport equations.
For the spheroidal solutions, there is a singularity in the appended Sturm- Liouville equation
one can obtain the quantum Liouville equation
for the Wigner distribution 
In classical mechanics, there are no equations that describe the evolution of the probability densities P(r, t) (or P(p,t), where p is momentum); only joint probability densities, [W.sub.cl](p,r,t), can be expressed by the Liouville equation
. Therefore, corresponding quantum equations for P(r, t), as used in (1), and P(p, t) cannot exist.
Sturm Liouville equation
has been extensively studied in both continuous and discrete cases [7, 20, 29, 33, 36].
(For [epsilon] = 0 this is usually called the Liouville equation
.) Suppose that [[mu].sub.0] is an invariant measure for (1.1) and the density [[rho].sub.0] of [[mu].sub.0] is a [C.sup.2] function.
One of the retrieved results is the Liouville Equation
, an equation Books24x7 has made "live." The user can then click on "Get MathML" to download the live equation and manipulate it.
where A is the normalization constant related to [eta] whose value is calculated with (14), whereas [chi](y) is obtained through the Liouville equation
. This latter gives