Topological Subordination in Quantum Mechanics

Abstract

An example of non-Markovian quantum dynamics is considered in the framework of a geometrical (topological) subordination approach. The specific property of the model is that it coincides exactly with the fractional diffusion equation, which describes the geometric Brownian motion on combs. Both classical diffusion and quantum dynamics are described using the dilatation operator xddx. Two examples of geometrical subordinators are considered. The first one is the Gaussian function, which is due to the comb geometry. For the quantum consideration with a specific choice of the initial conditions, it corresponds to the integral representation of the Mittag–Leffler function by means of the subordination integral. The second subordinator is the Dirac delta function, which results from the memory kernels that define the fractional time derivatives in the fractional diffusion equation.

Two Integral Representations for the Relaxation Modulus of the Generalized Fractional Zener Model

“…To obtain an alternative integral representation for G(t), which generalises (9), we use the relation of this function and the solution of a relaxation equation with a general fractional Caputo derivative with kernel k(t), and we apply the subordination principle for this equation [16,23,24]. The concept of subordination, originally introduced by Bochner in the theory of stochastic processes, has developed recently into a powerful tool in the study of anomalous relaxation and diffusion processes and the physics of complex systems (see, e.g., [25][26][27] and the recent review paper [28]).…”