#  In-Context Learning and Fine-Tuning GPT for Argument Mining

Jérémie Cabessa1 Hugo Hernault2 and Umer Mushtaq3
1DAVID Lab  UVSQ – University Paris-Saclay  78035 Versailles. France
2Playtika Ltd  CH-1003 Lausanne  Switzerland
3L3i  University of La Rochelle  17042 La Rochelle  France

###### Abstract

Large Language Models (LLMs) have become ubiquitous in NLP and deep learning.
In-Context Learning (ICL) has been suggested as a bridging paradigm between
the training-free and fine-tuning LLMs settings. In ICL, an LLM is conditioned
to solve tasks by means of a few solved demonstration examples included as
prompt. Argument Mining (AM) aims to extract the complex argumentative
structure of a text, and Argument Type Classification (ATC) is an essential
sub-task of AM. We introduce an ICL strategy for ATC combining $k$NN-based
examples selection and majority vote ensembling. In the training-free ICL
setting, we show that GPT-4 is able to leverage relevant information from only
a few demonstration examples and achieve very competitive classification
accuracy on ATC. We further set up a fine-tuning strategy incorporating well-
crafted structural features given directly in textual form. In this setting,
GPT-3.5 achieves state-of-the-art performance on ATC. Overall, these results
emphasize the emergent ability of LLMs to grasp global discursive flow in raw
text in both off-the-shelf and fine-tuned setups.

##  1 Introduction

Large Language Models (LLMs) have become ubiquitous in Deep Learning and have
shown impressive capabilities in most NLP tasks [1]. In the main, LLMs are
used in two distinct settings: (i) training-free, where the pre-trained LLM is
used for inference without any parameter adjustment, and (ii) fine-tuning,
where the parameters of the LLM are updated through supervised training to
enable transfer learning on a downstream task.

Recently, In-Context Learning (ICL) has been suggested as a bridging paradigm
between the training-free and fine-tuning settings. ICL is a prompt
engineering technique whereby an LLM is conditioned to solve tasks by means of
a few solved demonstration examples included as part of its input prompt [2].
Generally, the input prompt includes task instructions, the current input
sample to be solved as well as several solved input-output pair examples. In
this way, ICL maintains the training-free posture (parameters frozen) of the
LLM while at the same time providing it with some supervision through
demonstration examples. It also enables direct incorporation of selected
features inside the prompt template, thereby obviating the need for
architecture customization. Creative ICL strategies combining $k$NN-based
examples selection, generated chain-of-thought (CoT) prompting, and majority
vote ensembling have been proposed and shown to outperform fine-tuning
approaches [3, 4, 5, 6]. In the main, $k$NN-based examples selection optimizes
the process of learning from few examples and ensembling increases the
robustness of the predictions [7, 3, 5].

Argument Mining (AM) focuses on extracting the underlying argumentative and
discursive structure from raw text [8]. Argument Type Classification (ATC),
which involves classifying argumentative units in text according to their
argumentative roles, is the crucial sub-task in AM. Research has shown that
the argumentative role of a unit cannot be inferred solely for its text:
additional structural and contextual information is needed [9]. This
additional information can be incorporated via feature engineering [9],
memory-enabled neural architectures [10, 11] or LLM-based hybrid methods [12,
13].

We introduce an ICL strategy for ATC combining $k$NN-based examples selection
and majority vote ensembling. We experimented with various prompting templates
in order to reveal the proper contribution of different contextual elements.
In this training-free setting, we show that GPT-4 [14] is able to leverage
relevant information from only a few demonstration examples in order to
achieve very competitive classification accuracy. We further set up a fine-
tuning strategy incorporating well-crafted structural features input directly
in textual form. In this setting, GPT-3.5 achieves state-of-the-art
performance on the ATC task. Overall, these results emphasize the emergent
ability of LLMs to grasp global discursive flow in raw text in both off-the-
shelf and fine-tuned setups [15]. The code is available on GitHub here.

##  2 Methodology

###  2.1 Dataset

We consider the Persuasive Essays (PE) dataset which consists of 402
structured essays on various topics [9]. The train and test sets are composed
of 322 and 80 essays, respectively. The statistics of the PE dataset are given
in Table 1. The argument type classification (ATC) task consists of predicting
the type of each argument component (AC): ‘Major Claim’, ‘Claim’ or ‘Premise’.

Corpus Statistics | Component Statistics
---|---
Tokens | 147,271 | Major Claims | 751
Sentence | 7,116 | Claims | 1,506
Paragraphs | 1,833 | Premises | 3,832
Essays | 402 | Total | 6,089
Table 1: PE dataset statistics.

###  2.2 In-Context Learning (ICL)

In-context learning (ICL) refers to the emergent ability of LLMs to solve a
task based on a few demonstration examples given as contextual information
[2]. As the major advantage, the ICL paradigm does not require any further
fine-tuning of the model’s parameters (i.e. training-free framework).
Formally, let $x$ be a query input text, $Y=\\{y_{1},\dots,y_{k}\\}$ be a set
of candidate answers, and
$C=\left[I;t(x_{1},y_{i_{1}});\dots;t(x_{k},y_{i_{k}})\right]$ be a context
composed of instructions $I$ concatenated with text representations of example
input-output pairs $(x_{j},y_{i_{j}})$. Then, the LLM $\mathcal{M}$ predicts
the answer $\hat{y}$ such that $\hat{y}=\arg\max_{y_{i}\in
Y}P_{\mathcal{M}}(y_{i}\mid C;x)\,,$ where $P_{\mathcal{M}}(y_{i}\mid C;x)$ is
the probability that $\mathcal{M}$ generates $y_{i}$ when $C$ and $x$ are
given as prompt. The main rationale behind ICL is that the consideration of a
well-chosen context $C$ increases the probability of $\mathcal{M}$ predicting
the correct answer $y$ for input $x$, i.e. $P_{\mathcal{M}}(y\mid
C;x)>P_{\mathcal{M}}(y\mid x)$.

We consider a 2-step ICL strategy for argument type classification (ATC)
inspired by a recent study [3] (see Figure 1). More precisely, let $E$ be an
essay containing argument components (ACs) $c_{1},\dots,c_{m}$ with their
corresponding true classes $y_{1},\dots,y_{m}$, where each
$y_{i}\in\\{\text{Claim},\text{Major Claim},\text{Premise}\\}$. The LLM
generates class predictions $\hat{y}_{1},\dots,\hat{y}_{m}$ for
$c_{1},\dots,c_{m}$ as follows:

  1. (1)

$k$NN-based examples selection ($k=3,5$): First, $N$ neighboring essays
$E_{1},\dots,E_{N}$ of $E$ are selected according to some similarity measure.
The selection of $E_{1},\dots,E_{N}$ involves three options: (i) random
selection ($k$RN), (ii) selection based on essays having the same (or the
closest) number of ACs as $E$ ($k$NN${}^{\text{len}}$) and (iii) selection
based on essays whose title embeddings (with OpenAI’s text-embedding-ada-002)
are closest with respect to cosine similarity to the title embedding of $E$
($k$NN). Then, $k=\nicefrac{{N}}{{2}}$ essays, $E_{i_{1}},\dots,E_{i_{k}}$,
are randomly chosen from $E_{1},\dots,E_{N}$. Afterwards, a prompt containing
all the ACs and their corresponding classes in these $k$ essays is
constructed. Finally, the LLM predicts the classes
$\hat{y}_{1},\dots,\hat{y}_{m}$ of $c_{1},\dots,c_{m}$ based on this prompt.

  2. (2)

$n$-Ensembling ($n=3,5$): The $k$NN-based examples selection step, which
involves randomness, is repeated $n$ times ($n$Ens), leading to a set of $n$
class predictions sequences $\\{(\hat{y}_{i,1},\dots,\hat{y}_{i,m}):i=1,\dots
n\\}$. The final class predictions $\hat{y}_{1},\dots,\hat{y}_{m}$ of
$c_{1},\dots,c_{m}$ are obtained by applying a component wise majority vote to
the $n$ predictions sequences.

According to this ICL strategy, the classes $\hat{y}_{1},\dots,\hat{y}_{m}$
are predicted all-at-once (see Figure 1). A similar ICL strategy where the
classes $\hat{y}_{1},\dots,\hat{y}_{m}$ are inferred one-by-one (i.e. each
model inference leads to a single prediction $\hat{y}_{j}$) has been
considered, but shown to be significantly less efficient. Due to space
constraints, the latter results are omitted in this work.

###  2.3 Additional information and features

To help the LLM in its predictions, complementary information about the task
is generally added in the prompt. Here, the definitions of the classes (cf.
guidelines) together with their statistics in the train set can be included in
the prompt (info). Similarly, the whole text of the essay can be added in the
prompt (essay).

The text of an argument component alone does not suffice to predict its
argumentative class. In fact, the addition of contextual, structural and
syntactic features is indispensable for reaching at least satisfactory
prediction accuracy [9]. Based on this insight, an approach where contextual,
structural and syntactic features are given as text (instead of numerically)
to the LLM has been proposed [12, 13]. Here, structural features of the ACs
can be added in the prompt in the following textual form (fts): Is the AC
first in its paragraph: yes/no. Is the AC last in its paragraph: yes/no. Is
the AC in the introduction of the essay: yes/no. Is the AC in the conclusion
of the essay: yes/no.

Figure 1: 2-step ICL approach: a $k$NN-based example prediction ($k=3,5$) step
followed by an $n$-Ensembling ($n=3,5$) step (cf. text for further details).

###  2.4 Fine-tuning

Fine-tuning refers to the process of further training a pre-trained LLM on a
downstream task. We fine-tune GPT–3.5–turbo with and without injected textual
features (GPT–3.5–FeaTxt and GPT–3.5). As contextual features, we include the
title of the essay and the complete sentence in which the argument component
appears. As structural features, we include the paragraph number as well as
indications of whether the argument component is present in the
introduction/conclusion paragraph and whether it is the first and last
component in its paragraph, in the form given above.

##  3 Results

We used GPT–4 and GPT–3.5 on the ATC task with the 2-step ICL strategy
described in Section 2.2. We experimented with various prompting
configurations in order to assess the proper contribution of the different
contextual elements. The results are reported in Table 2.

Our 2-step ICL approach utilizing $5$NN and $5$-ensembling achieves a macro
F1-score of 83.6% (Line $5$). These results surpass those attained by the
fine-tuned BERT–FeaTxt and BERT–MINUS–FeaTxt models (see Table 3) [12, 13].
Therefore, our ICL approach is capable of capturing the global argumentative
flow of the essays through relevant contextual information and demonstration
examples, without requiring fine-tuning. Note that these results depend on the
complexity of the LLM, as GPT–3.5 with the very same prompting configuration
obtains markedly worse performance than GPT–4 (line $7$). This phenomenon
possibly points to an emergent ability acquired through model scaling.

In addition, lines $2$ and $4$ show that our two examples selection processes
($5$NN${}^{\text{len}}$ vs $5$NN) yield comparable results. Lines $1$ and $2$
emphasize the added value of the ensembling process, which increases the
robustness of the predictions. Lines $3$ and $5$ show the strong ability of
GPT to leverage additional information (class definitions and statistics). But
not all information is necessarily beneficial. In fact, although structural
features are known to be crucial for the ATC task [9], lines $5$ and $6$
reveal that adding them explicitly turns out to be counter-productive. Our
rationale is the the model is better able to infer these features from the raw
argumentative flow of the demonstration examples.

Prompt | MC | C | P | F1
---|---|---|---|---
(GPT–4) |  |  |  |
info + essay \+ fts \+ 5NN${}^{\text{len}}$ | 0.894 | 0.681 | 0.896 | 0.823
info + essay \+ fts \+ 5NN${}^{\text{len}}$ \+ 3Ens | 0.900 | 0.692 | 0.898 | 0.830
info + essay \+ fts \+ 5NN${}^{\text{len}}$ \+ 5Ens | 0.905 | 0.686 | 0.890 | 0.827
info + essay \+ fts \+ 5NN${}^{\text{len}}$ \+ 3Ens | 0.910 | 0.688 | 0.893 | 0.831
info + essay \+ fts \+ 5NN${}^{\text{len}}$ \+ 5Ens | 0.914 | 0.698 | 0.897 | 0.836
info + essay + fts + 5NN${}^{\text{len}}$ \+ 5Ens | 0.895 | 0.671 | 0.897 | 0.821
(GPT–3.5) |  |  |  |
info + essay + \+ fts \+ 5NN${}^{\text{len}}$ \+ 5Ens | 0.840 | 0.594 | 0.877 | 0.770
Table 2: Results for the ATC task using the 2-step ICL strategy.

We also fine-tuned GPT–3.5–turbo for $1$ and $2$ epochs, with and without
including the structural features described in previous works [12, 13]. The
results are reported in Table 3. In line with previous works [9], the addition
of structural features significantly improves GPT’s performance (lines
$7$-$9$). Most importantly, the fine-tuned GPT–3.5–FeaTxt achieves a macro
F1-score of $0.863$ which outperforms the SoTA ($0.857$) for the single ATC
task [11]. This shows that the fine-tuning approach outperforms the ICL
approach (see Tables 2 and 3). This suggests that, as opposed to simpler tasks
[3], ATC might not be solvable optimally without resorting to further task-
specific learning.

Model | MC | C | P | F1
---|---|---|---|---
Stab and Gurevych [9] | 0.891 | 0.682 | 0.903 | 0.826
Potash et al. [10] | 0.894 | 0.732 | 0.921 | 0.849
Kuribayashi et al. [11] |  |  |  | 0.856
BERT (4 ep.) [12] | 0.703 | 0.507 | 0.841 | 0.686
BERT–FeaTxt (4 ep.) [12] | 0.855 | 0.711 | 0.908 | 0.826
BERT–MINUS–FeaTxt (4 ep.) [13] | 0.900 | 0.687 | 0.903 | 0.831
GPT–3.5 (1 ep.) | 0.524 | 0.416 | 0.795 | 0.578
GPT–3.5–FeaTxt (1 ep.) | 0.905 | 0.745 | 0.915 | 0.855
GPT–3.5–FeaTxt (2 ep.) | 0.934 | 0.740 | 0.914 | 0.863
Table 3: Results of the ATC task achieved by previous non-LLM methods [9, 10,
11], by fine-tuned BERT-based models [13]) and by fine-tuned GPT–3.5 and
GPT–3.5–FeaTxt (ours).

##  4 Conclusion

We introduce a 2-step in-context learning (ICL) strategy for argument type
classification (ATC). Our strategy applied to GPT–4 implements training-free
learning by combining $k$NN-based examples selection and majority vote-based
ensembling mechanisms. To leverage the inherent transfer learning capabilities
of LLMs, we also fine-tune GPT–3.5 with and without selected well-crafted
features. Overall, our results confirm that the argument type of a component
can only be optimally captured through the discursive flow of the corpus text,
either given explicitly (ICL approach) or encoded into selected features
(fine-tuning approach). For future work, we plan to apply the ICL strategy to
all the Argument Mining sub-tasks so as to achieve an LLM-based, training-
free, end-to-end AM pipeline.

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# Flow Results and Hints of Incomplete Thermalization

###### Abstract

We classified $v_{2}$ measurements according to their sensitivities w.r.t. to
two planes, namely, reaction plane and participant plane. Likewise, in
$v_{2}/\epsilon$ scaling, we showed that one needs to choose a $\epsilon$ that
is sensitive to the same plane as that $v_{2}$ is sensitive to. We presented
our $v_{2}/\epsilon$ as a function of centrality and transverse momentum. We
studied the ratio of $v_{4}/v_{2}^{2}$. We discussed the applicable range for
hydrodynamics, as well as implications to an incomplete thermalization.

##  1 Introduction

In ultra-relativistic heavy ion collisions, spectators pass through each other
quickly, and the system begins its evolution with what is left behind - the
overlap region of two nuclei. The pressure gradient convert the spacial
anisotropy, quantified by eccentricity $\epsilon$, of the overlap region into
anisotropy in momentum space, quantified by azimuthal anisotropy $v_{2}$.
$v_{2}$ is defined as the second Fourier coefficient in the description of
particles distribution w.r.t. the reaction plane [1], and it is largely
determined by the collective motion in the in-plane direction. $v_{2}$ at
Relativistic Heavy Ion Collider (RHIC) is reported to be large and, for the
first time in heavy ion collisions, can be described by ideal hydrodynamics
[2]. Theoretical calculation shows that in order to explain the large $v_{2}$
observed at RHIC, one has to assume that the shear viscosity is extremely
small [3]. That is one of the important reasons for which scientists think
that a perfect liquid has been formed in relativistic heavy ion collisions
[4].

Since the announcement of the discovery of a perfect liquid, our understanding
of the matter created at RHIC has continued to advance. Elliptic flow analyses
have been extended to great details by all four experiments at RHIC. At the
same time, due to different techniques used, it becomes an increasingly amount
of work to understand/compare results across different experiments and,
sometimes, even different analyses within the same experiment. Therefore it
becomes important to understand what is the relation between various $v_{2}$
measurements. For the physics side, in order to quantify how perfect the
liquid is, it is necessary to re-examine the hydrodynamics limit. In this
paper, we try to study the hydrodynamic behavior under a more general context,
namely, the transport approach which recovers hydrodynamics when mean free
path is extremely small if compared to the system size [5]

##  2 Choosing the right $v_{2}$ and $\epsilon$ pairs

The ratio of $v_{2}/\epsilon$ reflects how well the initial anisotropy is
converted into momentum anisotropy [6]. This conversion process is directly
affected by dynamics of the system, e.g. Equation Of State, thermalization
etc. It is important to measure this quantity as accurate as possible.
However, there exist many $v_{2}$ measurements, for example, $v_{2}$ measured
by event plane method ($v_{2}\mathrm{\\{EP\\}})$, by cumulants
(${v_{2}\\{2\\}}$,${v_{2}\\{4\\}}$), by Lee-Yang zero method
($v_{2}\mathrm{\\{LYZ\\}}$), and by using event plane reconstructed with
Shower Maximum Detectors at Zero Degree Calorimeters
($v_{2}\mathrm{\\{ZDCSMD\\}}$), etc. Likewise there exist many $\epsilon$
calculations. In this section, we will try to make connections between various
$v_{2}$ and $\epsilon$ methods, and make the justification for the right
combination of them.

Define

$\mbox{\boldmath$\varepsilon$}=\\{\varepsilon_{x},\varepsilon_{y}\\}=\left\\{\left<{\frac{\sigma_{y}^{2}-\sigma_{x}^{2}}{\sigma_{x}^{2}+\sigma_{y}^{2}}}\right>_{part},\left<{\frac{2\sigma_{xy}}{\sigma_{x}^{2}+\sigma_{y}^{2}}}\right>_{part}\right\\},$
(1)

where $\sigma_{x}^{2}=\left<{x^{2}}\right>-\left<{x}\right>^{2}$,
$\sigma_{y}^{2}=\left<{y^{2}}\right>-\left<{y}\right>^{2}$, and
$\sigma_{xy}^{2}=\left<{xy}\right>-\left<{y}\right>\left<{x}\right>$, and the
average is taken over the coordinates of the participants in a given event.
With this definition, $\varepsilon_{x}$ is the eccentricity of reaction plane
(defined by the impact parameter), and $\varepsilon_{x}$ is also called
$\varepsilon_{RP}$. $\varepsilon_{y}$ has a distribution centered at zero with
finite width. The participant eccentricity measures the asymmetry in the
participant plane (defined by the principle axis of the ellipsoid), and is
given by
$\varepsilon_{part}=\sqrt{\varepsilon_{x}^{2}+\varepsilon_{y}^{2}}\equiv\varepsilon_{PP}$

When both $\varepsilon_{x}$ and $\varepsilon_{y}$ has a Gaussian distribution,
to the first order this is true and it is supported by Glauber Monte Carlo
simulations, then the probability density function for $\varepsilon_{PP}$ is

$\frac{dn}{d\varepsilon_{part}}=\frac{\varepsilon_{part}}{\sigma_{\varepsilon}^{2}}I_{0}\left(\frac{\varepsilon_{part}\left<{\varepsilon_{RP}}\right>}{\sigma_{\varepsilon}^{2}}\right)\exp\left(-\frac{\varepsilon_{part}^{2}+\left<{\varepsilon_{RP}}\right>^{2}}{2\sigma_{\varepsilon}^{2}}\right)\equiv\mathrm{BG}(\varepsilon_{part};\left<{\varepsilon_{RP}}\right>,\sigma_{\varepsilon}),$
(2)

With this p.d.f., one can show that
$\varepsilon_{part}\\{4\\}=\left<{\varepsilon_{RP}}\right>$ [7]. Similarly,
under the assumption that $v_{2}$ is proportional to the initial system
eccentricity (this is not true over a broad centrality range, but for a fine
centrality bin, it still ensures a good Gaussian for $v_{2}$ and
$s_{2}(\equiv\left<{\mathrm{sin2}(\phi-\Psi_{RP})}\right>)$, which is what is
required for the derivation of BG formula shown in Eq.( 2) ), one can find
that ${v_{2}\\{4\\}}=\left<{v_{2}\\{RP\\}}\right>$ [7, 8]. That means that,
${v_{2}\\{4\\}}$ and $\varepsilon_{part}\\{4\\}$ are measurements sensitive to
the reaction plane, not the participant plane. Indeed, that explains the
reason that STAR’s $v_{2}\mathrm{\\{ZDCSMD\\}}$ agrees with ${v_{2}\\{4\\}}$,
as shown in the left panel of Fig.1. $v_{2}\mathrm{\\{ZDCSMD\\}}$ is measured
with the first order event plane reconstructed by spectator neutrons thus is
more sensitive to $v_{2}$ in the reaction plane (not participant plane).

Fig. 1.: Left: $v_{2}$ measurements as a function of centrality. This plot is
from [9]. Right: $v_{2}$ of Au+Au collisions at
$\sqrt{s_{NN}}=130\mathrm{GeV}$. This plot is made based on datapoints from
[10]

Because the p.d.f. of the magnitude of the flow vector, the $q-$distribution
[10], shares an almost identical formula as Eq. (2), the $v_{2}$ obtained from
fitting the $q-$distribution should be equivalent to ${v_{2}\\{4\\}}$, as
confirmed by experimental data in the right panel of Fig.1.

To summarize this section, we find that ${v_{2}\\{4\\}}$,
$v_{2}\mathrm{\\{ZDCSMD\\}}$ and $v_{2}\mathrm{\\{q-dist\\}}$ are sensitive to
anisotropy in the reaction plane, they should be scaled with the standard
eccentricity or 4-particle cumulant eccentricity. Other $v_{2}$ measurements
that are based on two particle correlations, that includes $v_{2}\\{2\\}$,
$v_{2}\mathrm{\\{EP\\}}$ and $v_{2}\mathrm{\\{scalarProduct\\}}$ [10], etc.,
should be scaled with participant eccentricity or the 2-particle cumulant
eccentricity.

##  3 Hints of Incomplete Thermalization

The large data set from run IV Au+Au collisions at
$\sqrt{s_{NN}}=200\mathrm{GeV}$ allows us to extend ${v_{2}\\{4\\}}$
measurement to large $p_{t}$ and in fine centrality bins (see Fig. 2). For the
reason stated in the previous section, we scale ${v_{2}\\{4\\}}$ by initial
standard eccentricities. The left plot of Fig. 2 shows ${v_{2}\\{4\\}}$ scaled
by the eccentricity from Color Glass Condensate (CGC) [11], and the right plot
shows ${v_{2}\\{4\\}}$ scaled by the Monte Carlo Glauber eccentricity for
wounded nucleons. As expected, for the CGC case, the magnitude of
$v_{2}/\epsilon$ is lower if compared to the ratio in which a Glauber
eccentricity is used. For both cases, we see the ratio rises from peripheral
events to central events, indicating that stronger flow has been developed in
central collisions. We also notice that the $p_{t}$ where $v_{2}$ reaches its
maximum increases from peripheral collisions to central collisions, which is
consistent with the expectation that the applicable range for hydrodynamics
extends to large pt in central collisions. Note that $v_{2}/\epsilon$ shows
sign of saturation in central collisions for the CGC case but not much for the
Glauber case. This is explained by [11] as the following: in very peripheral
collisions, due to little asymmetry in the saturation scales, the CGC
eccentricity approaches the same value as that in the Glauber model, but in
central collisions, CGC predicts a larger eccentricity than the Glauber model
when there is a large asymmetry in the local saturation scales of the
collisions partners, along a path in impact-parameter direction away from the
origin. Note that $v_{2}\\{2\\}$ is not suitable for this study because it is
more susceptible to nonflow at large $p_{t}$.

Fig. 2.: $v_{2}$ scaled by initial CGC eccentricity (left) and Glauber
eccentricity (right) as a function of $p_{t}$. This plot is from [12].

To understand how well hydrodynamics describes STAR’s $v_{2}$, we investigated
the behavior of $v_{2}$ under the contex of the transport model, which will be
reduced to hydrodynamics when the mean free path is much smaller than the
system size [5]. In such approach, the dependence of $v_{2}/\epsilon$ on
particle’s density in the transverse plane ($1/SdN/dy$) can be described by:

$\frac{v_{2}}{\epsilon}=\left[\frac{v_{2}}{\epsilon}\right]_{hydro}\frac{1}{1+K/K_{0}}=\left[\frac{v_{2}}{\epsilon}\right]_{hydro}\frac{1}{1+\left(\sigma\frac{c_{s}}{c}\frac{1}{4S}\frac{dN}{dy}\right)^{-1}\frac{1}{K_{0}}}$
(3)

Where $K$ is Knudsen number defined by the mean free path divided by the
system size(sometimes it is more convenient to use $K^{-1}$ which means number
of collisions a particle encounters before it escapes), and $K_{0}$ is a
constant can be determined through transport calculations. In our study, we
take $K_{0}=0.7$ following the suggestion of Ollitrault [11]. A factor of 4 in
front of $S$ is to take into account the different definition of $S$ between
STAR ($S=\pi\sqrt{\left<{x^{2}}\right>\left<{y^{2}}\right>}$) and [11].
(($S=4\pi\sqrt{\left<{x^{2}}\right>\left<{y^{2}}\right>}$).
$\left[\frac{v_{2}}{\epsilon}\right]_{hydro}$ and $\sigma$ are free parameters
that have to be determined from fitting the data. In this approach
hydrodynamic limit of $v_{2}/\epsilon$ can be never reached, but can be only
asymptotic approached.

Fig. 3.: $v_{2}\\{4\\}$ scaled by initial eccentricities, as a function of
centrality. Fitted hydrodynamic limits for $v_{2}/\epsilon$ are indicated by
horizontal lines. CGC eccentricity and overlap area S are from [11].

Fig. 3 shows ${v_{2}\\{4\\}}$ scaled by CGC initial eccentricity and Glauber
initial eccentricity for wounded nucleons, as a function of particle density
in the transverse plane, with fits to Eq. 3. The fitted hydrodynamic limits
for the ratio of $v_{2}/\epsilon$ are $0.23$ and $0.36$, for CGC case and
Glauber case, respectively. For the same reason mentioned above, we see that
the curve shows a hint of saturation for the CGC case, but not for the Glauber
case, due to the relatively larger $\epsilon$ for CGC case in central
collisions. It is interesting to see that, for central Au+Au collisions, the
ratio of $v_{2}/\epsilon$ is about $20-30\%$ away from hydrodynamic limits. It
means that, there is still significant room for flow to grow before the system
saturates at hydrodynamic limits.

From the simple observation that both $v_{2}$ and $v_{4}$ are proportional to
$K^{-1}$ for small $K^{-1}$, one expects that $v_{4}/v_{2}^{2}$ decreases with
$K^{-1}$, reaching a minimum when the hydrodynamical regime is reached. For
this reason the ratio of $v_{4}/v_{2}^{2}$ has been argued as a probe to test
the degree of thermalization. Fig. 4 shows STAR’s measurement of
$v_{4}/v_{2}^{2}$ as a function of transverse momentum. The major systematic
uncertainty in this measurement comes from $v_{2}$ [12]. In this analysis, the
induced systematic error from $v_{2}$ uncertainty is estimated by studying the
difference between ${v_{2}\\{4\\}}$ and $v_{2}$ measured with event plane
constructed by tracks from STAR’s Forward Time Projection Chamber (FTPC). This
advanced study reduces the previously-reported systematic error at QM06
conference by $40\%$ relatively. The dashed lines are ratio come out of
calculations by solving Boltzmann equations with Monte Carlo simulation, with
different Knudsen number $K$. When the Knudsen number is small, it recovers
the hydrodynamic limit as indicated the solid line. The plot shows that the
system exhibits, again, significant deviation from ideal hydrodynamic limit
($K<<1$), and the data is consistent with a incomplete thermalized system with
$K>0.5$.

Fig. 4.: $v_{4}\mathrm{\\{EP_{2}\\}}/v_{2}^{2}\\{4\\}$ as a function of
$p_{t}$. This plot is from [12] with an advanced estimation on systematics
represented by half box-brackets.

##  4 Conclusion

To summarize, we found that ${v_{2}\\{4\\}}$, $v_{2}\mathrm{\\{ZDCSMD\\}}$ and
$v_{2}\mathrm{\\{q-dist\\}}$ are all sensitive to azimuthal correlation w.r.t
the reaction plane, not the participant plane,thus they should be scaled by
eccentricities that are sensitive to reaction plane too. That includes
standard eccentricity and 4-particle cumulant eccentricity. For $v_{2}$
methods that are based on two particle correlations, they are sensitive to the
azimuthal correlation w.r.t the participant plane, and they need to be scaled
by the corresponding eccentricities that are sensitive to the participant
plane. That includes participant eccentricity or 2-particle cumulant
eccentricity. We found that from peripheral to central Au+Au collisions flow
increases, and the applicable range for hydrodynamics extends to larger
$p_{t}$. However, $v_{2}/\epsilon$ and $v_{4}/v_{2}^{2}$ shows significant
deviation from ideal hydrodynamic limit, when that limit is extracted from
fitting the data itself with a Boltzmann equation motivated formula. Our study
shows that although in general hydrodynamic does a good job in terms of
describing $v_{2}$ at RHIC, there are features that are not consistent with a
complete thermalization and they cannot be easily dismissed.

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