Active Brownian particles in external force fields: field-theoretical models, generalized barometric law, and programmable density patterns

arXiv:2202.04423 · cond-mat.soft, cond-mat.mes-hall, cond-mat.stat-mech, nlin.AO · Submitted 2022-02-09 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Active Brownian particles in external force fields".

Mira: The gist: This work derives a local predictive field theory for interacting active Brownian particles in two and three spatial dimensions under space- and time-dependent external force fields,

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So, looking at the paper "Active Brownian particles in external force fields: field-theoretical models, generalized barometric law, and programmable density patterns," the core idea is using a local predictive field theory to understand how external forces shape active particle dynamics.

Mira: The authors show that this model allows them to derive a generalized barometric formula for gravity and demonstrates that an external harmonic trap can induce motility-induced phase separation in systems that are otherwise homogeneous.

Lev: What this means in simple terms is you can use an outside force, like a trap, to create specific density patterns you want in a system of active particles.

Kai: It moves the study of these systems from just looking at individual particle motion to understanding how those motions organize themselves collectively under external influence.

Mira: It gives us tools for designing active materials where we can program density structures by applying controlled external fields, which is a big step for controlling these systems.

Lev: The limitation they mention is that the model they present, specifically the 2nd-order-derivatives model, is not well suited to describe dense and interacting collectives of ABPs <ref:2202.04423#pg2,not well suited to describe dense and interacting collectives of ABPs>.

Kai: That limitation tells us that while this framework is useful for understanding the basics under certain conditions, it stops working when you get really dense or when you need to account for even more complex spatial variations.

Mira: So the path forward involves extending these phase-field models with higher order derivative terms, like up to sixth order, to study phenomena such as crystallization in high-gravity regions.

Lev: That would mean the computational cost goes way up, and you’d need better ways to handle those coupled equations on real hardware.

Conclusion: Kai: So we're wrapping up on this paper about active Brownian particles in external force fields and their models for barometric laws and density patterns, right?

Mira: Yeah, the authors are looking at how those external forces—like gravity or traps—actually change the collective movement of these particles.

Lev: From my side, I'm thinking about how much overhead that field theory would actually take to simulate on real hardware when you get those spatial and time-dependent force fields involved.

Kai: Right, but the main thing is they show you can build a predictive model for these things across two or three dimensions where you just need to know the external field details.

Mira: And they proved that with a harmonic trap, you can actually make the particles clump together in specific places where they wouldn't normally go on their own.

Lev: So it's not just passive diffusion anymore; you have active control over the resulting density structure because of what’s outside the system.

Kai: Exactly, and this means we can program these patterns, which is pretty wild for active matter research.

Mira: It suggests that by tuning the external field—whether it's a simple gravity pull or a specific trap shape—you can dictate the final density landscape.

Lev: That control over phase separation is where it gets interesting from an engineering standpoint because you’re moving from random dynamics to engineered states.

Kai: And their results, even in 2D simulations, actually line up pretty well with what we see in Brownian dynamics experiments when they're doing the measurements <ref:2202.04423#pg2>.

Mira: It validates the field theory approach for describing these collective behaviors under external influence.

Lev: But it’s important to remember that this model is based on certain approximations, and you can't just take those 2D or three dee results straight over to a system with much higher spatial complexity without re-deriving things <ref:2202.04423#pg2>.

Kai: That's true; the paper hints that if you want to look at things getting really crowded or high-gravity regions, you might need to add terms up to the sixth order in derivatives.

Mira: So the authors are laying out a roadmap for how this basic model can be extended into more complex physical regimes.

Lev: That extension would definitely increase the complexity of the equations, and we'd have to figure out how that translates into manageable error bounds for error correction schemes.

Jens Bickmann, *Stephan Bröker, *Raphael Wittkowski

Institut für Theoretische Physik, Center for Soft Nanoscience, Westfälische Wilhelms-UniversitƤt Münster

cond-mat.soft, cond-mat.mes-hall, cond-mat.stat-mech, nlin.AO

Submitted: 2022-02-09

Updated: 2022-02-09

Comments: 13 pages, 2 figures

Journal ref: Physical Review E 108, 044601 (2023)

DOI: 10.1103/PhysRevE.108.044601

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 77/100

The gist: The gist: This work derives a local predictive field theory for interacting active Brownian particles in two and three spatial dimensions under space- and time-dependent external force fields,

Key concepts

Active Brownian Particles (ABPs): These are particles that move randomly due to thermal noise but also possess self-propulsion. Their motion is governed by both random diffusion and a constant swimming speed, making their collective behavior complex and dependent on their interactions with each other.
Field Theory Derivation: The researchers created a mathematical model using the interaction-expansion method. This involves expanding the pair-distribution function and interaction forces into various orders (like gradients) to derive a simplified advection-diffusion equation that describes how the particle density field evolves over time.
Motility-Induced Phase Separation (MIPS): This is a phenomenon where active particles spontaneously cluster or separate into dense regions even when there are no external forces pushing them together. In this study, an external harmonic trap was shown to trigger MIPS, allowing researchers to create specific density patterns in the system.

Summary

The gist: This work derives a local predictive field theory for interacting active Brownian particles in two and three spatial dimensions under space- and time-dependent external force fields, showing that these models can describe dynamics up to fourth order in derivatives and predict phenomena like motility-induced phase separation (MIPS).

Theoretical Framework

The research develops a local, predictive field theory for interacting active Brownian particles (ABPs) in two (2D) and three (3D) spatial dimensions under external force fields that can be space- and time-dependent <ref:2202.04423#pg2> The translational motion of the ABPs is described by a Langevin equation involving translational Brownian noise, propulsion speed, particle-particle interaction force, and the external force field <ref:2202.04423#pg2>. The rotational motion is similarly governed by a Langevin equation depending on the dimensionality of the system <ref:2202.04423#pg2>.

Field Theory Derivation

A field-theoretical model is derived using the interaction-expansion method, involving integrating over all degrees of freedom except for one particle to obtain an equation for the one-particle density field <ref:2202.04423#pg2>. This process includes Fourier expansion (2D) or spherical harmonics expansion (3D) of the pair-distribution function, a gradient expansion of integrals containing the interaction force, an orientational expansion into Cartesian order-parameter tensors, and a quasi-stationary approximation (QSA) <ref:2202.04423#pg2>. The resulting model is described by an advection-diffusion equation for the density field <ref:2202.04423#pg2>.

Model Application and Results

The derived models are applied to two common types of external fields: gravity and harmonic traps <ref:2202.04423#pg2>. For gravity, the model yields a modified Boltzmann distribution that accounts for particle interactions and activity <ref:2202.04423#pg2>. For harmonic traps, the study shows that ABPs can be induced to undergo motility-induced phase separation (MIPS) at desired locations <ref:2202.04423#pg2>. The analytical predictions for the onset of MIPS and the sudden increase of interaction energy per particle in simulations are found to be in very good agreement <ref:2202.04423#pg2>.

Simulation Setup

Simulations were performed in two dimensions to keep computational effort moderate <ref:2202.04423#pg2>. For harmonic traps, Brownian dynamics simulations based on the Langevin equations were used with a purely repulsive Weeks-Chandler-Andersen potential as the interaction force <ref:2202.04423#pg2>. A circular harmonic trap was incorporated by prescribing the local change of particle velocity due to the trapping force <ref:2202.04423#pg2>. The initial particle distribution was uniform and random with an overall packing density of 0.2 <ref:2202.04423#pg2>.

Key Findings on External Fields

The study demonstrates that in a gravitational field, the density of ABPs decreases slower for increasing x than the density of passive Brownian particles at the same temperature T in the dilute limit <ref:2202.04423#pg2>. In higher densities, interactions lead to accumulation where activity creates an effective attractive interaction potential <ref:2202.04423#pg2>. For harmonic traps, ABPs can escape from weak or small traps due to self-propulsion, but a strong trap allows for MIPS clusters emerging inside the trap <ref:2202.04423#pg2>. The analytical prediction for the onset of MIPS and the sudden increase of interaction energy per particle in simulation data are found to be in very good agreement <ref:2202.04423#pg2>.

Conclusion and Implications

The results show that ABPs in external fields can exhibit interesting effects arising from the coupling of the external force, particle interactions, and activity <ref:2202.04423#pg2>. The occurrence of MIPS can be controlled quite arbitrarily by external fields, allowing for programmable density patterns in systems containing ABPs <ref:2202.04423#pg2>. This understanding is helpful for future applications where external fields are likely to be relevant <ref:2202.04423#pg2>. The models are applicable for small to large densities and activities of the particles <ref:2202.04423#pg2>. The paper suggests that extending the phase-field model by including terms up to sixth order in derivatives could be used to study ABPs in high-gravity regions that can induce crystallization <ref:2202.04423#pg2>. The research provides a local predictive field theory for interacting ABPs under external force fields <ref:2202.04423#pg2>.

Appendix Details

The appendix provides microscopic expressions for the coefficients occurring in the density current equations, including those for two spatial dimensions and three spatial dimensions <ref:2202.04423#pg2>. These expressions are given in Appendix A and Appendix B <ref:2202.04423#pg2>. The coefficients depend on microscopic properties of the system, such as the pair-distribution function <ref:2202.04423#pg2>.

References

[1] J. Elgeti, R. G. Winkler, and G. Gompper, Physics of microswimmers—single particle motion and collective behavior: a review, Reports on Progress in Physics 78, 056601 (2015).

[2] T. Speck, Collective behavior of active Brownian particles: from microscopic clustering to macroscopic phase separation, European Physical Journal Special Topics 225, 2287 (2016).

[3] A. Z¨ottl and H. Stark, Emergent behavior in active colloids, Journal of Physics: Condensed Matter 28, 253001 (2016).

[4] C. Bechinger, R. Di Leonardo, H. L¨owen, C. Reichhardt, G. Volpe, and G. Volpe, Active particles in complex and crowded environments, Reviews of Modern Physics 88, 045006 (2016).

[5] J. Katuri, K. D. Seo, D. S. Kim, and S. Sanchez, Artificial micro-swimmers in simulated natural environments, Lab on a Chip 16, 1101 (2016).

[6] H. Stark, Swimming in external fields, European Physical Journal Special Topics 225, 2369 (2016).

[7] Z. Wang, H.-Y. Chen, Y.-J. Sheng, and H.-K. Tsao, Diffusion, sedimentation equilibrium, and harmonic trapping of run-and-tumble nanoswimmers, Soft Matter 10, 3209 (2014).

[8] F. Ginot, I. Theurkauff, D. Levis, C. Ybert, L. Bocquet, L. Berthier, and C. Cottin-Bizonne, Nonequilibrium equation of state in suspensions of active colloids, Physical Review X 5, 011004 (2015).

[9] J. Vachier and M. G. Mazza, Dynamics of sedimenting active Brownian particles, European Physical Journal E 42, 11 (2019).

[10] A. I. Campbell and S. J. Ebbens, Gravitaxis in spherical Janus swimming devices, Langmuir 29, 14066 (2013).

[11] K. Wolff, A. M. Hahn, and H.-Y Stark, Sedimentation and polar order of active bottom-heavy particles, European Physical Journal E 36, 43 (2013).

[12] E. Locatelli, F. Baldovin, E. Orlandini, and M. Pierno, Active Brownian particles escaping a channel in single file, Physical Review E 91, 022109 (2015).

[13] A. I. Campbell, R. Wittkowski, B. ten Hagen, H. L¨owen, and S. J. Ebbens, Helical paths, gravitaxis, and separation phenomena for mass-anisotropic self-propelling colloids: experiment versus theory, Journal of Chemical Physics 147, 084905 (2017).

[14] F. Ginot, A. Solon, Y. Kafri, C. Ybert, J. Tailleur, and C. Cottin-Bizonne, Sedimentation of self-propelled Janus colloids: polarization and pressure, New Journal of Physics 20, 115001 (2018).

[15] B. ten Hagen, F. K¨ummel, R. Wittkowski, D. Takagi, H. L¨owen, and C.

Improvements for AI systems

Bold Header: Predictive Modeling for External Force Fields

This capability allows AI systems to derive predictive models that are applicable for space- and time-dependent external force fields, specifically for gravity and harmonic traps, enabling them to simulate particle dynamics under complex, non-uniform environments.

Bold Header: Programmable Density Pattern Realization

AI systems can be programmed to induce motility-induced phase separation (MIPS) in systems that, without external fields, remain in a homogeneous state using an external harmonic trap, leading to the realization of programmable density patterns.

Bold Header: High-Order Derivative Dynamics Modeling

The system can utilize a field theory that extends phase-field models towards external force fields, where the dynamic equation for the density field in this field theory is ρ˙ = −∇ · (J (int) + J (ext)). This allows for modeling further time-evolution of MIPS by truncating the gradient expansion at four orders.

Bold Header: Density-Dependent Effective Diffusion

The AI can incorporate a density-dependent diffusion coefficient, specifically D(ρ) = DT + a0 + a1ρ + a2ρ2, which allows for modeling how particle transport changes with local concentration, mimicking an effective attractive interaction potential.

Bold Header: High-Density Collective Dynamics Analysis

The system can analyze the dynamics of interacting ABPs in regimes where density dependence is critical, noting that the curve that considers interactions does not follow a Boltzmann distribution and that the density of the ABPs is larger than without interactions in this regime.