The dynamics of a cosmological model of dark matter and dark energy represented by a scalar field endowed with a cosh-like potential plus a cosmological constant is investigated in detail. By studying the appropriate phase space of the equations of motion, it is shown that a standard evolution of the Universe is recovered for appropriate values of the free parameters, and that the only late-time attractor is always the de Sitter solution. We also discuss the appearance of scalar field oscillations corresponding to dark matter behavior.

We show that a physical correspondence between Brownian motion and quantum mechanics can be established by formal analytic continuation if Wick rotation ($t\to it$) is replaced by the mirror symmetric, complex conjugate time transformation $t\to \pm it$. Invariance of the square modulus of the wave function under this transformation reveals a tight connection between Born's interpretation of the probability density and the proper time $t^2=(-it)(+it)$, as being both the result of a time symmetry breaking in the informational content of the microscopic (quantum) world, which takes place as we move to a macroscopic level. We find that under the transformation $t\to \pm it$, the Schrödinger equation and its complex conjugate conforms with a stochastic-mechanical model of the Brownian motion of a particle. From the present analysis, Nelson's osmotic velocity arises naturally and the quantum phase function admits a classical interpretation in terms of a continuous (scalar) potential field that influences the motion of the particle. The maximum phase variation defines the direction of the osmotic and current velocity. Whereas the paper focuses on the correlation between quantum mechanics and Ficks's second law for a constant diffusion coefficient, a discussion is also provided on the possibility of a quantum mechanical formulation for anomalous diffusion, where the diffusion coefficient is a function of space and/or time.

The dynamics of a cosmological model fueled by scalar field dark matter with a cosh-like potential plus a cosmological constant is investigated in detail. It is revealed that the late-time attractor is always the de Sitter solution, and that, depending on the values of the free parameters, the oscillating solution of the scalar field -- modeling cold dark matter -- mediates between some early stage (say, the radiation-dominated solution) and the accelerating de Sitter attractor.