This paper proposes a simple model of anomalous diffusion, in which a particle moves with the velocity field induced by a single "dipole" (a doublet or a pair of source and sink), whose moment is modulated randomly at each time step. A motivation to introduce such a model is that it may serve as a toy model to investigate an anomalous diffusion of fluid particles in turbulence. We perform a numerical simulation of the fractal dimension of the trajectory using periodic boundary conditions in two and three dimensions. For a wide range of the dipole moment, we estimate the fractal dimension of the trajectory to be 1.7 − −1.9 (2D) and 2.4 − −2.7 (3D).1 There are also literature discussing the relationship between turbulence and SLE using random fields [20,21,22], and it is expected that there may be a correspondence between these and our model.

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A general formalism is given in quantum optics within a ring cavity, in which a non-linear material is stored. The method is Feynman graphical one, expressing the transition amplitude or S-matrix in terms of propagators and vertices. The propagator includes the additional damping effect via the non-linear material as well as the reflection and penetration effects by mirrors. Possible application of this formalism is discussed, in estimating the averaged number of produced photons, Husimi function, and the observables to examine beyond the squeezing mechanism of photons.

The non-equilibrium thermodynamics of Onsager and Machlup and of Hashitsume is reformulated as a gravity analog model, in which thermodynamic variables, kinetic coefficients, and generalized forces form, respectively, coordinates and metric tensor and vector fields in a space of thermodynamic variables. The relevant symmetry of the model is the general coordinate transformation. Then, the entropy production is classified into three categories, when a closed path is depicted as a thermodynamic cycle. One category is time-reversal odd, and is attributed to the number of lines of magnetic flux passing through the closed path, having the monopole as a source. There are two time-reversal-even categories, one of which is attributed to the space curvature around the path, having the gravitational instanton as a source, which dominates for a rapid operation of the cycle. The last category is the usual one, which remains even for the quasi-equilibrium operation. It is possible to extend the model to include non-linear responses. In introducing new terms, dimensional counting is important, using two parameters, the temperature and the relaxation time. The effective action, being induced by the non-equilibrium thermodynamics, is derived. This is a candidate for the action that controls the dynamics of kinetic coefficients and thermodynamic forces. An example is given in a chemical oscillatory reaction in a solvent of van der Waals type. The fluctuation–dissipation theorem is examined à la Onsager, and a derivation of the gravity analog thermodynamic model from quantum mechanics is sketched, based on an analogy to the resonance problem.