text:
\section{Introduction}

The dynamics of many-body interacting systems is a long-standing investigation embracing various domains at the boundary between collective and chaotic processes.

	From a one-body modelling perspective, the dynamics
of fermionic systems~\cite{Ring1980,Maruhn2010} is efficiently described with the time-dependent Hartree-Fock (TDHF) framework, or time-dependent local density approximation (TDLDA), in condense-matter applications~\cite{Yabana1999,Reinhard2004}, as far as the variance of the involved observables is small and can be neglected.
	If this is not the case, additional beyond mean-field correlations should be included, depending on the degree of excitation~\cite{Abe1996,Chomaz2004,Simenel2010}.
	At low energy, as far as the system can be described in the small-amplitude limit,
a scheme of coherent-state propagation within the time-dependent generator coordinate method (TDGCM)~\cite{Reinhard1987,Goutte2005}, or a variational approach \`a la Balian-V\'en\' eroni~\cite{Balian1981,Simenel2012} is well suited.
	On the contrary, when the system experiences a violent dynamics, large fluctuations would spontaneously drive the system far away from the one-body TDHF evolution along many different directions, thus determining the shortcoming of the TDHF approximation  (and the above mentioned extensions).
	To address large-amplitude regimes, solutions beyond the single-particle picture may be needed,
for instance by propagating non-correlated states~\cite{Lacroix2015}.
	When even the low-energy regime is exceeded, dissipative behaviour results also from in-medium collisions, which are no more hindered by Pauli blocking.

In presence of mean-field instabilities, the collective dynamics may be driven to a chaotic regime; in a nuclear system, this would result in a highly non-linear process, leading to clusterisation from one-body density fluctuations, and oscillation of the neutron and proton fraction.
	The splitting of a composite system into fragments under violent perturbations signs the occurrence of the most catastrophic effect produced by large-amplitude fluctuations of the neutron and proton content.
	Such phenomenology, which is common in heavy-ion collisions at Fermi energies~\cite{EPJAtopicalWCI2006,EPJAtopicalNSE2014}, also characterises other fields, like solid-state physics (examples are metal clusters~\cite{Calvayrac2000,Fennel2010} or electrons in nanosystems~\cite{Chen2005}), ultracold atomic gases~\cite{Dalibard1998,Bloch2008} or some areas of astrophysics~\cite{Horowitz2006,Sebille2011,Schneider2013,Burrello2016}.
The description of the fragmentation process can only be afforded within approaches beyond the
mean-field approximation, incorporating the effect of many-body correlations, which induce
fluctuations in the evolution of the one-body density.
	Adapted to such situation, stochastic approaches typically propagate a bunch of mean-field trajectories within various orders of approximations, like stochastic TDHF (STDHF) formulations~\cite{Reinhard1992,Suraud2014,Slama2015,Lacombe2016}, or analogue semiclassical schemes within the Boltzmann-Langevin (BL) transport equation~\cite{Ayik1988,Reinhard1992bis}.

	In the following we exploit the last mentioned BL approach in the form of the recently introduced
Boltzmann-Langevin one body (BLOB) model~\cite{Napolitani2013,Rizzo2008} to undertake an exhaustive analysis of the interplay between mean-field and many-body correlations in
nuclear matter.
	We focus thereafter on crucial modelling issues, looking
in particular at the dynamics of fluctuations, both in stable systems and in unstable
conditions, leading to the disassembly of the system.
	The purpose is to examine virtues and limits of the BLOB approach, where the BL equation is solved in full phase space, and of corresponding approximations by carrying out a quantitative study of fluctuation amplitudes,
and comparing with some analytic expectations which characterise Fermi liquids.
	This analysis is important also in the spirit
of preparing reliable applications to heavy-ion collisions.
Indeed a good reproduction of the fluctuation dynamics is crucial for the predictions of features,
such as size and isotopic variances, of the products formed in nuclear reactions and to probe their link
to the properties of the nuclear effective interaction.

	In Sec.~\ref{sec_correlations} we survey some basic steps leading from a stochastic beyond-mean-field framework to the BLOB method, and related approximations.
	In Sec.~\ref{sec_BLNM}
the dynamics of fluctuations of the one-body density, as given by the linearized BL equation,
is discussed for nuclear matter initialized at moderate temperature and in several density conditions.
Owing to the presence of two components (neutrons and protons), one observes isovector fluctuation modes,
where  neutrons and protons oscillate out of phase, and isoscalar modes, with neutrons and protons moving
together.
Isovector fluctuations are of stable nature, reflecting the properties of the nuclear effective interaction
in the isovector channel. The performance of the BLOB model in reproducing analytic expectations for the
isovector variance and, in particular, its link to the nuclear symmetry energy is discussed in
Sec.~\ref{sec_ivfluctuations}.
For nuclear matter at suitable density and temperature conditions, isoscalar fluctuations may become
unstable, yielding a growth of the (isoscalar) fluctuation variance, which triggers
a process of clusterisation. 	Such situation is tested in Sec.~\ref{sec_isfluctuations}.
	In the spirit of connecting nuclear matter to open systems, sec.~\ref{sec_open} explores fluctuation observables related to blobs of matter, which correspond to emerging fragments in open systems.
	Conclusive statements from reviewing the results form Sec.~\ref{sec_conclusion}.

\section{Theoretical survey \label{sec_correlations}}

\subsection{N-body correlations in a stochastic one-body framework in dissipative regimes \label{sec_Nbody}}
It is usual to describe the evolution of an $N$-body system by replacing the Liouville-von Neumann equation
with the equivalent BBGKY hierarchy which, for a two-body interaction $V_{ij}$, yields the following chain of coupled equations
\begin{eqnarray}
i\hbar\frac{\partial\rho_1   }{\partial t} &=& [k_1           ,\rho_1   ] + \textrm{Tr}_2 [V_{12}       ,\rho_{12}] \notag\\
i\hbar\frac{\partial\rho_{12}}{\partial t} &=& [k_1+k_2+V_{12},\rho_{12}] + \textrm{Tr}_3 [V_{13}+V_{23},\rho_{123}] \notag\\
\dots&&\notag\\
i\hbar\frac{\partial\rho_{1^{\dots} k}}{\partial t} &=& \sum_{i=1}^{k}\Big[k_i+\sum_{j<i}^{k}V_{ij},\rho_{1^{\dots} k}\Big]  \notag\\
     &+&\sum_{i=1}^{k}\textrm{Tr}_{k+1}[V_{i k\!+\!1},\rho_{1^{\dots} k\!+\!1}] \notag\\
\dots&&,
	\label{eq:BBGKY}
\end{eqnarray}
where $\textrm{Tr}_{k}$ is a partial trace involving the many-body density matrix $\rho_{1^{\dots} k}$ (compact notation where the order corresponds to the number of indexes) and $k_i$ are kinetic energy operators.
	This is the avenue for constructing beyond-mean-field approximations, obtained through custom truncations of the hierarchy, or by reducing the complexity of the involved contributions at given orders.
	For instance, the inclusion of interactions beyond two bodies would be necessary to account for additional nuclear-structure features~\cite{Schuck2016}, or cluster correlations, and the explicit inclusion of correlations beyond the order $k=2$ would be necessary to describe high-coupling regimes~\cite{Lacroix2014}.

	If, on the other hand, a suited stochastic approach is adopted, simplified higher-order contributions can be introduced even though not explicitly implemented.
	Already in a first-order-truncation scheme ($k\!=\!1$) in a low-energy framework, it was found that a stochastic approach can be used to restore all the BBGKY missing orders approximately~\cite{Lacroix2016}, and generate large-amplitude fluctuations; in this case, a coherent ensemble of mean-field states is propagated along different trajectories from an initial stochastic distribution.
	Such scheme is however insufficient for our purpose, which is addressing dissipative regimes.
	In this case, it is necessary to introduce in-medium collisions in a second-order scheme ($k\!=\!2$), explicitly, i.e. the first two lines of the set~(\ref{eq:BBGKY}), from which kinetic equations are obtained~\cite{Balescu1976,Balian1991,Cassing1992,Bonasera1994}.
	If structure effects are neglected, it is then possible to propagate an incoherent ensemble of mean-field states, supplemented by a fluctuating term, in order to obtain a highly non-linear character of the dynamics.
	The stochastic treatment is not
obtained from exploiting a distribution of initial states
but, progressing from a single initial state. It acts intermittently all along the temporal evolution, producing successive splits of a given mean-field trajectory $\rho_1$ into subensembles $\rho_1^{(n)}$:
\begin{equation}
	\rho_1 \longrightarrow \{ \rho_1^{(n)}; n=1, \dots, \textrm{subens.} \}   \;.
\end{equation}
	This pattern then repeats for each element of the subensemble $\rho_1^{(n)}$ till eventually yielding trajectories ordered in bifurcating bundles, each one exhibiting a small variance around the mean trajectory of the corresponding envelope.
	In particular, in the time interval between two successive splits,
when fluctuations are built up, the system propagates keeping the mean trajectory unchanged within each envelope.
	Thus this time, $\tau$, has to be shorter than the time scales associated with the global effect of the collision integral  and with the mean-field propagation.

\subsection{Collisional correlations \label{sec_coll_correlations}}
	This stochastic scheme is equivalent to imposing that $\rho_1^{(n)}$ and $\rho_2^{(n)}$, i.e. the probabilities to find two nucleons, 1 and 2, at two configuration points, are not all the time decorrelated,
so that the two-body density matrix $\rho_{12}^{(n)}$ recovers some correlations of the upper orders of the BBGKY sequence in addition to the standard product of independent one-body densities which builds up the mean-field term.
	We can write at a time $t$:
\begin{eqnarray}
	&\rho_{12}^{(n)}(t) = \widetilde{\Omega}_{12}\mathcal{A}_{12}(\rho_1^{(n)}(t)\rho_2^{(n)}(t))\widetilde{\Omega}_{12}^{+}+\delta\rho_{12}^{(n)}(t)  \;;\quad&
	\label{eq:fluctuating_term}\\
	&\langle\delta\rho_{12}^{(n)}(t) \rangle_{\tau} = 0 \;;\;&
	\label{eq:fluctuating_mean}\\
	&\langle\delta\rho_{12}^{(n)}(t) \delta\rho_{12}^{(n)}(t) \rangle_{\tau} = \textbf{gain} + \textbf{loss} \;,&
	\label{eq:fluctuating_secmom}
\end{eqnarray}
	where $\widetilde{\Omega}_{12}$ is the M\o ller wave operator~\cite{Reed1979,Suraud1995} describing the diffusion of a particle with respect to another particle in the nuclear medium, related to a diffusion matrix $G_{12}=V_{12}\widetilde{\Omega}_{12}$, which is, in turn, related to the nucleon-nucleon differential cross section $|G_{12}|^2\!\sim\!\textrm{d}\sigma/\textrm{d}\Omega$.
	In this respect, the first term of the r.h.s. of eq.(\ref{eq:fluctuating_term}) contains collisional correlations, while the second term introduces a fluctuation of vanishing first moment
around the collision integral~\cite{Abe1996}.
	It should be noticed that the average in Eqs.(4,5) refers to an ensemble of one-body trajectories fluctuating, over the
time $\tau$, around a mean trajectory which follows the Boltzmann equation.
	Setting $\delta\rho_{12}^{(n)}(t)=0$
would then reduce to the (quantum) Boltzmann kinetic equation, which corresponds to a second-order truncation of the hierarchy without a fluctuation contribution.

	Finally, the description associated with one single mean-field trajectory $\rho_1^{(n)}$
yields the following form of the BL equation
containing an average collision contribution $\bar{I}_{\textrm{coll}}^{(n)}$ and a continuous source of fluctuation seeds $\delta I_{\textrm{coll}}^{(n)}$:
\begin{equation}
	i\hbar\frac{\partial\rho_1^{(n)}}{\partial t} \approx [k_1^{(n)}+V_1^{(n)} , \rho_1^{(n)}]
		+ \bar{I}_{\textrm{coll}}^{(n)} + \delta I_{\textrm{coll}}^{(n)}   \;.
\label{eq:STDHF}
\end{equation}

	It may be noted that Eq.~(\ref{eq:STDHF})
is similar to STDHF~\cite{Reinhard1992}, and it
transforms into an extended TDHF (ETDHF) theory~\cite{Wong1978,Wong1979,Lacroix2004} if the fluctuating term $\delta I_{\textrm{coll}}^{(n)}$ is suppressed;
	ETDHF can in fact efficiently describe the behavior
of some observable
related to dissipative processes, but it can not follow possible bifurcation paths deviating from the mean trajectory.
	Through a Wigner transform we can then replace Eq.~(\ref{eq:STDHF}) by a corresponding set of semiclassical BL trajectories:
\begin{equation}
	\frac{\partial f^{(n)}}{\partial t} = \{h^{(n)} , f^{(n)}\}
		+ I_{\textrm{UU}}^{(n)} + \delta I_{\textrm{UU}}^{(n)}   \;,
\label{eq:SMF}
\end{equation}
where the evolution of a statistical ensemble of Slater determinants is replaced by the evolution of an ensemble of distribution functions $f^{(n)}$, which at equilibrium correspond to a Fermi statistics.
	$h^{(n)}$ is the effective Hamiltonian acting on $f^{(n)}$.
	The residual average and fluctuating contributions of Eq.~(\ref{eq:STDHF}) are replaced by Uehling-Uhlenbeck (UU) analogue terms.
	$I_{\textrm{UU}}^{(n)}$ is related to the mean number of transitions
within a single phase-space cell $\Delta V_f$.
	While conserving single-particle energies, $\delta I_{\textrm{UU}}^{(n)}$ acts as a Markovian contribution expressed through its correlation~\cite{Colonna1994_a}
\begin{eqnarray}
	&\langle \delta I_{\textrm{UU}}^{(n)}(\mathbf{r},\mathbf{p},t) \delta I_{\textrm{UU}}^{(n)}(\mathbf{r}'\!,\mathbf{p}'\!,t')\rangle
	= {\bf gain + loss} =& \notag\\
	&= 2\mathcal{D}(\mathbf{r},\mathbf{p};\mathbf{r}'\!,\mathbf{p}'\!,t')\delta(t-t') \;,&
\label{eq:correlation}
\end{eqnarray}
where $\mathcal{D}$ is a
diffusion coefficient \cite{Ayik1988}.

\subsection{Obtaining the BLOB description: fluctuations in full phase space \label{sec_correlations_BLOB}}
From Eq.(4)
and from the procedure detailed in ref.~\cite{Ayik1990},
we assume
that the fluctuating term $\delta I_{\textrm{UU}}^{(n)}$ in Eq.~(\ref{eq:SMF}) should involve the same contributions composing the average collision term $I_{\textrm{UU}}^{(n)}$, i.e. the transition and the Pauli-blocking terms.
	This implies that also $\delta I_{\textrm{UU}}^{(n)}$ should be expressed in terms of one-body distribution functions.
This latter possibility can be exploited by replacing the residual terms $(I_{\textrm{UU}}^{(n)} + \delta I_{\textrm{UU}}^{(n)})$ by a similar UU-like term which respects the Fermi statistics both for the occupancy mean value and for the occupancy variance.
	In this case,
for a free Fermi gas,
the occupancy variance at equilibrium should be equal to $f^{(n)}(1-f^{(n)})$ in a phase-space cell $h^3$ and correspond to the movement of extended portions of phase space which have the size of a nucleon, i.e. the residual term $(I_{\textrm{UU}}^{(n)} + \delta I_{\textrm{UU}}^{(n)})$ should carry nucleon-nucleon correlations \cite{Bauer1987}.

	A natural solution to satisfy such requirement is to rewrite the residual contribution in the form of a rescaled UU collision term where a single binary collision involves extended phase-space portions of equal isospin $A$, $B$ to simulate wave packets, and Pauli-blocking factors act on the corresponding final states $C$, $D$, also treated as extended phase-space portions.
	The choice of defining each phase-space portion $A$, $B$, $C$ and $D$ so that its isospin content is either $1$ or $-1$ is necessary to preserve the Fermi statistics for both neutrons and protons,
and it imposes that blocking factors are defined accordingly in phase-space cells for the given isospin species.
	The above conditions lead to the BLOB equations~\cite{Napolitani2013}:
\begin{eqnarray}
	\frac{\partial f^{(n)}}{\partial t} - \{h^{(n)} , f^{(n)}\}
		= I_{\textrm{UU}}^{(n)} + \delta I_{\textrm{UU}}^{(n)}  = \notag\\
	= g\int\frac{\mathrm{d}\mathbf{p}_b}{h^3}\,
	\int
	W({\scriptstyle\mathrm{A}\mathrm{B}\leftrightarrow\mathrm{C}\mathrm{D}})\;
	F({\scriptstyle\mathrm{A}\mathrm{B}\rightarrow\mathrm{C}\mathrm{D}})\;
	\mathrm{d}\Omega\;,
\label{eq:BLOB}
\end{eqnarray}
where $g$ is the degeneracy factor.
	$W$ is the transition rate, in terms of relative velocity between the two colliding phase-space portions and differential nucleon-nucleon cross section
\begin{equation}
	W({\scriptstyle\mathrm{A}\mathrm{B}\leftrightarrow\mathrm{C}\mathrm{D}}) = |v_\mathrm{A}\!-\!v_\mathrm{B}| \frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}\;.
\label{eq:transition_rate}
\end{equation}
	$F$ contains the products of occupancies and vacancies of initial and final states over their full phase-space extensions.
\begin{equation}
	F({\scriptstyle\mathrm{A}\mathrm{B}\rightarrow\mathrm{C}\mathrm{D}}) =
	\Big[(1\!\!-\!\!{f}_\mathrm{A})(1\!\!-\!\!{f}_\mathrm{B}) f_\mathrm{C} f_\mathrm{D} - f_\mathrm{A} f_\mathrm{B} (1\!\!-\!\!{f}_\mathrm{C})(1\!\!-\!\!{f}_\mathrm{D})\Big]\;.
\label{eq:Pauli_scaled}
\end{equation}
	Details on the implementation of BLOB are given in appendix~\ref{sec_appendix_metrics}.
	In practice, if the test-particle method is employed, so that the system is sampled by $N_{\textrm{test}}$ test-particle per nucleon, phase-space portion $A$, $B$, $C$ and $D$ should be agglomerates of $N_{\textrm{test}}$ test-particles each, and the nucleon-nucleon cross section used in Eq.~(\ref{eq:transition_rate}) should be scaled by the same amount $N_{\textrm{test}}$ (Eq.~(\ref{eq:XSscaling}) ).
	Finally, the stochastic approach exploits the correlations carried in Eq.~(\ref{eq:BLOB}), recovering higher order than the $k\!=\!2$ truncation, and inducing the BL fluctuations-bifurcation scheme.

\subsection{Simplification through the SMF description: fluctuations projected}
\begin{figure}[b!]\begin{center}
	\includegraphics[angle=0, width=.95\columnwidth]{fig1}
\end{center}\caption
{
	Mean-value (left) and variance (right) of the number of effective in-medium collisions per nucleon and per unit time $\delta t=1$ fm/$c$ for SMF (full black circles) and BLOB, as a function of density.  The different open symbols correspond to BLOB calculations
with different number of test particles.
	The dashed-lines on the left panel represent fits to mean-value spectra and they
are replotted identically in the right panel, for comparison.
The crosses in the righ panel correspond to the SMF variance (black dots) multiplied by $N_{\textrm{test}}$.
	An asy-stiff parameterisation is used for the symmetry energy (the characteristics of the interactions are defined in Sec.~\ref{sec_ivfluctuations}).
}
\label{fig_collisionvariance}
\end{figure}

	At variance with the above description, the stochastic term in Eq.~(\ref{eq:SMF}) can be kept separate and treated as a stochastic force related to an external potential $U'$, like in the corresponding semi-classical stochastic mean field (SMF) model~\cite{Colonna1998}.
	This leads to treatments where fluctuations are implemented only in the coordinate space, i.e. they are projected on the spacial density.
	The difference between Eq.~(\ref{eq:BLOB}) and usual stochastic mean-field approaches is that those latter build fluctuations from introducing a well adapted external force or a distribution of initial conditions which should be accurately prepared in advance.
	On the contrary, Eq.~(\ref{eq:BLOB}) introduces fluctuations in full phase space and let them develop spontaneously and continuously over time in a dynamical process.

	Fig.~\ref{fig_collisionvariance} illustrates how this difference affects the statistics of effective in-medium collision (i.e. not Pauli blocked); it compares SMF and BLOB calculations performed in nuclear matter at $3$~MeV temperature and
at different densities in stable conditions, i.e. keeping the system uniform, and at equilibrium, i.e. when the average collision rate is constant in time (these conditions are described in details in Sec.~\ref{sec_ivfluctuations}).
	As a general trend, larger densities provide a larger number of collision candidates, so that, even if also the difficulty in relocating collision partners increases due to Pauli blocking, the resulting number of effective collisions per nucleon grows significantly with density.
	Despite the use of the same nucleon-nucleon cross section (which produces equal rates of attempted collisions per nucleon for all the employed approaches, not shown), the number of effective collisions per nucleon differs in the two models due to the different treatment of the Pauli blocking, which is more severe in BLOB, owing to the nucleon wave packet extension (for instance, at $\rho^0=0.16$~fm$^3$, large Pauli rejection rates, equal to 98\% in BLOB and to 95\% in SMF, result in different effective collision rates, see discussion in appendix~\ref{sec_appendix_metrics}).
	The main difference emerging from the comparison in Fig.~\ref{fig_collisionvariance} is that the variance of the number of effective collisions per nucleon
amounts to the mean value reduced by a factor $N_{\textrm{test}}$ in the SMF case, while it equals the mean without any dependence on $N_{\textrm{test}}$ in the BLOB case (several values of $N_{\textrm{test}}$ produce equal results).
	Such study confirms that, while in SMF fluctuations are strongly reduced in proportion to the quantity $N_{\textrm{test}}$, in BLOB fluctuations have large amplitude and exactly equal the mean value, according to the Poisson statistics~\cite{Burgio1991}.
	The quantification of such amplitude is the subject of the following sections.

\section{Strategy: comparing BL approaches in nuclear matter to the Fermi-liquid behaviour  \label{sec_BLNM}}
	The purpose of stochastic one-body approaches with collisional correlations like SMF or BLOB is introducing aspects of the Fermi liquid behaviour, including
fluctuations~\cite{Lifshitz1958,Pines1966}, in the description of heavy-ion collisions~\cite{Pethick1988}.

	In the following, we check how Eq.~(\ref{eq:BLOB}) handles isoscalar and isovector fluctuations of the one-body density, in equilibrated nuclear matter, with the aim of demonstrating that its implementation is better suited than approximate methods, like SMF,  to sample the development of inhomogeneities (equivalent process to fragment formation in finite open systems) and the related observables.
	We therefore compare results obtained with BLOB and SMF.

\subsection{Fluctuations in nuclear matter: \\analytic estimate}

	Let us consider nuclear matter at low temperature.
	Either from the stochastic fluctuating residual term of the BLOB treatment or from an external stochastic force in the SMF approach, we introduce a small disturbance in uniform matter
$\delta f(\mathbf{r},\mathbf{p},t) = f(\mathbf{r},\mathbf{p},t) - f^0(\mathbf{p},t)$
which lets fluctuations develop in time around the mean trajectory $f^0$.

	By considering neutron and proton distribution functions, we can further decompose fluctuations in isoscalar modes
$\delta f^\mathrm{s}$ and isovector modes $\delta f^\mathrm{v}$
\begin{eqnarray}
\delta f^\mathrm{s} &=& (f_\mathrm{n}-f_\mathrm{n}^0)+(f_\mathrm{p}-f_\mathrm{p}^0)\\
\delta f^\mathrm{v} &=& (f_\mathrm{n}-f_\mathrm{n}^0)-(f_\mathrm{p}-f_\mathrm{p}^0)\;,
\end{eqnarray}
corresponding to phase-space density modes where neutrons and protons oscillate in phase or out of phase, respectively.
	The temporal evolution of both those modes is obtained by applying the BL equation (\ref{eq:SMF}) to the phase-space fluctuations.
	For symmetric matter, and retaining only first-order terms in
$\delta f^\mathrm{q}$, one obtains:
\begin{equation}
	\frac{\partial\delta f^\mathrm{q}}{\partial t} + \frac{\mathbf{p}}{m}\cdot\nabla_\mathbf{r}\delta f^\mathrm{q} - \frac{\partial f^0}{\partial \epsilon}\frac{\partial\delta U^\mathrm{q}}{\partial \rho^\mathrm{q}}\frac{\mathbf{p}}{m}\cdot\nabla_\mathbf{r}\delta\rho^\mathrm{q} =   \frac{\partial f^0}{\partial \epsilon}\frac{\mathbf{p}}{m}\cdot\nabla_\mathbf{r} U'
	\;,
\label{eq:BLE_fluctuations}
\end{equation}
where the index $\mathrm{q}$ stands either for isoscalar ($\mathrm{q}=\mathrm{s}$) or isovector ($\mathrm{q}=\mathrm{v}$) modes, $f^0=f_\mathrm{n}^0+f_\mathrm{p}^0$, $U^q$ is the mean-field potential in the $q$ channel  and
$U'$ is an external stochastic force (SMF) or a fluctuating stochastic field (BLOB).
	We dropped the average collision term $I_{\textrm{UU}}$ because we consider small temperatures.

	To build our stochastic descriptions we assumed that, at least locally, fluctuations have a small amplitude around their mean trajectory so that $\delta f^\mathrm{q}\!\ll\!f^\mathrm{q}$.
	When the system is described as a periodic box,
collective modes are associated to plane waves of wave number $\mathbf{k}$.
	In this case, by expanding on plane waves expressed in Fourier components, we can study the evolution in time of phase space density fluctuations
\begin{equation}
	\delta f^\mathrm{q}(\mathbf{r},\mathbf{p},t)=\sum_k e^{(i\mathbf{k}\!\cdot\!\mathbf{r})} f_k^\mathrm{q}(\mathbf{p},t)
	= \sum_k e^{(i\mathbf{k}\!\cdot\!\mathbf{r})} e^{(i\omega_k t)} f_k^\mathrm{q}(\mathbf{p})
\end{equation}
and undulations in the density landscape
$\delta\rho^\mathrm{q}(\mathbf{r},t)=\sum_k e^{(i\mathbf{k}\!\cdot\!\mathbf{r})} \rho_k^\mathrm{q}(t)$.
	Rewritten in Fourier components, Eq.~(\ref{eq:BLE_fluctuations}) takes the form
\begin{equation}
	i\omega_k f_k^\mathrm{q}
+ i \mathbf{k}\cdot\frac{\mathbf{p}}{m}f_k^\mathrm{q} - i \frac{\partial f^0}{\partial \epsilon}\frac{\partial U_k^\mathrm{q}}{\partial \rho^\mathrm{q}} \mathbf{k}\cdot\frac{\mathbf{p}}{m}\rho_k^\mathrm{q} = i \frac{\partial f^0}{\partial \epsilon}\mathbf{k}\cdot\frac{\mathbf{p}}{m}\mathcal{F}_k^\mathrm{q}
	\;,
\label{eq:BLE_Fourier}
\end{equation}
where $U_k^\mathrm{q}$ and $\mathcal{F}_k^\mathrm{q}$ are Fourier components of the potential $U$ and of the stochastic fluctuating field $U'$, respectively.

{\it When the fluctuation modes are of stable nature,}
the response of the system to the action of the stochastic
fluctuating field $\mathcal{F}_k^\mathrm{q}$ determines the equilibrium variance $(\sigma_k^\mathrm{q})^2$ associated to the fluctuation $\rho_\mathbf{k}^\mathrm{q}$.
	The inverse Fourier transform of $(\sigma_k^\mathrm{q})^2$
gives the equilibrium variance of spacial density correlations
\begin{equation}
	(\sigma_{\rho^\mathrm{q}})^2 \equiv \langle (\delta\rho^\mathrm{q}(\mathbf{r}))^2 \rangle
	= (2\pi)^{-3}\sum_\mathbf{k}(\sigma_k^\mathrm{q})^2 d\mathbf{k}
\end{equation}
in a cell of volume $\Delta V$ at temperature $T$.
	At equilibrium, when the level density $\mathcal{N}\equiv(g/h^3)\int \partial_\epsilon f^0 \mathrm{d}\mathbf{p}$ for a degeneracy $g$ can be defined, these variances are related to the curvature of the free energy density
$F^\mathrm{q}(k)$
through the fluctuation--dissipation theorem so that
\begin{equation}
	(\sigma_k^\mathrm{q})^2 = \frac{T}{F^\mathrm{q}(k)}\;; \;\;\; (\sigma_{\rho^\mathrm{q}})^2 = \frac{T}{\Delta V}\Big\langle\frac{1}{F^\mathrm{q}(k)}\Big\rangle_\mathbf{k}
	\;,
\label{eq:fluctuation_dissipation}
\end{equation}
where $F^\mathrm{q}(k)= \partial_{\rho^\mathrm{q}} U_k^\mathrm{q} + 1/\mathcal{N}$, and for an average $\langle\cdot \rangle_\mathbf{k}$ extending over all $\mathbf{k}$ modes.

On the other hand, for unstable modes,
the diffusion coefficient $\mathcal{D}$, or rather its
projection
on a given unstable mode $k$,  $D_k$,
determines the following evolution for the intensity of response $(\sigma^\mathrm{q}_k)^2$ for the wave number $k$~\cite{Colonna1993,Colonna1994_a}:
\begin{equation}
	(\sigma^\mathrm{q}_k)^2(t) \approx  D_k\tau_k(e^{2t/\tau_k}-1)+(\sigma^\mathrm{q}_k)^2(t\!=\!0)e^{2t/\tau_k},
\label{eq:diffusion}
\end{equation}
where both the initial fluctuation seeds $(\sigma^\mathrm{q}_k)^2(t\!=\!0)$ and the fluctuation continuously introduced by the collisional correlations contribute to an exponential amplification of the disturbance, characterised by the growth time $\tau_k$.

\subsection{Scenarios for isovector and isoscalar fluctuations}

	In the following, starting from Eq.~(\ref{eq:BLE_Fourier}),
we concentrate on the propagation of isovector modes, which are always of stable nature, and
isoscalar modes, with a special focus on unstable conditions.

	Isovector fluctuations, based on Eq.~(\ref{eq:fluctuation_dissipation}) and studied in Sec.~\ref{sec_ivfluctuations} for nuclear matter at several density values, define how isospin distributes among different phases and portions of the system.
On the other hand, isoscalar fluctuations developing in mechanically unstable nuclear matter, which rely
 on Eq.~(\ref{eq:diffusion}), studied in Sec.~\ref{sec_isfluctuations}, coincide with the process of separation of those portions of the system into fragments.
	The latter scenario has been intensively investigated~\cite{Chomaz2004} foremost because in open dissipative systems, like heavy-ion collisions, it corresponds to a catastrophic process which can lead to the formation of nuclear fragments~\cite{Tabacaru2003,Borderie2008}.
	The size-distribution of fragments and their formation time are
ruled by the dispersion relation for wavelengths related to unstable $k$ modes so that, when unstable modes succeed to get amplified, inhomogeneities develop and eventually lead to mottling patterns at later times.
	Then, in this case, isovector fluctuations define the isotopic features of fragments
and their connection to the symmetry energy~\cite{Colonna2013}.

\section{Results on isovector fluctuations
\label{sec_ivfluctuations}}

	Isovector effects in nuclear processes may arise from different mechanisms~\cite{Baran2012,DiToro2003}, like the interplay of isospin and density gradients in the reaction dynamics,
or nuclear cluster formation, or the decay scheme of a compound nucleus.
	In systems undergoing a nuclear liquid-gas phase transition, a role is played also by isospin distillation~\cite{Chomaz2004,Baran2005}, a mechanism which consists in producing a less symmetric nucleon fraction in the more volatile phase of the system along the direction of phase separation in a $\rho_n$--$\rho_p$ space, as an effect of the potential term in the symmetry energy~\cite{Ducoin2007,Colonna2008}.

	Along with these analyses, it is particularly instructive to investigate the developing of isovector fluctuations, around the mean trajectory, in two-component nuclear matter.

\subsection{Preparation of a stable and uniform system}

	We consider nuclear matter with periodic boundary conditions.
We refer the reader to the appendix~\ref{sec_appendix_parameters} for details on the parameters chosen for the calculations in the following sections.

	Selecting isovector modes ($\mathrm{q}\rightarrow \mathrm{v}$) in Eq.~(\ref{eq:BLE_Fourier}), the phase-space density corresponds to $f^\mathrm{v}=f_\mathrm{n}-f_\mathrm{p}$.
	In order to select the isovector behaviour,
we keep only the isovector contribution in the nuclear potential.
Indeed, in absence of isoscalar terms, the system is stable at all density values $\rho_0$
and one can investigate how isovector fluctuations depend on $\rho_0$.
\begin{equation}
	U^\mathrm{q}\rightarrow U^\mathrm{v} = 2[(\rho_n-\rho_p)/\rho^0]E_{\textrm{sym}}^{\textrm{pot}}\;,
\end{equation}
where $\rho^0$ is the uniform-matter density and $E_{\textrm{sym}}^{\textrm{pot}}$ is the potential term in the symmetry energy.
	Following the procedure of ref.~\cite{Colonna2013}, $U_k^\mathrm{v}$ is obtained from the above quantity by introducing an interaction range through a Gaussian smearing $g_\sigma$ of width $\sigma$, and by taking the Fourier transform; its derivative with respect to $(\rho_n\!-\!\rho_p)$ yields
\begin{equation}
	F^\mathrm{v}(k) = (2/\rho^0)E_{\textrm{sym}}^{\textrm{pot}}(\rho^0)g_\sigma(k) + 1/\mathcal{N}\;.
\end{equation}
Thus, from Eq.~(\ref{eq:fluctuation_dissipation}), we obtain the relation:
\begin{equation}
F_{\textrm{eff}}^\mathrm{v} =
	\frac{T}{2\Delta V} \frac{\rho^0}{(\sigma_{\rho^\mathrm{v}})^2} = \frac{T}{2\Delta V} \frac{\rho^0}{\langle [\delta\rho_n(\mathbf{r})-\delta\rho_p(\mathbf{r})]^2 \rangle}
	\;,
\label{eq:isovector_variance}
\end{equation}
where $F_{\textrm{eff}}^\mathrm{v}$ can be assimilated to an effective symmetry free energy which, at zero temperature and neglecting surface effects, coincides with the symmetry energy  $E_{\textrm{sym}}(\rho^0)$.

	In conventional BUU calculations, however, the smearing effect of the test particles introduces a corresponding scaling factor~\cite{Colonna1994_a}, so that
$F_{\textrm{eff}}^\mathrm{v} \approx N_{\textrm{test}} E_{\textrm{sym}}$.
	Such scaling actually reduces drastically the isovector fluctuation variance produced by the UU collision term.
	In the following, we investigate how the collision term used in the BLOB approach differs from the UU treatment.
	Since the former is not an average contribution and it acts independently of the number of test particles, we expect a larger isovector fluctuation variance.

	To prepare a transport calculation, the system is initialised with a Fermi-Dirac distribution at a temperature $T=3$~MeV; it is then sampled for several values of $\rho^0$ and the potential, restricted to the only isovector contribution, is tested for a stiff and a soft density dependence of the symmetry energy for symmetric matter (see appendix~\ref{sec_appendix_parameters}).

\subsection{isovector fluctuation variance and symmetry energy}
\begin{figure}[b!]\begin{center}
	\includegraphics[angle=0, width=1\columnwidth]{fig2}
\end{center}\caption
{
The effective symmetry energy,  scaled by $N_{\textrm{test}}$,
as extracted from SMF simulations, using  Eq.~(\ref{eq:isovector_variance}), for nuclear matter at temperature $T$ = 3 MeV and at several density values.
The different symbols correspond to the three edge sizes $l$ of the cells over which the isovector variance is evaluated.
Two parametrizations of the symmetry energy are considered in the calculations:
asy-soft (left panel) and asy-stiff (right panel).
In both panels, the dot-dashed line represents the analytical expression of the symmetry energy.
	For the asy-stiff case (right panel), an SMF calculation where the collision term is suppressed is also shown for the edge size  $l$ = 2.05 fm
(full violet line).
Error bars indicate the standard deviation of the $F_{\textrm{eff}}^\mathrm{v}/N_{\textrm{test}}$ distribution obtained considering several time instants within a large interval at equilibrium.
}
\label{fig_Fivrho}
\end{figure}
\begin{figure}[b!]\begin{center}
	\includegraphics[angle=0, width=.9\columnwidth]{fig3}
\end{center}\caption
{
	Isovector variance as a function of time calculated, in a single event,
with SMF (left) and BLOB (right),
for different values of the system density, corresponding to the different
colors and ranging from 0.02 fm$^{-3}$ (blue line) to 0.2 fm$^{-3}$ (red line).
Cells of different size $l$ are considered in the calculations:
$l$ = 1 fm (top panels), $l$ = 2.05 fm (middle panels) and
$l$ = 3.55 fm (bottom panels).
An asy-stiff form of the symmetry energy is considered.
}
\label{fig_densityvariance_versus_t}
\end{figure}

	From a set of calculations for different densities ranging from $\rho^0=0.02$ to $\rho^0=0.2$~fm$^{-3}$ we obtain a numerical solution of the r.h.s. of
Eq.~(\ref{eq:isovector_variance}) for SMF. We use $N_{\textrm{test}}$ = 40.
	We consider
an equilibrium temperature extracted for each density bin from the slope of the Fermi-Dirac distribution evolved in time.
	The isovector variance $(\sigma_{\rho^\mathrm{v}})^2$, calculated in cells of edge size $l=1$, $l=2.05$ and $l=3.55$~fm, is multiplied by $N_{\textrm{test}}$, in order to extract $F_{\textrm{eff}}^\mathrm{v}$ and to compare it with
the symmetry energy $E_{\textrm{sym}}$.
	The comparison, shown in Fig.~\ref{fig_Fivrho}, is satisfactory and it is the closest in shape to $E_{\textrm{sym}}$ for larger cells than $l=1$~fm but, however, the large scaling factor $N_{\textrm{test}}$ has to be taken into account.
	The better agreement in larger cells reflects the decreasing importance of surface effects, so that the calculation gets close to the (volume) symmetry energy.

	We notice that an equivalent calculation where the collision term is suppressed yields identical distributions; such collisionless calculation corresponds to switching off the collision term.
	The need of scaling by $N_{\textrm{test}}$ to recover the expected fluctuation value reflects the fact that isovector fluctuations
are not correctly implemented in SMF.
Indeed, much attention is paid in the model to a good reproduction of isoscalar fluctuations and amplification
of mean-field unstable modes, by introducing an appropriate external field \cite{Colonna1998}.
On the other hand, explicit fluctuation terms are not injected in the isovector channel in SMF.
	In this case, one just obtains the fluctuations related to the use
of a finite number of test particles which, as far as the Fermi statistics
is preserved, amount to the physical ones divided by  $N_{\textrm{test}}$.
	These results completes the study of ref.\cite{Colonna2013} concerning SMF.

\begin{figure}[t!]\begin{center}
	\includegraphics[angle=0, width=1\columnwidth]{fig4}
\end{center}\caption
{
The isovector variance, at equilibrium,  as a function of the system density,
calculated with SMF (full circles) and BLOB (open circles), for two
parametrizations of the symmetry energy: asy-soft (left panels) and
asy-stiff (right panels)
Cells of different size $l$ are considered in the calculations:
$l$ = 1 fm (top panels), $l$ = 2.05 fm (middle panels) and
$l$ = 3.55 fm (bottom panels).

	For one case (asystiff, $l=2.05$~fm) other calculations are shown: SMF with the collision term suppressed (full violet line), and BLOB calculations for different constant values of $\sigma_{\textrm{NN}}$ (green lines).
	Error bars indicate the standard deviation of the isovector variance distribution obtained considering several time instants within a large interval at equilibrium.
}
\label{fig_densityvariance_versus_rho}
\end{figure}

	We now turn to BLOB calculations.
	Fig.~\ref{fig_densityvariance_versus_t} shows that the isovector variance in BLOB results larger than in SMF.
	Such difference is therefore the effect of the treatment of collisional correlations in BLOB, which displays a dependence with the system density.
	In particular, the low-density limit of the spectrum corresponds to a situation where the collision rate is vanishing.
	In this case,
the BLOB procedure is practically ineffective (see also the discussion
in Section V.A)
and all approaches converge to the same isovector variance, just related to the finite number of test particles employed.
	At larger density than saturation ($\rho\ge 0.18$~fm$^{-3}$), BLOB displays a longer path to convergence which is due to the difficulty of relocating large portions of phase space in binary collisions without violating Pauli blocking.

	Fig.~\ref{fig_densityvariance_versus_rho} condenses and extends the information of Fig.~\ref{fig_densityvariance_versus_t} by displaying the density evolution of the isovector variance attained at equilibrium as evaluated in cells of different size $l$, for asy-stiff and asy-soft forms of the symmetry energy.
	The SMF data correspond to those analysed in Fig.~\ref{fig_Fivrho}.
	The BLOB spectra progressively deviate from SMF data for increasing density.
	Such deviation
increases for larger cell sizes indicating that the isovector fluctuations are better built in large volumes~\cite{Rizzo2008}.
	This is related to the variety of configurations, concerning
shape and extension of the nucleon wave
packet, which occur in the implementation of the fluctuating collision
integral. This introduces a smearing of fluctuations on a scale comparable
to the wave-packet extension in phase space.
	However, the gain in isovector variance exhibited by the BLOB approach, indicates that the dependence on $N_{\textrm{test}}$ is partially reduced with respect to the SMF scheme.

\subsection{Interference between mean-field propagation,
collisional and numerical correlations \label{sec_ivfluctuations_noise}}

	According to Eq.~(\ref{eq:BLOB}), the BLOB approach should introduce and revive  fluctuations continuously.
	The agglomeration procedure employed in BLOB is actually able to construct agglomerates of test particles of the same isospin species and which are located around local density maxima in random selected phase-space cells: this technique should preserve at least partially the isovector correlations in the system, contrarily to the usual BUU technique which smears them out.
	This advance with respect to BUU is however not sufficient because of the concurrent effects associated with the
mean-field dissipation.
Indeed, fluctuations are propagated according to a total inverse relaxation time
\begin{equation}
	1/\tau_k' = 1/\tau_k^{\textrm{coll}} + 1/\tau_k^{\textrm{m.f.}}\;,
\end{equation}
	so that, if the collisional rate is too small, they are damped by the mean-field dynamics
before they can reach a sizeable amplitude.

Moreover, even in absence of any explicit fluctuation seed, the dynamics is actually affected by a numerical noise, due to the use of a finite number of test particles in the numerical resolution of the transport
equations; such spurious contribution imposes the dependence of $(\sigma_{\rho^\mathrm{q}})^2$ on $N_{\textrm{test}}$~\cite{Colonna1993}.
	If this latter effect may be negligible with respect to the large isoscalar fluctuations
developing in presence of mean-field instabilities (see next Section),
it becomes a highly interfering contribution for the isovector modes.
In other words, the numerical noise leads to an effective
diffusion coefficient $D_k' = D_k + D_k^{\textrm{noise}}$.
If a small number of test particles is considered,  and two-body collisions
are not so frequent, then $D_k^{\textrm{noise}}$ prevails over $D_k$, causing a deviation of the fluctuation
amplitude from the correct value.

For these reasons, though in principle the fluctuation
equilibrium value, as deduced from BLOB, should not depend on the details of the nucleon-nucleon cross section $\sigma_{\textrm{NN}}$ and on the number of test particle employed,
our results depend significantly on both ingredients.

	Two ways can be tested to get a deeper insight into
this problem: either the collision term should be considerably enhanced, or fluctuations generated by
test particles should be prevented.

	The first solution can be achieved by simply multiplying $\sigma_{\textrm{NN}}$ by a large factor, with the drawback of then handling incorrect collision rates.
	Of course, this is not a problem if one is interested in equilibrated
matter, as in the present case,
but it would be crucial when dealing with non-equilibrium processes,
such as nuclear reactions.
	Some tests in the first direction are proposed in Fig.~\ref{fig_densityvariance_versus_rho}, by employing a constant $\sigma_{\textrm{NN}}$ with progressively larger values, showing that the isovector variance grows with the collision rate, as we expect on the basis of the arguments discussed above.

\begin{figure}[b!]\begin{center}
	\includegraphics[angle=0, width=.95\columnwidth]{fig5}
\end{center}\caption
{
	Left:
 The isovector variance, at equilibrium, as a function of the number of test
particles, $N_{\textrm{test}}$, as obtained in SMF simulations (open circles) and in BLOB calculations (open squares), using a constant $\sigma_{\textrm{NN}}$=50mb, for nuclear
matter at density $\rho^0$ = 0.08 fm$^{-3}$ and temperature $T=3$~MeV.
BLOB calculations are also shown for $\sigma_{\textrm{NN}}$=100mb (green full circles).
Cells of different size $l$ are considered, as indicated on the figure.
Dashed lines represent a fit of SMF simulations, assuming a trend proportional
to the inverse of the test particle number.
	Right:
Time evolution of the isovector variance, as obtained in a single event, for
BLOB calculations with constant $\sigma_{\textrm{NN}}$=100mb and $l=2.05$fm.
The different curves correspond to different number of test particles employed
in the simulations.
The asy-stiff form of the symmetry energy is considered.
}
\label{fig_Ntest}
\end{figure}
	The second check would consist in employing the largest possible number of test particles per nucleon.
	In this case, the collisionless transport model would ideally correspond to the Vlasov approach and, when collisional correlations are introduced, interferences with spurious stochastic sources can be highly reduced.
	However, interference effects with the mean-field propagation can still be important,
depending on the collision rate.
	As far as numerical complexity can be handled, Fig.~\ref{fig_Ntest}, left, illustrates such situation:
SMF calculations show a behaviour $\propto 1/N_{\textrm{test}}$, independently of $\sigma_{\textrm{NN}}$.
	On the other hand, in the BLOB case
one observes that,  especially in the
largest cells, where fluctuations are more effective,
the corresponding variance deviates more and more, for large test particle numbers,
from the SMF results, reaching a kind of saturation value.  The latter depends on the cell size, $l$,
and on the cross section employed (see also the discussion above).
	Fig.~\ref{fig_Ntest}, right, shows the time evolution of the fluctuation variance. It appears that
for a small number of test particles (up to 100) the variance oscillates around its initial value, which is
essentially associated with the numerical noise and scales as $1/N_{\textrm{test}}$.  Increasing the number of test particles,
the numerical noise gets smaller and the BLOB fluctuation source prevails on it, building up a fluctuation variance
which is larger than the initial value.
	However, since	we are considering systems at low
temperature,
the number of nucleon-nucleon collisions is extremely low and insufficient to rapidly introduce a pattern of
isovector correlations, unless one employes very high values for the cross section: the isovector variance shows in fact a very gentle growth.

	In conclusion, the BLOB fluctuation source term works well in conditions
where the collision rate is large enough, as compared to the mean-field propagation and to the spurious
dissipative terms associated with the finite number of test particles.
	These conditions are likely reached in the first, non equilibrated stages of heavy ion collisions,
but not necessarily for equilibrated nuclear matter at low temperature. In the latter case, the variance associated with the
fluctuating collision integral can be recovered by artificially increasing the employed $\sigma_{\textrm{NN}}$.
Indeed, as shown in Fig.~\ref{fig_densityvariance_versus_rho} and Fig.~\ref{fig_Ntest},
we observe that the fluctuation variance built
by BLOB may deviate significantly from the SMF results, being up to a factor ten larger, especially when considering
fluctuations in larger cells ($l\approx$ 2 to 3 fm).

\section{Results on isoscalar fluctuations \label{sec_isfluctuations}}

	If fluctuation seeds are introduced in homogeneous neutral nuclear matter at low temperature, Landau zero-sound~\cite{Landau1957} collective modes should stand out and propagate in the system.
	In the present section we analyse whether the BLOB approach is able to develop, as aimed, isoscalar fluctuations of correct amplitude in nuclear matter spontaneously, and not from an external contribution, when the system is placed in a dynamically unstable region of the equation of state~\cite{Belkacem1994}, like the spinodal zone.
	In this circumstance, as soon as fluctuation seeds are generated, unstable zero-sound waves should be amplified in time.
	In the opposite situation, in conditions of mechanical stability, undamped stable zero-sound waves propagate.
	Then, for stable configurations, the same argumentations of Sec.~\ref{sec_ivfluctuations} hold and, in this case, the fluctuation variance is linked to matter incompressibility.

\subsection{Sampling zero-sound propagation in mechanically stable and unstable nuclear matter}

	The  propagation
of fluctuations in
nuclear matter can be described in a linear-response approximation~\cite{Colonna1994_a} as far as deviations from the average dynamical path are small.
	In Eq.~(\ref{eq:BLE_Fourier}), by selecting isoscalar modes ($\mathrm{q}\rightarrow \mathrm{s}$, we drop the $\mathrm{s}$ index in the following),
and setting residual contributions to zero, we obtain a linearised Vlasov equation in terms of frequencies $\omega_k$ to describe stable matter with isoscalar contributions:
\begin{equation}
	\omega_k f_k + \mathbf{k}\cdot\frac{\mathbf{p}}{m}f_k - \frac{\partial f^0}{\partial \epsilon}\frac{\partial U_k}{\partial \rho} \mathbf{k}\cdot\frac{\mathbf{p}}{m} \rho_k = 0
	\;.
\label{eq:linearised_Vlasov}
\end{equation}
	Different wave numbers $\mathbf{k}$ are decoupled, each linked to a collective solution $f_k$ given by the Fourier-transformed equation of motion.
	By applying the self-consistency condition $\rho_k = (g/h^3)\int f_k(\mathbf{p}) \mathrm{d}\mathbf{p}$, we obtain the dispersion relation for the propagation of density waves in Fermi liquids at $T=0$ :
\begin{equation}
	1 = \frac{g}{h^3}  \frac{\partial U_k}{\partial\rho} \int \frac{\partial f^0}{\partial\epsilon}  \frac{\mathbf{k}\cdot\mathbf{p} /m}{\omega_k+\mathbf{k}\cdot\mathbf{p} /m} \mathrm{d}\mathbf{p}
	\;.
\label{eq:dispersion_relation_primitive}
\end{equation}
where $\omega_k$ and $-\omega_k$ are pair solutions due to the invariance $\mathbf{p} \leftrightarrow -\mathbf{p}$.
	As well documented in the literature, at $T=0$ eigenmodes $f_k$ depend
only on states near the Fermi level.
	Indeed, the momentum integral is
restricted to the Fermi surface because
$\partial_\epsilon f^0 \approx -\delta(\epsilon-\epsilon_{\textrm{F}})$, being $\epsilon_{\textrm{F}}$ the Fermi energy.
The dispersion relation reduces to an expression where solutions correspond to sound velocities $s=\omega_k / (k v_\textrm{F})$ in units of Fermi velocity  $v_\textrm{F} = p_\textrm{F}/m$.
	In this case, introducing the Landau parameter
\begin{equation}
	F_0(\mathbf{k})=\mathcal{N}_0\partial_\rho U_k= (3/2)(\rho^0/\epsilon_{\textrm{F}})\partial_\rho U_k\;,
\end{equation}
where $\mathcal{N}_0 = \mathcal{N}(T=0)$ is linked to the number of levels at Fermi energy $\epsilon_{\textrm{F}}$, the dispersion relation takes the form~\cite{Kalatnikov1958}:
\begin{equation}
	1 + \frac{1}{F_0} = L(s) = \frac{s}{2} \textrm{ln} \left(\frac{s+1}{s-1} \right)
	\;,
\label{eq:disprel_T0}
\end{equation}
where the  Lindhard function, $L(s)$ has been introduced.
	In correspondence to the effective interaction employed, reflected in the potential shown in Fig.~\ref{fig_disp_rel_analytical}a,
	the Landau parameter is illustrated in Fig.~\ref{fig_disp_rel_analytical}b, while Fig.~\ref{fig_disp_rel_analytical}c presents the roots of the dispersion relation, corresponding to the effective interaction.

\begin{figure}[t!]\begin{center}
	\includegraphics[angle=0, width=1\columnwidth]{fig6}
\end{center}\caption
{
(a) The isoscalar mean-field potential as a function of the density. The dots indicate
the two density values considered in SMF/BLOB simulations.
(b) The Landau parameter $F_0$ as a function of the density.
(c) real (for $F_0>0$) and imaginary (for $F_0<0$) roots of the dispersion relation; the spinodal region corresponds to $F_0<-1$, the region for $-1<F_0<0$ corresponds to Landau damping and positive values of $F_0$ define stable modes.
Blue dots indicate the $\gamma$ values
corresponding to the density $\rho^0$ = 0.053 fm$^{-3}$.
The $\gamma$ value corresponding to $\rho^0$ = 0.14 fm$^{-3}$ is beyond the
interval shown in the figure (in the direction of the green arrow).
(d) The chemical potential, divided by the Fermi energy, as a function of the temperature $T$, for the two density values considered in the calculations:
 $\rho^0$ = 0.053 fm$^{-3}$ (blue line) and  $\rho^0$ = 0.14 fm$^{-3}$ (green line).
The dots correspond to the temperature value ($T=3$~MeV) considered
in the simulations.
}
\label{fig_disp_rel_analytical}
\end{figure}
\subsection{Warm systems and interaction range}

	Eq.~(\ref{eq:disprel_T0}) is only valid at zero temperature.
	When the temperature is significant, two-body collision rates become prominent and these mean-field dominated zero-sound waves are absorbed and taken over by hydrodynamical first-sound collective modes.
	Since our approach exploits two-body collisions to introduce fluctuations in a self-consistent mean field, we expect the possible occurrence of a zero-to-first-sound transition which, at variance with other Fermi liquids~\cite{Abel1966}, should be even smeared out due to the small values taken by the Landau parameter $F_0$ in nuclear matter.
	It was found that, depending on how the system is prepared and on the type of collective motion, such transition should arise in a range of temperature from 4 to 5 MeV and occur as late as 200~fm/$c$~\cite{Larionov2000, Kolomietz1996}.
	In practice, zero-sound modes associated to wave vectors $\mathbf{k}$ characterise the system as long as the corresponding phase velocity exceeds the velocity of a particle on the Fermi surface $v_{\textrm{F}}$ or, equivalently, as long as the corresponding frequency $\omega_k$ is much higher then the two-body collision frequency $\nu$.
	These premises imply that, after defining a homogeneous initial configuration at a suited finite and not so large temperature, we should study early intervals of time to extract properties of the response function which can be compared with zero-sound conditions.

	Temperature effects can be included, in an approximate manner, considering the low temperature Sommerfeld expansion of the chemical potential $\mu$(T):
\begin{equation}
	\frac{\mu(T)}{\epsilon_{\textrm{F}}} \approx 1-\frac{\pi^2}{12} \left(\frac{T}{\epsilon_{\textrm{F}}}\right)^2
	\;,
\label{eq:wall_dissipation}
\end{equation}
which is  illustrated in Fig.~\ref{fig_disp_rel_analytical}d.

	As a further modification, we consider that zero-sound conditions also present a strong dependence on the interaction range.
	This latter can be included in the dispersion relation by applying a Gaussian smearing factor of the mean-field potential
which is related to the nuclear interaction range in configuration space~\cite{Colonna1994,Kolomirtz1999}.

\begin{equation}
	U \rightarrow U \otimes g(k), \;\;\textrm{with} \;\; g(k)= \textrm{e}^{-\frac{1}{2} (k\sigma)^2}
	\;,
\label{eq:Gaussian_smearing}
\end{equation}

From Eq.~(\ref{eq:wall_dissipation}) and Eq.~(\ref{eq:Gaussian_smearing}),
the dispersion relation, Eq.~(\ref{eq:disprel_T0}), involves
an effective Landau parameter \cite{Colonna1994},
\begin{equation}
	\widetilde{F}_0(k,T) = \frac{\mu(T)}{\epsilon_{\textrm{F}}}{F}_0g(k)
	\;.
\label{eq:effectiveF}
\end{equation}

	Mechanically unstable conditions are experienced when the evolution of local density $\rho$ and pressure $P$ implies that the incompressibility
is negative.
	This situation is reflected by an effective Landau parameter $\widetilde{F}_0(k=0,T)$
smaller than $-1$, so that
\begin{equation}
	\frac{\partial P}{\partial\rho} \approx \frac{2}{3}\epsilon_{\textrm{F}}
[1+\widetilde{F}_0(k=0,T)] \;\; <\,0
	\;,
\label{eq:chi_Landau}
\end{equation}
and it corresponds to imaginary solutions of the dispersion relation~\cite{Pomeranchuk1959}.
	By replacing $s\rightarrow i\gamma$, the relation yielding imaginary solutions can be put in the form:
\begin{equation}
	1 + \frac{1}{\widetilde{F}_0(k,T)}
= \gamma\,\textrm{arctan}\frac{1}{\gamma}
	\;.
\label{eq:disprel_gamma}
\end{equation}

The growth rate $\Gamma_k=1/\tau_k$ is obtained from the solutions of the dispersion relation
\begin{equation}
|\gamma| = \frac{|\omega_k|}{kv_{\textrm{F}}} =
\frac{1}{\tau_k k v_{\textrm{F}}}
	\;,
\label{eq:gamma}
\end{equation}

	As far as the Fermi statistics is kept in a sufficiently large periodic portion of mechanically unstable nuclear matter, and a fluctuation source term
is acting,
the expectation is that the intensity of the response should be amplified with the growth rate
$\Gamma_k$
imposed by the mean-field potential $U$ as a function of the unstable mode $\mathbf{k}$.

\subsection{Obtaining the dispersion relation \label{sec_disprel}}

	To check such expectation numerically through a BL transport approach
	we keep the same scheme for the definition of the box metrics as in Sec.~\ref{sec_ivfluctuations}; the isoscalar density variance is calculated over cells of edge size $l=1$~fm.
	We now use the full parametrisation of the energy potential per nucleon Eq.~(\ref{eq:pot}), where we use a stiff density dependence of
	the symmetry energy
(the same parametrisation was analysed in ref.~\cite{Colonna1997}).
	Nuclear matter is isospin symmetric and it is initially uniform and prepared at a temperature $T=3$~MeV and at densities equal to $\rho^0\!=\!0.053$ and $0.14$~fm$^{-3}$.
	Fig.~\ref{fig_disp_rel_analytical} illustrates the values taken by the potential and by the dispersion relation related to these choices.
	The collision term involves the usual isospin- and energy-dependent free nucleon-nucleon cross section with an upper cutoff at $\sigma_{\textrm{NN}}=50$~mb.

	Within the dynamical calculation
we should register at each interval of time $t$ the density in all cells of edge size $l$ of the lattice which constitutes the periodic system of edge size $L$.
	A specific cell can be identified by the vector $\mathbf{n}'$.
Having introduced such a lattice, the perturbation wave number $k$
can be expressed as $k=2\pi n /L$, where $n$ is the modulus of
a vector ranging from 1 to $n_{\textrm{max}} = L/l$ along each of the three spacial directions.
	Then the amplitude of the isoscalar fluctuation of a mode $k$ is obtained from the Fourier transform, $F_\mathbf{k}$,
of the space density
\begin{eqnarray}
	&&\sigma^2_k(t) = \langle F_\mathbf{k}^2(t)\rangle =
	\frac{1}{l^3}\Big\langle\Big[\sum_{\mathbf{n}'}\rho_{\mathbf{n}'}(t)\,\mathrm{exp}\big(ia\mathbf{n}\!\cdot\!\mathbf{n}'\big)\Big]^2\Big\rangle \\\notag
	&&\propto \Big\langle\Big[\sum_{\mathbf{n}'}\rho_{\mathbf{n}'}(t)\textrm{cos}(a\mathbf{n}\!\cdot\!\mathbf{n}')\Big]^2 \!+\! \Big[\sum_{\mathbf{n}'}\rho_{\mathbf{n}'}(t)\textrm{sin}(a\mathbf{n}\!\cdot\!\mathbf{n}')\Big]^2\Big\rangle
	\;,
\label{eq:isoscalar_amplitude}
\end{eqnarray}
where $a=2\pi  l/L$ and the average is extended over all orientations of $\mathbf{k}$.

	The distribution of ratios $\tilde{\sigma}^2_k(t) = \sigma^2_k(t)/ \sigma^2_k(t=0)$, averaged over several dynamical paths, is shown at different time intervals in Fig.~\ref{fig_isoscalar_variance} for the two density choices.
It should be noticed that the initial fluctuation amplitude is due
to the finite number of test particles employed in the calculations.
However, as soon as the BLOB term starts to act, fluctuations of larger amplitude are built up (see also the discussion in Section V.A), and further amplified by the unstable mean-field.
	The system prepared at $\rho^0\!=\!0.053$~fm$^{-3}$, inside the spinodal region, exhibits a clear  growth of instabilities as a function of time for some $k$ waves, while the system prepared at $\rho^0\!=\!0.14$~fm$^{-3}$, outside the spinodal region, presents an evolution of the response intensity corresponding to a more gentle growth in a Landau-damping regime.
	However, even in this latter case, fluctuations reach a significant amplitude, owing to the small compressibility value in the density region considered, but cluster formation is not observed.

\begin{figure}[b!]\begin{center}
	\includegraphics[angle=0, width=1\columnwidth]{fig7}
\end{center}\caption
{
	BLOB calculation. Response intensity $\tilde{\sigma}^2_k(t) = \sigma^2_k(t)/ \sigma^2_k(t=0)$, averaged over 100 dynamical paths,
as a function of the wave number $k$. The different curves correspond to different time instants, as indicated on the figure.
Left panel: results for $\rho^0\!=\!0.053$~fm$^{-3}$ (spinodal).
Right panel: results for $\rho^0\!=\!0.14$~fm$^{-3}$ (Landau damping).
}
\label{fig_isoscalar_variance}
\end{figure}
\begin{figure}[t!]\begin{center}
	\includegraphics[angle=0, width=.95\columnwidth]{fig8}
\end{center}\caption
{
	Early time evolution of the response intensity $\tilde{\sigma}^2_k(t)$ for several modes ($n=j$ stands for all $k$ modes within $2\pi(j-1)/L
\le k< 2\pi j/L$).
	(a) BLOB calculations for $\rho^0\!=\!0.053$~fm$^{-3}$ (spinodal).
	(b) SMF calculations for $\rho^0\!=\!0.053$~fm$^{-3}$.
	(c) BLOB calculations for $\rho^0\!=\!0.14$~fm$^{-3}$ (Landau damping).
}
\label{fig_is_variance_early}
\end{figure}
\begin{figure}[t!]\begin{center}
	\includegraphics[angle=0, width=1\columnwidth]{fig9}
\end{center}\caption
{
BLOB calculations for nuclear matter at $\rho_0$ = 0.053 fm$^{-3}$ and
temperature $T$ = 3 MeV: The growth rate $\Gamma_k$ , as extracted from the
simulations (see Eq.(35)), is plotted as a function of the wave number $k$ (full
line with dots).
	Uncertainties are evaluated from the linear-response fit.
The dashed lines correspond to analytical predictions, obtained solving Eq.(32)
with different values of the width $\sigma$ of the Gaussian smearing factor.
	For comparison, a calculation in one-dimension within the approach of ref.~\cite{Rizzo2008} is also shown (green diamonds).
}
\label{fig_growth_rate}
\end{figure}

	Correspondingly, the early evolution in time of $\tilde{\sigma}^2_k(t)$ is analysed in Fig.~\ref{fig_is_variance_early} for the two density choices.
For the leading modes, one can consider a linear fit of
the quantity plotted in Fig.~\ref{fig_is_variance_early}a,
in intervals ranging from around 20fm/$c$ to time instants close to saturation
(see Eq.~\ref{eq:diffusion}).
	The very initial path is excluded from the fit because, as previously mentioned, the fluctuation mechanism sets in spontaneously after that a sufficient number of collisions has occurred, and does not emerge from suited initial conditions.
	Differences from the ideal linear response in the growing side of single modes indicate a more complex
behaviour, resulting from the coupling of different wavelengths and the tendency toward a chaotic evolution~\cite{Baldo1995};
	A SMF calculation is also presented for the unstable system, where the linear growth of the leading modes is initially comparable to the BLOB approach and deviates at later times.
	This behaviour is due to the efficiency of the collision term in the BLOB model in reviving fluctuations of correct amplitude, compared to SMF, where fluctuations are not introduced by the collision term.
	As a consequence, in SMF fluctuations decay by the combination of small wavelengths into larger ones, while in BLOB higher fluctuation amplitudes can be attained before reaching the saturation regime.

	The numerical extraction of the growth rate $\Gamma_k$,
i.e., of the quantity given by
the analytic relation of Eq.~(\ref{eq:gamma}) is obtained from the time derivative
(at early time instants) of the amplitude of the isoscalar fluctuation for a given mode $k$ as
\begin{equation}
\Gamma_k = \frac{1}{2} \frac{\partial}{\partial t} {\textrm ln}\prec \tilde{\sigma}^2_k(t) \succ
	\;,
\label{eq:tau_numerical}
\end{equation}
where the average $\prec \cdot \succ$ is taken over several stochastic dynamical trajectories.
	Such analysis is presented in Fig.~\ref{fig_growth_rate}, where the numerical calculation, averaged over 100 events, is compared to the analytic result of Eq.~(\ref{eq:gamma}).
	The range of the interaction, as an effect of the implemented surface term, would correspond to a Gaussian smearing of around $\sigma=0.8$~fm to $\sigma=0.9$~fm.
	We infer that BLOB reproduces consistently the expected dispersion relation within the uncertainties of the linear regression.
	Another calculation, also based on the same mean-field, but which employs the earlier approach of ref.~\cite{Rizzo2008}, also solved in three dimensions but with fluctuations developing along one axis of configuration space, produces a similar result.
	While BLOB keeps the different unstable modes decoupled for a more extended interval of time during their early growth, also resulting into a larger ultraviolet cutoff, the other
approach (green points) presents some alterations due to the combining of unstable modes, where small-wavelength ($n\!>\!7$) are gradually absorbed by large-wavelength ($n\!=\!0,n\!=\!1$).
	The effect in this case is an increase of the growth rate for small $k$ values and it signs the entrance of the chaotic behaviour which characterises larger times~\cite{Jacquot1996}.
	The largest $k$ modes, corresponding to wavelengths which drop below the Gaussian smearing width $\sigma$ are meaningless.
	As a final result of this study, the leading modes are found in a wavelength range from 8 to 9fm, and for a growth time $\tau_k$ of around 30fm$/$c.

\section{Connecting nuclear matter to open systems \label{sec_open}}
	The aim
of this work is studying the effect of isovector and isoscalar fluctuations.
	The ultimate purpose of the transport approaches discussed therein is describing the formation of nuclear fragments in a fermionic system and their properties through the combination of these two types of fluctuating modes, as will be detailed more diffusely in forthcoming works.
	In particular, isovector fluctuations, on top of other isospin transport effects, impose that the isospin content is distributed through a density-dependent process of distillation, supplemented by an isotopic variance.
	The onset of isoscalar modes is then responsible for breaking the uniformity of the density landscape and eventually partitioning it into nuclear fragments, where the isospin properties of the initial nesting sites are preserved.
	The isoscalar and isovector mechanisms should therefore be intimately connected in order to describe fragment formation.

\subsection{Fragment formation: patterns and time scales}
	Qualitatively, we may underline some connection between the wavelengths involved in the dispersion relation analysed in Fig.~\ref{fig_growth_rate}, and fragment formation~\cite{Matera2000}, considering that at the system density $\rho^0$ the leading modes correspond to fragments of mass $A\approx \rho^0 \lambda^3$; for the leading wavelengths, this corresponds to a distribution of sizes peaked around Neon.
	These results are also in agreement with other previous studies where quantum effects were taken into considerations explicitly~\cite{Ayik1995,Jacquot1997,Colonna1998bis,Norenberg2000,Ayik2008} despite a more schematic treatment of fluctuations, or of the whole dynamics (2-dimension treatments, fluctuations propagated from an initial state, spherical geometries).
	In this respect, BLOB extends these previous attempts to a model that can be applied at the same time to nuclear matter and, rather successfully, to heavy-ion collisions in three dimensions and without any preliminary initialisation of fluctuation seeds \cite{Napolitani2013,Napolitani2015}.

	From the growth time of the leading modes in Fig.~\ref{fig_growth_rate}, we infer that the corresponding process of fragment formation would be rather short, progressing from when the system has been largely diluted.
	This suggests that the scenario studied in nuclear matter can be quite directly translated to the phenomenology of open systems~\cite{Chomaz2004}.
	As an example, Fig.~\ref{fig_mottling} illustrates the correspondence between a portion of nuclear matter (simulated for $T=3$~MeV and $\rho^0\!=\!0.053$~fm$^{-3}$ for an interaction defined as in Eq.~(\ref{eq:pot}) and a hot system formed in the collision $^{136}$Xe$+^{124}$Sn at 32~$A$MeV for a central impact parameter $b=0$ (such system was studied in an experimental campaign~\cite{Moisan2012,Ademard2014}).
	In particular, we observe some analogy between the early time when inhomogeneities emerge in nuclear matter (20fm/$c$) and when fragments start forming in an open system (around 100fm/$c$) right after accessing low-density spinodal conditions (around 80fm/$c$).
	In both systems, a spinodal signal stands out by exhibiting equal-size inhomogeneities in configuration space within a similar time scale~\cite{Napolitani2013}, and it is smeared out by fragment recombination later on.
	At even later times, the evolution is different, in the box calculation clusters continue interacting with each other while in the open system they split apart.
\begin{figure}[t!]\begin{center}
	\includegraphics[angle=0, width=.85\columnwidth]{fig10.eps}
\end{center}\caption
{
	Density-landscape at several time instants
for nuclear matter in a periodic box at $\rho^0\!=\!0.053$~fm$^{-3}$
and T = 3 MeV (left) and in a hot open nuclear system formed in a head-on $^{136}$Xe+$^{124}$Sn collision at 32~$A$MeV (right); arrows indicate the beam direction. Both simulations employ the BLOB approach with the same mean-field properties.
	The big arrow proposes an analogy between nuclear matter and the open system in correspondence with the arising of the spinodal instability.
}
\label{fig_mottling}
\end{figure}

	Experimental investigations of heavy-ion collisions at Fermi energies already pointed out that the range of masses given by the dispersion relation of Fig.~\ref{fig_growth_rate} is actually favoured in multifragmentation mechanisms; the kinematics of the process was also found to be rather explosive.
	The spinodal mechanism was therefore proposed as a suited description~\cite{Borderie2001,Desesquelles2002};
	BLOB has already been adapted successfully to nuclear collisions and tested over various systems which experience spinodal instability~\cite{Napolitani2013,Napolitani2015,Colonna2016}.

\subsection{Isospin content in fragments}

	Fig.~\ref{fig_isotopicdistributions} completes the survey, investigating the isospin content in potential ripples containing $N'$ neutrons and $Z'$ protons for the system $^{136}$Xe$+^{124}$Sn at 32~$A$MeV.
	Distributions of isotopic variances are calculated for the most probable mass range around a forming Carbon and a forming Neon (two upper rows).
	The distributions are studied in an early time span (before that
fragments are clearly formed,
around 130fm/$c$) and in a late time interval (during fragment formation around 200fm/$c$).
	They are compared with the analytic distributions obtained at a temperature $T=5.5$~MeV, as extracted from the calculation, and at the local density $\rho$.
	The isotopic variance (see Eq.~(\ref{eq:fluctuation_dissipation})) can be studied as the probability of variation $\delta$ around the mean value of $N'-Z'$ and for a given $A'$ yielding the distribution
\begin{equation}
	Y \approx \textrm{exp}[-(\delta^2/A')\,C_{\textrm{sym}}(\rho)/T]\;.
\end{equation}
	The local density is evaluated either in the (denser) centroid of the potential ripples $\rho_{\textrm{centroid}}$, or averaged all over the volume of the emerging fragments $\widetilde{\rho}$, or, more significantly, corresponding to the matter contained in the volume of the potential ripples $\rho_{\textrm{well}}$.
	We deduce that, as expected from the calculation in stable nuclear matter discussed above, the isotopic width results underestimated with respect to the analytic prediction of Eq.~(\ref{eq:fluctuation_dissipation}).
The difference is still acceptable due to the following two effects.
	First of all, fluctuations are built out of equilibrium: this implies that the collision rate is higher, generally leading to larger variances.
	Secondly, in open systems, particle evaporation may contribute in widening the isotopic spectra.

	The bottom row of Fig.~\ref{fig_isotopicdistributions} investigates the average isospin content measured in potential ripples in the corresponding early and late time intervals as a function of the $\rho_{\textrm{well}}$.
	The
mass $A'$ grows with the density and the corresponding isospin content decreases, signing a process of isospin distillation.

\subsection{Extent of different clusterisation processes in heavy-ion collisions}
\begin{figure}[b!]\begin{center}
	\includegraphics[angle=0, width=.85\columnwidth]{fig11a}\\
	\includegraphics[angle=0, width=.85\columnwidth]{fig11b}
\end{center}\caption{
	BLOB simulation of head-on $^{136}$Xe+$^{124}$Sn collisions at 32~$A$MeV.
	Upper row:
Isotopic distribution (full dots) in potential ripples containing $N'$ neutrons and $Z'$ protons for forming-cluster configurations with masses around $A'=15$ (left)
and $A'=24$ (right)  at t = 130 fm/c.
The lines correspond to analytic distributions (see Eq.(36)), corresponding to
the density extracted from different portions of potential ripples (see text).
Middle row: The same as in the upper row, but at t = 200 fm/c.
	Bottom. Average isospin content measured in potential ripples at two different times, as indicated on the figure,  as a function of $\rho_{\textrm{well}}$. Larger density values correspond to pre-fragments of larger mass.
The dot-dashed line with an arrow indicates the asymmetry of the projectile-target composite system.
}
\label{fig_isotopicdistributions}
\end{figure}

	The processes discussed in this work, related to phase-space fluctuations and mechanical instabilities, involve a rich phenomenology of phase transitions and thresholds between very different reaction mechanisms.
	They may therefore also present some similarities in their outcome with other rather different processes and, in some situations, combine with them.
	For instance, the onset of instabilities of Rayleigh type is a common process in macroscopic hydrodynamic systems, like classical fluids with a non-negligible surface tension~\cite{Ashgriz1990}, which has also been proposed as a possible additional scenario for nuclear multifragmentation in heavy-ion collisions~\cite{Moretto1992,Lukasik1997}, in some specific situations.
	Such process occurs in systems where a dilute core expands into a denser shell (Rayleigh-Taylor instability), or it acts on very deformed systems involving cohesional forces which respond to external perturbations (Plateau-Rayleigh instability).
	The system in such hydrodynamic scenario develops hole nucleation, evolving into a sponge-like or a filamented configurations which then relax into compact droplets.
	The mechanism is faster than ordinary fission and density variations of bound matter along the process do not need to be significant.

	On the other hand, the spinodal process described above can only occur if the system traverses a specific region of the equation of state, characterised by negative incompressibility where nucleation progresses from a dilute phase, letting blobs of larger density gradually emerge.
	From a microscopic point of view, it is rather associated to the nuclear liquid-gas phase transition and it requires a time comparable to the equilibration time of the system
in reactions at Fermi energies.
	We point out that, in simulations of heavy-ion collision, the BLOB approach is actually able to describe the interplay between spinodal processes and the above mentioned hydrodynamic effects~\cite{Napolitani2016}.

	Heavy-ion collisions and nuclear matter also involve processes of nuclear cluster formation, from light charge particles to heavier nuclear molecules, but those products emerge from an even different mechanisms~\cite{Typel2014}, which would require the explicit inclusion of additional correlations in the hierarchy of Eq.~(\ref{eq:BBGKY}).
	Light charged particles related to nuclear clustering have too small size, exceeding the ultraviolet cutoff of the dispersion relation, so that they can not belong to the unstable multipole modes which characterise spinodal fragmentation.
	Solutions for an explicit treatment of cluster formation are proposed in refs.~\cite{Danielewicz1991,Kuhrts2001,Ono2016}.
	Connections between nuclear clustering and (spinodal) multifragmentation might be proposed, considering that multifragmentation might act on defining nuclear sources with rather complex shape from which clustered structures might eventually emerge.

\section{Conclusions  \label{sec_conclusion}}

	This work presents crucial steps to validate BL transport models applied to a fermionic system,
both in stable and mechanically unstable conditions, as far as the development of isoscalar and isovector
fluctuations at various densities is concerned.
	In particular, the amplitude of fluctuations
is investigated in relation with the corresponding properties of the nuclear effective interaction.
	A transport approach constructed by requiring to satisfy the dispersion relation of
mean-field unstable modes proves to be suited for the description of nuclear multifragmentation.

	In practice, even though technically demanding, the BLOB approach constitutes a conceptually  straightforward solution of the BLE in three dimensions.
	Through a simple renormalisation of the collision term, the Fermi statistics is in fact preserved for a long time and a correct isoscalar fluctuation amplitude is obtained independently of the ingredients of the numerical implementation (like the number of test particles), just associated with the collision rate and the growth time of the unstable modes.
	Also the fluctuation variance of isovector observables are better treated than in conventional semiclassical approaches, even though the expected variance is still not achieved.
	Indeed, in the implementation of the fluctuating collision integral, different configurations, varying
in shape and extension, are possible to represent the nucleon wave packet.
	This induces smearing effects on the fluctuation amplitude.
	Moreover, if the collision rate is very low, the characteristic time scales associated with the construction of collisional two-body correlations are larger than the typical mean-field time scale and the fluctuations are damped by the propagation in the stable mean-field.
	This explains why, even for equilibrated nuclear matter, the model yields a dependence of the isovector fluctuation amplitude on the nucleon-nucleon cross section.
	It is also observed that, if two-body collisions are too rare, the numerical noise dominates the dynamics and one obtains a fluctuation variance reduced by a factor $1/N_{\textrm{test}}$ with respect to the expected value, as obtained in standard transport approaches where fluctuations are neglected~\cite{Bonasera1994}.
	On the other hand, if the collision rate is large enough, the fluctuation amplitude does not depend on the number of test particles employed in the simulations.

	It is worth noting that the dynamical approach presented in this work does not imply any thermodynamic hypothesis (equilibration for instance) in the implementation of the fluctuation source, so that the characteristic thermodynamic features of multifragmentation, like the occurrence of a nuclear liquid-gas phase transition, are obtained as a result of the transport dynamics~\cite{Napolitani2013}: this finding makes the present dynamical description and alternative statistical approaches for multifragmentation mutually consistent.
	Finally, this approach can easily connect nuclear matter to heavy-ion collisions in the same framework.

\begin{acknowledgments}

	This project has received funding from the European Unions Horizon 2020 Research and Innovation
Programme under Grant Agreement No. 654002.

	Research conducted in the scope of the International Associated Laboratory (LIA) COLL-AGAIN.

\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


text:
Q: What is to the word subside as procession is to that of proceed What is the noun form of "subside"?

The process of a migraine subsiding could be called the migraine's ___

A: Subsidence
/səbˈsaɪdəns; ˈsʌbsɪdəns/
noun
1.
the act or process of subsiding or the condition of having subsided
2.
(geology) the gradual sinking of landforms to a lower level as a result of earth movements, mining operations, etc
Unfortunately the rest of your sentence doesn't make much sense as written. Maybe you mean "the process of the migraine subsiding could be described as the migraine's subsidence."?

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Running a windjammer schooner is not all clear sailing for John Finger and Anne Mahle.
NEW YORK(CNN/Money) - Making money sailing a two-masted wooden schooner through the sublime waters of Maine's Penobscot Bay sounds like anything but work, and Jon Finger and Anne Mahle, owners of the J&E Riggin, do love it.
Turning a love of sailing into a living can be tough, though.
High overhead: "We didn't expect there to be so much maintenance," says Mahle. The cost of keeping a 90-foot, wooden-hulled sailboat in trim rises every year, usually faster than inflation. Insurance costs have spiked and many other expenses -- hiring a crew, advertising, and promotion -- are up.
Spotty bookings: Changes in the economy, politics, and the price of gas can have a major impact on how many people sign up for a voyage.
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Price resistance: With few luxuries, the Riggin is a far cry from the Queen Mary. Cabins are tiny, berths narrow, and the two phone-booth-sized heads are on the deck, a climb up from the cabins. The lack of amenities does affect how much you can charge. A typical sail costs between $140 and $170 per person per day, depending on length. It's difficult to squeeze much more out of passengers.
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Mahle, who also has her skipper's license, keeps guests happy by plying her advanced cooking skills on the galley's big, wood-burning range. She trained in restaurants, at home, and at the Culinary Institute of America.
Their talents keep clients coming back: 60 percent of their bookings are returns or referrals, according to Mahle.
Finger is low-key while Mahle is enthusiastic and bubbly. Her personality recently played out on national TV with a spot on "The Today Show," in which she cooked up one of the main dishes she features aboard the Riggin.
Evenings, they often entertain guests with songs and stories. Finger plays guitar and Mahle sings well.
They have also experimented with theme cruises. This year, they'll have a knitting trip, where participants can learn new techniques from an expert.
They bought the Riggin in 1997 -- paying 20 percent down and financing about $320,00. At first, business was ship-shape; they had five consecutive years of increasing bookings of 10 percent annually. But the stock market meltdown, followed by 9/11, took some of the wind out of their sales (sorry), and business has been a bit spotty since.
The couple coped by extending the season. "It used to run from late June through shortly after Labor Day," says Finger. Now, they schedule trips from before Memorial Day until early October.
Mahle practices minor kitchen economies. She doesn't skimp on ingredients, buying prime meats and excellent fruits and veggies, all in larger quantities than usually get eaten. She reworks the leftovers into what she calls "resurrection food." Lobster becomes a lobster dip appetizer or bisque. Corned beef becomes part of a breakfast hash.
She also says she has been "slowly figuring out how to buy things wholesale, not so much the food, but the other supplies."
Other contributions to the business income come from selling Riggin-logoed clothing items, caps, and pitchers. Mahle also wrote a cookbook. She markets it on the company Web site and eBay and 85 percent of its sales come from non-passengers.
Eventually, the couple would like to expand their income from these sources, so they don't have to work quite as hard on the boat. To that end, Mahle has started writing a cooking/food column for the Portland Press Herald.
They don't expect to sail forever. "The limit for most people seems to be about 20 years," says Finger.
The Riggin, fortunately, has gone up in value, Finger believes, although by how much is hard to say. A bigger boat, the Victory Chimes, is on the market for $1.5 million.
One of the best things about the windjammer business is that it can involve the whole family. Finger and Mahle not only spend most of their professional lives together, they take their two daughters, Chloe, who's eight and Ella, four, on the job as well.
Some day the couple would like to cut back on their sailing schedule, which consists, basically, of up to 16-hour days, seven days a week. "I'd like to have the boat income for fun," says Mahle, "and the family income from writing books and columns."
While they may never make a killing in the business, the couple should be able to make a good living, and they feel they have made a good life.
John and Karen Lewis took a similar road to business success, opening an ecolodge in Costa Rica. For more, click here.
Jacquelyn Tran also integrated her professional and family lives. Click here for her story.


text:
Electronic: Pop Crossover Moods: Solo Male Artist
by A Zillion Reasons
© Copyright - Patrik Larsson / A Zillion Reasons (634479151842)
A unique mix of pop, dance and electronic music with strong melodies.
Genre: Pop: Today's Top 40
Share Time Download
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4. I dont think that this is the place to be
5. Commercialization
6. Have no fear
8. Love is a beautiful thing
9. Fire
10. Beauty
A Zillion Reasons is a one man group, Patrik Larsson.
Patrik is a talented songwriter, who started writing songs in 2002.
So far, two full albums and a short album are released. The newly released second album, i.e. Continue, is now available here at CD Baby.
The goal for A Zillion Reasons is to make music with strong melodies, something which often is forgot in contemporary music. Most of the songs are based on 80ies and 90ies style groups such as Depeche Mode and Soft Cell.
More modern influences are found in Russian pop groups such as Ruki Vverh, Japanese pop (or J-pop) such as BOA and Belgian Dance such as Silver. The cross over is something that can be described as pop-dance-electronic.
A Zillion Reasons is primarily a studio project, but the songs are too good to stay there. Here is your chance to listen to the music, to buy the music and to play it for others. Will you be among the first to get your hands on the new album?
AZR is willing to support talented artists that wants to perform on stage, but lacks strong songs. AZR is also actively searching for female singers who want to make it in the dance industry.

redpajama_set_name: RedPajamaCommonCrawl


Antique jewellery is anything over 100 years old, also called estate jewellery.
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It can be done through email and on the market. In addition to your insurer about the world Wide Web. When something this happens, the question is, what will happen to them from doing some of the vehicle at a decision. For instance, would have very straight forward or even just a few payments in the end of the top-rated insurance companies. There are some things you weren't aware of?
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It seems like almost every day that a new structure of work is presented to the modern workforce. With options like freelancing and utilizing the "gig economy," workers and employers have more options than ever when it comes to how we work, and how businesses can stay competitive in recruiting top talent as market demands increase.
One of the newest methods of work and in obtaining advanced skills is through crowdsourcing.


I think the answer to this is...maybe. Lots of factors to consider.
>have a higher or lower rate for offsite vs onsite rates?
deal mostly with very price-conscious clients, it might make a difference.
you can't just jigger and test and optimize the way you might like to.
>difference -- $5? $10? a percentage?


Here, we'll post everything from fashion to favorite products to our pups. This section will be light and give a fun inside look at our lives away from the track.
From a personal standpoint, I have to say that 2016 was, without a doubt, the BEST for me. I became a mother and got to experience all the joys and emotions that used to just seem like cliches for how people described motherhood. Now, I actually get to FEEL these emotions (not just hear about them) and let me tell you, they are totally amazing. Yes, love IS blind and yes, I do believe that Abigail is the cutest, greatest, most talented little human ever.
Categories: Baby, Lifestyle, Running | Tags: 2017, achilles, Goals, surgery | Permalink.


text:
tags: Catholic Church, Catholicism, Christianity, Jesus Christ, Joey Valencia, Joey Velasco, JOEY VELASCO SICK, Philippine art, Philippine society, religion, The Last Supper
Last Supper With The Street Children
The 12 children in the painting are real people the painter, Joey Velasco, discovered in poor areas of Metro Manila and Quezon City.
After treating them to meals, Velasco took their pictures and retreated to his room to start working on the painting.
Velasco said, the children, aged 4-14, reveal a story of a greater hunger than a plate of rice could satisfy.
He said, "It was they who touched my soul. Through them, God spoke to me and moved me to paint their stories and tell others about their lives."
The young girl standing at the extreme left, where Judas appears in the da Vinci painting, is 10-year-old Nene.
Velasco met her at the Manila North Cemetery, where she and her family lived as squatters among the graves.
Onse, 9, sits at the table, his plate cleaned to the last crumb, he listens to Jesus to feed his other hungers.
The child, who scavenges with a push cart, has a father addicted to drugs and a mother who works as a strip dancer.
Itok, another scavenger who at 11 is the family breadwinner, sits at the right hand of Jesus.
According to Velasco, Itok spent time in jail after being caught in a number of robberies.
Another child in the painting does not live in Quezon City.
Velasco placed a small Sudanese boy under the table eating the fallen scraps with the cats.
The artist explained, "The skinny child is not one of the hungry kids who roam our busy streets at night. He is "an imaginary symbolic figure" who in the past "had satisfied himself with unnecessary food, (but) now finds himself under the table seeking spiritual crumbs."
The children featured in the painting are no longer in the areas where Velasco originally found them.
Through his partnership with Gawad Kalinga, an organization dedicated to sheltering the homeless, the 12 children and their families now have homes at Romeo Cabrera Village in Quezon City.
The children's stories are featured in the book "They Have Jesus: The Stories of the Children of the 'Hapag ng Pag-asa (Table of Hope).'"
Note: This piece was emailed to At Midfield by Danny Gagelonia, my Kuya
Breaking news update:
This writer has just become aware of a report that the fragile health of Maestro Joey, who lives in Fairview, Quezon City, took a turn for the worse some 12 hours ago.
The online report at http://diyeydi.tumblr.com/post/823895250 quotes unidentified Salesian Brothers as having said that Joey Velasco suffered "a heart attack and experienced seizures yesterday." It also shared the information that "cancer has spread to Velasco's brain."
The unidentified poster of the news did not mention were Joey is confined but sought prayers for the acclaimed painter and his family. Joey is an ex-seminarian of the Salesians of Don Bosco.
I last had a chance to have merienda with Joey some two years ago during which he shared his abiding faith in the Lord as he struggled with his health condition.
We join the prayers for his recovery.
Here's a look-back at Probe Profile's feature on Joey two years ago:
from → Philippine news
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beckya8388 permalink
Where can I get reprints/posters of Joey Velasco's various renditions of the Last Supper?
Fernando Gagelonia permalink*
I have digital copies of some of his works. But it's best that I relay your query to him and get his nod. Will get back to you. Are you in the Philippines po. I think h also already has a web site buy am not sure… Will check…
I live in the Chicago area and just returned a few days ago from a vacation in Manila. I saw three posters of his work while visiting a professor at Ateneo and he told me about Joey's exhibit there. For the rest of my visit I was hunting for the reprints and went to several St. Paul Libreria branches but they all said they were out of stock. Aside from wanting them for some friends at my local parish I also particularly wanted to give a poster of the Last Supper with the street children to my neighbor's child for his First Communion next month.
mattlaurel710 permalink
June 4, 2010 11:00 am
Heart breaking, Ding. I shared this story with my son William and he was very touched by the portrait of the young Sudanese boy eating underneath the table with the cats. He asked how could a Sudanese boy live on his own in Manila, i said that this little boy is only a symbolic figure in the painting. When we lived back home, he had playmates who lived in the shanties near Don Bosco and he remembers how boys his age would take care of their siblings and cook rice, clean their homes and not go to school at such a young age. William said that he could not understand how others could have so much in life and others could not have anything at all. William said that the first thing that came to his mind after studying the painting was this: we should never take anything for granted in life. That he should always be grateful. Thank you for sharing this with us. My boy appreciated reading this and seeing the beautiful painting.
This post has been in my blog archive for quite a while, Matt dearest but it does continue to touch hearts. Glad it did yours. Stay well. 🙂
Prayers For Joey Velasco « At Midfield
Paalam, Joey Velasco, Isang Dakilang Pintor « At Midfield


1972 House Bill 6263. An Act Authorizing The City Of Lynn To Make Certain Ordinances Granting Salary Increases Retroactive To The First Day Of The Calendar Year.
Title 1972 House Bill 6263. An Act Authorizing The City Of Lynn To Make Certain Ordinances Granting Salary Increases Retroactive To The First Day Of The Calendar Year.


that includes exquisite and unusual jewellery, glassware, ceramics, textiles, fine furniture, prints and metal work from the regions finest designer makers. With a wide variety of handmade designs and individual artworks on offer, you'll find a wealth of Christmas gift ideas, from magnificent one-off designer pieces to affordable goods.
Organised by the Shropshire Guild of Contemporary Craft and featuring guest craftspeople and artisans from across the West Midlands, Gorgeous is the perfect opportunity to get ahead of your Christmas list or find the perfect gift to give yourself. And there's more to Gorgeous than just shopping… Throughout the weekend many of the exhibitors will be demonstrating their craft, giving you an insight to their work and an opportunity to ask questions. And, if all of that works up an appetite, you'll find tasty treats from some of the regions food providers available to eat or take home.
Furniture maker Robert Shelton combines traditional design with a contemporary twist to create furniture of real character. Using locally sourced wood he crafts individual pieces of great character – and specialises in Windsor chairs.
Jeweller Sarah Piper Heap works in silver and other natural materials to create jewellery and tableware from her studio in the borderland of Shropshire and North Wales.
Ceramicist and sculptor Caroline Bennett creates unique figurative and free-flowing sculptures for the house and garden using a range of techniques, from coil and slab building to throwing and slip casting.
Now in its 13th year, Gorgeous is a celebration of creative skills, promoting the work of designers and craftsmen in the region, many of whom enjoy status as internationally recognised artists. It also builds on The Gorge's rich heritage of creativity; the place where early industrialists mined for raw materials to make iron, tiles and porcelain goods.
Gorgeous is open 10am to 5pm on Saturday and 10am-4pm on Sunday and admittance is free.
For further information, call Caroline Bennett on 01746 765078, email caroline@carolinebennett.co.uk or visit www.shropshireguild.co.uk. The Gorge is easily reached via the M6 and M54 motorways exiting at Telford (M54 junction 4 or 6).