# Look at Tempered Subdiffusion in a Conjugate Map: Desire for the Confinement

^{*}

## Abstract

**:**

## 1. Introduction

- –
- Trapping, crowded environment (CTRW, FFPE, subordinated BM);
- –
- Labyrinthine environment (OD, percolation, RWRW);
- –
- Viscoelastic system (FBM, FLSM, FLE, ARFIMA);
- –
- System with time-dependent diffusion (scaled BM, scaled FBM, ARFIMA);
- –
- System with transient diffusion (BM with transient subordinators).

## 2. Conjugate Laplace Exponents and Stochastic Representation of Anomalous Diffusion

## 3. Tempered $\mathit{\alpha}$-Stable Process and Its Conjugate Partner

## 4. Confined Distributions for Infinitely Divisible Motion

**Confinement Principle:**Any subordinated infinitely divisible motion, in which the subordinator is characterized by the Laplace exponent conjugate to a tempered $\alpha $-stable process, has a confined probability distribution. By the infinitely divisible motion we mean a wide class of infinitely divisible processes, including Brownian motion (as a marginal case), Lévy stable motion (Lévy flight) and many other processes with jumps. It is important that each case of characteristic exponents in such an infinitely divisible motion determines its confined probability distribution. For the pure Brownian motion this is the Laplace distribution whereas for the Lévy flights its generalization is the Linnik distribution. This procedure covers a class of geometrically infinitely divisible distributions as a confined case of the infinitely divisible motion subordinated by a special subordinator responsible for the confinement.

## 5. Conditionally Non-Exponential Decay of Relaxation

## 6. G-Proteins vs. a2AR Receptors from the Analysis of the SPT Data

- ${H}_{0}$—an observed trajectory $X=\left\{{X}_{1},{X}_{2},\cdots ,{X}_{n}\right\}$ comes from Brownian motion,
- ${H}_{1}$—the trajectory looks like a confined or directed diffusion.

**Competition Principle**between parent processes: Brownian motion, Lévy motion or other infinitely divisible process even for any fixed subordinator conjugated one to tempered α-stable responsible for confinement. If Brownian motion is parent, the confined distribution from our subordination approach can have only the Laplace form. In the above data sets any feature, for example, typical for Lévy motion, is not detected. If this was true, it would be a chance for the play of generalized Laplace distributions as a confined distribution. Another case is the Ornstein–Uhlenbeck process leading to the normal statistics in confined trajectories, it has the same (Brownian) roots too. Therefore, the presence of normal and Laplace distributions together into confined trajectories is quite logical and justified physically.

## 7. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(Color online) Mean squared displacement of tempered subdiffusion (

**a**) and its conjugate partner (

**b**) with $\alpha =0.6$ and $\delta =1$ (for $D=1$). The dashed red and dash-dot green lines show asymptotic behavior of the values. If the panel (

**a**) indicates a transition of the subdifussion into normal diffusion at long times, whereas the panel (

**b**) shows the emergence of diffusion-limited aggregation.

**Figure 2.**(Color online) Propagator $p(x,t)$ for the tempered subdiffusion (

**a**) and its conjugate partner (

**b**), tending to the confinement, with a constant potential, $\alpha =0.5$ and $\delta =1$, drawn for consecutive dimensionless instances of time. Starting with the Dirac delta-function and passing to the subdiffusive PDF, for $t\to \infty $ the value $p(x,t)$ becomes the normal distribution, shown by black dotted line on the panel (

**a**), and the Laplace distribution (black dotted line) on the panel (

**b**).

**Figure 3.**(Color online) Propagators $p(x,t)$ from the parent processes, having the $\beta $-stable Lévy distribution: (

**a**) $1<\beta <2$; (

**b**) $0<\beta \le 1$; under the subordinator, conjugate to a tempered random process in the sense of Bernstein functions, for $t\to \infty $. The value $A={D}^{*}\alpha {\delta}^{\alpha -1}/2$ is taken equal to 1.

**Figure 4.**(Color online) Relaxation functions, caused by the inverse tempered subordinator (

**a**) and its conjugate partner (

**b**) respectively, with $\alpha =0.6$, $\delta =1$ and $b=1$. The first represents the tempered relaxation, and the second is confined. The dashed red line shows a conditionally non-exponential decay due to the confinement effect (${lim}_{t\to \infty}{\varphi}_{\mathrm{conf}}\left(t\right)=\mathrm{const}\ne 0$).

**Figure 5.**(Color online) Analysis of the experimental data as applied to G-protein and receptor random-walk trajectories along the coordinates x and y with the cutoff length of trajectories more and equal to 50.

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Stanislavsky, A.; Weron, A.
Look at Tempered Subdiffusion in a Conjugate Map: Desire for the Confinement. *Entropy* **2020**, *22*, 1317.
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Stanislavsky A, Weron A.
Look at Tempered Subdiffusion in a Conjugate Map: Desire for the Confinement. *Entropy*. 2020; 22(11):1317.
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**Chicago/Turabian Style**

Stanislavsky, Aleksander, and Aleksander Weron.
2020. "Look at Tempered Subdiffusion in a Conjugate Map: Desire for the Confinement" *Entropy* 22, no. 11: 1317.
https://doi.org/10.3390/e22111317