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]


[**Sensitivity of the transverse flow towards symmetry energy**]{}\
, [**Aman D. Sood$^b$**]{}, [**Rajeev K. Puri$^a$**]{}[^1][**and J. Aichelin $^b$**]{}\
$^a$[*Department of Physics, Panjab University, Chandigarh -160 014, India.\
*]{} $^b$[*SUBATECH, Laboratoire de Physique Subatomique et des Technologies Associées, Université de Nantes - IN2P3/CNRS - EMN 4 rue Alfred Kastler, F-44072 Nantes, France.\
*]{}

We study the sensitivity of transverse flow towards symmetry energy in the Fermi energy region as well as at high energies. We find that transverse flow is sensitive to symmetry energy as well as its density dependence in the Fermi energy region. We also show that the transverse flow can address the symmetry energy at densities about twice the saturation density, however it shows the insensitivity towards the symmetry energy at densities $\rho/\rho_{0}$ $>$ 2. The mechanism for the sensitivity of transverse flow towards symmetry energy as well as its density dependence is also discussed.

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

The heavy-ion collisions (HIC) are the only method presently available in the laboratory to produce large volume of excited nuclear matter. The production of such state is essential to investigate not only the gross characteristics of nuclear matter but also in exploring the explosion mechanism of supernova and cooling rate of neutron stars. After about three decades of extensive efforts in both nuclear experiments and theoretical calculations, equation of state (EOS) of isospin symmetric matter is well understood by experiments of collective flow [@sch] and subthreshold kaon production [@klak93; @kaon1]. Nowadays, the nuclear EOS of asymmetric nuclear matter has attracted a lot of attention. The EOS of asymmetric nuclear matter can be described approximately by $$E(\rho, \delta) = E_{0}(\rho,
\delta=0)+E_{\textrm{sym}}(\rho)\delta^{2}$$ where $\delta$ = $\frac{\rho_{n}-\rho_{p}}{\rho_{n}+\rho_{p}}$ is isospin asymmetry, E$_{0}$($\rho$, $\delta$) is the energy of pure symmetric nuclear matter and E$_{\textrm{sym}}$($\rho$) is the symmetry energy with E$_{\textrm{sym}}$($\rho$$_{0}$) = 32 MeV is the symmetry energy at normal nuclear matter density. The symmetry energy is E($\rho$,1) - E$_{0}$($\rho$,0), ie. the difference of the energy per nucleon between pure neutron matter and symmetric nuclear matter. The symmetry energy is important not only to the nuclear physics community as it sheds light on the structure of radioactive nuclei, reaction dynamics induced by rare isotopes but also to astrophysicists since it acts as a probe for understanding the evolution of massive stars and the supernova explosion [@kubis07]. The existing and upcoming radioactive ion beam (RIB) facilities led a way in understanding nuclear symmetry energy. Experimentally, symmetry energy is not a directly measurable quantity and has to be extracted from observables which are related to symmetry energy. Therefore, a crucial task is to find such observables which can shed light on symmetry energy. A large number of studies on the symmetry energy of nuclear matter have been done during the past decade [@zhang05; @qli06; @tsang09; @bali97; @baran98; @li02; @li04]. These studies reveal that in heavy-ion collisions induced by neutron-rich nuclei, the effect of nuclear symmetry energy can be studied via the pre equilibrium *n/p* ratio [@zhang05; @qli06; @tsang09], isospin fractionation [@bali97; @baran98], *n-p* differential transverse flow [@li02; @li04] and so on. These observables have their relative importance depending upon the region of density one wants to explore. For e.g. below saturation density (0.3 $\rho_{0} \leq
\rho \leq \rho_{0}$), observables like fragment yield, isoscaling parameter, isospin diffusion, double *n*-*p* ratio have been used to extract symmetry energy. On the other extreme, $\pi^{+}$/$\pi^{-}$ ratio, relative and differential collective flow between triton/He$^{3}$, *n*-*p* differential collective flow act as probe of symmetry energy at high densities. In the low density region, Shetty *et al.* [@shetty04] extracted the symmetry energy by comparing the isoscaling parameter from $^{40}$, $^{40}$+$^{58}$, $^{58}$ and $^{58}$, $^{58}$+$^{58}$, $^{58}$ reactions with dynamical Antisymmetrized Molecular Dynamics (AMD) calculations [@ono03] and found it to be of the form E$_{\textrm{sym}} = 31.6
(\frac{\rho}{\rho_{0}})^{\gamma}$, with $\gamma$ = 0.69. Famiano *et al*. [@fami06] studied the symmetry energy by comparing the experimental double neutron to proton ratio in $^{112}$+$^{112}$ and $^{124}$+$^{124}$ reactions Isospin-dependent Boltzmann-Uehling-Uhlenbeck (IBUU) calculations [@li97] and obtained the form E$_{sym} = 32
(\frac{\rho}{\rho_{0}})^{\gamma}$, $\gamma$ = 0.5. Recently Tsang *et al*. [@tsang09] compared the isospin diffusion and n-p double ratio for $^{124}$+$^{112}$ reaction with Isospin-dependent Quantum Molecular Dynamics (IQMD) calculations [@zhang08] and obtained a similar form of symmetry energy with $\gamma$ = 0.4-1.05. The situation is worse at higher densities. The results are model dependent and contradicting also. The FOPI collaboration at GSI studied $\pi^{+}$/$\pi^{-}$ ratio in $^{40}$+$^{40}$, $^{96}$+$^{96}$, $^{96}$+$^{96}$, and $^{197}$+$^{197}$ reactions [@reis07]. A comparison of this data [@xiao09] with IBUU calculations showed a softer form of the density dependence of symmetry energy, which is in contrast to those obtained from the low density studies where stiffer form of symmetry energy reproduced the data well. $\pi^{+}$/$\pi^{-}$ ratio was also compared with Improved Quantum Molecular Dynamics (ImQMD) calculations by Feng *et al*. [@feng10] and favored a stiffer form of symmetry energy and in contradiction with IBUU calculations. At densities higher than saturation density, collective flow has also been proposed as a novel mean to probe the high density behavior of symmetry energy [@li02]. In this paper, we aim to see the sensitivity of collective transverse in-plane flow towards symmetry energy and also to see the effect of different density dependence of symmetry energy on the same. The various forms of symmetry energy used in present study are: E$_{\textrm{sym}} \propto F_{1}(u)$, E$_{\textrm{sym}} \propto F_{2}(u)$, and E$_{\textrm{sym}} \propto
F_{3}(u)$, where *u* = $\frac{\rho}{\rho_{0}}$, F$_{1}(u)
\propto u$, F$_{2}(u) \propto u^{0.4}$, F$_{3}(u) \propto u^{2}$, and F$_{4}$ represents calculations without symmetry energy. The different density dependences of symmetry energy are shown in fig. 1. The various lines are explained in the caption of the fig. 1. Section 2 describes the model in brief. Section 3 explains the results and discussion and Sec. 4 summarizes the results.

-1cm ![(Color online) The density dependence of symmetry energy for various forms of symmetry energy : F$_{1}$ ($\gamma$ = 1.0) (solid), F$_{2}$ ($\gamma$ = 0.4) (dashed), and F$_{3}$ ($\gamma$ = 2.0) (dotted).[]{data-label="fig2"}](fig1.eps "fig:"){width="12cm"}

The model
=========

The present study is carried out within the framework of isospin-dependent quantum molecular dynamics (IQMD) model [@hart98]. The IQMD model treats different charge states of nucleons, deltas, and pions explicitly, as inherited from the Vlasov-Uehling-Uhlenbeck (VUU) model. The IQMD model has been used successfully for the analysis of a large number of observables from low to relativistic energies. The isospin degree of freedom enters into the calculations via symmetry potential, cross sections, and Coulomb interaction.

In this model, baryons are represented by Gaussian-shaped density distributions $$f_{i}(\vec{r},\vec{p},t) =
\frac{1}{\pi^{2}\hbar^{2}}\exp(-[\vec{r}-\vec{r_{i}}(t)]^{2}\frac{1}{2L})
\times \exp(-[\vec{p}- \vec{p_{i}}(t)]^{2}\frac{2L}{\hbar^{2}})$$ Nucleons are initialized in a sphere with radius R = 1.12 A$^{1/3}$ fm, in accordance with liquid-drop model. Each nucleon occupies a volume of *h$^{3}$*, so that phase space is uniformly filled. The initial momenta are randomly chosen between 0 and Fermi momentum ($\vec{p}$$_{F}$). The nucleons of the target and projectile interact by two- and three-body forces, potential,and interactions. In addition to the use of explicit charge states of all baryons and mesons, a symmetry potential between protons and neutrons corresponding to the Bethe-Weizsäcker mass formula has been included. The hadrons propagate using Hamilton equations of motion: $$\begin{aligned}
\frac{d\vec{{r_{i}}}}{dt} = \frac{d\langle H
\rangle}{d\vec{p_{i}}};& & \frac{d\vec{p_{i}}}{dt} = -
\frac{d\langle H \rangle}{d\vec{r_{i}}}\end{aligned}$$ with $$\begin{aligned}
\langle H\rangle& =&\langle T\rangle+\langle V \rangle
\nonumber\\
& =& \sum_{i}\frac{p^{2}_{i}}{2m_{i}} + \sum_{i}\sum_{j>i}\int
f_{i}(\vec{r},\vec{p},t)V^{\textrm{ij}}(\vec{r}~',\vec{r})
 \nonumber\\
& & \times f_{j}(\vec{r}~',\vec{p}~',t) d\vec{r}~ d\vec{r}~'~
d\vec{p}~ d\vec{p}~'.\end{aligned}$$ The baryon potential V$^{\textrm{ij}}$, in the above relation, reads as $$\begin{aligned}
  \nonumber V^{\textrm{ij}}(\vec{r}~'-\vec{r})& =&V^{\textrm{ij}}_{\textrm{Skyrme}} + V^{\textrm{ij}}_{\textrm{Yukawa}} +
  V^{\textrm{ij}}_{\textrm{Coul}} + V^{\textrm{ij}}_{\textrm{sym}}
    \nonumber\\
   & =& [t_{1}\delta(\vec{r}~'-\vec{r})+t_{2}\delta(\vec{r}~'-\vec{r})\rho^{\gamma-1}(\frac{\vec{r}~'+\vec{r}}{2})]
   \nonumber\\
   &  & +t_{3}\frac{\exp(|(\vec{r}~'-\vec{r})|/\mu)}{(|(\vec{r}~'-\vec{r})|/\mu)}+
    \frac{Z_{i}Z_{j}e^{2}}{|(\vec{r}~'-\vec{r})|}
   \nonumber \\
      &  & +t_{4}\frac{1}{\varrho_{0}}T_{\textrm{3i}}T_{\textrm{3j}}\delta(\vec{r_{i}}~'-\vec{r_{j}}).
 \end{aligned}$$ Here *Z$_{i}$* and *Z$_{j}$* denote the charges of *ith* and *jth* baryon, and *T$_{3i}$* and *T$_{3j}$* are their respective *T$_{3}$* components (i.e., $1/2$ for protons and $-1/2$ for neutrons). The parameters *$\mu$* and *t$_{1}$,...,t$_{4}$* are adjusted to the real part of the nucleonic optical potential. For the density dependence of the nucleon optical potential, standard type parametrization is employed.

![ (Color online) Time evolution of root mean square radius of a single $^{40}$, $^{48}$ and $^{60}$ nuclei in coordinate (top panel) and momentum space (bottom) obtained with IQMD for EOS used in the present study for symmetry energy F$_{1}$(u).[]{data-label="fig1"}](fig2.eps "fig:"){width="8cm"} -0cm

0.5cm ![(Color online) Left panel: $<\frac{p_{x}}{A}>$ as a function of Y$_{\textrm{c.m.}}/Y_{\textrm{beam}}$ for different energies of 100, 400, and 800 MeV/nucleon for different forms of symmetry energy. Right panel: The time evolution of $<p_{x}^{\textrm{dir}}>$ at 100, 400, and 800 MeV/nucleon for different forms of symmetry energy at b/b$_{\textrm{max}}$=0.2-0.4. Lines are explained in the text.[]{data-label="fig4"}](fig3.eps "fig:"){width="12cm"} 0.5cm

-1cm ![(Color online) The time evolution of rapidity distribution for the calculations with no symmetry energy for various bins at b/b$_{\textrm{max}}$=0.2-0.4. Lines are explained in the text.[]{data-label="fig2"}](fig4.eps "fig:"){width="12cm"}

Results and discussion
======================

We simulate several thousands events for the neutron-rich system of $^{48}\textrm{Ca}$+$^{48}\textrm{Ca}$ and $^{60}\textrm{Ca}$+$^{60}\textrm{Ca}$ at energies of 100, 400, and 800 MeV/nucleon at impact parameter of b/b$_{\textrm{max}}$=0.2-0.4. We use a soft equation of state and isospin- and energy-dependent cross section reduced by 20$\%$, i.e. $\sigma$ = 0.8 $\sigma_{nn}^{\textrm{free}}$. The details about the elastic and inelastic cross sections for proton-proton and proton-neutron collisions can be found in [@hart98; @cug]. The cross section for neutron-neutron collisions is assumed to be equal to the proton-proton cross section.

Since $^{60}$ has a very high asymmetry, so to ensure the stability of the nuclei in the present study, we display in fig. 2 the time evolution of root mean square radius of single nucleus of $^{40}$ (solid line), $^{48}$ (dashed), and $^{60}$ (dotted) in the coordinate (fig. 2(a)) and momentum space (fig. 2(b)). The results are displayed for nuclei intialized with symmetry energy F$_{1}$(u). We find that the stability is of same order for all the three nuclei. There are several methods used in the literature to define the nuclear transverse in-plane flow. In most of the studies, one uses ($p_{x}/A$) plots where one plots ($p_{x}/A$) as a function of $Y_{\textrm{c.m.}}/Y_{\textrm{beam}}$. Using a linear fit to the slope, one can define the so-called reduced flow (F). Alternatively, one can also use a more integrated quantity “directed transverse in-plane flow $\langle{p_{x}^{\textrm{dir}}}\rangle$” which is defined as [@hart98]: $$\langle p_{x}^{\textrm{dir}}\rangle~=~\frac{1}{A}\sum_i {\rm
sign}\{Y(i)\}~p_{x}(i),$$ where $Y(i)$ and $p_{x}(i)$ are the rapidity distribution and transverse momentum of the $ith$ particle. In this definition, all rapidity bins are taken into account. It, therefore, presents an easier way of measuring the in-plane flow rather than complicated functions such as ($p_{x}/A$) plots.

In fig. 3 we display the $<\frac{p_{x}}{A}>$ as a function of Y$_{\textrm{c.m.}}/Y_{\textrm{beam}}$ at final time (left panels) and time evolution of $<p_{x}^{\textrm{dir}}>$ (right panels) calculated at 100 (top panel), 400 (middle) and 800 MeV/nucleon (bottom) for different density dependence of symmetry energy. Solid, dash dotted, and dotted lines represent the symmetry energy proportional to $\rho$, $\rho^{0.4}$ and $\rho^{2}$ whereas dashed line represents calculations without symmetry energy. Comparing left and right panels in fig. 3, we find that both the methods show similar behavior to symmetry energy. For eg. at incident energy of 100 MeV/nucleon for E$_{\textrm{sym}} \propto
\rho^{0.4}$, $<p_{x}^{\textrm{dir}}>$ = 0. Similarly, the slope of $<\frac{p_{x}}{A}>$ at midrapidity is zero. We also find that the transverse momentum is sensitive to symmetry energy as well as its density dependence F$_{1}(u)$, F$_{2}(u)$ and F$_{3}(u)$ in the low energy region (100 MeV/nucleon). At energies above Fermi energy, both the methods show insensitivity to the different symmetry energies. This is because the repulsive nn scattering dominates the mean field at high energies.

To understand the sensitivity of transverse momentum to the symmetry energy as well as its density dependence in the Fermi energy region, we calculate the transverse flow as well as rapidity distribution of particles having $\frac{\rho}{\rho_{0}}$ $<$ 1 (denoted as bin 1) and particles having $\frac{\rho}{\rho_{0}}$ $\geq$ 1 (bin 2) separately at all the time steps. Since both the methods show similar behavior to symmetry energy, so for simplicity the following discussions will be in terms of $<p_{x}^{\textrm{dir}}>$.

In fig. 4 we display the rapidity distributions at 100 MeV/nucleon of all the particles (dotted line), particles corresponding to bin 1 (solid) and to bin 2 (dashed) at 0, 10, 20, 30, 40, and 60 fm/c. We have calculated rapidity distributions for different forms of symmetry energy used in this paper. We find that it is insensitive towards the symmetry energy [@indra; @sanjeev]. In fig. 4, we display the rapidity distribution calculated without symmetry energy. During the initial stages we see the two Gaussians at projectile and target rapidities for all the 3 bins. The peaks of Gaussians will be more prominent at higher energies. The interest for our discussion is in bin 1 and bin 2. During the start of the reaction (0 fm/c) more number of particles lie in bin 1 i.e. more number of particles have $\frac{\rho}{\rho_{\textrm{0}}}$ $<$ 1. As the nuclei begin to overlap, the density increases in the overlap zone. Now, the number of particles increases in bin 2 (at 10 fm/c). From 10 fm/c to 20 fm/c, the number of particles keep on increasing in bin 2 at midrapidity, i.e particles from large rapidity keep on shifting to the bin 2 in the midrapidity region. This is expected since at incident energies in the Fermi energy region dynamics is governed by the attractive mean field. The dominance of attractive mean field will prompt the deflection of particles into negative angles i.e. towards participant zone. After 20-30 fm/c, the expansion phase of the reaction begins and number of particles keep on increasing in bin 1 and by 60 fm/c most of the particles lie in bin 1.

-1cm ![(Color online) The time evolution of $<p_{x}^{\textrm{dir}}>$ for different forms of symmetry energy for different bins at b/b$_{\textrm{max}}$=0.2-0.4 . Lines have same meaning as in fig. 4.[]{data-label="fig2"}](fig5.eps "fig:"){width="12cm"}

In fig. 5 we display the time evolution of $<p_{x}^{\textrm{dir}}>$ for different symmetry energies used in this paper at 100 MeV/nucleon for particles lying in the different bins. Lines have the same meaning as in fig. 4. Panels a, b, and c are for E$_{\textrm{sym}} \propto \rho, \rho^{0.4}$, and $\rho^{2}$, respectively. Panel d is for calculations without symmetry energy. The total $<p_{x}^{\textrm{dir}}>$ is negative during the initial stages and keep on decreasing till 30 fm/c which indicates dominance of attractive interaction. In panels a and b, it becomes positive whereas in panel c and d it remains negative during the course of the reaction. If we look at $<p_{x}^{\textrm{dir}}>$ of particles lying in bin 1 for F$_{1}
(u)$ (fig. 5a) and F$_{2} (u)$ (fig. 5b) in the time interval 0 to about 20-25 fm/c, we see that it remains positive. It increases with time upto 15 fm/c and reaches a peak value. This is because in the spectator region (where high rapidity particles lies) the repulsive symmetry energy will accelerate the particles away from the overlap zone in the transverse direction. After 15 fm/c, $<p_{x}^{\textrm{dir}}>$ (of particles in bin 1) begins to decrease. This is because these particles will now be attracted towards the central dense zone. As shown in fig. 4, from 10 to 20 fm/c particles in bin 2 keep on increasing in the midrapidity region. In case of F$_{1} (u)$ and F$_{2} (u)$, particles which enter the central dense zone (bin 2) have already a high positive value of $<p_{x}^{\textrm{dir}}>$ (i.e. going away from dense zone). So attractive mean field have to decelerate the particles first, make them stop and then accelerate the particles back towards the overlap zone. At about 20-25 fm/c particles from bin 1 have zero $<p_{x}^{\textrm{dir}}>$ (see shaded area in fig. 5a and 5b). Up to 30 fm/c, particles feel the attractive mean field potential after which the high density phase is over, i.e. in case of F$_{1} (u)$ and F$_{2} (u)$ between 0-30 fm/c particles from bin 1 are accelerated towards the overlap zone only for a short time interval of about 5 fm/c, whereas for the case of F$_{3} (u)$ (fig. 5c) and F$_{4}$ (fig. 5d) between 0-30 fm/c, particles from bin 1 are accelerated towards the overlap zone for a longer time interval of about 20 fm/c between 10-30 fm/c. Moreover, the $<p_{x}^{\textrm{dir}}>$ of particles lying in the bin 1 (for F$_{3} (u)$ and F$_{4}$) follows a similar trend. This is because, for $\rho/\rho_{0} < $ 1 the strength of symmetry energy F$_{3}
(u)$ will be small and so there will be less effect of symmetry energy on the particles which is evident from fig. 5c where one sees that the $<p_{x}^{\textrm{dir}}>$ remains about zero during the initial stages between zero to about 10 fm/c.

0.5cm ![(Color online) (a) The time evolution of $<p_{x}^{\textrm{dir}}>$ at 100 MeV/nucleon for different forms of symmetry energy for $^{48}$+$^{48}$. (b) The time evolution of $<\rho_{\textrm{max}}/\rho_{\textrm{0}}>$ and (c) $<\rho_{\textrm{avg}}/\rho_{\textrm{0}}>$ for F$_{1} (u)$ and F$_{4}$. Lines have same meaning as in fig. 3.[]{data-label="fig4"}](fig6.eps "fig:"){width="10cm"} 0.5cm

The $<p_{x}^{\textrm{dir}}>$ due to particles in bin 2 (dashed line) decreases in a very similar manner for all the four different symmetry energies between 0-10 fm/c. Between 10-25 fm/c, $<p_{x}^{\textrm{dir}}>$ for F$_{3} (u)$ and F$_{4}$ decreases more sharply as compared to in case of F$_{1} (u)$ and F$_{2} (u)$. This is because in this time interval particles from bin 1 enters into bin 2. As discussed earlier, $<p_{x}^{\textrm{dir}}>$ of particles entering bin 2 from bin 1 in case of F$_{1} (u)$ and F$_{2} (u)$ will be less negative due to stronger effect of symmetry energy as compared to in case of F$_{3} (u)$ and F$_{4}$.

Since the reaction $^{60}$+$^{60}$ is an extreme case with large isospin asymmetry. To check if the above predicted effects survive in reactions which are experimentally accessible, we simulate the reaction of $^{48}$+$^{48}$ for all the different forms of symmetry energy used in the present study. The reaction $^{48}$+$^{48}$ has been used in many previous studies [@wang09]. The results are shown in fig. 6(a). We find that even for this reaction the transverse flow shows sensitivity to symmetry energy as well as its density dependence. In fig. 6(b) and 6(c), we display the time evolution of maximum density $<\rho_{\textrm{max}}/\rho_{0}>$ and average density $<\rho_{\textrm{avg}}/\rho_{0}>$, respectively at 100 MeV/nucleon. We find that the density reached is about 2.0 times the saturation density. Moreover, the maximal density is reached in the time interval 0-30 fm/c and the effect of symmetry energy on $<p_{x}^{\textrm{dir}}>$ of particles during this interval decides the fate of the final value of $<p_{x}^{\textrm{dir}}>$. Thus transverse flow can address the symmetry energy at densities about 2.0 times than saturation density.

Summary
=======

We have checked the sensitivity of transverse flow towards symmetry energy in the Fermi energy as well as at high energies. We have found that transverse flow is sensitive to symmetry energy as well as its density dependence in the Fermi energy region. We have also shown that the transverse flow can address the symmetry energy at densities about twice the saturation density, however it shows the insensitivity towards the symmetry energy at densities $\rho/\rho_{0}$ $>$ 2. We have also discussed the mechanism for the sensitivity of transverse flow towards symmetry energy as well as its density dependence.

This work has been supported by a grant from Centre of Scientific and Industrial Research (CSIR), Govt. of India and Indo-French Centre For The Promotion Of Advanced Research (IFCPAR) under project no. 4104-1.

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