Social Behaviour of Agents: Capital Markets and Their Small Perturbations

O. Hudak

Faculty of Finance, Matej Bel University,

Cesta na amfiteater 1, SK-974 15 Banska Bystrica, Slovak Republic,

e-mail<EMAIL_ADDRESS>
## Abstract

We study social behaviour of agents on capital markets when these are
perturbed by small perturbations. We use the mean field method. Social
behaviour of agents on capital markets is described: volatility of the market,
aversion constant and equilibrium states are discussed. Relaxation behaviour
of agents on the capital market is studied. Equation of motion for the agent
average number is of the relaxation type. Development of the group of agents
in the states corresponding to minimum of the aim function is either linear
either exponentially damped. There exist characteristic volatility constants
$V_{c3}$ and $V_{c3}$. The constant b of verification of information
contribution to the aversion constant A and the $A_{0}$ constant of aversion
are distinguishing three types of dependencies of the minimum of the aim
function on the expected volatility EV and on the expected returns E.
Arbitrage trades and group forces lead the group into the equilibrium state.
Verification of information intensity influences return back to the
equilibrium state. The linear in time damping to the equilibrium state is
characterized with the characteristic time $T_{3}$ and $T_{6}$, the
exponential with a characteristic time $\tau$. Their dependence on the
expected volatility, on the expected profit and characteristics of agents is
discussed.

##  1 Introduction

The group of agents on the capital market is characterized as a social group,
[1] \- [6]. We study information which is private [7], there is present an
information asymmetry. Unverifiability is a special kind of information
problem [7]. Then there exist subgroups of interacting agents on the capital
market the interaction of which has the following property: they are
nonreducibile [5]. Nonreducibile subgroups form cells. Configuration of
interactions between agents in the group forms a macroscopic structure. The
statistical equilibrium due to microreversibility is characterized by the
maximum of entropy and by the minimum of energy (costs of information exchange
minus a return). In [5] we used methods of statistical physics to study social
behaviour of agents. Three observed dependencies of personal radius of an
agent on other characteristics enabled us to characterize the quantities of
cells, faces (nonreducibile subgroups), vertices (agents) and bonds
(interactions) [2]. There exist constrains in this structure, a fixed number V
of agents in the group present, a number E of interactions within the group is
constant, a number F of subgroups which are nonreducibile (faces) is constant,
and a number C of cells is constant for a given conditions as concerning the
expected return and expected risk. This structure is equivalent to random
cellular networks. Such networks and their evolution were studied in [3] and
[4]. Random cellular networks describe structure in the group of agents in
some approximation, this approximation is expected to be sufficient to
described main characteristics of the group as concerning the exchange of the
information. The capital market and social behaviour of agents was studied
recently, [6]. The quantity which characterizes verification of information
contributes to aversion of agent with respect to a risk. The mix of
investments of an agent in a given cell with an average measure A of risk
aversion in the cell is found from extremes of the function FM, the aim
function. We have found the minimum of the average (per cell) function $<FM>$.
An absolute minimum corresponds to such a state in which there is an optimal
mix. The number of nonreducible subgroups F per cell $<f>$ contributes
linearly to aversion constant A, $(A=A_{0}+(<f>-4).b)$. The larger quantity
which characterizes verification of information, the larger expected profit is
and the larger accepted expected risk is. We have formulated model describing
behaviour of agents on the capital market [6]. A pyramidal (hierarchical)
structure appears in the group for $<f>=4$. In this case the aversion constant
is $A=A_{0}$, which corresponds to hierarchical economy systems. Better
verification of information leads to expectation of higher returns and thus
the acceptable risk is larger. A total averse constant A is increasing with
verification of information and market-based economy system is present in
larger extend. We have found that there exists co-existence of both economy
systems in real economy. In our model the market-based system corresponds to a
state with $<n>=6$ and the hierarchy-based to $<n>=3$. We assumed that in our
model the individual investors and agents from financial intermediary firms do
not differ.

The aim of this paper is to study social behaviour of agents on capital
markets when these are perturbed. We study small perturbations of the market
using mean field method. Social behaviour of agents on capital markets is
described firstly. Model and results of its study, which were done recently by
the author [5] and [6] for this behaviour are shortly discussed for reference
purposes. Volatility of the market, aversion constant and equilibrium states
are discussed then in the next section. Relaxation behaviour of agents on the
capital market in the fourth section is studied. Equation of motion for the
agent average number is of the relaxational type, and is described in the next
section. Development of the group of agents in the states corresponding to
minimum of the aim function is linear and exponentially damped for the case
when the extreme of the aim function (which is found from the first derivative
of the aim function equal to zero) is minimum and when it is maximum. This
development is studied in the sixth section. Conclusions are in the last
section.

The main results of the paper are the following. There exist characteristic
volatility constants $V_{c3}$ and $V_{c3}$. The constant b of verification of
information contribution to the risk constant A and the $A_{0}$ constant of
aversion are distinguishing three types of dependencies of the minimum of the
aim function on the expected volatility EV and on the expected return E. These
are $A_{0}>4b$, $2b<A_{0}<4b$ and $A_{0}<2b$. The minimum in $3<<n><6$. in
$<n>=3$ and in $<n>=6$ is found. We assumed in this paper that profits are
positive. The law of one price is violated by arbitrage opportunities.
Arbitrage trades and group forces lead the group into the equilibrium state.
The other groups interacting with the group of agents on the capital market
are not studied here directly. This group consists of nonreducibile subgroups
[6]. Nonreducibile subgroups are also groups with their properties. Real
markets contain many agents acting on them. We will assume that there is a
large number of agents in the group of agents on the capital market. In the
group of agents on the capital market there are formal groups and informal
groups allianced in response to the need for social contact concerning the
capital market. Thus interactions of agents described by topology are
interactions of agents in formal and informal groups. On the capital market
such groups as its subgroups are forming. We mentioned nonreducibile subgroups
above, other subgroups are cells which belong to formal type groups and cells
which belong to informal type groups. Thus there exist different time periods
during which such subgroups exist due to different nature of these subgroups.
There are different time periods characteristic for the group of agents on the
capital market due to different time to perform different tasks. We assume
that processes in the group of agents on capital market are of the relaxation
type. There are such reasons for the induced fluctuation from the equilibrium
(or metastable) state as events in economy and on the market. We assume that
these events do not change the equilibrium (or metastable) state. Thus the
group of agents has social behaviour for which some of the events in economy
and on the market lead to a fluctuation around the minimum state, we assume
that this fluctuation is a perturbation of the market of the short time nature
at some time t = 0. The equation of motion for the agent average number $<n>$
per nonreducibile subgroup is found, and is of the relaxation type. We further
assume that the average number $<n>$ of agents per nonreducibile subgroup is
perturbed from the equilibrium state $<n>$. The fluctuation from this state is
$\delta<n>$. The perturbation $\delta<n>(t)$, which has at the time t = 0
value $\delta<n>_{0}$, is evolving to zero after some time. We study
characteristics of this evolution when the extreme of the aim function is
maximum and when this extreme is minimum. We then consider two limiting cases,
when the aversion constant $A_{0}$ is near to 2b from below, $A_{0}\rightarrow
2b$, and when the aversion constant $A_{0}$ tends to zero, $A_{0}\rightarrow
0$. In the first case we find the relaxation constant $\tau$ to the lowest
order in $(A_{0}-2b)$. When the expected volatility EV increases the
relaxation constant $\tau$ in this limit decreases. When the expected
volatility EV decreases the relaxation constant $\tau$ in this limit
increases. When the aversion constant $A_{0}$ tends to 2b, e.i. when it
increases in this limit, the relaxation constant $\tau$ decreases to zero in
the fourth order in $(A_{0}-2b)$ . The constant $(A_{0}-4b)$ is called in our
paper as the contribution to the total aversion constant from the crowd. It
increases when the aversion constant $A_{0}$ tends to 2b. Increasing the
aversion constant $A_{0}$ leads to larger acceptance of the crowd of expected
risk, and thus of expected return. Verification of information with its
decreasing intensity in this limit leads to slower return back to the
equilibrium state. The expected return in this limit and in the first term of
the expansion in small parameter is not present in the expression for the
relaxation constant $\tau$. We obtain also the relaxation constant $\tau$ to
the lowest order in $(A_{0}$ tending to zero. When the aversion constant
$A_{0}$ tends to 0, e.i. when it decreases, the relaxation constant $\tau$
remains finite. When the expected volatility EV increases the relaxation
constant $\tau$ in this limit increases. The state with the fluctuation at
time $t=0$ returns back to the minimum less quickly. When the expected
volatility EV decreases the relaxation constant $\tau$ in this limit
decreases. The state with the fluctuation at time $t=0$ returns back to the
minimum more quickly. The larger expected return, the smaller the relaxation
constant and the return to the equilibrium is again more quickly. The larger
expected volatility, the larger the relaxation constant and the return to the
equilibrium is more slowly. The smaller expected return, the larger the
relaxation constant and the return to the equilibrium is more slowly. The
larger the intensity b of verification of information the smaller the
relaxation constant $\tau$. The state with a fluctuation at time $t=0$ returns
back to the equilibrium state more quickly. The smaller the intensity b of
verification of information the larger the relaxation constant $\tau$. The
state with fluctuation returns back to the equilibrium state more slowly.
Decreasing the aversion constant $A_{0}$ leads to smaller acceptance of the
expected risk, and thus of the expected return. The equation of motion leads
in the second case of minimum to the linear in time damping to the equilibrium
state with the characteristic time $T_{3}$, and $T_{6}$. For $A_{0}>2b$ the
characteristic constant $V_{c6}<0$ is negative. The characteristic time
$T_{6}$ increases with decrease of risk EV, with decrease of profits, and with
decrease of the risk constant $A_{0}$ toward $4b$ . This increase is however
to a finite value. For $A_{0}<2b$ the characteristic constant $V_{c6}>0$ is
positive. The characteristic time $T_{6}$ increases with decrease of risk EV
to the $V_{c6}$ value or with increase of risk EV to the $V_{c6}$ value. The
increase of the characteristic time $T_{6}$ is now large. The characteristic
constant $V_{c3}$ is defined as $V_{c3}=2A_{0}E\frac{A_{0}-2b}{A_{0}-4b}$.
With decrease of risk EV toward $V_{c3}$ the characteristic time $T_{3}$
increases to large values.

##  2 Social Behaviour of Agents: Capital Markets

We know [5] that personal diameter r increases when F, the measure of
verification of information, decreases. We generalized this relation to every
cell: assuming that the personal diameter r increases when $<f>$, the average
number of nonreducIbile groups per cell, decreases. In the topological
structure $<f>$ is an average number of faces per cell. Thus generalized
quantity $<f>$ is also the quantity which characterizes verification of
information. Aversion of an agent to risk on capital markets contains [6] a
contribution proportional to the verification of information. We assume that
the number of nonreducible subgroups F per cell, $<f>$, contributes linearly
to this aversion constant A:

$A=A_{0}-4.b+<f>.b.$ (1)

$A_{0}$ is the risk aversion constant for $<f>=4$, then $A=A_{0}$. b is a
constant which characterizes contribution of the quantity $<f>$ to the risk
aversion, it is an intensity of contribution of verification of information to
the aversion constant. Note that the larger quantity $<f>$ the better
verification of information, the larger expected return contribution of the
investment and thus the larger acceptable risk. The pyramidal (hierarchical)
structure appears in the group when $<f>=4$, then $A=A_{0}$, [6]. Such a type
of structure corresponds to hierarchical economy systems [7]. Due to very low
risk price signals do not work in these systems. Information is flowing
through the hierarchy to the decision makers in such structure, however we do
not consider the direction of flow of interaction between agents in [5] and
[6]. When the structure contains topologically only one cell then $<n>=6$ and
A tends to infinity. Better verification of information leads to expectation
of higher returns and thus the acceptable expected risk is larger. Such a type
of structure corresponds to market-based economy systems [7]. Price signals
are essential in these systems. We interpreted $(<f>.b)$ as an average measure
of risk aversion of a personal (individual) contribution to the risk aversion.
It is different from the crowd contribution of a person in the group, because
$<f>$ is a characteristic quantity of verification of information. Agents
supply or demand goods and services (on the capital market shares or bonds) if
the market price of goods and services exceeds or undervalues his/her own
valuation of goods and services (shares or bonds). To make own valuation of
shares or bonds the agent individually verifies information about them. In
practice, [7], both types of economy system involve interaction of people: in
the market system they interact as traders, in the hierarchy system as agents
within a hierarchy organization. Both market- and hierarchy- based systems co-
exist in modern economy [7].

In market economies intermediation process is understood quite well in a
hypothetical market economy by considering how it might function without
financial intermediaries [8]. In our model of capital markets [6] every point
(vertex) corresponds to an agent acting on the market. In the simplified model
economy described in [8] these agents would be households. Financial
intermediaries which lead to lower costs were described in our model [6] as
cell with a given risk aversion constant of the firm to which risk aversion
constants of agents from this firm - cell should not be too far. Some other
cells are cells with zero risk tolerance constant, agents in these cells
correspond to individual savers - agents. We assume that every agent has its
objective to maximize its expected utility of wealth [9]. If returns are
normally distributed and the investor has constant absolute risk aversion,
then expected utility can be written in an exponential form [10]. Let us
denote by A the aversion constant which when increasing will prefer E, the
expected value of end-of-period wealth. EV, the expected variance of end-of-
period wealth has then increasing acceptable value. The expected utility of an
agent is found as large as possible by the maximization procedure, [11],
assuming that the risk aversion is constant. In our model [6] we assume that
every agent has the same E (the expected value of end-of-period wealth) and
the same EV (the expected variance of end-of-period wealth). We considered in
[6] not only maximization of expected utility of an agent, but also
optimization of interactions of agents in such a way in which interacting
agents maximize their expected utility. Optimization of the corresponding aim
function for given E and EV leads to optimization of the number of agents in
the cell and to optimization of verification of information which is
characterized by $<f>$. Then structure of exchange of information in the group
of agents will be such that we obtain the highest value of the aim function
fM, [6].

The mix of investment of a single cell in the mean field approximation is
found from the function:

$<FM>=(-c.E_{B}+\frac{c^{2}}{2}EV_{B})\frac{<f>.<n>}{3}+\gamma_{B}c_{B}^{2}\frac{<f>.<n>}{3},$
(2)

where $E_{B}=E+2\gamma c_{B}$ is expected return for a given investment period
modified by contribution of the return from the cell (firm) financial
intermediary, $EV_{B}=EV+2\gamma$ is expected risk for a given investment
period modified by contribution of the risk from the cell (firm) financial
intermediary, E and EV is expected return and risk respectively for a given
investment period. Here $c_{B}$ is a risk constant given for the cell. Note
that $c=\frac{1}{A}$. We study the case $\gamma=0$, which does not distinguish
between individual investors (for which $\gamma=0$ ) and a financial
intermediary firm (for which $\gamma>0$) .

Thus the equation for the aim function FM is found from (2):

$<FM>=(-c.E+\frac{c^{2}}{2}EV)\frac{<f>.<n>}{3},$ (3)

here $\frac{<f>.<n>}{3}$ is an average number of agents per cell. From (1) and
(3) we find:

$<FM>=(-\frac{E}{A_{0}-4.b+<f>.b}+\frac{EV}{2.(A_{0}-4.b+<f>.b)^{2}})\frac{<f>.<n>}{3}$
(4)

For an equilibrium structure with a given number of cells C, of faces F, of
interactions E, and of agents V, see [3] and [4], the relation holds:

$<f>=\frac{12}{(6-<n>)},$ (5)

We use $A_{1}=A_{0}-4.b$. From the aim function FM (4) we have found number of
cell C, of faces F and of interactions E which minimizes it, in the mean field
approximation. The number of agents V is given. It means that there is a
minimization of FM with respect to $<n>$, number of interaction in an
nonreducibile subgroup e.i. with respect to $<f>$. We have found [6] such a
structure of agents on the capital market which gives the lowest value of the
aim function FM taking into account their interaction exchanging information.

The function FM has extremes. An absolute minimum corresponds to such a state
in which there is an optimal mix, there is an equilibrium state of the market.
There exists an extreme of FM, $<n>^{\prime}$, which is given by [6]:

$<n>^{\prime}=6.\frac{(EV(A_{0}-2b)-2E(A_{0}-2b)^{2})}{(EV.A_{0}-2E(A_{0}-4b)(A_{0}-2b))}.$
(6)

The extreme may be an absolute minimum for $A_{0}<2b$. This is the case when
the tendency to verify information is stronger with respect to the aversion
constant $A_{0}$. The extreme (6) is minimum if the inequalities hold:

$2E(2b-A_{0})>EV>E\frac{A_{0}}{b}(2b-A_{0})$ (7)

Here the state with $<n>=6$ has lower value of the aim function than the state
$<n>=3$.

The extreme is also minimum if the inequality:

$E\frac{A_{0}}{b}(2b-A_{0})>EV\geq 0,$ (8)

holds. Here the state with $<n>=3$ has now lower value of the aim function
than the state $<n>=6$.

In both these cases the extreme is a state in which some type of hierarchical
structures exists. Either locally, the group is not hierarchical as a whole,
in the first case, or it is of the hierarchical structure type in the second
case. The minimum gives the average number of agents per nonreducibie subgroup
which is less than 6 and larger than 3, in the first case, and is equal to 3
in the second case.

The extreme (6) is also minimum if:

$EV>2E(2b-A_{0}).$ (9)

Here the state with $<n>=6$ is minimum. In this case the minimum of the aim
function with $<n>=6$ is a state in which there is no type of hierarchical
structure present in the group.

The extreme may be an absolute maximum for $A_{0}>2b$. In this case the state
$<n>=6$ has always lower value of the aim function than the state with
$<n>=3$.

The extreme (6) is maximum if, for $A_{0}>4b$, the inequality holds:

$EV>2E\frac{A_{0}(A_{0}-2b)}{A_{0}-4b}$ (10)

There exist two states which minimize the aim function FM. The first one is
with $<n>=3$ (hierarchical) structure.

The second one is for $<n>=6$. The value of the aim function FM is such that
this state is much more stable than the state above with $<n>=3$, which is a
metastable state. The number $<n>=6$ means that there is a one cell structure.
Efficient capital markets are ideal markets which are of the this type (strong
stability, conservative and aggressive agents are present on the market,
return is high) in our model.

The extreme (6) is maximum if, for $A_{0}>4b$, inequalities hold:

$2E\frac{A_{0}(A_{0}-2b)}{A_{0}-4b}>EV>2E\frac{(A_{0}-4b)(A_{0}-2b)}{A_{0}}$
(11)

There exists one state which minimizes function FM for $3\leq<n>\leq 6$. The
state with $<n>=3$, which was a metastable state, is now not the metastable
state. The minimum of the aim function FM is for $<n>=6$.

The extreme (6) is maximum if, for $4b>A_{0}>2b$, the inequality holds:

$EV<2E\frac{A_{0}(A_{0}-2b)}{A_{0}-4b}$ (12)

There exist two states which minimize function FM. The first one is with
$<n>=3$ and the second one is with $<n>=6$. This later state has the value of
the aim function FM such that this state is much more stable than the state
corresponding to $<n>=3$ which is a metastable state.

The extreme (6) is maximum if, for $4b>A_{0}>2b$, the inequality holds:

$EV>2E\frac{A_{0}(A_{0}-2b)}{A_{0}-4b}$ (13)

The state with $<n>=6$ is now a state corresponding to the minimum of the aim
function. There is no metastable state.

##  3 Volatility of the Market, Aversion Constant and Equilibrium State

Let us discuss the relations between the volatility of the market, the
aversion constant and the equilibrium state. When the constant b of
verification of information contribution to the risk constant A is such that
$A_{0}>4b$, than for the volatility EV of the price return on the capital
market higher that the characteristic constant $V_{c3}$, e.i. for $EV>V_{c3}$,
there is an extreme as maximum present for $<n>$. The state $<n>=3$, has
always higher value of the aim function than the state $<n>=6$. The
characteristic constant $V_{c3}$ is defined as
$V_{c3}=\frac{-A_{0}V_{c6}}{-A_{1}}$. Here the characteristic constant
$V_{c6}$ is defined as $V_{c6}=-2E(A_{0}-2b)$. It depends on the expected
return E. The characteristic constant $V_{c3}$ can be rewritten as
$V_{c3}=\frac{2A_{0}E(A_{0}-2b)}{-A_{1}}=\frac{2A_{0}E(A_{0}-2b)}{-(A_{0}-4b)}$.
For $A_{0}>4b$ the characteristic constant $V_{c6}<0$. For $0\leq EV\leq
V_{c3}$ there is a maximum at $<n>=3$. The $<n>=6$ state is the minimum state.

When the constant b of verification of information contribution to the risk
constant A is such that $2b<A_{0}<4b$, than for the volatility EV, which is
always positive $EV>0$ (only in the economy with fixed prices the expected
volatility is zero $EV=0$ ) the state $<n>=6$ is always the minimum state. The
state $<n>=3$ is a maximum. The characteristic constant $V_{c3}$ and the
characteristic constant $V_{c6}$ are negative in this case.

When the constant b of verification of information contribution to the risk
constant A is such that $0\leq A_{0}\leq 2b$, then we find that two
characteristic constants $V_{c3}$ and $V_{c6}$ obey the inequalities $0\leq
V_{c3}\leq V_{c6}$. We must distinguish then several cases. The $0\leq
V_{c3}\leq V_{c6}\leq EV$ case is such that the state $<n>=6$ is the minimum
state. The state $<n>=3$ is the maximum state. The $0\leq V_{c3}\leq EV\leq
V_{c6}$ case is such that the state $3<<n><6$ is the minimum state, it
minimizes the aim function. The state $<n>=3$ is the maximum state, and also
the state $<n>=6$ is the maximum state. Depending on whether $EV\leq V_{cm}$
or whether $V_{cm}\leq EV$ the state $<n>=6$ is higher than the state $<n>=3$,
and the state $<n>=3$ would be higher than the state $<n>=6$, respectively.
Here $V_{cm}$ is an constant defined as $V_{cm}=\frac{A_{0}V_{c6}}{2b}$. The
$0\leq EV\leq V_{c3}\leq V_{c6}$ case is such that the state $<n>=3$ is
minimum, the state $<n>=6$ is maximum.

We assumed in this paper and in our previous work [6] that profits are
positive. The case of negative profits does not correspond to a capital
markets on which there is proportional relation between the risk and the
positive profit. Here we consider longer term relations.

##  4 Relaxation Behaviour of Agents on the Capital Market

Relaxation behaviour is observed in many social groups. Let us consider
arbitrage processes for the group of agents on the capital market. Arbitrage
is a way to earn guaranteed profits without investing money [7]. The law of
one price says that the same good cannot sell at two different prices on the
market. Arbitrage opportunities violate this law. If two assets have identical
payoffs and different prices on the market then this fluctuation of the market
will not sustained as an equilibrium state [12]. A profit without risk will
not be earned for long time: the buyers and sellers will drive the prices
together by trading. Group norms create a force toward behavioural
regularities [13]. Thus this driving price process is a behavioural
regularity. Group cohesiveness creates a force toward behavioural uniformity
[13]. Thus arbitrage trades and mentioned forces lead the group into the
equilibrium state. We will consider the relaxation process, which after
perturbing the group of agents on the market for example by an arbitrage
opportunity to the market state with fluctuation, will lead to the equilibrium
state.

One of the organizational behaviour characteristics and aims is productivity
[13]. Productivity is a performance measure including effectiveness and
efficiency. Effectiveness of the organizational behaviour is achievements of
goals. Efficiency of the organizational behaviour is the ratio of effective
output to the input required to achieve it. An organization is productive when
it achieves its goals and does so by transferring inputs into outputs with the
lowest cost. A group of agents on the capital market is a part of an
organization. We consider here agents interacting together exchanging
information for given conditions on the capital market. The behaviour of
agents in the general group is more than the sum of individual actings in his
or her own way. The agents in the group behave in an other way than when they
are alone. To understand dynamics of behaviour of the group of agents one
should know how an individual, an agent in the group, is influenced by the
pattern behaviour he/she is expected to exhibit, what the group considers to
be acceptable standard of behaviour, and the degree to which group members are
attracted to each other. Organizational system behaviour reaches its highest
level of sophistication, [13], when we add formal structure to knowledge of
individual and group behaviour. Organization is more that the sum of their
member groups. We study here behaviour of the group of agents on the capital
market as mentioned above. The other groups interacting with this group are
not studied here directly. This group consists of nonreducibile subgroups [6].
nonreducibile subgroups are also groups with their properties. A group is in
general formed by two or more individuals, interacting and interdependent, who
have come together to achieve some goals. The group of agents on the capital
market may have two members, a buyer and a seller. This would be a very
elementary case of a market. Real markets contain many agents acting on them.
We will assume that there is a large number of agents in the group of agents
on the capital market. A group may be formal or informal, [13]. Formal groups
are defined by the organizational structure. Informal groups are alliances
that are neither formally structured nor organizationally determined. These
groups are naturally created groups in the environment that appear in response
to the need for social contacts [13]. In the group of agents on the capital
market there are formal groups (financial institutions, firms, other
institutions) and informal groups allianced in response to the need for social
contact concerning the capital market (agents as individual investors, agents
as members of financial institutions, agents as members of firms, agents as
members of other institutions, all of them forming also informal groups).
There exist in general such groups as command, task, interest, or friendship
groups. The last type of the group develops because the individual members
have some common characteristics. Social aliances, which may extend outside
the work situation, can be based on such common characteristics as is interest
about development on the capital market, about development of the price of a
share, about development of a firm projects, etc. Thus interactions of agents
described by topology in our papers are interactions of agents in formal and
informal groups. Agents belong to a number of groups in general. Different
groups provide different benefits to their members. Agents are joining into
groups due to such reasons as needs for security, status, self-esteem,
affiliation, power and goal achievement [13]. This is true in general and this
is true also for agents on the capital market. There is no standardized way
how a group develops. The Five-Stage Model and the Punctuated-Equilibrium
Model are some models describing this development, [13]. In the group of
agents on the capital market such groups as its subgroups are forming. We
mentioned nonreducibile subgroups above, other subgroups are cells which
belong to formal type groups and cells which belong to informal type groups.
Thus there exist different time periods during which such subgroups exist due
different nature of these subgroups. Some of the groups may be more successful
some of them less. It depends on several variables. Major components that
determine a group performance and satisfaction are, [13], external conditions
imposed on the group, group member resources, group structure, the group
process-performance/satisfaction relationship is dependent on the task which
the group is working on. These group properties and characteristics hold also
for subgroups in the group of agents on capital market. Within a group there
are processes such as the communication pattern used by its members to
exchange information, group decision processes, leader behaviour, power
dynamics, conflict interactions etc [13]. The task uncertainty has the
property that nonroutine tasks require considerably more processing of
information. Tasks with low uncertainty tend to be standardized. The groups
performing first type of the task need to interact with other groups to obtain
more and better information. The groups performing the tasks of the second
type do not need to interact with other groups so much. Performing task means
that after some time this task will be fulfilled.

As we can see there are different time periods characteristic for the group of
agents on the capital market due to different time to perform different tasks.
While there may be present some oscillation type processes of exchange of
information between groups, relaxation of this interaction is expected when
the task is fulfilled. Exchange of information within a group may be more
oscillatory than for the intergroup case, nevertheless when the task is
fulfilled, these interactions will also relax to zero. There are many
subgroups in the group of agents on the capital market ending their task in
different times. We will assume that processes in the group of agents on
capital market are of the relaxation type.

##  5 Equation of Motion for the Agent Average Number

The social group of buyers and sellers is such that its state corresponds to
the absolute minimum of the aim function. The social group of buyers and
sellers may be in such a state which corresponds to a metastable minimum of
the aim function. Due to many reasons the group may shift from the state
corresponding to a minimu of the aim function to a very near state. There are
such reasons for the induced fluctuation from the equilibrium (or metastable)
state as events in economy and on the market. Arbitrage process is of this
type. We will assume that these events do not change the equilibrium (or
metastable) state. Thus the group of agents has social behaviour for which
some of the events in economy and on the market lead to a fluctuation around
the minimum state, we assume that this fluctuation is a perturbation of the
market of the short time nature at some time t = 0. We would like to know how
the social behaviour of agents return back to the equilibrium (or metastable)
minimum state. Thus type of events which may lead to change of the minimum
state to other minimum state is not considered in this paper. Also large
amplitude perturbations which after some time vanishes are not considered
here.

The equation of motion for the agent average number $<n>$ per nonreducibile
subgroup is in this case assumed, as we discussed, of the relaxation type:

$\frac{\partial}{\partial
t}\delta<n>=-\lambda\frac{\delta<FM>(<n>+\delta<n>)}{\delta\delta<n>}$ (14)

Here we assume that the average number $<n>$ of agents per nonreducibile
subgroup is perturbed from the equilibrium state $<n>$. The fluctuation from
this state is $\delta<n>$. Time t is in general in the interval $0\leq
t\leq\infty$. The perturbation $\delta<n>(t)$, which has at the time t = 0
value $\delta<n>_{0}$, is evolving to zero after some time. We will study
characteristics of this evolution when the extreme of the aim function is
maximum and when this extreme is minimum. In the first case we have found the
minimum at $<n>=3$ and at $<n>=6$. In the second case the minimum is such that
$3<<n><6$, other minimum is at $<n>=3$, or at $<n>=6$

##  6 Development of the Group of Agents: The Extreme is Minimum

When the extreme (6) is minimum, then the state $3<<n><6$ or the state with
$<n>=3$ which has lower value of the aim function than the state $<n>=6$ or
the state with $<n>=6$ which has lower value of the aim function than the
state $<n>=3$ , are equilibrium states. The equation of motion leads in the
first case to the exponential damping of the fluctuation to the equilibrium
state with the relaxation time $\tau$:

$\delta<n>=\delta<n>_{0}\exp(-\frac{t}{\tau})$ (15)

here $\delta<n>_{0}$ is a fluctuation of the state at time zero , and where:

$\tau=\frac{144(EVb)^{2}(A_{0}-2b)}{\lambda(EV.A_{0}-2E(A_{0}-2b)(A_{0}-4b))^{2}}.S$
(16)

and where the quantity S is defined as:

$S=-(EV+2E(A_{0}-4b))+\frac{(A_{0}-4b)^{2}}{192EV.b^{2}(A_{0}-2b)^{2}}[96b(A_{0}-2b)(-EV^{2}+2E^{2}(A_{0}-2b)^{2})$
(17)

$+(EV.A_{0}-2E(A_{0}-2b)(A_{0}-4b))^{2}]+\frac{E(A_{0}^{2}-(4b)^{2})}{2b}$

Let us consider two limiting cases, when the aversion constant $A_{0}$, is
near to 2b from below, $A_{0}\rightarrow 2b$, and when the aversion constant
$A_{0}$ tends to zero, $A_{0}\rightarrow 0$.

In the first case we obtain that the relaxation constant $\tau$ is, to the
lowest order in $(A_{0}-2b)$, given by:

$\tau=\frac{486b^{2}(A_{0}-2b)^{4}}{\lambda.EV.A_{0}^{4}}$ (18)

When the expected volatility EV increases the relaxation constant $\tau$ in
this limit decreases. The state with the fluctuation at time $t=0$ returns
back to the minimum more quickly. When the expected volatility EV decreases
the relaxation constant $\tau$ in this limit increases. The state with the
fluctuation at time $t=0$ returns back to the minimum less quickly. The
expected volatility EV influences how quickly the state with fluctuation
returns back to the equilibrium state. When the aversion constant $A_{0}$
tends to 2b, e.i. when it increases in this limit, the relaxation constant
$\tau$ decreases to zero in the fourth order in $(A_{0}-2b)$ . The constant
$(A_{0}-4b)$ is called in our paper as the contribution to the total aversion
constant from the crowd. It increases when the aversion constant $A_{0}$ tends
to 2b. The state with fluctuation decreases more quickly to the equilibrium
state. The ability of the crowd to remain self-aware and think logically is
suppressed [1]. Increasing the aversion constant $A_{0}$ leads to larger
acceptance of the crowd of expected risk, and thus of expected return. When
the intensity of personal contribution to the aversion constant, b, decreases
in this limit,then the relaxation constant $\tau$ firstly increases, and then
for $b<\frac{A_{0}}{6}$ decreases. This later inequality is however such that
the expansion in small parameter $A_{0}-2b\propto\frac{2A_{0}}{3}$ is not
valid in general. Thus verification of information with its decreasing
intensity in this limit leads slower return back to the equilibrium state.
Note that the expected return in this limit and in the first term of the
expansion in small parameter is not present in the expression for the
relaxation constant $\tau$.

In the second case we obtain that the relaxation constant $\tau$ is, to the
lowest order in $A_{0}$, given by:

$\tau=\frac{27.EV^{3}}{32\lambda E^{4}b^{2}}$ (19)

When the expected volatility EV increases now the relaxation constant $\tau$
in this limit increases. The state with the fluctuation at time $t=0$ returns
back to the minimum less quickly. When the expected volatility EV decreases
the relaxation constant $\tau$ in this limit decreases. The state with the
fluctuation at time $t=0$ returns back to the minimum more quickly. The
expected volatility EV influences how quickly the state with fluctuation
returns back to the equilibrium state also in this case. When the aversion
constant $A_{0}$ tends to 0, e.i. when it decreases, the relaxation constant
$\tau$ remains finite. The constant $(A_{0}-4b)$,the contribution to the total
aversion constant from the crowd, becomes larger in amplitude and negative.
The crowd behaves more irrationally. As we mentioned above the smaller
expected volatility, the smaller the relaxation constant and the return to the
equilibrium is more quickly. And the larger expected return, the smaller the
relaxation constant and the return to the equilibrium is again more quickly.
The larger expected volatility, the larger the relaxation constant and the
return to the equilibrium is more slowly. The smaller expected return, the
larger the relaxation constant and the return to the equilibrium is more
slowly. The ability of the crowd to remain self-aware and think logically is
suppressed [1] as we mentioned. The dominance of the crowd s belief system
limits severe quality of data which crowd recognizes as information [1]. The
larger the intensity b of verification of information the smaller the
relaxation constant $\tau$. The state with a fluctuation at time $t=0$ returns
back to the equilibrium state more quickly. The smaller the intensity b of
verification of information the larger the relaxation constant $\tau$. The
state with fluctuation returns back to the equilibrium state more slowly.
Decreasing the aversion constant $A_{0}$ leads to smaller acceptance of the
expected risk, and thus of the expected return.

The equation of motion leads in the second case of minimum to the linear in
time damping to the equilibrium state with: the characteristic time $T_{3}$:

$\delta<n>=\delta<n>_{0}(1-\frac{t}{T_{3}})$ (20)

where:

$T_{3}=\frac{\mid\delta<n>_{0}\mid 3A_{0}^{2}}{\lambda\mid(EV-
V_{c3})(A_{0}-4b)\mid}$ (21)

Here $\delta<n>_{0}$ is again a fluctuation of the state at time zero.

When the state $<n>=6$ is an equilibrium state, the equation of motion lead in
this case to the linear in time damping with the characteristic time $T_{6}$:

$\delta<n>=\delta<n>_{0}(1-\frac{t}{T_{6}})$ (22)

where:

$T_{6}=\frac{\mid\delta<n>_{0}\mid 12\delta^{2}}{\lambda\mid V_{c6}-EV\mid}$
(23)

Discussion of dependencies of characteristic times $T_{3}$ and $T_{6}$ are
formally the same as in the case when the extreme is maximum, see below.

##  7 Development of the Group of Agents: The Extreme is Maximum

When the extreme (6) is maximum, then the state $<<n>=6$ or the state with
$<n>=3$ are the states corresponding to minimum of the aim function. In this
case the state $<n>=6$ has always lower value of the aim function than the
state with $<n>=3$. Thus the state $<n>=6$ is an equilibrium state, the state
with $<n>=3$ is a metastable state. As discussed above the state $<n>=6$ has
always lower value of the aim function than the state with $<n>=3$. In the
case discussed above the state $<n>=6$ is an equilibrium state, and the state
with $<n>=3$ is not a metastable state, it is a maximum.

The equation of motion lead in the first case to the linear in time damping
with the characteristic time $T_{6}$:

$\delta<n>=\delta<n>_{0}(1-\frac{t}{T_{6}})$ (24)

where:

$T_{6}=\frac{\mid\delta<n>_{0}\mid 12\delta^{2}}{\lambda\mid V_{c6}-EV\mid}$
(25)

Here $\delta<n>_{0}$ is again a fluctuation of the state at time zero. For
$A_{0}>2b$ the characteristic constant $V_{c6}<0$ is negative. The
characteristic time $T_{6}$ increases with decrease of risk EV, with decrease
of profits, and with decrease of the risk constant $A_{0}$ toward $4b$ . This
increase is however to a finite value. For $A_{0}<2b$ the characteristic
constant $V_{c6}>0$ is positive. The characteristic time $T_{6}$ increases
with decrease of risk EV to the $V_{c6}$ value or with increase of risk EV to
the $V_{c6}$ value. The increase of the characteristic time $T_{6}$ is now
large.

The equation of motion leads in case of the metastable state case $<n>=3$ also
to the linear in time damping to the locally minimum state with the
characteristic time $T_{3}$:

$\delta<n>=\delta<n>_{0}(1-\frac{t}{T_{3}})$ (26)

where:

$T_{3}=\frac{\mid\delta<n>_{0}\mid 3A_{0}^{2}}{\lambda\mid(EV-
V_{c3})(A_{0}-4b)\mid}$ (27)

Here $\delta<n>_{0}$ is again a fluctuation of the state at time zero. The
characteristic constant $V_{c3}=2A_{0}E\frac{A_{0}-2b}{A_{0}-4b}$. With
decrease of risk EV toward $V_{c3}$ the characteristic time $T_{3}$ increases
to large values,

The state $<n>=6$ has always lower value of the aim function than the state
with $<n>=3$. In the case discussed above when the state $<n>=6$ is an
equilibrium state, and the state with $<n>=3$ is not a metastable state, the
characteristic time $T_{6}$ discussion above is the same. There is no linear
in time damping to the state $<n>=3$, it is a maximum.

##  8 Conclusions

We studied capital market and social behaviour of agents. Quantity F, and in
general $<f>$, is the quantity which characterizes verification of information
and thus contributes to aversion of an agent with respect to a risk. We
generalized this relation to every cell: we assumed that personal diameter r
increases when $<f>$, the average number of nonreducibile groups per cell,
decreases, $<f>$ is an average number of faces per cell. Aversion of an agent
to risk on capital markets also contains characterization of verification of
information. We assumed that the number of nonreducible subgroups F per cell,
$<f>$, contributes linearly to this aversion constant A.

When the pyramidal structure appears in the group, $<f>=4$, then $A=A_{0}$.
This type of structure corresponds to hierarchical economy systems. When the
structure contains topologically only one cell then $<n>=6$ and A tends to
infinity. Better verification of information leads to expectation of higher
returns and thus the acceptable risk is larger. This type of structure
corresponds to market-based economy systems. They work via price signals. We
interpret $A_{0}-4.b$ as a kind of measure of risk aversion of crowd and we
interpret $(<f>.b)$ as an average measure of risk aversion of a personal
/individual/ contribution to risk aversion different from the crowd
contribution of this person in the cell.

In practice both types of system involve interaction of people: in the market
system they interact as traders, in the hierarchy as agents within an
organization. Both market- and hierarchy- based systems co-exist in modern
economy: we have open markets and we also have large organizations. This
coresponds to our state which is minimum for $3<<n><6$. In our model the
market-based system corresponds to a state with $<n>=6$ and the hierarchy-
based to $<n>=3$.

The mix of investments of an agent in a given cell with an average measure A
of risk aversion in the cell is found from extremes of the function FM, the
aim function. We find the minimum of the average /per cell/ function $<FM>$.
We assume that in our model the individual investors and agents from financial
intermediary firms do not differ. More general case will studied elsewhere.

When $<f>=4$ then the pyramidal (hierarchical) structure exists in which there
are only nonreducibie subgroups with $n=3$. Thus we consider this structure as
a structure with a uniform risk aversion constant $A_{0}$. Then a personal
contributions to the risk aversion constant of an agent are those
contributions which are corresponding to structures with $<n>>3$. Thus there
is a linear contribution to the risk aversion constant which is proportional
to $<f>-4$. This corresponds to nonuniform risk aversion constants in the
structure. This contains a personal contribution to the risk aversion
constant. We use $\frac{<f>.<n>}{3}$ as an average number of agents per cell.

The extreme of $<FM>$ is minimum for some conditions on the market, and in the
group of agents, see above. This minimum corresponds to $<n>$ between 3 to 6.
This is a stable state. The extreme may be a maximum. Then there are two
minima of the function FM, one of them is that which corresponds to $<n>=3$
(the pyramidal hierarchical structure) which is a metastable state, and the
other one corresponds to $<n>=6$ (the structure with hexagons in average and
with one cell) which is a stable state. Efficient capital markets are markets
which are of the second type (strong stability, conservative and aggressive
agents are present on the market, returns are high). Aboav’s law describes how
many, $m(f,n)$, agents are present in average in nonreducibie subgroups in a
cell neighbouring to a cell with f nonredducibile subgroups with agent average
number n. This enables us to characterize variance of n for the distribution
of nonreducibile subgroups. This variance diverges to infinity with $<n>$
tending to 6.

The financial intermediaries play a key role to reduce the risks faced by
individual savers, by pooling their savings and using these to assemble
diversified portfolios of assets. This diversification requires specialized
knowledge and expertise, and a large portfolio. This will studied using our
model with nonzero $\gamma$ constant coresponding to a firm with a given
tolerance risk constant.

The aim of this paper was to study social behaviour of agents on capital
markets when these are perturbed. We studied small perturbations of the market
using mean field method. Social behaviour of agents on capital markets was
described firstly. Model and results of its study, which were done recently by
the author [5] and [6], for this behaviour were shortly discussed for
reference purposes. Volatility of the market, aversion constant and
equilibrium states are discussed then in the next section. Relaxation
behaviour of agents on the capital market in the fourth section is studied.
Equation of motion for the agent average number is of the relaxational type,
and is described in the next section. The oscillation type of motion is not
considered here. Development of the group of agents in the states
corresponding to minimum of the aim function is linear and exponentially
damped for the case when the the extreme of the aim function is minimum and is
maximum. This development was studied in the sixth section.

We conclude that there exist characteristic volatility constants $V_{c3}$ and
$V_{c3}$. The constant b of verification of information contribution to the
risk constant A and the $A_{0}$ constant of aversion are distinguishing three
types of dependencies of the minimum of the aim function on the expected
volatility EV and on the expected returns E. These are $A_{0}>4b$,
$2b<A_{0}<4b$ and $A_{0}<2b$. The minimum in $3<<n><6$. in $<n>=3$ and in
$<n>=6$ is found. Thus estimating the average values of the aversion constant
$A_{0}$ and of the constant b of the intensity of the verification of
information we should distinguish which type of dependence of the equilibrium
market state occurs. The other groups interacting with the group of agents on
the capital market are not studied here directly. Their inclusion would lead
to study of the intergroup interactions. On the capital market subgroups are
forming. We mentioned nonreducibile subgroups above, other subgroups are cells
which belong to formal type groups and cells which belong to informal type
groups. We assume that processes in the group of agents on capital market are
of the relaxation type. There are such reasons for the induced fluctuation
from the equilibrium (or metastable) state as events in economy and on the
market. We assume that these events do not change the equilibrium (or
metastable) state. Thus the group of agents has social behaviour for which
some of the events in economy and on the market lead to a fluctuation around
the minimum state, we assume that this fluctuation is of a perturbation of the
market of the short time nature at some time t = 0. The equation of motion for
the agent average number $<n>$ per nonreducibile subgroup is found, and is of
the relaxation type. We studied characteristics of evolution of perturbation
on the market when the extreme of the aim function is maximum and when this
extreme is minimum. We consider two limiting cases, when the aversion constant
$A_{0}$, is near to 2b from below, $A_{0}\rightarrow 2b$, and when the
aversion constant $A_{0}$ tends to zero, $A_{0}\rightarrow 0$. In the first
case we obtain that the relaxation constant $\tau$ is, to the lowest order in
$A_{0}-2b)$. When the expected volatility EV increases the relaxation constant
$\tau$ in this limit decreases. When the expected volatility EV decreases the
relaxation constant $\tau$ in this limit increases. When the aversion constant
$A_{0}$ tends to 2b, e.i. when it increases in this limit, the relaxation
constant $\tau$ decreases to zero in the fourth order in $(A_{0}-2b)$ . The
constant $(A_{0}-4b)$ is called in our paper as the contribution to the total
aversion constant from the crowd. It increases when the aversion constant
$A_{0}$ tends to 2b. Increasing the aversion constant $A_{0}$ leads to larger
acceptance of the crowd of expected risk, and thus of expected return.
Verification of information with its decreasing intensity in this limit leads
slower return back to the equilibrium state. The expected return in this limit
and in the first term of the expansion in small parameter is not present in
the expression for the relaxation constant $\tau$. We obtain the relaxation
constant $\tau$ to the lowest order in $(A_{0}$ tending to zero. When the
expected volatility EV increases now the relaxation constant $\tau$ in this
limit increases. The state with the fluctuation at time $t=0$ returns back to
the minimum less quickly. When the expected volatility EV decreases the
relaxation constant $\tau$ in this limit decreases. The state with the
fluctuation at time $t=0$ returns back to the minimum more quickly. When the
aversion constant $A_{0}$ tends to 0, e.i. when it decreases, the relaxation
constant $\tau$ remains finite. The larger expected return, the smaller the
relaxation constant and the return to the equilibrium is again more quickly.
The larger expected volatility, the larger the relaxation constant and the
return to the equilibrium is more slowly. The smaller expected return, the
larger the relaxation constant and the return to the equilibrium is more
slowly. The larger the intensity b of verification of information the smaller
the relaxation constant $\tau$. The state with a fluctuation at time $t=0$
returns back to the equilibrium state more quickly. The smaller the intensity
b of verification of information the larger the relaxation constant $\tau$.
The state with fluctuation returns back to the equilibrium state more slowly.
Decreasing the aversion constant $A_{0}$ leads to smaller acceptance of the
expected risk, and thus of the expected return. The equation of motion leads
in the second case of minimum to the linear in time damping to the equilibrium
state with the characteristic time $T_{3}$ and $T_{6}$. For $A_{0}>2b$ the
characteristic constant $V_{c6}<0$ is negative. The characteristic time
$T_{6}$ increases with decrease of risk EV, with decrease of profits, and with
decrease of the risk constant $A_{0}$ toward $4b$ . This increase is however
to a finite value. For $A_{0}<2b$ the characteristic constant $V_{c6}>0$ is
positive. The characteristic time $T_{6}$ increases with decrease of risk EV
to the $V_{c6}$ value or with increase of risk EV to the $V_{c6}$ value. The
increase of the characteristic time $T_{6}$ is now large. The characteristic
constant $V_{c3}=2A_{0}E\frac{A_{0}-2b}{A_{0}-4b}$. With decrease of risk EV
toward $V_{c3}$ the characteristic time $T_{3}$ increases to large values, Our
model thus, under defined conditions describes development on the market, when
small perturbations are present. The developed capital markets are different
from emerging capital markets, [14]. It is interesting, that we have found and
exponentially damped behaviour of the index on the Bratislava Stock Exchange
and Prague Stock Exchange, [15], when the trading started on these markets
after coupon privatization ended. The prices of the most liquid shares were in
that moment still not in equilibrium, they developed after several months to
an equilibrium state. It is well known property of prices (of commodities
physical and financial) that when after increase of the price due to some
perturbation these prices return back to the equilibrium price more slowly
than when after decrease of the price of the commodity due to some
perturbation these prices return back to the equilibrium price. Thus the
relaxation constant of the first return back is larger than of the second one.
While the expected price is in both cases the same, we can say within our
model, assuming that we are in the limit of low $A_{0}$, that the expected
volatility is lower in the second case than in the first second one. The
quantitative and qualitative dependencies described above may be thus compared
with those observed on real markets.

## Acknowledgment

The paper represents a part of results of the VEGA project 1/0495/03.

## References

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of Agents, http://arXiv.org/abs/cond-mat/0312723 , 2003

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# Impact of fragment formation on shear viscosity in the nuclear liquid-gas
phase transition region

X. G. Deng (邓先概) ID Key Laboratory of Nuclear Physics and Ion-beam Application
(MOE), Institute of Modern Physics, Fudan University, Shanghai 200433, China
Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai
201800, China  National Superconducting Cyclotron Laboratory, Michigan State
University, East Lansing, MI 48824-1321, USA  P. Danielewicz ID
111Corresponding author<EMAIL_ADDRESS>National Superconducting
Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824-1321,
USA  Department of Physics and Astronomy, Michigan State University, East
Lansing, MI 48824-1321, USA  Y. G. Ma (马余刚) ID 222Corresponding author:
<EMAIL_ADDRESS>Key Laboratory of Nuclear Physics and Ion-beam
Application (MOE), Institute of Modern Physics, Fudan University, Shanghai
200433, China  Shanghai Research Center for Theoretical Nuclear Physics， NSFC
and Fudan University, Shanghai 200438, China  H. Lin (林豪)  National
Superconducting Cyclotron Laboratory, Michigan State University, East Lansing,
MI 48824-1321, USA  Department of Physics and Astronomy, Michigan State
University, East Lansing, MI 48824-1321, USA  Y. X. Zhang (张英逊) ID China
Institute of Atomic Energy, Beijing 102413, China

###### Abstract

Within the improved quantum molecular dynamic (ImQMD) model we follow the
evolution of nuclear matter for planar Couette flow in a periodic box. We
focus on the region of liquid-gas phase transition and extract the shear
viscosity coefficient from the local stress tensor, directly following
viscosity definition. By switching on and off the mean field and thus inducing
the phase transition, we are able to observe the impact of clumping in the
phase-transition region onto the viscosity.

###### pacs:

25.70.-z, 24.10.Lx, 21.30.Fe

##  I Introduction

Shear viscosity is not only one of the crucial bulk properties of the liquids
and gases in our surroundings, but also it is important for the study of
strongly interacting matter, such as quark gluon plasma (QGP) in early
Universe and matter inside neutron stars [1, 2, 3, 4, 5, 6, 7, 8]. The
strongly interacting matter can also undergo the liquid-gas (LG) phase
transitions in lower energy heavy-ion collisions [9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20, 21]. Compilation of data for different substances [3]
demonstrates a drop of the shear viscosity in the region of a phase
transition, when the entropy density is employed as a universal reference. One
reason for the drop could be long-range correlations developing in the
transition region. When scaled with entropy density, the reduced shear
viscosity of strongly interacting matter appears particularly low in the dense
stage of ultra-relativistic collisions [3, 4, 23, 22, 24, 6], which is often
tied to the proximity of the system to the QGP phase transition.

In the hadronic regime, the shear viscosity of nuclear matter around normal
density was recently inferred from $\gamma$ decay of the isovector giant
dipole resonance populated in a fusion-evaporation reaction [25, 26]. In
intermediate energy heavy-ion collisions, shear viscosity for nuclear matter
was assessed [27] by interpreting stopping data in terms of the Boltzmann-
Uhlenbeck-Uehling (BUU) equation [28, 29], rooted in the Landau theory, with
adjustments in the BUU pertaining to supranormal densities and moderate
temperatures. Viscosity in the LG region was addressed with a combination of
Maxwell construction and relaxation time approximation in [30]. Quantum
Molecular Dynamics (QMD) models have also been used [31, 32, 33] to provide
background circumstances of intermediate-energy collisions, in correlating
entropy density and shear viscosity there, with the viscosity estimated either
in terms of the mean free path or in kinetic theory [28].

In the previous investigations, the peculiar impact of the LG phase-
transition, of growing long-range correlations, on the shear viscosity, was
never explored. In the BUU approach, uniform density is imposed and momentum
is transported by moving of nucleons independently [28, 29, 34, 35]. In the
Maxwell construction, the shear viscosity coefficient in the phase coexistence
region is a linear combination of the coefficients for the two uniform phases
[30], higher at a given density and temperature, than without the transition.
In this work, we set out to the determine the shear viscosity in the LG phase
transition region, while accounting for correlations that underlie the
fragment formation in low-density matter. We employ the so-called an Improved
QMD (ImQMD) model [36, 37], which provides a natural framework for the study
of correlations. We enclose a neutron-proton symmetric system in a periodic
box, establish a planar Couette flow inside, and determine the shear viscosity
coefficient from the momentum flux across the box, following the definition of
the coefficient. This approach is then similar to that in experiments and it
works without equilibrium assumptions. Nonetheless, QMD-type models struggle,
more than the BUU-type, in enforcing Pauli principle for nucleon-nucleon
($N$-$N$) collisions [38]. This can lead to undesirable effects in long-term
stationary calculations of transport coefficients, as the latter are quite
sensitive to Pauli blocking. Benefiting from the proximity of the system to
equilibrium, we know though the blocking factors better than they can be
estimated in a QMD model and we exploit that knowledge to circumvent the
difficulty in the ImQMD model.

The present article is organized as follows: In Sec. II, we introduce our
simulation model and the analysis framework for extracting the viscosity. In
Sec. III.1, we present results for the shear viscosity coefficient of nuclear
matter, obtain with and without the mean field interactions turned on, and
extract using the SLLOD algorithm which will be explained in next section.
Without the mean field interactions, the LG phase transition disappears. Sec.
III.2 supplements the results of III.1 with more discussion of the LG phase-
transition and with analysis of the correlations giving rise to the fragments.
Sec. IV is dedicated to conclusions.

##  II Nuclear system and analysis method

###  II.1 ImQMD model

Two types of models are employed in practice to simulate heavy-ion collisions
and to extrapolate conclusions to the limit of infinite nuclear matter. One,
BUU type [39], solves directly the BUU equation. Another, QMD type [40, 41,
37], follows molecular dynamics that incorporates elements of the BUU
equation. The BUU models follow nucleon single-particle distributions only,
while the QMD models include some multi-nucleon correlations and are thus
better suited for the goals of the present work. From the latter type, we
choose the improved quantum molecular dynamic (ImQMD) model [36, 37] for our
calculations.

In the ImQMD, we employ a potential energy density [36, 37], with the spin-
orbit term dropped, specifically,

$\begin{split}V_{loc}&=\frac{\alpha}{2}\frac{\rho^{2}}{\rho_{0}}+\frac{\beta}{\gamma+1}\frac{\rho^{\gamma+1}}{\rho_{0}^{\gamma}}+\frac{g_{sur}}{2\rho_{0}}\,(\bigtriangledown\rho)^{2}\\\
&+g_{\tau}\frac{\rho^{\eta+1}}{\rho_{0}^{\eta}}+\frac{g_{sur,iso}}{\rho_{0}}\,[\nabla(\rho_{n}-\rho_{p})]^{2}+\frac{C_{s}}{2\rho_{0}}\rho^{2}\,\delta^{2}\,,\end{split}$
(1)

where $\rho=\rho_{\text{n}}+\rho_{\text{p}}$ is the net nucleon density,
$\rho_{\text{n}}$ and $\rho_{\text{p}}$ are the proton and neutron densities,
respectively, and $\delta=(\rho_{n}-\rho_{p})/\rho$ is the normalized
asymmetry of neutron and proton. The normal density is
$\rho_{0}=0.16~{}\text{fm}^{-3}$.

The Skyrme density functional with interactions such as Eq. (1), as well as
other approaches to nuclear systems that aim at a level of realism, yield the
aforementioned liquid-gas phase transition at the subnormal densities for the
near-symmetric nuclear matter. In the model of nuclear-matter system, the
Coulomb interactions are switched off and the near symmetric matter has
comparable numbers of neutrons and protons. But in this work, we also check
the effect of Coulomb interactions. When uniform fermionic liquid and gas
phases are assumed, they may coexist in equilibrium with each other. At the
ground state ($T$ = 0), the two phases are nuclear matter at normal density
and vacuum. Difference in the densities of the model phases in equilibrium
shrinks as the temperature increases and the difference disappears as both
phases reach critical density and temperature, $\rho_{c}\sim 0.4\,\rho_{0}$
and $T_{c}\sim 18\,\text{MeV}$ [10, 11], respectively. Since the fluctuations
exist in the non-equilibrium nuclear system, the phase transition picture
becomes more complex, with fragments forming at lower temperatures in the
phase-transition region and the fragment formation probability representing a
$U$-shape as a function of fragment mass, e.g., Ref. [42]. The light-mass arm
of the $U$-shape represents the gas phase and the heavy-mass arm – the liquid
phase.

With our focus on the shear viscosity of infinite symmetric nuclear matter, we
set up a periodic cubic box as in Ref. [38]. In our simulations, we use a
fixed number of nucleons $A=600$. The box size is adjusted to yield a desired
average density. The width of the Gaussian wave-packets that represent the
nucleons, which has some impacts on the fragments that are formed in dilute
matter, is set at $\sigma=2\,\text{fm}$. For simplicity, the $N$-$N$ cross
section is fixed at $40\,\text{mb}$ in this paper. Even though the value of
shear viscosity increases with the decreasing of $N$-$N$ cross section as
shown in our previous work [32] as well as our check in this work, the choice
of $40\,\text{mb}$ for the $N$-$N$ cross section does not change our
conclusion of the present work . In QMD models, the Pauli blocking for $N$-$N$
collisions is normally calculated according to the wave-packets, with the
blocking fluctuating at low temperatures and the momentum distributions
gradually evolving towards Boltzmann, rather than Fermi-Dirac form [38].
Benefiting from the proximity of our systems to a local equilibrium, we
circumvent the problem using the local equilibrium Fermi-Dirac distributions,
corresponding to the local nucleon and kinetic energy density for which
collective motion is deducted, in the blocking for the collisions. A similar
procedure was employed in the transport-code comparison [38]. And in the
following, the temperature is an input parameter of the ImQMD model, and
initial momentum distributions of the nucleon is determined by the Fermi-Dirac
distribution at finite temperature as performed in our earlier Thermal IQMD
(ThIQMD) model [31, 43].

In this work, we consider both of cases without and with mean field. In the
case without mean field and in the cascade mode, the temperature of the system
would not change during the process of simulation. However, when one turns on
the mean field, the nucleon potential and kinetic energy will change due to
the fragment formation. Then the system would heat up, and we need to adjust
this transient state temperature to the initial temperature by introducing the
factor ‘$h$’ which can be nominally positive and less than ‘1’. Here the
transient state temperature is determined by fitting the momentum distribution
with Fermi-Dirac function at certain times. If one can find that the transient
state temperature increases as time, we need a friction factor ‘$h$’ to cool
the system. Also the ‘$h$’ is only implant in a period of time (the length of
time depends on the initial density and temperature). One needs to adjust it
to be an appropriate value, eg. ‘$h$’ = 0.999 for density of 0.1$\rho_{0}$ and
$T$ = 6 MeV. In practice, the ‘$h$’ is implemented when the box system is
close to the equilibrium, and then the box system cools down to initial
temperature and reaches equilibrium again after some time evolution. One
problem for our calculations is that the term can lead to phase space
occupations inconsistent with the Pauli principle for fermions. We resolve
that by employing the term only at high momenta and/or low spatial densities,
under the conditions of low phase-space density.

###  II.2 Gaussian thermostated SLLOD algorithm

The SLLOD algorithm belongs to common methodologies of the Non-Equilibrium
Molecular Dynamics (NEMD) calculations. It was named by Evans and Morriss [44]
and related to the dynamics with (artificial) strain rate algorithm for non-
equilibrium molecular dynamics calculation and has been extensively applied to
predict the rheological properties of real fluids [44, 45].

In the SLLOD algorithm, a planar Couette flow field is applied to the system,
with a shear rate $\dot{\gamma}$ = $\partial v_{x}/\partial y$. When the shear
is presented, the periodic boundary conditions are modified to those of Lees
and Edwards [46], consistent with constant shear throughout space. The shear
viscosity coefficient is computed from [45]

$\begin{split}\eta=-\frac{\langle P_{xy}\rangle}{\dot{\gamma}}\,,\end{split}$
(2)

where $\langle\cdots\rangle$ denotes ensemble average and the stress tensor is
given by [47]

$P_{\alpha\beta}=P_{\alpha\beta}^{\rm cont}+P_{\alpha\beta}^{\rm coll}\,.$ (3)

There are two contributions to the stress tensor, the continuous one and the
collision impulse one. The continuous contribution to the stress tensor is
given by

$\begin{split}P_{\alpha\beta}^{\rm cont}(t)=\frac{1}{V}\int{\rm
d}^{3}r\,P_{\alpha\beta}(\vec{r},t)\;\;\;(\alpha,\beta=x,y,z)\,.\end{split}$
(4)

Here, $V$ is the volume of the simulation box and $P_{\alpha\beta}(\vec{r},t)$
originates from summing over the wave packets and is formally defined as [48]

$\begin{split}&P_{\alpha\beta}(\vec{r},t)=\sum_{i}^{A}\frac{p_{{i}\alpha}\,p_{{i}\beta}}{m_{i}}\,\rho_{i}(\vec{r},t)\\\
&+\frac{1}{2}\sum_{i}^{A}\sum_{i\neq
j}^{A}F_{ij\alpha}\,R_{ij\beta}\,\rho_{j}(\vec{r},t)\\\
&+\frac{1}{6}\sum_{i}^{A}\sum_{i\neq j}^{A}\sum_{i\neq j\neq
k}^{A}(F_{ijk\alpha}\,R_{ik\beta}+F_{jik\alpha}\,R_{jk\beta})\rho_{k}(\vec{r},t)\\\
&+\cdots\,.\end{split}$ (5)

Here, $\vec{F}$ are two- and three-body forces that are continuous as a
function of time. The Latin indices pertain to the particles and the first
index in the force represents the particle on which the force acts and the
remaining particle indices indicate the particles that give rise to that
force. The relative positions of two particles are
$\vec{R}_{ij}=\vec{r}_{i}-\vec{r}_{j}$. In QMD-type models, contribution to
the density associated with particle $i$ can be written in terms of a wave
packet as

$\begin{split}\rho_{i}(\vec{r})=\frac{1}{(2\pi\sigma^{2})^{3/2}}\exp\Big{[}-\frac{(\vec{r}-\vec{r}_{i})^{2}}{2\sigma^{2}}\Big{]}\end{split}$
(6)

During collisions between nucleons, momentum is transported through action of
impulse forces. All collisions within time interval $\Delta t$ yield a time-
averaged contribution to the stress tensor equal to [47]

$\begin{split}P_{\alpha\beta}^{coll}(t)=\frac{1}{V}\sum_{i}^{Ncoll}\frac{\Delta
p_{i1\alpha}}{\Delta t}\cdot(r_{i1\beta}-r_{i2\beta})\,.\end{split}$ (7)

Here, the summation is over nucleon-nucleon collisions, with the two nucleons
in the collision labelled as ’1’ and ’2’. The factor ${\Delta p_{i1\alpha}}$
is the $\alpha$ component of the impulse on nucleon 1 in the collision. The
factor $(r_{i1\beta}-r_{i2\beta})$ is the $\beta$ component of the
displacement vector over which the momentum is moved in the collision.

Fig. 1 shows the evolution of the off-diagonal element of the stress tensor
for the system, first allowing the system to equilibrate during the first
$400\,\text{fm}/c$ and then applying the shear of the Couette flow within the
box, still to be discussed. Without shear, the element fluctuates around zero.
After the shear is switched on, the element quickly rises and stabilizes after
about $100\,\text{fm}/c$. The collision contribution to the element is very
small, as illustrated for different densities in Fig. 2. The fluctuations are
stronger at the higher density, but still very much near zero.

Figure 1: The $xy$ element of the stress tensor, as function of time at the
density of $0.1\,\rho_{0}$, for different temperatures in the simulations.
Figure 2: Collision contribution to the off-diagonal stress tensor element, as
function of time at the densities of $0.1\,\rho_{0}$ (a) and $0.8\,\rho_{0}$
(b), for different temperatures in the simulations. Figure 3:  Deduced shear
viscosity as function of the shear rate $\dot{\gamma}$ applied within the
SLLOD algorithm, in the system at the density of $0.2\,\rho_{0}$ and
temperature $T=10\,\text{MeV}$, when no mean field is employed.

With the applied planar Couette flow field, the centroids of nucleonic wave-
packets satisfy the SLLOD equations [45]:

$\displaystyle\frac{d\vec{r}_{i}}{dt}=\frac{\vec{p}_{i}}{m_{i}}+\dot{\gamma}y_{i}\hat{x}\,,$
(8)
$\displaystyle\frac{d\vec{p}_{i}}{dt}=\vec{F}_{i}-\dot{\gamma}p_{yi}\hat{x}\,.$
(9)

The construction is that of introducing a local velocity with the component in
the $x$ direction changing linearly in the $y$ direction. The $\dot{\gamma}$
factor regulates the pace of the velocity change in the $y$ direction. This,
together with the Lees-Edwards periodic boundary condition, allows to combine
a uniform shear with periodicity.

One should notice that the shear rate $\dot{\gamma}$ can neither be too weak,
nor too strong [49, 50]. If it is too weak, the induced off-diagonal stress
tensor elements will have hard time competing with numerical inaccuracies. If
it is too strong, the response of the system will cease to be linear in the
shear. In either case, the extraction of the shear viscosity is likely to
fail. In Fig. 3 we show the extracted viscosity as a function of the shear
rate in an exemplary system at the density of $0.2\,\rho_{0}$ and temperature
$T=10\,\text{MeV}$, when no mean field is applied. In most of the
calculations, we employ the rate of $\dot{\gamma}$ = 0.002 $c$/fm when the
system reaches equilibrium.

Figure 4: Relaxation time as function of temperature at different densities in
the calculations without (a) and with (b) mean field. Figure 5: Same as Fig. 4
but for shear viscosity. Figure 6: The ratio of the shear viscosity with mean
field to that without as function of temperature at different average density.

##  III Results and discussion

###  III.1 Impact of fragment formation on shear viscosity

For a viscoelastic system, relaxation time is required for the establishment
of the stationary state. For our simulations, shear rate $\dot{\gamma}$ is
implemented into the system at 400 fm/c as shown in Fig. 1. Stress tensor
$P_{xy}$ responds to the shear rate and $-\langle P_{xy}\rangle$ increases as
time and reaches constant for some times. For responding to shear rate
$\dot{\gamma}$ (t) which actually is set to be constant, stress tensor
$P_{xy}(t)$ can be expressed as [45]

$\begin{split}P_{xy}(t)=-\int_{t_{0}}^{t}\eta_{M}(t-s)\dot{\gamma}(s)ds\,,\end{split}$
(10)

where $\eta_{M}(t)$ is a memory function and $t_{0}$ is the time for
implementation of shear rate. For the Maxwell model of viscoelasticity, the
memory function is identified as [45, 51, 52]

$\begin{split}\eta_{M}(t)=G\exp(-t/\tau)\,,\end{split}$ (11)

where $G$ is the infinite frequency shear modulus and $\tau$ is relaxation
time. With Eq. 10 and Eq. 11, one can get simple expression for the stress
tensor:

$\begin{split}P_{xy}(t)=A+B\exp(-t/\tau)\,.\end{split}$ (12)

By fitting the curves as in Fig. 1 with Eq. 12, one can get the relaxation
time which is shown in Fig. 4. Fig. 4 (a) and (b) display a decreasing of
relaxation time with the increasing of temperature. In Fig. 4 (b), it is found
that at low $T$ and density of low energy nuclear LG phase transition domain
the relaxation time is reduced. For the hadron gas or hadron mixture as shown
in Refs. [53, 54, 55], the relaxation time has the same order of magnitude but
within different temperature region.

In Fig. 5 we show our major results, i.e., the shear viscosity coefficient
calculated for symmetric nuclear matter at different densities, using the
SLLOD algorithm, with and without mean field. Our treatment of the Pauli
principle allows us to push the calculations down to relatively low
temperatures, but it does not eliminate completely the problems around
immediate vicinity of zero temperature and we restrict ourselves to $T\geq
4\,\text{MeV}$.

Without mean field, the viscosity is that of a gas of nucleons. Comparing with
the results elsewhere, ours include collisional contributions, but these are
small in the density region we consider, cf. Fig. 2. Because of this, our
results in Fig. 5(a), obtained without MF, are similar to those elsewhere,
such as in Ref. [28]. In low temperature region, the nucleonic gas becomes
degenerate and the Pauli blocking begins to play a role for collisions, the
shear viscosity begins to increase with the decreasing of temperature in the
degenerate gas. The latter is due to the increase in the mean free path
because collisions become increasingly blocked as temperature decreases. And
in high temperature region, the purely nucleonic shear viscosity coefficient
slowly increases with temperature due to the increase in nucleon velocities
with temperature.

Figure 7: Shear viscosity as function of temperature in the calculations
without and with the Coulomb interaction. Figure 8: Same as Fig. 4 but for the
ratio of shear viscosity over entropy density ($\eta/s$).

With MF included, our results for shear viscosity in Fig. 5(b) are similar to
those without MF at higher temperatures and densities. To facilitate the
comparison, we show in Fig. 6 the ratio of the shear viscosity obtained with
mean field over that obtained without, as a function of temperature at given
average densities. The ratio is close to unity at temperatures in excess of
$\sim 12\,\text{MeV}$ and at densities closer to the normal density
$\rho_{0}$. At low densities, below $0.4\,\rho_{0}$ and temperatures less than
$10\,\text{MeV}$, we observe a strong reduction in the shear viscosity when
the mean field is included. In our system, a strong reduction in the shear
viscosity occurs in the region of the phase transition and can be understood
in terms of formation of separated fragments that move slowly, absorb most of
the mass in the system, and stall the transport of momentum. By moving slowly,
the fragments play only a passive role in transporting the momentum. The
latter gets primarily transported by the nucleons in the gas phase, but these
rather collide with the fragments than with each other, which shortens their
mean free path. In addition, with most mass in the fragments, the role of the
Pauli principle in the gas is reduced, additionally reducing the path. At
higher densities, the fragments connect and the low density gas phase is
contained to voids in the liquid, so momentum transport can progress. Any role
of the Pauli principle is restored. Moreover, while the fragments forming, the
long-range Coulomb interaction may have effect on shear viscosity. For
checking, in Fig. 7, Coulomb interaction is taken into account. It is found
that in the region of low temperature, shear viscosity is reduced by the
Coulomb interaction. Imagining two fluid layers passing from one to another,
Coulomb interaction of among protons which is repulsive will reduce momentum
transformation or momentum flux between two layers that will decrease shear
viscosity. In low temperature region, Coulomb interaction could be important
in comparison with thermal motion. While with increasing temperature, Coulomb
interaction effect becomes smaller since nuclear matter at higher temperature
is more uniform and the weight of Coulomb interaction becomes negligible.
Based on the above arguments, we can explain the temperature dependent Coulomb
correction on shear viscosity.

It is interesting to note that for the quark gluon plasma by taking into
account interaction, shear viscosity faces a strong suppression in low
temperature region of quark hadron phase transition domain going from $T$ =
0.300 GeV to 0.120 GeV in contrast with the noninteracting QGP case [56]. Our
results for the reduction of shear viscosity are quite similar to this QGP
phase transition even though our system is for LG system.

For a simplified picture, by utilizing the relation [57],

$\begin{split}s=\frac{g}{(2\pi\hbar)^{3}V}\int[f\ln
f+(1-f)\ln(1-f)]d^{3}pd^{3}r\,,\end{split}$ (13)

one can get the entropy density ($s$), where $g=4$ is spin-isospin degeneracy
and $f$ is the distribution function, which can be given,

$\begin{split}f(\epsilon)=\frac{1}{\exp(\frac{\epsilon(\vec{r})-\mu(\vec{r})}{T})+1}\end{split}$
(14)

where $\mu$ is chemical potential, and the particle energy $\epsilon$ =
$p^{2}/(2m)+$$U(\rho(\vec{r}))$ in which $U(\rho(\vec{r}))$ represents single
particle potential. Here, by taking into account fragment effects in Eq. (14),
we can obtain local entropy densities with the local chemical potentials.
After integration local entropy densities over the whole space, the total
entropy density can be got.

Figure 9: Reduced density moments for different powers $N$ vs temperature at
the average density of 0.2$\rho_{0}$ from the calculations with mean field.

With shear viscosity we have as in Fig. 5, the ratio of shear viscosity over
entropy density ($\eta/s$) which is a hot agenda and relates to a bound value
of $\hbar/4\pi$, known as the Kovtun-Son-Starinets (KSS) bound [58] can be
obtained. The hot quark gluon plasma (QGP) has very low $\eta/s$ which is very
close to $\hbar/4\pi$ and behaves as a nearly perfect fluid [59]. When the QGP
cools down, in phase of relativistic hadron gas, $\eta/s$ increases and
becomes significantly higher than the KSS bound [60, 61]. For the finite
colder nuclear matter around the saturation density, it is 2.5$-$6.5 times of
$\hbar/4\pi$ within the temperature range 0.8$-$2.1 MeV [25, 26]. For the hot
finite nuclear matter, $\eta/s$ is around 3.0 - 70 times of $\hbar/4\pi$ for
the density of 0.2 - 1.25 $\rho_{0}$ [33, 31, 32], in which the minimum
$\eta/s$ value corresponds to the liquid-gas phase transition in previous
model calculations. And here we get $>$ 2 times of $\hbar/4\pi$ for the
infinite nuclear matter which is consistent with the results in Ref. [32, 33,
35, 30, 62]. To briefly summarize the values of ratios of shear viscosity over
entropy density for different systems, Table 1 and Table 2 are made. From
tables, we can see the QGP matter around 170 MeV has the lowest $\eta/s$ close
to the KSS bound ($1/4\pi$), however, the nuclear matter around a few to a few
tens MeV which is in a liquid gas phase transition range has also relative low
$\eta/s$, i.e. about several times of KSS bound. The above low $\eta/s$ may
reflect the universal property of strong interaction matter regardless of
partonic and nucleonic level. In contrary, for atomic and molecular substances
of He, Ni and water, they have relative large $\eta/s$ which are dominated by
electromagentic interaction.

Table 1: $\eta/s$ of different atomic and molecular systems: Helium, Nitrogen and H2O in certain ranges of temperature and pressure [3].  Systems [Ref.] |  T (K) |  Pressure (MPa) |  $\eta/s~{}(\hbar/4\pi)$
---|---|---|---
Helium [3] |  2$-$20 |  0.1$-$1 |  8.8$-$126
Nitrogen [3] |  50$-$ 600 |  0.1$-$10 |  11.9$-$7000
H2O [3] |  300$-$1200 |  10$-$100 |  25.5$-$377
Table 2: $\eta/s$ of different quark matter (QGP) and finite nuclear matter (FNM) as well as infinite nuclear matter (INM) in certain ranges of temperature and density. Here Tc$\approx$ 170 GeV is transition temperature from the hadronic phase to QGP phase [56].  Systems [Refs.] |  T (MeV) |  $\rho$/$\rho_{0}$ |  $\eta/s~{}(\hbar/4\pi)$
---|---|---|---
QGP1 [56, 63, 64, 65] |  $\leqslant$$\rm T_{c}$ |  — |  3.77$-$25.1
QGP2 [56, 63, 64, 65, 66] |  $>$$\rm T_{c}$ |  — |  1.00$-$6.91
FNM1 [26, 25] |  0.8$-$2.1 |  $\sim$1.0 |  2.5$-$6.5
FNM2 [33] |  3.5$-$16 |  0.9$-$1.25 |  3.0$-$70.0
FNM3 [31] |  1$-$30 |  0.3$-$1.0 |  7.0$-$60.0
FNM4 [32] |  6$-$16 |  0.2$-$0.3 |  9.5$-$20.0
INM1 [30] |  8$-$14 |  0.01$-$0.3 |  4$-$30
INM2 $[$this work$]$ |  4$-$25 |  0.1$-$0.8 |  2$-$55
Figure 10: Density moment as function of temperature with (the lines with
symbols) and without (the dash lines) mean field for different densities with
$N$ = 2 (a) and $N$ = 6 (b).

###  III.2 Density moments

The LG phase-transition region and fragment formation are characterized by
enhanced density fluctuations [67]. Similar fluctuations would be there for
the anticipated QCD phase transition in baryon-rich matter [68]. The density
fluctuation can be quantified in terms of average density moment [69],

$\begin{split}\langle\rho^{N}\rangle=\frac{1}{A}\int\rho(\vec{\bf{r}})^{N}\rho(\vec{\bf{r}})d^{3}\vec{\bf{r}}\,.\end{split}$
(15)

Here, $A$ is the nucleon number and $N$ is the power for the moment.

The reduced moments $\langle\rho^{N}\rangle/\langle\rho\rangle^{N}$ for our
simulations are presented in Figs. 9 and 10. Fig. 9 shows the moments at the
average density of $0.2\,\rho_{0}$, as function of temperature for different
$N$ order. The moments rise when temperature drops to $10\,\text{MeV}$ or
lower, with the rise in the moments becoming more pronounced with the rise in
$N$, reflecting the clumping of the matter. Similar clumping signatures have
been effectively employed in the simulations for the vicinity of the high-
energy QCD phase transition [69, 70]. Fig. 10 displays results for the moments
both with and without MF at different average densities. Without MF, the
moments change little with temperature, no matter what average density, as no
clumping occurs. With MF the enhancement in the moments occurs for the same
temperatures and densities that the relative drop in shear viscosity occurs in
Fig. 6, underscoring the connection of fragment formation to the change in
viscosity.

##  IV Conclusions

In summary, we examined the stress tensor for a system of nucleons governed by
the ImQMD version of the nuclear molecular dynamics, either employing or not
the mean field, within the thermodynamic region of the liquid-gas phase
transition. The nucleons were enclosed in a periodic box with boundary
conditions representing constant shear throughout space. Following the SLLOD
algorithm, we extracted the shear viscosity coefficient for the system, after
it reached a stationary equilibrium. We found that the mean field has little
impact on the coefficient outside of the phase transition. However, for lower
average densities and temperatures in the region of the transition, the shear
viscosity coefficient drops significantly when the mean field is included. The
drop can be attributed to the formation of fragments, or long-range
correlations, of which presence can be revealed, in the box configuration,
with average values of the moments of density. Also it is found that Coulomb
interaction as repulsive interaction among protons reduces shear viscosity in
the low temperature region.

The shear viscosity can be tested in nuclear collisions through examinations
of momentum transfer between different momentum regions [27]. The region of
the liquid-gas phase transition [42] is crossed in nuclear collisions at tens
of MeV/nucleon.

###### Acknowledgements.

X.G.D. expresses his gratitude to the University of Chinese Academy of
Sciences for the scholarship awarded in 2017 (Grant No. N201709) that made
this work possible. P. Danielewicz acknowledges support from the U.S.
Department of Energy Office of Science under Grant No. DE-SC0019209. This work
further received partial support from the National Natural Science Foundation
of China under Contract No. 11890714, No. 12147101, No. 11947217 and No.
11875323, the Key Research Program of the CAS under Grant No. XDB34030000,
China Postdoctoral Science Foundation Grant No. 2019M661332 and Postdoctoral
Innovative Talent Program of China No. BX20200098.

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