# Closed Formula for Transport across Constrictions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}segregation [34] rely on the transport of (charged) chemical species across nanoporous materials.

## 2. Model

#### 2.1. Transport across Free Energy Barriers

#### 2.2. Piecewise Linear Potential and Homogeneous Diffusion Coefficient

## 3. Discussion

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Lighthill, M.J.; Whitham, G.B. On kinematic waves II. A theory of traffic flow on long crowded roads. Proc. R. Soc. Lond. Ser. A. Math. Phys. Sci.
**1955**, 229, 317–345. [Google Scholar] [CrossRef] - Wang, C.; Quddus, M.A.; Ison, S.G. The effect of traffic and road characteristics on road safety: A review and future research direction. Saf. Sci.
**2013**, 57, 264–275. [Google Scholar] [CrossRef] - Vermuyten, H.; Belian, J.; De Boeck, L.; Reniers, G.; Wauters, T. A review of optimisation models for pedestrian evacuation and design problems. Saf. Sci.
**2016**, 87, 167–178. [Google Scholar] [CrossRef] - Jeong, H.Y.; Jun, S.C.; Cheon, J.Y.; Park, M. A review on clogging mechanisms and managements in aquifer storage and recovery (ASR) applications. Geosci. J.
**2018**, 22, 667–679. [Google Scholar] [CrossRef] - Jäger, R.; Mendoza, M.; Herrmann, H.J. Clogging at pore scale and pressure-induced erosion. Phys. Rev. Fluids
**2018**, 3, 074302. [Google Scholar] [CrossRef] [Green Version] - Marin, A.; Lhuissier, H.; Rossi, M.; Kähler, C.J. Clogging in constricted suspension flows. Phys. Rev. E
**2018**, 97, 021102. [Google Scholar] [CrossRef] - Kusters, R.; van der Heijden, T.; Kaoui, B.; Harting, J.; Storm, C. Forced transport of deformable containers through narrow constrictions. Phys. Rev. E
**2014**, 90, 033006. [Google Scholar] [CrossRef] [Green Version] - Bielinski, C.; Aouane, O.; Harting, J.; Kaoui, B. Squeezing multiple soft particles into a constriction: Transition to clogging. Phys. Rev. E
**2021**, 104, 065101. [Google Scholar] [CrossRef] - Garcimartín, A.; Pastor, J.M.; Ferrer, L.M.; Ramos, J.J.; Martín-Gómez, C.; Zuriguel, I. Flow and clogging of a sheep herd passing through a bottleneck. Phys. Rev. E
**2015**, 91, 022808. [Google Scholar] [CrossRef] [Green Version] - Altshuler, E.; Ramos, O.; Nuñez, Y.; Fernandez, J.; Batista-Leyva, A.; Noda, C. Symmetry Breaking in Escaping Ants. Am. Nat.
**2005**, 166, 643–649. [Google Scholar] [CrossRef] - Zuriguel, I.; Echevería, I.; Maza, D.; anésar Martín-Gómez, R.C.H.; Garcimarín, A. Contact forces and dynamics of pedestrians evacuating a room: The column effect. Saf. Sci.
**2020**, 121, 394–402. [Google Scholar] [CrossRef] - Squires, T.M.; Quake, S.R. Microfluidics: Fluid physics at the nanoliter scale. Rev. Mod. Phys.
**2005**, 77, 977. [Google Scholar] [CrossRef] [Green Version] - Dressaire, E.; Sauret, A. Clogging of microfluidic systems. Soft Matter
**2017**, 13, 37–48. [Google Scholar] [CrossRef] [PubMed] - Douféne, K.; Tourné-Péteilh, C.; Etienne, P.; Aubert-Pouëssel, A. Microfluidic Systems for Droplet Generation in Aqueous Continuous Phases: A Focus Review. Langmuir
**2019**, 35, 12597–12612. [Google Scholar] [CrossRef] - Convery, N.; Gadegaard, N. 30 years of microfluidics. Micro Nano Eng.
**2019**, 2, 76–91. [Google Scholar] [CrossRef] - Weatherall, E.; Willmott, G.R. Applications of tunable resistive pulse sensing. Analyst
**2015**, 140, 3318–3334. [Google Scholar] [CrossRef] - Saleh, O.A.; Sohn, L.L. Direct detection of antibody–antigen binding using an on-chip artificial pore. Proc. Natl. Acad. Sci. USA
**2003**, 100, 820–824. [Google Scholar] [CrossRef] [Green Version] - Ito, T.; Sun, L.; Bevan, M.A.; Crooks, R.M. Comparison of Nanoparticle Size and Electrophoretic Mobility Measurements Using a Carbon-Nanotube-Based Coulter Counter, Dynamic Light Scattering, Transmission Electron Microscopy, and Phase Analysis Light Scattering. Langmuir
**2004**, 20, 6940–6945. [Google Scholar] [CrossRef] - Heins, E.A.; Siwy, Z.S.; Baker, L.A.; Martin, R.C. Detecting Single Porphyrin Molecules in a Conically Shaped Synthetic Nanopore. Nano Lett.
**2005**, 5, 1824–1829. [Google Scholar] [CrossRef] - Arjm, I.N.; Van Roy, W.; Lagae, L.; Borghs, G. Measuring the Electric Charge and Zeta Potential of Nanometer-Sized Objects Using Pyramidal-Shaped Nanopores. Anal. Chem.
**2012**, 84, 8490–8496. [Google Scholar] [CrossRef] [Green Version] - Robards, K.; Ryan, D. Principles and Practice of Modern Chromatographic Methods; Elsevier: Amsterdam, The Netherlands, 2022. [Google Scholar]
- Reithinger, M.; Arlt, W. Prediction of the Partitioning Coefficient in Liquid-Solid Chromatography using COSMO-RS. Chem. Ing. Tech.
**2011**, 83, 83–89. [Google Scholar] [CrossRef] - Michaud, V.; Pracht, J.; Schilfarth, F.; Damm, C.; Platzer, B.; Haines, P.; Harreiß, C.; Guldi, D.M.; Spiecker, E.; Peukert, W. Well-separated water-soluble carbon dots via gradient chromatography. Nanoscale
**2021**, 13, 13116–13128. [Google Scholar] [CrossRef] - Seidel-Morgenstern, A.; Keßler, L.C.; Kaspereit, M. New Developments in Simulated Moving Bed Chromatography. Chem. Eng. Technol.
**2008**, 31, 826–837. [Google Scholar] [CrossRef] - Soni, G.V.; Singer, A.; Yu, Z.; Sun, Y.; McNally, B.; Meller, A. Synchronous optical and electrical detection of biomolecules traversing through solid-state nanopores. Rev. Sci. Instrum.
**2010**, 81, 014301. [Google Scholar] [CrossRef] [Green Version] - Carvalho, M.S. Flow of Complex Fluids through Porous Media: Application in Oil Recovery. In Proceedings of the Offshore Technology Conference, Rio de Janeiro, Brazil, 27–29 October 2015; p. 6. [Google Scholar] [CrossRef]
- Foroozesh, J.; Kumar, S. Nanoparticles behaviors in porous media: Application to enhanced oil recovery. J. Mol. Liq.
**2020**, 316, 113876. [Google Scholar] [CrossRef] - Farhadian, H.; Nikvar-Hassani, A. Water flow into tunnels in discontinuous rock: A short critical review of the analytical solution of the art. Bull. Eng. Geol. Environ.
**2019**, 78, 3833–3849. [Google Scholar] [CrossRef] - Boon, N.; Roij, R.V. Blue energy: From ion adsorption and electrode charging in sea and river water. Mol. Phys.
**2011**, 109, 1229–1241. [Google Scholar] [CrossRef] [Green Version] - Preuster, P.; Papp, C.; Wasserscheid, P. Liquid organic hydrogen carriers (LOHCs): Toward a hydrogen-free hydrogen economy. Acc. Chem. Res.
**2017**, 50, 74–85. [Google Scholar] [CrossRef] [PubMed] - Solymosi, T.; Geißelbrecht, M.; Mayer, S.; Auer, M.; Leicht, P.; Terlinden, M.; Malgaretti, P.; Bösmann, A.; Preuster, P.; Harting, J.; et al. Nucleation as a rate-determining step in catalytic gas generation reactions from liquid phase systems. Sci. Adv.
**2022**, 8, eade3262. [Google Scholar] [CrossRef] - Suter, T.A.M.; Smith, K.; Hack, J.; Rasha, L.; Rana, Z.; Angel, G.M.A.; Shearing, P.R.; Miller, T.S.; Brett, D.J.L. Engineering Catalyst Layers for Next-Generation Polymer Electrolyte Fuel Cells: A Review of Design, Materials, and Methods. Adv. Energy Mater.
**2021**, 11, 2101025. [Google Scholar] [CrossRef] - Du, N.; Roy, C.; Peach, R.; Turnbull, M.; Thiele, S.; Bock, C. Anion-Exchange Membrane Water Electrolyzers. Chem. Rev.
**2022**, 122, 11830–11895. [Google Scholar] [CrossRef] [PubMed] - Hepburn, C.; Adlen, E.; Beddington, J.; Carter, E.A.; Fuss, S.; Mac Dowell, N.; Minx, J.C.; Smith, P.; Williams, C.K. The technological and economic prospects for CO
_{2}utilization and removal. Nature**2019**, 575, 87–97. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Alberts, B.; Johnson, A.; Lewis, J.; Raff, M.; Roberts, K.; Walter, P. Molecular Biology of the Cell; Garland Science: Oxford, UK, 2007. [Google Scholar]
- Pethig, R. Ion, Electron, and Proton Transport in Membranes: A Review of the Physical Processes Involved. In Modern Bioelectrochemistry; Gutmann, F., Keyzer, H., Eds.; Springer: Boston, MA, USA, 1986; pp. 199–239. [Google Scholar] [CrossRef]
- Dubyak, G.R. Ion homeostasis, channels, and transporters: An update on cellular mechanisms. Adv. Physiol. Educ.
**2004**, 28, 143–154. [Google Scholar] [CrossRef] [PubMed] - Calero, C.; Faraudo, J.; Aguilella-Arzo, M. First-passage-time analysis of atomic-resolution simulations of the ionic transport in a bacterial porin. Phys. Rev. E
**2011**, 83, 021908. [Google Scholar] [CrossRef] [Green Version] - Peyser, A.; Gillespie, D.; Roth, R.; Nonner, W. Domain and Interdomain Energetics Underlying Gating in Shaker-Type KV Channels. Biophys. J.
**2014**, 107, 1841–1852. [Google Scholar] [CrossRef] [Green Version] - Lee, H.; Segets, D.; Süß, S.; Peukert, W.; Chen, S.C.; Pui, D.Y. Liquid filtration of nanoparticles through track-etched membrane filters under unfavorable and different ionic strength conditions: Experiments and modeling. J. Membr. Sci.
**2017**, 524, 682–690. [Google Scholar] [CrossRef] - Melnikov, D.V.; Hulings, Z.K.; Gracheva, M.E. Electro-osmotic flow through nanopores in thin and ultrathin membranes. Phys. Rev. E
**2017**, 95, 063105. [Google Scholar] [CrossRef] [Green Version] - Bacchin, P. Membranes: A Variety of Energy Landscapes for Many Transfer Opportunities. Membranes
**2018**, 8, 10. [Google Scholar] [CrossRef] [Green Version] - Berezhkovskii, A.M.; Dagdug, L.; Bezrukov, S.M. Two-site versus continuum diffusion model of blocker dynamics in a membrane channel: Comparative analysis of escape kinetics. J. Chem. Phys.
**2019**, 151, 054113. [Google Scholar] [CrossRef] - Nipper, M.; Dixon, J. Engineering the Lymphatic System. Cardiovasc. Eng. Technol.
**2011**, 2, 296–308. [Google Scholar] [CrossRef] [Green Version] - Wiig, H.; Swartz, M. Interstitial fluid and lymph formation and transport: Physiological regulation and roles in inflammation and cancer. Physiol. Rev.
**2012**, 92, 1005–1060. [Google Scholar] [CrossRef] - Yoganathan, A.P.; Cape, E.G.; Sung, H.W.; Williams, F.P.; Jimoh, A. Review of hydrodynamic principles for the cardiologist: Applications to the study of blood flow and jets by imaging techniques. J. Am. Coll. Cardiol.
**1988**, 12, 1344–1353. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Jensen, K.H.; Berg-Sørensen, K.; Bruus, H.; Holbrook, N.M.; Liesche, J.; Schulz, A.; Zwieniecki, M.A.; Bohr, T. Sap flow and sugar transport in plants. Rev. Mod. Phys.
**2016**, 88, 035007. [Google Scholar] [CrossRef] [Green Version] - Shimmen, T.; Yokota, E. Cytoplasmic streaming in plants. Curr. Opin. Cell Biol.
**2004**, 16, 68–72. [Google Scholar] [CrossRef] - Zwanzig, R. Diffusion past an entropy barrier. J. Phys. Chem.
**1992**, 96, 3926–3930. [Google Scholar] [CrossRef] - Reguera, D.; Rubi, J.M. Kinetic equations for diffusion in the presence of entropic barriers. Phys. Rev. E
**2001**, 64, 061106. [Google Scholar] [CrossRef] [Green Version] - Kalinay, P.; Percus, J.K.P. Projection of two-dimensional diffusion in a narrow channel onto the longitudinal dimension. J. Chem. Phys.
**2005**, 122, 204701. [Google Scholar] [CrossRef] - Kalinay, P.; Percus, J.K. Extended Fick-Jacobs equation: Variational approach. Phys. Rev. E
**2005**, 72, 061203. [Google Scholar] [CrossRef] [PubMed] - Kalinay, P.; Percus, J.K. Approximations of the generalized Fick-Jacobs equation. Phys. Rev. E
**2008**, 78, 021103. [Google Scholar] [CrossRef] - Martens, S.; Schmid, G.; Schimansky-Geier, L.; Hänggi, P. Entropic particle transport: Higher-order corrections to the Fick-Jacobs diffusion equation. Phys. Rev. E
**2011**, 83, 051135. [Google Scholar] [CrossRef] [Green Version] - Chacón-Acosta, G.; Pineda, I.; Dagdug, L. Diffusion in narrow channels on curved manifolds. J. Chem. Phys.
**2013**, 139, 214115. [Google Scholar] [CrossRef] [PubMed] - Malgaretti, P.; Pagonabarraga, I.; Rubi, J. Entropic transport in confined media: A challenge for computational studies in biological and soft-matter systems. Front. Phys.
**2013**, 1, 21. [Google Scholar] [CrossRef] [Green Version] - Malgaretti, P.; Pagonabarraga, I.; Rubi, J.M. Entropic electrokinetics. Phys. Rev. Lett.
**2014**, 113, 128301. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Malgaretti, P.; Pagonabarraga, I.; Rubi, J.M. Geometrically Tuned Channel Permeability. Macromol. Symp.
**2015**, 357, 178–188. [Google Scholar] [CrossRef] [Green Version] - Malgaretti, P.; Pagonabarraga, I.; Miguel Rubi, J. Entropically induced asymmetric passage times of charged tracers across corrugated channels. J. Chem. Phys.
**2016**, 144, 034901. [Google Scholar] [CrossRef] [Green Version] - Chinappi, M.; Malgaretti, P. Charge polarization, local electroneutrality breakdown and eddy formation due to electroosmosis in varying-section channels. Soft Matter
**2018**, 14, 9083–9087. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Malgaretti, P.; Janssen, M.; Pagonabarraga, I.; Rubi, J.M. Driving an electrolyte through a corrugated nanopore. J. Chem. Phys.
**2019**, 151, 084902. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Reguera, D.; Schmid, G.; Burada, P.S.; Rubi, J.M.; Reimann, P.; Hänggi, P. Entropic Transport: Kinetics, Scaling, and Control Mechanisms. Phys. Rev. Lett.
**2006**, 96, 130603. [Google Scholar] [CrossRef] [Green Version] - Reguera, D.; Luque, A.; Burada, P.S.; Schmid, G.; Rubi, J.M.; Hänggi, P. Entropic Splitter for Particle Separation. Phys. Rev. Lett.
**2012**, 108, 020604. [Google Scholar] [CrossRef] [PubMed] - Marini Bettolo Marconi, U.; Malgaretti, P.; Pagonabarraga, I. Tracer diffusion of hard-sphere binary mixtures under nano-confinement. J. Chem. Phys.
**2015**, 143, 184501. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Malgaretti, P.; Pagonabarraga, I.; Rubi, J. Rectification and non-Gaussian diffusion in heterogeneous media. Entropy
**2016**, 18, 394. [Google Scholar] [CrossRef] [Green Version] - Puertas, A.; Malgaretti, P.; Pagonabarraga, I. Active microrheology in corrugated channels. J. Chem. Phys.
**2018**, 149, 174908. [Google Scholar] [CrossRef] [Green Version] - Malgaretti, P.; Harting, J. Transport of neutral and charged nanorods across varying-section channels. Soft Matter
**2021**, 17, 2062–2070. [Google Scholar] [CrossRef] - Bianco, V.; Malgaretti, P. Non-monotonous polymer translocation time across corrugated channels: Comparison between Fick-Jacobs approximation and numerical simulations. J. Chem. Phys.
**2016**, 145, 114904. [Google Scholar] [CrossRef] [Green Version] - Malgaretti, P.; Oshanin, G. Polymer Translocation Across a Corrugated Channel: Ficks-Jacobs Approximation Extended Beyond the Mean First-Passage Time. Polymers
**2019**, 11, 251. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bodrenko, I.V.; Salis, S.; Acosta-Gutierrez, S.; Ceccarelli, M. Diffusion of large particles through small pores: From entropic to enthalpic transport. J. Chem. Phys.
**2019**, 150, 211102. [Google Scholar] [CrossRef] [PubMed] - Ledesma-Durán, A.; Hernández-Hernández, S.I.; Santamaría-Holek, I. Generalized Fick–Jacobs Approach for Describing Adsorption–Desorption Kinetics in Irregular Pores under Nonequilibrium Conditions. J. Phys. Chem. C
**2016**, 120, 7810–7821. [Google Scholar] [CrossRef] [Green Version] - Chacón-Acosta, G.; Núñez-López, M.; Pineda, I. Turing instability conditions in confined systems with an effective position-dependent diffusion coefficient. J. Chem. Phys.
**2020**, 152, 024101. [Google Scholar] [CrossRef] - Burada, P.S.; Schmid, G.; Reguera, D.; Rubi, J.M.; Hänggi, P. Biased diffusion in confined media: Test of the Fick-Jacobs approximation and validity criteria. Phys. Rev. E
**2007**, 75, 051111. [Google Scholar] [CrossRef] [Green Version] - Malgaretti, P.; Puertas, A.M.; Pagonabarraga, I. Active microrheology in corrugated channels: Comparison of thermal and colloidal baths. J. Colloid Interface Sci.
**2022**, 608, 2694–2702. [Google Scholar] [CrossRef] - Yang, X.; Liu, C.; Li, Y.; Marchesoni, F.; Hänggi, P.; Zhang, H.P. Hydrodynamic and entropic effects on colloidal diffusion in corrugated channels. Proc. Natl. Acad. Sci. USA
**2017**, 114, 9564–9569. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Malgaretti, P.; Stark, H. Model microswimmers in channels with varying cross section. J. Chem. Phys.
**2017**, 146, 174901. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sandoval, M.; Dagdug, L. Effective diffusion of confined active Brownian swimmers. Phys. Rev. E
**2014**, 90, 062711. [Google Scholar] [CrossRef] - Kalinay, P. Transverse dichotomic ratchet in a two-dimensional corrugated channel. Phys. Rev. E
**2022**, 106, 044126. [Google Scholar] [CrossRef] - Antunes, G.C.; Malgaretti, P.; Harting, J.; Dietrich, S. Pumping and Mixing in Active Pores. Phys. Rev. Lett.
**2022**, 129, 188003. [Google Scholar] [CrossRef] - Berezhkovskii, A.M.; Pustovoit, M.A.; Bezrukov, S.M. Diffusion in a tube of varying cross section: Numerical study of reduction to effective one-dimensional description. J. Chem. Phys.
**2007**, 126, 134706. [Google Scholar] [CrossRef] [PubMed] - Berezhkovskii, A.M.; Dagdug, L.; Bezrukov, S.M. Range of applicability of modified Fick-Jacobs equation in two dimensions. J. Chem. Phys.
**2015**, 143, 164102. [Google Scholar] [CrossRef] [Green Version] - Kalinay, P.; Percus, J.K. Corrections to the Fick-Jacobs equation. Phys. Rev. E
**2006**, 74, 041203. [Google Scholar] [CrossRef] [PubMed] - Pineda, I.; Alvarez-Ramirez, J.; Dagdug, L. Diffusion in two-dimensional conical varying width channels: Comparison of analytical and numerical results. J. Chem. Phys.
**2012**, 137, 174103. [Google Scholar] [CrossRef] - García-Chung, A.A.; Chacón-Acosta, G.; Dagdug, L. On the covariant description of diffusion in two-dimensional confined environments. J. Chem. Phys.
**2015**, 142, 064105. [Google Scholar] [CrossRef] - Lifson, S.; Jackson, J.L. On the Self-Diffusion of Ions in a Polyelectrolyte Solution. J. Chem. Phys.
**1962**, 36, 2410–2414. [Google Scholar] [CrossRef] - Reimann, P.; Van den Broeck, C.; Linke, H.; Hänggi, P.; Rubi, J.M.; Pérez-Madrid, A. Giant Acceleration of Free Diffusion by Use of Tilted Periodic Potentials. Phys. Rev. Lett.
**2001**, 87, 010602. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Berezhkovskii, A.M.; Bezrukov, S.M. Intrinsic diffusion resistance of a membrane channel, mean first-passage times between its ends, and equilibrium unidirectional fluxes. J. Chem. Phys.
**2022**, 156, 071103. [Google Scholar] [CrossRef] [PubMed] - Carusela, M.F.; Malgaretti, P.; Rubi, J.M. Antiresonant driven systems for particle manipulation. Phys. Rev. E
**2021**, 103, 062102. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Sketch of a channel with varying-section $h\left(x\right)$. The minimal ${h}_{min}$ and maximal ${h}_{max}$ amplitudes are marked. The channel was periodic along the x direction with period L.

**Figure 2.**Transport across porous media. (upper left) Permeability $\chi $ as obtained form Equation (46) (solid lines), Equation (26) with constant diffusion coefficient (dashed lines), and Equation (26) with a diffusion coefficient as given by Equation (40) (dashed-dotted lines) normalized by the one across a constant-section channel ${\chi}_{o}=D\beta /4L$, as a function of the geometry of the channel $\Delta S=ln\frac{{h}_{0}+{h}_{1}}{{h}_{0}-{h}_{1}}=ln\frac{{h}_{max}}{{h}_{min}}$ for different values of the particle radius. (upper right) Ratio of $\tilde{\chi}$ over $\chi $ normalized by $\chi $ for the datasets shown in the left panel. (bottom left) Permeability $\chi $ normalized by the one across a constant-section channel ${\chi}_{o}=D\beta /4L$ as a function of the radius of the particle, R, normalized by the average channel width, ${h}_{0}$, for different channel geometries captured by $\Delta S$. (bottom right) Ratio of $\tilde{\chi}$ over $\chi $ normalized by $\chi $ for the datasets shown in the left panel.

**Figure 3.**Dependence of approximated channel permeability $\tilde{\chi}$ (as defined in Equation (46)) normalized by that of a constant section channel ${\chi}_{o}$ as function of the amplitude of the dimensionless free energy barrier $\beta \Delta A$ that encodes the physical properties of the confined system.

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Malgaretti, P.; Harting, J.
Closed Formula for Transport across Constrictions. *Entropy* **2023**, *25*, 470.
https://doi.org/10.3390/e25030470

**AMA Style**

Malgaretti P, Harting J.
Closed Formula for Transport across Constrictions. *Entropy*. 2023; 25(3):470.
https://doi.org/10.3390/e25030470

**Chicago/Turabian Style**

Malgaretti, Paolo, and Jens Harting.
2023. "Closed Formula for Transport across Constrictions" *Entropy* 25, no. 3: 470.
https://doi.org/10.3390/e25030470