You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: |
    We use a large cell Monte Carlo Renormalization procedure to compute the critical exponents of a system of growing linear polymers. We simulate the growth of non-intersecting chains in large MC cells. Dense regions where chains get in each others’ way, give rise to connected clusters under coarse graining. At each time step, the fraction of occupied bonds is determined in both the original and the coarse grained configurations, and averaged over many realizations. Our results for the fractal dimension on three dimensional lattices are consistent with the percolation value.

    PACS numbers: 05.10.Cc, 64.60Ak
author:
- 'Duygu Balcan$^{1}$ and Ayşe Erzan$^{1,2}$'
title: Monte Carlo Renormalization Group for Entanglement Percolation
---

Introduction
============

The gelation process in crosslinked polymers has been studied within the context of percolation theory [@Stauffer1; @Stauffer2; @Herrmann1; @Herrmann2], but the rheokinetics [@rheokinetics] of bulk linear polymers as a function of chain length has recieved less theoretical attention. As the density of chains, and their length increases, the viscosity starts to increase well before the onset of vitrification. [@Cicerone] It has been proposed [@deGennes; @Doi] that this is due to the entanglement of the polymer chains.

We would like to pose the question of whether the percolation of entanglement clusters is in the same universality class as percolation. We define entanglement clusters starting from ordinary sets of connected bonds. We will consider two such sets connected, as long as they come with a lattice constant of each other, i.e., share the end points of an empty bond. Since the length of the chains, or the average length between entanglements, could introduce a second length scale into the problem, this could potentially lead to a crossover to a different universality class than percolation.

It has been found in two dimensions [@ConiglioDaoud79] that the vulcanization process, which involves the crosslinking of long chains, is in the percolation universality class, and that there is a crossover between Self Avoiding Walk and percolation behaviour as a function of the fugacity of the crosslinkers. Similarly, Jan et al. [@Jan] find in three dimensions that the crossover from SAW to percolation exponents already occurs for any finite value of the concentration of initiators (from which the chainlike structures grow), for a mixture of monomers with functionalities $\ge 2$, i.e., again in the presence of crosslinkers. A Monte Carlo simulation in two dimensions reveals [@Bunde] that a growth model can cross over from SAW like behaviour to percolation, as a function of the cluster mass.

The percolation of clusters which are not necessarily connected but linked to each other by loops, has also been studied by very large MC computations [@Kantor]. These authors find that the new critical point is very close to the ordinary percolation threshold on the cubic lattice, with the eigenvalue of the renormalized occupation probability being indistinguishable from that of the ordinary percolation problem.

In this paper we introduce a special Monte Carlo (MC) Renormalization Group procedure to investigate the universality class for the entanglement phase transition of a linear polymer system, in three dimensions and with no crosslinkers (or monomers with functionality greater than 2) present. The polymerization process is modeled by growing non-intersecting chains from a set of randomly chosen sites on a cubic lattice. Chains which occupy nearest neighbor sites on the lattice are considered to be part of the same connected cluster, and coarse grain to occupied bonds.

In the next section, we present the simulations and the MC renormalization group procedure. In section 3 we present an analysis of the results. We conclude with a discussion in section 4.

MC simulations and the Renormalization Group
============================================

In this section we describe a Monte Carlo renormalization group procedure, to deal with the percolation of entangled clusters of linear polymers. Since the long linear chains of the growing bulk polymer cannot be accomodated in small cells, we start by simulating the polymer growth on relatively large lattices. However we make a different choice than the one made by Swendsen [@Swendsen1; @Swendsen2; @Swendsen3] in the way that the MC renormalization group is introduced, as illustrated schematically in Fig.1.

The MC Renormalization Group procedure which we use consists of paving the large MC cell by small cells on which one performs a coarse graining transformation. [@Reynolds1] This induces an RG transformation on the bond occupation probability, averaged over many independent realizations.

In the conventional MC renormalization group procedure, the coarse graining is done in one step [@Swendsen1; @Swendsen2; @Swendsen3]. In the case of percolation, this would mean coarse graining the large MC cell to one occupied or empty bond - i.e., one asks the question whether the cell is spanned by a connected cluster or not. Adopting the conventional definition of a connected cluster, this one-step coarse graining procedure is clearly not appropriate in our case, since for non-intersecting linear chains, the MC cell will be spanned in case there is at least one chain that grows across it, and otherwise not. This does not describe the physical problem at hand.

The effect which we are trying to model has to do with linear chains that do not intersect, but nevertheless get physically close to each other in dense regions where they constrain each others’ motion, at least on short time scales, and thus form an effective three-dimensional network. We look, therefore, for a way in which we can capture this effectively three dimensional behaviour, and the answer lies in making a coarse graining with a scale factor $b$ smaller than the linear size $L$ of the large cell. We will illustrate below, that chains which are non intersecting can, under coarse graining, go to branched structures, which behave very much like ordinary percolation clusters.

The Kinetic Growth Model for Linear Polymers
--------------------------------------------

We have generated our configurations on a cubic lattice with linear dimension $L$ and periodic boundary conditions. We start from an empty lattice, and then place $C$ initiators at randomly chosen lattice sites. As the chains grow, the initiators are displaced in the growth direction and mark the “active end" of the chains. We don’t allow a chain to intersect itself or the other chains during this process. If the chain hits a dead end, it stops growing.

The growth rules are as follows. During one time step, from time $t$ to $t+1$, we consider each chain one by one, look at the active end of the chain and count the number of empty nearest neighbor sites. If this number is zero, the chain is trapped, and we proceed to the next chain. If the number is greater than zero, with [*growth probability*]{} $k$ we extend the chain to one of these empty sites, chosen at random. (This extra parameter $k$ gives us the possibility of probing arbitrarily small increments in the occupation probability on our finite cell.)We mark this new location as the active end of the chain, and place a bond between the old and the new location of this active end. After this process has been applied to all the chains, one gets the configuration at time $t+1$.

We use this set of kinetic growth rules, to grow a mass of non-intersecting linear chains in relatively large cells, consisting of cubic lattices of linear size $L=8,16,32$.

Compactification and Coarse-Graining
------------------------------------

In Fig. 2, we display the small cells with which our lattice will be paved, in order to perform the coarse graining transformation. This choice for the coarse graining rule has been used by Reynolds et al. [@Reynolds2] and Bernasconi [@Bernasconi]. Under our coarse-graining rule each cell (with $3 b^3$ bonds, where $b$ is the rescaling factor, here chosen to be 2) goes to at most three bonds in the coarse-grained lattice.

Before we coarse grain, we perform a compactification of our randomly connected clusters by connecting nearest neighbor occupied sites, which are not already connected by a bond.(See Fig. 3) This is motivated by the fact that in dense regions, chains which physically get in each others’ way are not chemically connected, but nevertheless contribute to the instantaneous shear modulus of the effective network.

We determine the state of the bonds in the coarse-grained lattice by considering the configurations in each small cell, as illustrated in Fig. 4. We look for a path which spans the cell in each Cartesian direction. If there is at least one such path this cell goes to an occupied bond in this direction in the coarse-grained lattice. Otherwise, it goes to an empty bond. By applying this rule to each small cell in the original lattice we get the new coarse-grained lattice.

It can be noticed in Fig. 4 that the occupied bonds spanning the small cell in the horizontal and the vertical directions are not chemically connected, and yet, (even without the bonds added for compactification) they go, under coarse graing to an occupied junction.

We now compute the fraction of occupied bonds in both the original and coarse-grained lattices, $p(t)$ and $p'(t)$, where we also count the bonds added in the compactification process. These are averaged over many independent runs, to find $<p(t)>$ and $<p'(t)>$, for each time $t$. Eliminating $t$, the coarse graining procedure leads to a renormalization group transformation on the average occupation probability, as shown in Figure 5. The fixed point of the RG transformation and the RG eigenvalue are computed from the MC data, leading to the value of the critical exponent we are interested in.

Finite size scaling and the fractal dimension
=============================================

The correlation length, $$\xi \sim{(p-p^\ast)}^{-\nu}\;\;\;, \label{xi}$$ for $p$ very close to the critical point $p^\ast$ (the fixed point of the RG transformation) exceeds $L$. From then on, $\xi \sim L$, i.e., it behaves like a constant with respect to $(p-p^\ast)$. In other words, the relationship in Eq.(\[xi\]) breaks down and a finite size scaling analysis is in order. [@Stauffer1]

The mass contained in the incipient infinite cluster, $\Delta M \sim 
L^{D_f}$, where $D_f$ is the fractal dimension of the percolation cluster,so that $P_\infty = \Delta M / V \sim L^{-\beta / \nu} \sim L^{ 
D_f - d }$. We may also show that the mass $M_0$ contained in the rest of the finite clusters scales like $M_0/V \sim L^{(2-\tau)D_f}$, with $(2-\tau)D_f=D_f-d$. Thus, the total concentration $p= (\Delta M + M_0)/V 
\equiv M/V$ of occupied bonds scales like $$p\sim L^{D_f-d}\;\;\;. \label{fractal}$$ We will make use of this fact to compute the fractal dimension of the percolation cluster, which can be directly related to the other known critical exponents, e.g. via $D_f=(\beta+\gamma)/\nu$. [@Herrmann2]

From (\[fractal\]), we see that under a rescaling transformation $L \to L'$=$L/b$, $M \to M' \sim b^{-D_f} M$, so that $$p'=\frac{M'}{V'} \sim b^{d-D_f}p\;\;\;.$$ At the fixed point of the RG transformation, $p^\ast$, with $\lambda\equiv
dp^\prime / dp \vert_{p^\ast}$, one has, $\lambda=b^{d-D_f}$, or $$D_f=d-{\frac{\ln \lambda}{\ln b}}\;\;\;. \label{Df}$$ In Fig. 5, we plot $<p'(t)>$ versus $<p(t)>$ for $L$=32 where the averages have been performed over $10^4$ independent runs, with $C=0.02 \times L^3$ and $k=0.01$. Notice that $p^\ast \approx 0.02$ is much smaller than the expected value for percolation on a cubic lattice (0.2488) [@Stauffer1]. We have varied $C$ between 1 to 9$\%$ of the lattice points, and a fixed point $p^\ast$ has been found for all these different $C$ values, although the value of $p^\ast$ descreases with decreasing $C$. This can be seen to result from an effective length scale introduced by the increasingly long chains growing from fewer and fewer initiators, i.e., an effective lattice spacing of the order of $C^{-1/3}/L$ [@Jan].

We obtain $\lambda$ by making a linear fit to the data around this point, and we obtain the value of the critical exponent $D_f$ from Eq. (\[Df\]). Our results are given in Table 1. Within our error bars we have not found any dependence of the RG eigenvalue on the concentration of initiators, within the range of values we have considered. The errors reported in Table 1 are those arising from taking one standard deviation of each of our data points reported in the inset of Fig. 5. The best value of $D_f$ for percolation clusters in three dimensions, obtained from simulation results [@Stanley] is $2.53\pm 0.03$. Given that we are performing a finite cell real space renormalization group calculation, our results are consistent with the percolation fractal dimension.

     [ $L$ ]{}       8      16      32
  --------------- ------- ------- -------
     $\lambda$     1.476   1.479   1.472
       $D_f$       2.438   2.436   2.442
   $\Delta$$D_f$   0.004   0.002   0.001

  : $\lambda$ and $D_f$ values obtained by our simulations.

Discussion
==========

The physics of bulk linear polymers is an extremely interesting and rapidly growing field. As the average molecular weight (or chain length) grows, bulk linear polymers are known to exhibit many of the properties of ordinary gels, such as resistance to shear and the capacity to take up solvent and swell, while retaining their original shape. [@Yilmaz] This behaviour is present even in linear polymers like PMMA (Poly-methyl methacrylate) where inter-chain interactions [@Saiani; @Erukhimovich] are extremely weak, and therefore must arise from purely geometrical effects, in other words entanglements. [@Zhu; @Tian] In this context we use “entanglement" to mean a dense region where chains physically impede the motion of other chains, and not in the mathematically precise sense of knots [@Kantor; @Holroyd1; @Holroyd2; @Grimmet].

The transition to a regime where the presence of an entanglement network can be ascertained via dilatometric techniques [@Okay1] takes place before or around the onset of the “gel effect," namely the characteristic rapid growth in the rate of polymerization [@Okay2; @Okay3]. This effect arises due to the transition to a diffusion-limited regime, where end to end termination of chains is essentially suppressed, so that single chains grow rapidly [@Okay1; @Okay2; @Okay3; @Zhu]. Its onset has been conjectured to be due to the entanglement of the linear chains [@deGennes; @Doi].

Defining entanglement clusters as chains which will be considered connected if they happen to pass through nearest neighbor sites, we set out to study the scaling behaviour in the vicinity of the entanglement percolation threshold in three dimensions. We have demonstrated, via a modified MC Renormalization Group, that for realistic concentrations of initiatiors, entangled clusters of linear chains have the same fractal dimension as ordinary percolation, although the entanglement percolation threshold is much lower than for standard bond percolation on a cubic lattice.

[**Acknowledgements**]{}

We would like to thank Nihat Berker for useful comments. One of us would like to acknowledge partial support from the Turkish Academy of Sciences.

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