# Cusp of Non-Gaussian Density of Particles for a Diffusing Diffusivity Model

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## Abstract

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## 1. Introduction

## 2. Results

#### 2.1. The Model

#### 2.2. The General Case: Arbitrary Distribution of Waiting Times

#### 2.2.1. Short Time Regime

#### 2.2.1.1. $P(x,t)$ for Arbitrary Waiting Times

#### 2.2.2. Long Time Regime

#### 2.2.3. Simulations

#### 2.3. Exponentially Distributed Waiting Times

#### 2.3.1. Equal Mean Waiting Times ${\langle \tau \rangle}_{+}={\langle \tau \rangle}_{-}$

#### Positional Distribution Function

#### 2.3.2. Different Mean Waiting Times ${\langle \tau \rangle}_{+}\ne {\langle \tau \rangle}_{-}$

## 3. Discussion

#### 3.1. The Histogram of the Diffusion Coefficient as Extracted from Experimental Data

#### 3.1.1. Super-Statistics

#### 3.1.2. Time Average MSD

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CTRW | Continuous Time Random Walk |

TAMSD | Time Average Mean Squared Displacement |

## Appendix A. A Two State Model with ${\mathit{D}}_{\mathbf{+}}\mathbf{>}{\mathit{D}}_{\mathbf{-}}\mathbf{>}\mathbf{0}$

#### Appendix A.1. P(x,t) for Arbitrary Waiting Times

**Figure A1.**Distribution of displacements $P(x,t)$ obtained by simulations of a two state system with ${D}_{+}>{D}_{-}>0$ and gamma distributed waiting times $\tau \sim Gamma(3,1)$ at ${D}_{+}$ and $\tau \sim Gamma(6,1)$ at ${D}_{-}$ following Equation (30). We compare with Equation (A4) (solid lines) with $t=0.5$, ${\langle \tau \rangle}_{+}=3$, ${\langle \tau \rangle}_{-}=6$, ${D}_{+}=10$. For ${D}_{-}=0.1$ (red triangles), ${D}_{-}=5$ (cyan squares) and ${D}_{-}=9$ (magenta circles). Exponential like decaying is present at small values for x, when ${D}_{+}=10>>{D}_{-}=0.1$ (red solid line). In the cases when ${D}_{-}\u27f6{D}_{-}$ (cyan and magenta solid lines), $P(x,t)$ follows a full Gaussian distribution.

#### Appendix A.1.1. Short Time Regime

#### Appendix A.1.2. Long Time Regime

#### Appendix A.2. P(x,t) for Exponentially Distributed Waiting Times with ${\langle \tau \rangle}_{+}\ne {\langle \tau \rangle}_{-}$

## Appendix B. A Complementary Deduction of ${\mathit{f}}_{\mathit{t}}^{\mathbf{\pm}}\mathbf{\left(}{\mathit{T}}_{\mathbf{+}}\mathbf{\right|}\mathbf{1}\mathbf{)}$

#### Appendix B.1. Non-Equilibrium Initial Conditions

## Appendix C. P(x,t) from Simulations with Uniform and Gamma Distributed Waiting Times within the Complete Range of x

**Figure A2.**Distribution of displacements $P(x,t)$ in semi-log scale, obtained from simulations, of a two state system with uniform and gamma distributed waiting times within the short time limit and displaying the whole span of x. $P(x,t)$ for uniformly distributed waiting times is shown in red triangles. In addition, the case of gamma distributed waiting times is shown in blue squares. We employed the same set of parameters as those used in Figure 3 in the left panel. Both cases fit with Equation (26) (red and blue solid lines).

## Appendix D. PDF of Occupation Times for Exponentially Distributed Waiting Times and Non-Equilibrium Initial Conditions

**Figure A3.**Left: ${g}_{t}\left({p}_{+}\right)$ Equation (A34) for $\langle \tau \rangle =1$ and $t\in \{0.1,0.5,1,2,5,10\}$ and non-equilibrium initial conditions (starting from state “+”). The uniform approximation of ${g}_{t}\left({p}_{+}\right)$ Equation (A35) for $t=0.1$ is shown in black circles. Right: ${g}_{t}\left({p}_{+}\right)$ Equation (A39) for ${\langle \tau \rangle}_{+}=1$, ${\langle \tau \rangle}_{-}=5$ and $t\in \{0.1,0.5,2,5,10,20\}$ and non-equilibrium initial conditions (starting from state “+”). The uniform approximation of ${g}_{t}\left({p}_{+}\right)$ Equation (A40) for $t=0.1$ is shown in black circles.

## Appendix E. Deduction of ${\mathit{g}}_{\mathit{t}}\mathbf{\left(}{\mathit{p}}_{\mathbf{+}}\mathbf{\right)}$ for Waiting Times with Similar Mean Waiting Times

## Appendix F. Deduction of ${\mathit{g}}_{\mathit{t}}\mathbf{\left(}{\mathit{p}}_{\mathbf{+}}\mathbf{\right)}$ for Waiting Times with ${\mathbf{\langle}\mathit{\tau}\mathbf{\rangle}}_{\mathbf{+}}\mathbf{\ne}{\mathbf{\langle}\mathit{\tau}\mathbf{\rangle}}_{\mathbf{-}}$

## Appendix G. Deduction of the MSD in a Two State Model with ${\mathbf{\langle}\mathit{\tau}\mathbf{\rangle}}_{\mathbf{+}}\mathbf{\ne}{\mathbf{\langle}\mathit{\tau}\mathbf{\rangle}}_{\mathbf{-}}$

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**Figure 1.**Typical trajectory of $x\left(t\right)$ given by Equation (2) with ${D}_{+}=10$ (blue regions), and ${D}_{-}=0$ (red regions). For this trajectory, exponential waiting times with ${\langle \tau \rangle}_{+}=1$ and ${\langle \tau \rangle}_{-}=5$ were used.

**Figure 2.**Alternating process for the diffusivity, starting from the state ‘+’ and $N=2k+1$. For the case of equilibrium initial conditions exposed in Section 2.2, for $N=1$, ${\tau}_{1}$ works as the forward recurrence time with PDF Equation (10).

**Figure 3.**Distribution of displacements $P(x,t)$ in semi-log scale, obtained by simulations, for a two state system with uniform distributed waiting times and gamma distributed waiting times. The left panel presents short time results where a tent like shape is clearly visible and a non-analytical feature is obvious, while the right panel exhibits Gaussian statistics for long times. Left: $P(x,t)$ for $t=1$ for $\tau \sim U(0,5)$ at ${D}_{+}$ and $\tau \sim U(0,10)$ at ${D}_{-}$ (red triangles)—with ${\langle \tau \rangle}_{+}=2.5<{\langle \tau \rangle}_{-}=5$. In addition, $t=2$ with $\tau \sim Gamma(0.5,8)$ at ${D}_{+}$ and $\tau \sim Gamma(0.5,12)$ at ${D}_{-}$ (blue squares), such that ${\langle \tau \rangle}_{+}=4<{\langle \tau \rangle}_{-}=6$. Both cases fit with Equation (26) (red and blue solid lines) with a tent like shape. In both normalized histograms at $x=0$, there is a peak representing the Dirac delta function in Equation (26). Right: $P(x,t)$ for $t=30$ and waiting times uniformly distributed (green triangles) with the same parameters as above and for gamma distributed waiting times (orange squares) with $\tau \sim Gamma(2,1)$ at ${D}_{+}$, $\tau \sim Gamma(8,1)$ at ${D}_{-}$, and ${\langle \tau \rangle}_{+}=2<{\langle \tau \rangle}_{-}=8$. We employed the last set of parameters in the gamma distributed waiting times in order to avoid an overlapping between curves. $P(x,t)$ converges to the Gaussian statistics Equation (29) (green and orange solid lines). In all the presented cases, ${D}_{+}=10$ and ${D}_{-}=0$ were used.

**Figure 4.**Left: Comparison between ${g}_{t}\left({p}_{+}\right)$ Equation (34) (red solid line) and the short time uniform approximation Equation (35) (black asterisks) for exponentially distributed waiting times Equation (31) with ${\langle \tau \rangle}_{\pm}=\langle \tau \rangle =1$ and $t=0.1$. Right: ${g}_{t}\left({p}_{+}\right)$ Equation (34) for $\langle \tau \rangle =1$ and $t\in \{0.1,0.5,1,2,5,10\}$.

**Figure 5.**$P(z,t)$ in semi-log scale, with $z=x/\sqrt{t}$. Left: For short times $t=0.1$ (red circles) and $t=0.5$ (blue crosses), $P(z,t)$ is represented by Equation (37) (black solid line) with a tent like shape. Right: The same for large times $t=5$ (orange circles) and $t=10$ (green crosses), $P(z,t)$ converges to the Gaussian distribution Equation (41) (magenta solid line). In all the cases, ${D}_{+}=10$, ${D}_{-}=0$, and $\langle \tau \rangle =1$ were used.

**Figure 6.**Left: Comparison between ${g}_{t}\left({p}_{+}\right)$ Equation (45) (red solid line) and the uniform approximation Equation (46) (black asterisks) for ${\langle \tau \rangle}_{+}=1$, ${\langle \tau \rangle}_{-}=5$ and $t=0.1$. Right: ${g}_{t}\left({p}_{+}\right)$ Equation (45) for ${\langle \tau \rangle}_{+}=1$, ${\langle \tau \rangle}_{-}=5$ and $t\in \{0.1,0.5,2,5,10,20\}$.

**Figure 7.**For a system with, ${\langle \tau \rangle}_{+}=1$ and ${\langle \tau \rangle}_{-}=5$, $P(z,t)$ in semi-log scale, with $z=x/\sqrt{t}$. For short times $t=0.1$ (red circles) and $t=0.5$ (blue crosses), $P(z,t)$ is represented by Equation (47) (black solid line) with a tent like shape. For large times $t=20$ (orange circles) and $t=30$ (green diamonds), $P(z,t)$ converges to the Gaussian statistics Equation (29) (magenta solid line). In all the cases, ${D}_{+}=10$ and ${D}_{-}=0$ were used. Compared with Figure 5, in this case, the Gaussian curve is above the tent curve, contrary to the case with equal mean waiting times. This is because the coefficient of the Gaussian curve Equation (29) is bigger compared with the weight of the delta peak in Equation (47). In Figure 5, we have the opposite, and the weight of the corresponding delta function in Equation (37) is bigger compared with the Gaussian Equation (41).

**Figure 8.**Distribution of diffusion coefficients $P\left(D\right)$ obtained via TAMSD analysis of simulated trajectories of a two state system with ${D}_{+}=10$, ${D}_{-}=0$ and exponentially distributed waiting times. From the linear plots of the TAMSD versus the lag time estimates of D were extracted. We show two cases, the first for a system with the same mean waiting times ${\langle \tau \rangle}_{+}={\langle \tau \rangle}_{-}=\langle \tau \rangle =1$ (red boxes). In addition, the PDF of D for a system with different mean waiting times with ${\langle \tau \rangle}_{+}=1$ and ${\langle \tau \rangle}_{-}=5$ is also shown (blue boxes). For the system with the same mean waiting times, the average diffusivity found in the simulations is $\langle D\rangle =4.98$, and, for the case of different mean waiting times, we have $\langle D\rangle =1.69$. In both cases, we used $t=1000$ and 1000 trajectories.

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**MDPI and ACS Style**

Hidalgo-Soria, M.; Barkai, E.; Burov, S. Cusp of Non-Gaussian Density of Particles for a Diffusing Diffusivity Model. *Entropy* **2021**, *23*, 231.
https://doi.org/10.3390/e23020231

**AMA Style**

Hidalgo-Soria M, Barkai E, Burov S. Cusp of Non-Gaussian Density of Particles for a Diffusing Diffusivity Model. *Entropy*. 2021; 23(2):231.
https://doi.org/10.3390/e23020231

**Chicago/Turabian Style**

Hidalgo-Soria, M., E. Barkai, and S. Burov. 2021. "Cusp of Non-Gaussian Density of Particles for a Diffusing Diffusivity Model" *Entropy* 23, no. 2: 231.
https://doi.org/10.3390/e23020231