# Non-Normalizable Quasi-Equilibrium Solution of the Fokker–Planck Equation for Nonconfining Fields

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## Abstract

**:**

## 1. Introduction

## 2. The System

## 3. Non-Normalizable Quasi-Equilibrium

## 4. Time-Dependent Solution

**I**; and second, for $x\gg 1$ (region

**III**). These two asymptotic limits must be matched in the overlap region $1\ll x\ll 1/k$ (region

**II**).

**I**, the term $-{k}^{2}{\mathsf{\Psi}}_{k}\left(x\right)$ is negligible, due to the smallness of k. To leading order, we have the homogeneous equation,

**II**, where $1\ll x\ll 1/k$, based on Equation (19),

**III**, since $x\gg 1$, the ${v}^{\u2033}\left(x\right)$ and ${v}^{\prime 2}\left(x\right)$ terms are negligible and, therefore, Equation (19) now reads

**II**solution, we get

**III**, we have

**I**to region

**III**. By considering a point $x=\ell \sim \varphi $, as given by Equation (36), where regions

**I**and

**III**overlap, the whole probability in region

**III**can be written as

**I**, we need the small $\xi $ approximation to the function $g\left(x\right)$, see Equation (26). This works similarly to our small $\xi $ approximation for $\mathcal{A}$. We have

**III**result. For sufficiently large t, Equation (46) becomes

**I**is where most of the probability is found and, hence, this expression captures the behavior of the majority of the particles.

**I**(center) and

**III**(tail). (The upper insets replicate Figure 2, to complete the portrait.) The results from numerical simulations are represented by solid lines, while the theoretical results for regions

**I**(short dashed line) and

**III**(long dashed lines) are also plotted, in good agreement in the respective regions. In the insets, all of the curves are plotted together, in a zoom of the matching region.

## 5. Regularization Procedure

**I**concentrates most of the probability and ${P}^{I}$ becomes nearly time-independent, then Equation (49) allows for the predicting of the NQE value. It is noteworthy that the NQE regime of different observables is related to different timescales.

## 6. Final Remarks

**I**corresponds to the central part of the PDF and region

**III**to the tails, while region

**II**is where both solutions overlap, as can be seen in Figure 8; moreover, $2{\ell}_{0}$, as defined in Equation (25), is a lengthscale that plays the role of an effective partition function, and the shift $\varphi $ can be estimated through Equation (36). Region

**I**concentrates most of the probability, out of fluctuations in region

**III**, where the erfc function acts as an effective cutoff blocking free diffusion. The shift is related to the region of the well that has to be overcome to escape. To see this note that ${l}_{0}$ is large, but ${e}^{-v\left(0\right)/\xi}={e}^{1/\xi}$ is similarly large (while the erfc is of order one or less), hence the small x solution in region

**I**exponentially overwhelms the solution in region

**III**. Subsequently, the shift $\varphi $ decreases with increasing scaled temperature $\xi $, as can be seen in Figure 6. Equation (61) shows how nearly time-independent solutions can emerge. They last exponentially long times, for sufficiently low temperatures, and they can be associated to the NQE regime.

**I**decays in time, while, in our case, it remains nearly constant. Additionally, notice that the current approach differs from the calculation of ensemble averaged observables performed in the limit of $t\to \infty $ [12,13,20]. Here, we avoid the limit of infinite time by considering the upper bound on the measurement time, namely, the escape time, ${e}^{1/\xi}$, which allows us to isolate the dominant Boltzmann-like behavior at the center of the packet.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Defaveri, L.; Anteneodo, C.; Kessler, D.A.; Barkai, E. Regularized Boltzmann-Gibbs statistics for a Brownian particle in a nonconfining field. Phys. Rev. Res.
**2020**, 2, 043088. [Google Scholar] [CrossRef] - Fermi, E. Uber die Wahrscheinlichkeit der Quantenzustande. Z. Phys.
**1924**, 26, 54. [Google Scholar] [CrossRef] - Plastino, A.; Rocca, M.C.; Ferri, G.L. Resolving the partition function’s paradox of the hydrogen atom. Physica A
**2019**, 534, 122169. [Google Scholar] [CrossRef] - Sabhapandit, S.; Majumdar, S.N. Freezing Transition in the Barrier Crossing Rate of a Diffusing Particle. Phys. Rev. Lett.
**2020**, 125, 200601. [Google Scholar] [CrossRef] [PubMed] - Dechant, A.; Lutz, E.; Barkai, E.; Kessler, D.A. Solution of the Fokker–Planck Equation with a Logarithmic Potential. J. Stat. Phys.
**2011**, 145, 1524–1545. [Google Scholar] [CrossRef] - van Kampen, N.G. Stochastic Processes in Physics and Chemistry; North-Holland Personal Library: New York, NY, USA, 1981. [Google Scholar]
- Risken, H. The Fokker–Planck Equation; Springer: Berlin, Germany, 1989. [Google Scholar]
- Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. Numerical Recipes: The Art of Scientific Computing, Third Edition (C++); Cambridge University Press: Cambridge, UK, 2007. [Google Scholar]
- Redner, S. A Guide to First-Passage Processes; Cambridge University Press: Cambridge, UK, 2001. [Google Scholar]
- Hänggi, P.; Talkner, P.; Borkovec, M. Reaction-rate theory: Fifty years after Kramers. Rev. Mod. Phys.
**1990**, 62, 251. [Google Scholar] [CrossRef] - Arrhenius, S. Über die Reaktionsgeschwindigkeit bei der Inversion von Rohrzucker durch Säuren. Z. Phys. Chem. (Leipzig)
**1889**, 4, 226. [Google Scholar] - Aghion, E.; Kessler, D.A.; Barkai, E. From NonNormalizable Boltzmann-Gibbs Statistics to Infinite-Ergodic Theory. Phys. Rev. Lett.
**2019**, 122, 010601. [Google Scholar] [CrossRef] [PubMed][Green Version] - Aghion, E.; Kessler, D.A.; Barkai, E. Infinite ergodic theory meets Boltzmann statistics. Chaos Solitons Fractals
**2020**, 138, 109890. [Google Scholar] [CrossRef] - Kessler, D.; Ner, Z.; Sander, L. Front propagation: Precursos, cutoffs, and structural stabilitty. Phys. Rev. E
**1998**, 58, 107–114. [Google Scholar] [CrossRef][Green Version] - Brunet, E.; Derrida, B. Shift in the velocity of a front due to a cutoff. Phys. Rev. E
**1997**, 56, 2597–2604. [Google Scholar] [CrossRef][Green Version] - Mayer, J.E.; Montroll, E. Molecular distribution. J. Chem. Phys.
**1941**, 9, 2. [Google Scholar] [CrossRef] - Mathematica, Version 11; Wolfram Research: Champaign, IL, USA, 2016.
- Aaronson, J. An Introduction to Infinite Ergodic Theory; American Mathematical Society: Providence, RI, USA, 1997. [Google Scholar]
- Akimoto, T.; Barkai, E. Aging generates regular motions in weakly chaotic systems. Phys. Rev. E
**2013**, 87, 032915. [Google Scholar] [CrossRef][Green Version] - Rebenshtok, A.; Denisov, S.; Hänggi, P.; Barkai, E. Non-Normalizable Densities in Strong Anomalous Diffusion: Beyond the Central Limit Theorem. Phys. Rev. Lett.
**2014**, 112, 110601. [Google Scholar] [CrossRef] [PubMed][Green Version] - Cooper, A.; Covey, J.P.; Madjarov, I.S.; Porsev, S.G.; Safronova, M.S.; Endres, M. Alkaline-Earth Atoms in Optical Tweezers. Phys. Rev. X
**2018**, 8, 041055. [Google Scholar] [CrossRef][Green Version] - Campagnola, G.; Nepal, K.; Schroder, B.W.; Peersen, O.B. Superdiffusive motion of membrane-targeting C2 domains. Sci. Rep.
**2013**, 5, 17721. [Google Scholar] [CrossRef] [PubMed][Green Version] - Debets, V.E.; Janssen, L.M.C.; Šarić, A. Characterising the diffusion of biological nanoparticles on fluid and cross-linked membranes. Soft Matter
**2020**, 16, 10628–10639. [Google Scholar] [CrossRef] [PubMed] - Wang, D.; Wu, H.; Schwartz, D.K. Three-Dimensional Tracking of Interfacial Hopping Diffusion. Phys. Rev. Lett.
**2017**, 119, 268001. [Google Scholar] [CrossRef] [PubMed][Green Version] - Metzler, R.; Klafter, J. The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep.
**2000**, 339, 1–77. [Google Scholar] [CrossRef]

**Figure 1.**Dimensionless potentials (

**a**) ${v}_{\mu}\left(x\right)$ and (

**b**) ${v}_{\kappa ,\mu}\left(x\right)$, (

**c**,

**d**) the respective forces, plotted for three different values of $\mu $ and $\kappa =5$. Notice that the potential becomes flat and the force falls to zero at large distances from the origin and, therefore, ineffective, in the sense that these fields are non-binding and the normalization factor Z in Equation (4) diverges.

**Figure 2.**Probability density function $P(x,t)$ for different times, from the numerical integration of the FPE (7) with the potentials (

**a**) ${v}_{4}\left(x\right)$ and (

**b**) ${v}_{5,4}\left(x\right)$, for $\xi =0.05$. For comparison, in each case we also plot (red dotted line) the Boltzmann expression that is given by Equation (47), ${e}^{-v\left(x\right)/\xi}/\left(2{\ell}_{0}\right)$, where ${\ell}_{0}$ is defined in Equation (25). The maxima in the plots correspond to minima in the potential field. Note that, as we increase time, the approximation given by Equation (47) works better; however, for times much longer than the escape time, a different behavior will be found.

**Figure 3.**The MSD $\langle {x}^{2}\left(t\right)\rangle $ versus time t, obtained from numerical solutions of the FPE (solid lines) with the potential fields given by (

**a**) ${v}_{4}\left(x\right)$ and (

**b**) ${v}_{5,4}\left(x\right)$, and different values of $\xi $ indicated in the legend. The dashed lines correspond to the expression derived along this work for NQE states, as given by Equation (57). Being drawn for comparison, the dotted lines correspond to the harmonic approximation ($\langle {x}^{2}\rangle =\xi /\mu $ and $\langle {x}^{2}\rangle =2\xi /({\kappa}^{2}+2\mu )$), showing the improvement of the presented theory.

**Figure 4.**Pre-factor of the PDF, $C(x,t)$ defined in Equation (11), at different times, for the field ${v}_{4}$ and scaled temperature $\xi =0.1$. (

**a**) For very small times, (

**b**) for an intermediate time scale. The behavior of the PDF at the origin $P(0,t)$ is shown in (

**c**), with its linear ordinate axis representation in the inset.

**Figure 5.**The integrand $a\left(x\right)$ of $\mathcal{A}$, for different values of $\xi $ using the field ${v}_{\mu}\left(x\right)=-1/{(1+{x}^{2})}^{\mu /2}$, with several values of $\mu $. Solid lines represent a direct numerical evaluation of $a\left(x\right)$ and the dotted line represents our approximation in Equation (29).

**Figure 6.**The ratio $\varphi =-\mathcal{A}/{\ell}_{0}$ vs. $\xi $, the scaled temperature of the system, where $\mathcal{A}$ and ${\ell}_{0}$ are given by Equations (27) and (25), respectively. We plot the direct solution (points) with the leading order (

**right panel**, dotted lines), Equation (30), and the leading order with the first correction (

**left panel**, solid line), Equation (32).

**Figure 7.**The ratio $x-g\left(x\right)/{\ell}_{0}$ vs. x calculated directly (solid line) from Equation (22), together with our deep well approximation (dotted line), Equation (45) for ${v}_{\mu}\left(x\right)=-1/{(1+{x}^{2})}^{\mu /2}$ with several values of $\mu $ and two values of $\xi $.

**Figure 8.**Central region

**I**(

**a**,

**b**) and tail region

**III**(

**c**,

**d**) of the PDF, for the potentials ${v}_{4}$ (

**a**,

**c**) and ${v}_{4,5}$ (

**b**,

**d**), at different times t (increasing from light to dark color) indicated in the legend. PDF from the numerical integration of the FPE (black solid lines) and theoretical predictions (dashed lines): ${P}^{I}(x,t)$ (blue short dashed) given by Equation (46), and ${P}^{III}(x,t)$ (green long dashed) that is given by (39). The upper insets include the Boltzmannian (red dashed) curve for comparison (duplicating Figure 2). The intermediate matching region is amplified in the lower insets.

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**MDPI and ACS Style**

Anteneodo, C.; Defaveri, L.; Barkai, E.; Kessler, D.A. Non-Normalizable Quasi-Equilibrium Solution of the Fokker–Planck Equation for Nonconfining Fields. *Entropy* **2021**, *23*, 131.
https://doi.org/10.3390/e23020131

**AMA Style**

Anteneodo C, Defaveri L, Barkai E, Kessler DA. Non-Normalizable Quasi-Equilibrium Solution of the Fokker–Planck Equation for Nonconfining Fields. *Entropy*. 2021; 23(2):131.
https://doi.org/10.3390/e23020131

**Chicago/Turabian Style**

Anteneodo, Celia, Lucianno Defaveri, Eli Barkai, and David A. Kessler. 2021. "Non-Normalizable Quasi-Equilibrium Solution of the Fokker–Planck Equation for Nonconfining Fields" *Entropy* 23, no. 2: 131.
https://doi.org/10.3390/e23020131