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Classical statistical mechanics is a ' crypto-deterministic' theory, where each element of the probability distribution of the dynamical variables specifying a given system evolves with time according to deterministic laws of motion; the whole uncertainty is contained in the form of the initial distributions. A theory based on such concepts could not give a satisfactory account of such non-deterministic effects as radioactive decay or spontaneous emission (cf. Whittaker (2)). Classical statistical mechanics is, however, only a special case in the general theory of dynamical statistical (stochastic) processes. In the general case, there is the possibility of 'diffusion' of the probability 'fluid', so that the transformation with time of the probability distribution need not be deterministic in the classical sense. In this paper, we shall attempt to interpret quantum mechanics as a form of such a general statistical dynamics.
Quantum Mechanics as a statistical theory by J. E. Moyal
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Time Evolution
The time evolution in the phase space formulation of quantum mechanics is described by the vonNeumann equation
$$ \frac{\partial f}{\partial t} = -\frac{1}{i \hbar} (f \star H - H \star f ). $$
For the Wigner function, this equation reads
$$ \frac{\partial W}{\partial t} = - \{\{ W,H \}\} = \frac{2}{ \hbar} W \left( \frac{\hbar}{2 } ( \overleftarrow \partial_x \overrightarrow \partial_p - \overleftarrow \partial_p \overrightarrow \partial_x)\right) H = - \{ W,H\} + \mathcal{O}(\hbar^2) , $$
where $\{\{ ,\}\}$ denotes the Moyal bracket and $\{ ,\}$ the Poisson bracket.
In the limit, $\hbar \to 0$ the von Neumann equation reduces to the Liouville equation.
The difference between the von Neumann equation and the Liouville equation is that in the former the density of points in phase space is not conserved. Formulated differently, the probability fluid is diffusive and compressible.
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Important Papers
In the phase space formulation, the transition from classical mechanics to quantum mechanics is known as deformation quantization.
The map between function in the phase space formulation of quantum mechanics and the operators in the corresponding Hilbert space is known as Wigner–Weyl transform.
The chief advantage of the phase space formulation is that it makes quantum mechanics appear as similar to Hamiltonian mechanics as possible by avoiding the operator formalism, thereby "'freeing' the quantization of the 'burden' of the Hilbert space".[5] This formulation is statistical in nature and offers logical connections between quantum mechanics and classical statistical mechanics, enabling a natural comparison between the two (cf. classical limit). https://en.wikipedia.org/wiki/Phase_space_formulation