Variational Principle for Pattern Selection in Laplacian Growth Without Surface Tension

Core Concepts

A new variational principle based on maximizing the area spanned by a growing pattern successfully predicts pattern selection in Laplacian growth without surface tension, agreeing with experimental observations and complementing previous non-variational approaches.

Abstract

• Bibliographic Information: Mineev-Weinstein, M., & Alekseev, O. (2024). Variational Pattern Selection. arXiv preprint arXiv:2411.03001v1.
• Research Objective: This paper aims to introduce a novel variational principle for pattern selection in Laplacian growth, specifically focusing on scenarios where surface tension is negligible. The authors apply this principle to classical problems like the Saffman-Taylor finger in a channel and a wedge, as well as the universal fjord opening angle.
• Methodology: The authors leverage a stochastic growth theory, where tiny Brownian particles attach to a growing domain. By maximizing the entropy associated with this stochastic process, they derive a deterministic equation of growth. This equation, equivalent to the classical Hadamard formula, forms the basis for their variational principle.
• Key Findings: The study demonstrates that maximizing the area spanned by the growing pattern, which is equivalent to maximizing the entropy of the process, accurately predicts the selected pattern in various Laplacian growth scenarios. This principle successfully reproduces the selected finger width in a channel and shows excellent agreement with experimental observations for finger selection in a wedge. Additionally, the principle confirms the existence of a universal fjord opening angle.
• Main Conclusions: The proposed variational principle provides a new perspective on pattern selection in Laplacian growth, demonstrating that surface tension is not a necessary factor for selection. This approach offers a simpler and potentially more fundamental understanding of pattern formation in these systems.
• Significance: This research significantly contributes to the field of pattern formation and interfacial dynamics by introducing a novel variational principle for pattern selection in Laplacian growth without surface tension. This principle offers a new avenue for understanding and predicting pattern selection in various physical systems.
• Limitations and Future Research: While the variational principle shows promise, further investigation is needed to address some open questions. The authors highlight the need to understand the unexpected loss of selection accuracy at very large times in the wedge geometry and the surprising consistency of the critical value of a(t) for different wedge angles. Additionally, exploring the applicability of this principle to other pattern-forming systems beyond Hele-Shaw cells could be a fruitful avenue for future research.

Stats

The selected finger width in a channel is λ = 1/2.
The universal fjord opening angle is approximately 11.7° for wedge angles between 35° and 90°.
The empirical law for finger selection in a wedge, λ(θ) = 1 - 10°/θ, was observed in experiments.

Quotes

"Selection problems are challenging both in physics (to identify the selection mechanism) and in mathematics (to handle a small singular term). The desire to find a functional, whose extremal describes the selected pattern, is understandable."
"Since Nature always favors the largest entropy scenario, then in view of (8) to solve the selection problem is to find a value of a parameter to select from (it is λ in the STF case), which maximizes the area spanned by the growing domain D(t)."
"These selections came from differential equations, but not from variation of a functional, so Langer’s question in [13] about variational formulation still persists. After applying the entropy functional defined below to the Saffman-Taylor selection problems in the next section, we believe the answer is “yes”."

Key Insights Distilled From

Variational Pattern Selection

by Mark Mineev-... at arxiv.org 11-06-2024

https://arxiv.org/pdf/2411.03001.pdf

Deeper Inquiries

How can this variational principle be generalized and applied to other pattern-forming systems beyond Laplacian growth and Hele-Shaw cells?

This variational principle, grounded in maximizing the entropy of the most probable scenario, holds promising potential for generalization to other pattern-forming systems beyond Laplacian growth and Hele-Shaw cells. Here's how:

Identifying a Conserved Quantity: The first crucial step involves identifying a relevant conserved quantity analogous to the area enclosed by the interface in Laplacian growth. This quantity should meaningfully characterize the evolving pattern and be tied to the system's underlying physics or dynamics.

Formulating the Entropy Functional: Once the conserved quantity is established, an entropy functional needs to be constructed. This functional should reflect the statistical distribution of possible pattern configurations, with the conserved quantity acting as a constraint. The specific form of the entropy functional will depend on the nature of the pattern-forming system and the stochastic processes involved.

Maximizing Entropy: The variational principle then dictates that the selected pattern, among a family of possible solutions, will be the one that maximizes this entropy functional. This maximization process, potentially involving techniques from calculus of variations, will yield the relationship between the system's parameters and the selected pattern's characteristics.

Examples of Potential Applications:

Dendritic Growth: In dendritic growth, the conserved quantity could be the total surface area or volume of the dendrite. The entropy functional could be formulated based on the probability distribution of different branching angles and lengths.

Reaction-Diffusion Systems: For reaction-diffusion systems, the conserved quantity might involve concentrations of reacting species. The entropy functional could be based on the spatial distribution of these concentrations, potentially incorporating concepts from statistical mechanics.

Biological Systems: In biological pattern formation, such as morphogenesis, the conserved quantity could be related to cell density or nutrient distribution. The entropy functional could be formulated based on the probabilities of different cell arrangements or signaling pathways.
The key challenge lies in creatively identifying the appropriate conserved quantity and formulating a meaningful entropy functional that captures the essential stochasticity of the specific pattern-forming system under investigation.

Could the inclusion of noise or fluctuations in the stochastic growth model potentially explain the observed loss of selection accuracy at very large times?

Yes, incorporating noise or fluctuations into the stochastic growth model could plausibly explain the observed decline in selection accuracy at very large times. Here's why:

Perturbations to the Conserved Quantity: Noise and fluctuations can introduce random perturbations to the conserved quantity (e.g., area in Laplacian growth) over time. While these perturbations might be small initially, their cumulative effect can become significant over long periods, especially in systems exhibiting sensitivity to initial conditions.

Deviations from Optimal Path: The variational principle identifies the most probable path by maximizing entropy. However, noise can cause the system to deviate from this optimal path. At large times, these deviations can accumulate, leading to a less well-defined or "blurred" selection.

Exploration of Wider Solution Space: Noise allows the system to explore a broader range of possible configurations beyond the immediate vicinity of the most probable path. At large times, this exploration can lead to the system sampling configurations that are less consistent with the selection mechanism based on the conserved quantity.
Incorporating Noise:
To model this explicitly, one could introduce noise terms into the stochastic growth equations or modify the entropy functional to account for a distribution of possible values for the conserved quantity. Numerical simulations with noise could then be used to investigate whether the observed loss of selection accuracy at large times can be reproduced.
Further Research:
Investigating the role of noise in this context could provide valuable insights into the limitations of deterministic selection mechanisms and the importance of considering stochastic effects, especially in the long-time behavior of pattern-forming systems.

What are the implications of this variational principle for understanding the role of entropy and information theory in pattern formation and self-organization processes in nature?

This variational principle, centered on maximizing the entropy of the most probable scenario, carries profound implications for understanding the intricate roles of entropy and information theory in shaping pattern formation and self-organization processes across the natural world.

Entropy as a Driving Force: The principle suggests that entropy, often associated with disorder, can act as a driving force towards order and organization. By maximizing entropy within the constraints imposed by conserved quantities, systems naturally select specific patterns from a vast array of possibilities.

Information Content of Patterns: The selected patterns, maximizing entropy, can be viewed as carrying the highest information content about the system's constraints and underlying dynamics. This perspective links pattern formation to information encoding and transmission, suggesting that nature might utilize patterns to organize and convey information efficiently.

Robustness and Stability: Patterns selected through entropy maximization are likely to be more robust and stable against perturbations. This robustness stems from the fact that these patterns represent a "peak" in the probability landscape, making them less susceptible to random fluctuations.

Emergence from Simplicity: The principle hints at the possibility of complex patterns emerging from relatively simple rules based on maximizing entropy under constraints. This resonates with the concept of self-organization, where intricate structures arise spontaneously from local interactions without centralized control.

Broader Implications:

Understanding Biological Systems: This principle could shed light on the development of complex structures in biological systems, from the intricate patterns on butterfly wings to the organization of cells within tissues.

Designing Artificial Systems: It offers potential inspiration for designing self-assembling materials and robust, adaptive systems by incorporating entropy-driven principles into their design.

Fundamental Physics: Exploring the connections between entropy, information, and pattern formation might even hold clues for understanding fundamental physical phenomena, such as the emergence of structure in the early universe.
In essence, this variational principle encourages a paradigm shift in how we view entropy and information in the context of pattern formation, highlighting their potential as fundamental organizing principles in nature.