Characteristic Time of Diffusive Mixing in Cube with Reflecting Edges

V. V. Uchaikin suggested a mathematical model of an anomalous diffusion in a space. These model origins in an investigation of processes in complex systems with variable structure: glasses, liquid crystals, biopolymers, proteins and a turbulence in a plasma. Here a coordinate of diffusing particle has stable distribution and so its density satisfies diffusion equation with partial derivatives. In this paper, the anomalous diffusion with periodic initial conditions on an interval with reflecting edges, important for example in technical mechanics, is considered and analyzed.


Introduction
In [1] a mathematical model of an anomalous diffusion in a space was suggested. These model origins in an investigation of processes in complex systems with variable structure: glasses, liquid crystals, biopolymers, proteins and a turbulence in a plasma [2].
In this model a coordinate of diffusing particle has stable distribution (not normal one). As a result a density of its distribution satisfies an analogy of diffusion equation in which second derivative by coordinate is replaced by partial derivative.
In this paper the anomalous diffusion with periodical initial conditions on an interval with reflecting edges is considered. Such problem is important for example in technical mechanics for an analysis of fuel mixing in straight flow engine [3] too.
Suppose that ( ), 0, y t t ≥ is homogeneous random process with independent increments and an initial condition (0) = 0.
δ are delta-functions by variables , y t accordingly. It is impossible to use the method of Fourier series to analyze anomalous diffusion on segment with reflecting edges. To get over this difficulty in this paper an analogy with the wave equation for finite string with fixed edges is used. Some approaches to generalize one-dimensional results are considered. Main analytical results of this paper have been obtained in [4]. But in this paper these results are supplemented by physical interpretation and consideration of diffusion in multidimensional cube with reflecting boundaries.

Geometric Representation
Each realization of random process ( ), 0, y t t ≥ may be considered as a curve Γ on the plane ( , ).
y t Suppose that the curve Γ is reflected from the lines = 1, = 1. y y − For this aim represent the plane ( , ) y t as a transparent and infinitely thin sheet of paper with the curve . Γ Bend this sheet of the paper along the lines = 1, = 3, y y ± ± into transparent strip 1 1 y − ≤ ≤ with fragments of the initial curve Γ . The curve Γ converted into the curve γ described by the process Similar [1] the random process ( ), 0 Y t t ≥ may be interpreted as a model of the anomalous diffusion on the interval [ 1,1] − with reflecting edges. It is clear that if the curve Γ coincides with some straight line then the curve γ constructing in an accordance with the law of geometrical optics: falling angle equals to reflecting angle.
Remark that Formula (3) which gives distribution density of diffusion reflected process is analogous to reflection method formula which gives solution of wave equation for finite string with fixed edges [5, chapter III, §13, points 5, 6].

Rate Convergence to Uniform Distribution
Define auxiliary function has period 4 and is even.
z Lemma 1. For arbitrary 1 t ≥ the following inequality is true: Remark that Lemma 1 is true for any even distribution density ( ) t p u , which satisfies Formula (4).

Self-similarity of Anomalous Diffusion
Next consideration closely connecting with a concept of a self-similar random process (see for example [6]) has a large application in modern physics [7][8][9][10].
Assume that > 0, r consider Markov process Formula (6) is based on self-similarity property of the process ( ) Y t and may be interpreted as an increasing (or a decreasing dependently on r ) in a r times of characteristic mixing time of anomalous diffusion on segment [ , ] r r − in a comparison with diffusion on segment [ 1,1]. − For < 1 r anomalous diffusion "works" slower and for > 1 r works faster than normal diffusion.

Periodical Initial Conditions
Diffusion process with periodical initial conditions origins for example in fuel mixing at straight flow engine [3, chapter 7, § 7.1]. For its modelling, take natural n and define Markov process Here random process ( ) y t and random variable The equality (7) means that the diffusion (normal or anomalous) on the segment [ 1,1] − with periodical initial conditions and reflecting edges leads to the same result as a diffusion on isolated (by reflecting edges) sub segments 2 1 1 2 3 1 1 , 1 , = 0, , 1, k k k n n n n n of the segment [ 1,1]. − Remark that the equality (7) is true for each self-similar random process ( ) y t with independent and symmetrically distributed increments. Theorem 1. For 1 a t n ≥ the following inequality is true: Formula (8) is interpreting as a decreasing in a n times of characteristic mixing time of anomalous diffusion on the segment [ 1,1] − with periodical initial conditions.

Numerical Experiment
Obtained analytical results comparing here with results of numerical experiment and a closeness of densities , t π π for different t are estimated. For this aim, independent random variables with distributions of ( ), Y t coinciding with a distribution of random variable Analogously with statistics of Chi-square we calculated 2 Table 1 shows qualitative coincidence of numerical results with estimates of Formula (8). If 2 a → then rate convergence of ( ) Y t to uniform distribution density increases. However, it is necessary to remark that an increasing of t demands an increasing of N that complicates numerical experiment.

Multidimensional Diffusion on Square with Reflecting Boundaries and Periodical Initial Conditions
In this section, we consider as normal so anomalous diffusion on k -dimensional square with reflecting boundaries. Normal diffusion is analyzed by the method of Fourier series but for anomalous diffusion it is more convenient to use the reflection formula.

Multidimensional Normal Diffusion
Consider a model of k -dimensional normal diffusion in the cube [ 1,1] Consequently, it is possible to choose n -periodical initial condition so that in the cube [

Multidimensional Anomalous Diffusion
Consider with independent components distributed as ( )  In an analysis of impurity blowing in an airflow, we simplify its picture to divide the most important parameters influencing on a mixing time. Therefore, we assume that a process of an impurity mixing consists of two stages. In first stage, an impurity consists of separate particles, which move as Stokes particles and do not diffuse before their merger with the airflow. In second period impurity, particles diffuse in cross direction to the airflow.

Merger of Injected Stokes Particle with Airflow
Consider first stage when impurity particles move independently each other's. To simplify considered problem we believe that the airflow is uniform and different ripples do not influence on particles behavior, physical properties of matters are constant.
Assume that Stokes particle with a mass m is injected with a velocity v in direction x perpendicularly air flow moving with a velocity V in plane channel which has a width l in direction y (see Figure 3). Define particle coordinates ( , ) x y to moment when impurity particle mergers with the airflow. Related motion equations have the following form = , = , (0) = 0, (0) = 0.