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PACS-Codes: 05.20.-y, 02.30.Mv, 66.10.-x, 78.70.Nx, 05.60.Cd
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Single particle diffusion, non-Gaussian effects, cumulants, time
expansion, Van Hove self-correlation function, Green's functions,
supercooled liquids
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\begin{document}
\title{Theorem on the Distribution of Short-Time Single Particle
Displacements with Physical Applications}
\author{Ramses van Zon and E. G. D. Cohen\vspace{2mm}\\
\emph{The Rockefeller University, New York, NY 10021, USA}}
\date{30 May 2005}
\maketitle
\begin{abstract}
The distribution of the initial short-time displacements of a single
particle is considered for a class of classical systems of particles
under rather general conditions. This class of systems contains
canonical equilibrium of a multi-component Hamiltonian system as a
special case. We prove that for this class of systems the $n$th order
cumulant of the initial short-time displacements behaves as the
$2n$-th power of time for all $n>2$, rather than exhibiting a general
$n$th power scaling. This has direct applications to the initial
short-time behavior of the Van Hove self-correlation function, to its
non-equilibrium generalizations the Green's functions for mass
transport, and to the non-Gaussian parameters used in supercooled
liquids and glasses. Moreover, in the context of the Green's functions
this theorem is expected to be relevant for mass transport at
(sub)picosecond time~scales.
\noindent\textbf{\footnotesize KEY WORDS:}
Single particle diffusion, non-Gaussian effects, cumulants, time
expansion, Van Hove self-correlation function, Green's functions,
supercooled liquids.
\noindent\textbf{\footnotesize PACS:}
05.20.-y, 02.30.Mv, 66.10.-x, 78.70.Nx, 05.60.Cd
\end{abstract}
\section{Introduction}
\label{introduction}
This paper concerns a universal property of correlations of the
initial short-time behavior of the displacements of single particles
for a rather general class of ensembles of classical systems, both in
and out of equilibrium. Among other things, these correlations
(expressed in terms of so-called cumulants) have applications to
neutron
scattering,\cite{VanHove54,Schofield61,Rahmanetal62,Rahman64,NijboerRahman66,LevesqueVerlet70,Sears72,BoonYip80,HansenMcDonald86,Squires,Arbeetal02,Arbeetal03}
to the description of non-equilibrium systems on picosecond time
scales and nanometer length
scales\cite{Kincaid95,KincaidCohen02a,KincaidCohen02b,CohenKincaid02,VanZonCohen05b}
and to heterogeneous dynamics in supercooled liquids and
glasses.\cite{OdagakiHiwatari91,KobAndersen95,HurleyHarrowell96,Kobetal97,ZangiRice04}
Our motivation to consider the displacements of single particles comes
from trying to describe the behavior of non-equilibrium systems on all
time scales using the so-called Green's function theory. This theory
aims to describe, among other things, the time evolution of the number
densities, momentum density and energy density, by expressing them in
terms of Green's functions. It has so far been successfully applied to
self-diffusion\cite{Kincaid95} and to heat
transport\cite{KincaidCohen02a,KincaidCohen02b,CohenKincaid02} while a
study of mass transport in binary (isotopic)
mixtures\cite{VanZonCohen05b} is in progress. The main advantage of
the Green's functions over hydrodynamics is that they can in principle
describe the system on all time and length scales, in particular on
time scales of the order of picoseconds and on length scales of the
order of nanometers. The connection between the picosecond and
nanometer scale can be understood by realizing that with typical
velocities of 500 m/s, a particle in a fluid at room temperature moves
about 0.5 nm in 1 ps. Hence it could also be relevant for
nanotechnology, although at present the Green's function theory exists
in a classical formulation only and does not yet take
quantum-mechanical effects into account.
For mass transport in multi-component
fluids,\cite{Kincaid95,VanZonCohen05b} the Green's functions
$G_\lambda(r,r',t)$ have the physical interpretation of being the
probability that a single particle of component $\lambda$ is at a
position $r$ at time $t$ given that it was at a position $r'$ at time
zero. At time zero the system is not in equilibrium, in fact,
far-from-equilibrium situations have been studied in this
context.\cite{KincaidCohen02a,KincaidCohen02b,CohenKincaid02,VanZonCohen05b}
Given this interpretation, it is clear that the displacements of single
particles are the central quantity.
Apart from the non-equilibrium aspect, the above interpretation of the
Green's functions is the same as that of the classical equilibrium Van
Hove self-correlation function
$G_s(r-r',t)$.\cite{VanHove54,BoonYip80,HansenMcDonald86} In the
literature on the classical Van Hove self-correlation function in the
context of neutron scattering on an equilibrium
fluid,\cite{VanHove54,BoonYip80,Squires} it has been noted by
Schofield\cite{Schofield61} and Sears\cite{Sears72} that the cumulants
$\kappa_n$ (certain combinations of the moments defined below) of the
displacement of a particle in a time $t$ in a fluid with a smooth
interparticle potential behave for small $t$ as $\kappa_2 =
\order{t^2}$, $\kappa_4 = \order{t^8}$, $\kappa_6 = \order{t^{12}}$,
while all odd cumulants~vanish.
The relevance of the cumulants for the Van Hove self-correlation
function can be seen from its Fourier transform, the incoherent
intermediate scattering
function\cite{VanHove54,BoonYip80,Squires,HansenMcDonald86}
\begin{equation}
F_s(k,t) = \baverage{e^{\mathrm{i} k
\hat{\mathbf{k}}\cdot[\mathbf{r}_1(t)-\mathbf{r}_1(0)]} }_\text{eq} =
\baverage{e^{\mathrm{i} k \Delta x_1(t)}}_\text{eq},
\label{Fsdef}
\end{equation}
where $k$ is a wave vector, $\hat{\mathbf{k}}$ its direction (a unit
vector), $\mathbf r_1(t)$ the position of a (single) particle at
time $t$ and $\saverage{}_\text{eq}$ an equilibrium average.
In the last equality we have chosen
$\hat{\mathbf{k}}=\hat{\mathbf{x}}$ (in an isotropic equilibrium fluid
the result is independent of the direction $\hat{\mathbf k}$) and
defined $\Delta x_1(t)$ as the displacement of the single particle in
the $x$ direction: $\Delta x_1(t)=\hat{\mathbf x}\cdot[\mathbf r_1(t)-
\mathbf r_1(0)]$. In probability theory\cite{Cramer46,VanKampen}
$\log\saverage{\exp[\mathrm{i} k X]}$ is the cumulant generating
function for the stochastic variable $X$, so we see from \Eq{Fsdef}
that $\log F_s(k,t)$ is the cumulant generating function of $\Delta
x_1(t)$. Note that $\Delta x_1(t)$ is a random variable here as it depends
on an initial phase point drawn from a probability distribution (here
the equilibrium distribution). The cumulant generating function is,
by definition, equal to $\sum_{n=1}^\infty\kappa_n (\mathrm{i}
k)^n/n!$ where $\kappa_n$ is the $n$th
cumulant.\cite{Cramer46,VanKampen} The relation of the incoherent
intermediate scattering function and the cumulants of the displacement
is thus expressed
by\cite{Rahmanetal62,NijboerRahman66,Sears72,BoonYip80,DeSchepperetal81}
\begin{equation}
F_s(k,t) =
\exp \sum_{n=1}^\infty \frac{\kappa_n}{n!}(\mathrm{i} k)^n.
\label{Fsrelation}
\end{equation}
This connection with the incoherent scattering function (and thus with
neutron
scattering\cite{VanHove54,Schofield61,Rahmanetal62,Rahman64,NijboerRahman66,LevesqueVerlet70,Sears72,BoonYip80,HansenMcDonald86,Squires,Arbeetal02,Arbeetal03})
explains the early interest in the cumulants of displacements from a
physical perspective. In section~\ref{discussion} we will discuss also
more recent physical applications of the cumulants such as the
non-equilibrium Green's functions and non-Gaussian parameters used in
the theory of supercooled liquids and glasses.
The results for $\kappa_n$ mentioned above suggested for equilibrium
systems with smooth potentials a behavior as \textorder[power]{t}{2n}
for $\kappa_n$ when $n>2$ but to the best of our knowledge no proof of
this property is available at present.
We note that an $\order{t^{2n}}$ behavior would be in stark contrast
to results obtained for hard disk and hard sphere fluids in
equilibrium.\cite{Sears72,DeSchepperetal81} Sears\cite{Sears72}
considered results up to \textorder{t}{8} and \textorder{t}{12} for
$\kappa_4$ and $\kappa_6$, respectively, for smooth potentials and
he found by using a limit in which the smooth potential reduces to a hard
core potential that for hard spheres $\kappa_4=\order{|t|^5}$ and
$\kappa_6=\order{|t|^7}$, while the odd cumulants were still zero. An
alternative approach was followed by De Schepper \emph{et
al.}\cite{DeSchepperetal81} consisting of directly evaluating the Van
Hove self-correlation function for short times for hard spheres based on
pseudo-Liouville operators (which replace the usual ones for smooth
potentials). De Schepper \emph{et al.}\cite{DeSchepperetal81} obtained
$\kappa_n = \order{|t|^{n+1}}$ for all even $n>2$, with corrections of
$\order{|t|^{n+2}}$.
The $\order{t^{2n}}$ result for smooth potentials is the more
remarkable in that a naive estimate of the short time behavior of
$\kappa_n$ based on its connection with the moments would predict a
behavior as \textorder{t}{n}. Hence \emph{all} terms from
$\order{t^n}$ up to $\order{t^{2n-1}$},\footnote{Everywhere in this
paper `up to $\order{t^\alpha}$' means `up to and including
$\order{t^\alpha}$'.} should \emph{vanish}. The question we address
here is whether this is indeed general for smooth potentials.
In fact in this paper we shall show that under quite general
conditions, the main one being that the velocities of the particles
are Gaussian (`normal') distributed and independent of each other and of their
positions, one can prove the following:
\begin{maintheorem}
Consider a classical mechanical system of $N$ degrees of freedom
(`particles'), described by positions $r_i$ and velocities $v_i$
($i=1\ldots N$), collectively denoted by $r^N$ and $v^N$,
respectively, and whose time evolution is given by the
equations of motion
\begin{eqnarray}
\dot{r}_i &=& v_i
\label{motion1}
\\
m_i\dot{v}_i &=& F_i(r^N,t)
\label{motion2}
\end{eqnarray}
with arbitrary masses $m_i$ and velocity-independent and smooth forces
$F_i$. If the initial ensemble is described by a probability
distribution in which each velocity is Gaussian and independent of the
other velocities and the positions of all the particles, \emph{i.e.},
it is of the form
\begin{equation}
P(r^N,v^N)=
f(r^N)\prod_{i=1}^N
\frac{\exp\big[-\frac12\beta_im_i(v_i-u_i)^2\big]}
{\sqrt{2\pi/( m_i\beta_i)}}.
\label{distribution}
\end{equation}
then the cumulants $\kappa_n$ (defined below) of the displacements
\begin{equation}
\Delta r_i(t) \equiv r_i(t)-r_i(0)
\label{displacement}
\end{equation}
obey
\begin{equation}
\kappa_n
= \begin{cases}
\,c_nt^n\,+\order{t^{n+1}} &\quad \text{for $n\leq2$}\\
c_nt^{2n}+\order{t^{2n+1}} &\quad \text{for $n> 2$}
\end{cases}
\label{Theorem}
\end{equation}
for sufficiently short initial times $t$ (and where the coefficients
$c_n$ will be given later in \Eq{cn}).
\end{maintheorem}
\noindent
We stress that in this Theorem the forces are independent of the
velocities, but may depend on the positions $r^N$ and on the time $t$
in any way as long as they are smooth. An example of smooth forces
would be infinitely differentiable forces $F_i(r^N,t)$, but also
Lennard-Jones forces are allowed provided the distribution of the
the positions $f(r^N)$ assigns a vanishing probability for the particles
to be at zero distance of one another, which is the singular point of
the Lennard-Jones potential at which it is not smooth.
Furthermore,
we note that the initial distribution in \Eq{distribution} is not an
equilibrium distribution, but valid for a system consisting of
particles with different individual masses $m_i$, mean velocities
$u_i$ and ``temperatures'' $\beta_i$. Furthermore, there is no
restriction to the distribution of positions $f(r^N)$. The
distribution in \Eq{distribution} shares, however, with the
equilibrium distribution its Gaussian dependence on the
velocities, which is crucial for the Theorem to hold.
Canonical equilibrium for a single or multi-component fluid is just a
special case of the systems covered by the Theorem. In that case, one
has $F_i=-\partial U/\partial r_i$, $f(r^N)\propto\exp[-\beta
U(r^N)]$, $u_i=0$ and $\beta_i=\beta$.
The probability distribution functions of the form in
\Eq{distribution}, in which each particle has its own mean velocity
$u_i$ and `temperature' $\beta_i$, may seem of a mathematical
generality which has little physical relevance. Note however that this
is a convenient way to describe mixtures of any arbitrary number of
components. In such a mixture, the mean velocities and temperatures of
the different components could be selected physically \emph{e.g.}\ by
means of a laser (or perhaps even a neutron beam), tuned to a
resonance of one of the components only, which would give the
particles of that component a nonzero average momentum as well as a
different initial `temperature' $\beta_i$ due to the recoil
energy.\cite{Squires}
The proof we will present below is a ``physicists' proof'', meaning that
the proof does not claim to have complete mathematical rigor but has every
appearance of being correct, perhaps under mild and reasonable
additional conditions such as a finite radius of convergence of
various series as well as the existence of the moments and cumulants.
The paper is structured as follows. In section~\ref{moments and
cumulants}, we give some definitions to be able to treat the cumulants
in more detail. In section~\ref{proof}, we give the steps needed to
prove the main Theorem. In it, we need an auxiliary theorem concerning
general Gaussian distributed variables whose proof is postponed to the
Appendix. The coefficients $c_n$ in \Eq{Theorem} of the Theorem are
determined in section~\ref{proof} as well. We conclude with a
discussion of the results and their broader physical relevance in
section \ref{discussion}.
\section{Moments and cumulants}
\label{moments and cumulants}
In this paper, when we speak of a `variable' we mean a function of the
positions $r^N$ and the velocities $v^N$ and possibly the time
$t$. All physical quantities are variables of this kind. For such
variables $X(r^N,v^N,t)$, the average with respect to the distribution
function $P(r^N,v^N)$ will be denoted by
\begin{equation}
\saverage{X} = \int\! dr^Ndv^N\, P(r^N,v^N) X(r^N,v^N,t).
\label{average}
\end{equation}
The displacement of each degree of freedom or particle $i$ in time $t$
was defined in~\Eq{displacement}. Note that $\Delta r_i(t)$ is the
displacement of \emph{one} particle~$i$, so throughout, $i$ will
stay fixed, although it may have any value between $1$~and~$N$. Its
$n$th moment\footnote{Although the moments $\mu_n$ and cumulants
$\kappa_n$ are really the `moments and cumulants of the probability
distribution function of $\Delta r_i(t)$,' we will refer to them
simply as the `moments and cumulants of $\Delta r_i(t)$.'} is the
average of the $n$th power of the displacement, \emph{i.e.},
\begin{equation}
\mu_n \equiv \saverage{\Delta r^n_i(t)},
\label{moments}
\end{equation}
where the dependence on $i$ and $t$ on the left hand side (\lhs) was
suppressed. The cumulants of $\Delta
r_i(t)$,\footnotemark[\value{footnote}] denoted by $\kappa_n$, are
equal to these moments with certain factorizations of them subtracted
and thus sensitive to the correlations of $\Delta r_i(t)$. Their
precise definition is via the cumulant generating
function\cite{Cramer46,VanKampen}
\begin{equation}
\Phi(k) \equiv \log\average{\exp[\mathrm{i} k \Delta r_i(t)]}
=
\sum_{n=1}^\infty \frac{\kappa_n}{n!}(\mathrm{i} k)^n.
\label{cumulantgenerator}
\end{equation}
from which the $\kappa_n$ follow as
\begin{equation}
\kappa_n \equiv \dd[n]{\Phi}{(\mathrm{i} k)}\bigg|_{k=0}\:.
\label{kappan}
\end{equation}
An alternative notation for the cumulants, which is more analogous to
\Eq{moments} and which is especially convenient in the case of several
variables, is\cite{VanKampen}
\begin{equation}
\cumulant{\Delta r_i^n(t)}\equiv \kappa_n.
\label{cumulantnot}
\end{equation}
The cumulant generating function in \Eq{cumulantgenerator} can be
expressed in terms of the moments $\mu_n$ since
$\Phi(k)=\log\saverage{\exp[\mathrm{i} k \Delta
r_i(t)]}=\log[1+\sum_{n=1}^\infty (\mathrm i k)^n\mu_n/n!]$. The
relations between the cumulants and moments can then be found
by Taylor expanding the logarithm around $1$ and using a multinomial
expansion for the resulting power series. This gives, somewhat
formally,
\newcommand{\sofrac}[2]{#1/#2}
\begin{equation}
\kappa_n \:=\: -\,n!
\mathop{\sum_{p_\ell\geq 0}}_{\sum_{\ell=1}^\infty \ell p_\ell = n}
\Big(\sum_{\ell=1}^\infty p_\ell-1\Big)!
\prod_{\ell=1}^\infty \frac{\big(\sofrac{-\mu_\ell}{\ell!}\big)^{p_\ell}
}{p_\ell!}
\label{kappaintermsofmu}
\end{equation}
For instance, for the first few $\kappa_n$, \Eq{kappaintermsofmu} becomes
$\kappa_1=\mu_1$, $\kappa_2=\mu_2-\mu_1^2$ and
$\kappa_3=\mu_3-3\mu_1\mu_2+2\mu_1^3$.\cite{Cramer46,VanKampen,NumericalRecipes}
In general, $\kappa_n=\mu_n\pm$ factored terms, where the factored
terms contain all ways of partitioning the moment $\mu_n$ into a
product of lower moments $\mu_\ell$ such that all $\ell$ values
(taking into account the `frequencies of occurrence' $p_\ell$) add up
to $n$.
Apart from the cumulants $\kappa_n$ of the single variable $\Delta
r_i(t)$, we will also need the general definition of cumulants which
applies to any number of variables. These are defined as
follows.\cite{Cramer46,VanKampen} For a set of variables $X_\gamma$
$(\gamma=1\ldots\alpha)$ the cumulant generating function is defined
as
\begin{equation}
\Phi(k_1,\ldots k_\alpha) \equiv
\log\baverage{\exp
\sum_{\gamma=1}^\alpha \mathrm{i} k_\gamma X_\gamma}.
\label{GXdef}
\end{equation}
With the help of this generating function, the cumulants can be
expressed similarly as in \Eq{kappan} by
\begin{equation}
\cumulant{X_1^{n_1} \ldots X_\alpha^{n_\alpha}}
\equiv
\left(
\prod_{\gamma=1}^\alpha \dd[n_\gamma]{\ }{(\mathrm{i} k_\gamma)}\right)
\Phi(k_1,\ldots k_\alpha)\Bigg|_{k_{1\ldots\alpha}=0}.
\label{defcumvar}
\end{equation}
(Note that $k_{1\ldots \alpha}=k$ is a short-hand notation for $k_1=k$,
$k_2=k$, \dots, $k_\alpha=k$.)
The cumulants can be expressed in terms of moments
analogously to \Eq{kappaintermsofmu}:
\begin{multline}
\cumulant{X_1^{n_1}\ldots X_\alpha^{n_\alpha}}
\\ \:=\:
- n_1!\cdots n_\alpha!\!
\mathop{\sum
_{p_{\{\ell\}}\geq 0}}
_{\sum_{\{\ell\}} \ell_\gamma p_{\{\ell\}} = n_\gamma}
\Big(\sum_{\{\ell\}} p_{\{\ell\}}-1\Big)!
\prod_{\{\ell\}} \frac{1}{p_{\{\ell\}}!}
\bigg(-\frac{\saverage{X_1^{\ell_1}\ldots X_\alpha^{\ell_\alpha}}}{\ell_1!\cdots\ell_\alpha!}\bigg)^{p_{\{\ell\}}}\!\!,
\label{cumintermsofmom}
\end{multline}
where $\{\ell\}=\{\ell_1,\ldots,\ell_\alpha\}$ denotes a set of
nonnegative $\ell_\gamma$ values, $\gamma=1\ldots\alpha$ and
$p_{\{\ell\}}$ gives the frequency of occurrence of that set. In
\Eq{cumintermsofmom} the sum is over the frequencies of all possible
sets $\{\ell\}$ (and thus all possible moments
$\saverage{X_1^{\ell_1}\ldots X_\alpha^{\ell_\alpha}}$) for which
$\sum_{\ell=1}^\infty \ell_\gamma p_{\{\ell\}} = n_\gamma$. This is
nothing else but factorizing the expression $X_1^{n_1}\ldots
X_\alpha^{n_\alpha}$ in all possible ways. Hence,
the cumulants $\cumulant{X_1^{n_1}\ldots
X_\alpha^{n_\alpha}}=\average{X_1^{n_1}\ldots X_\alpha^{n_\alpha}}\pm$
factored terms that are products of the moments $\saverage{X_1^{\ell_1}\ldots
X_\alpha^{\ell_\alpha}}$ such that the $\ell_{\gamma}$ values for
fixed $\gamma$ add up (taking into account their frequencies
$p_{\{\ell\}}$) to $n_\gamma$.
\section{Proof of the main Theorem}
\label{proof}
\subsection*{Strategy based on Gaussian velocities}
The Theorem formulated in Eqs.~(\ref{motion1}--\ref{Theorem}) in
section~\ref{introduction} will be proved in this section, although we
will defer the details of the proof of a required auxiliary theorem to
the Appendix for greater clarity.
To obtain the initial, short time behavior of the moments and
cumulants of the displacement, $\Delta r_i(t)$ may be Taylor-MacLaurin
expanded around $t=0$ as
\begin{equation}
\Delta r_i(t) =
\sum_{\gamma =1}^{\infty}
\frac{t^\gamma}{\gamma!} \frac{d^\gamma r_i}{dt^\gamma}\Big|_{t=0}
=\sum_{\gamma=1}^\infty \frac{t^\gamma}{\gamma!} \ddh[\gamma-1]{v_i}{t}\Big|_{t=0}
\label{expansion}
\end{equation}
where we used \eq{motion1}. Because of the equations of motion
\eq{motion1} and \eq{motion2}, the $d^\gamma v_i/dt^\gamma$, viewed as
functions of $r^N$, $v^N$ and $t$, are recursively related by
\begin{equation}
\ddh[\gamma+1]{v_i}{t}
= \sum_{j=1}^N
\left[
\dd{\ }{r_j}\Big(\ddh[\gamma]{v_i}{t}\Big)
v_j
+\dd{\ }{v_j}\Big(\ddh[\gamma]{v_i}{t}\Big)
\frac{F_j}{m_j}
\right]
+\dd{\ }{t}\Big(\ddh[\gamma]{v_i}{t}\Big)
\label{recursion}.
\end{equation}
To show that $\kappa_n=\order{t^{2n}}$, it is of course possible to
straightforwardly work out $\kappa_n$, using \Eqs{kappaintermsofmu},
\eq{moments}, \eq{expansion}, \eq{recursion}, \eq{average} and
\eq{distribution}, in that order. Such a procedure was essentially
followed by Schofield\cite{Schofield61} for $n\leq6$ and
Sears\cite{Sears72} for $n\leq8$ for equilibrium fluids. In their
expressions many cancellations occurred before $\kappa_n$ could be
seen to be, for these cases, of \textorder{t}{2n} instead of
\textorder{t}{n}. These cancellations seemed to happen as a
consequence of equilibrium properties. However, by carrying out the
same straightforward procedure for the more general class of
non-equilibrium initial conditions in \Eq{distribution}, we have found
that while odd moments are no longer zero still
$\kappa_n=\order{t^{2n}}$ for $n=3$, $4$, $5$ and $6$. These results
for $\kappa_3$, $\kappa_4$, $\kappa_5$ and $\kappa_6$ naturally led us
to propose the Theorem. Because the straightforward calculations for
$\kappa_3$, $\kappa_4$, $\kappa_5$, and $\kappa_6$ for this
non-equilibrium case are very lengthy, they will not be presented
here. In any case this procedure is not very suited to determine the
order in $t$ of $\kappa_n$ for general $n$, because with increasing
$n$ an increasing number of terms have to be combined (taking together
equal powers of $t$ from the various products of moments) before they
can be shown to be zero.
\emph{Our strategy for proving that the $n$th cumulant
$\cumulant{\Delta r_i^n}=\order{t^{2n}}$ for all $n>2$, will be to
exploit the Gaussian distribution of the velocities as much as
possible.} But we can only hope to use the Gaussian nature of the
velocities if we succeed in bringing out explicitly the dependence of
the coefficients of the power series in $t$ of the cumulants of the
velocities. This dependence has so far only been given implicitly ---
the cumulants are related to the moments by \Eq{kappaintermsofmu}, the
moments contain $[\Delta r_i]^\ell$, $\Delta r_i$ is expanded in the
time $t$ in \Eq{expansion}, and the coefficients in that expansion are
the derivatives of the velocity $v_i$ whose dependence on the
positions, the velocities and the time can be found by using
\Eq{recursion} recursively. To make this more explicit, the first
part of the proof will be to expand the cumulants as a powers series
in the time $t$ and the second part will be to express this series
more explicitly in the velocities. In the third and last part we will
then use the properties of Gaussian distributed variables,
\emph{i.e.}, the velocities, to complete the proof of the main
Theorem.
It turns out that the properties of Gaussian distributed variables
that we will require in the third part of the proof are formulated
for Gaussian variables whose mean is zero, while in \Eq{distribution} the
velocities are Gaussian but do not have zero mean.
For this reason, it is convenient to introduce already at
this point new velocity variables whose mean is zero (cf. \Eq{distribution}):
\begin{equation}
V_i \equiv v_i-\saverage{v_i} = v_i-u_i.
\label{Vdef}
\end{equation}
The $V_i$ are generalizations of the peculiar velocities used in local
equilibrium situations and will be referred to in the general case
treated in this paper as \emph{peculiar velocities} as well.
Substituting \Eq{Vdef} into the probability distribution
\Eq{distribution} and the time expansion in \eq{expansion}, gives them
in their peculiar velocity form:
\begin{eqnarray}
P(r^N,V^N)&=&
f(r^N)\prod_{i=1}^N
\frac{\exp\big[-\frac12\beta_i m_i V_i^2\big]}
{\sqrt{2\pi/( m_i\beta_i)}}
\label{newdistribution}
\\
\Delta r_i(t)
&
=&
u_i t +
\sum_{\gamma=1}^{\infty} \frac{t^\gamma}{\gamma!} \ddh[\gamma-1]{V_i}{t}.
\label{neexpansion}
\end{eqnarray}
In order to make future expressions less complicated we introduce for
the coefficients in \Eq{neexpansion} the notation ($\gamma=1,$ $2,$
$\ldots$)
\begin{equation}
X_\gamma =\frac{1}{\gamma!} \ddh[\gamma-1]{V_i}{t}.
\label{wdef}
\end{equation}
Thus \emph{e.g.}\ $X_1=V_i$ and \Eq{neexpansion} becomes
\begin{equation}
\Delta r_i(t)
=
u_i t +
\sum_{\gamma=1}^{\infty} X_\gamma t^\gamma,
\label{newexpansion}
\end{equation}
The recursion relation between the coefficients $X_\gamma$ is, from
\Eqs{recursion} and \eq{wdef}:
\begin{eqnarray}
\frac{X_{\gamma+1}}{\gamma+1} &
=&
\sum_{j=1}^N
\left[
\dd{X_\gamma}{r_j}
(u_j+V_j)
+\dd{X_\gamma}{V_j}
\frac{F_j}{m_j}
\right]
+\dd{X_\gamma}{t}
.
\label{newerrecursion}
\end{eqnarray}
We will now start the actual proof of the Theorem for general $n$.
\subsection*{First part: Expanding the cumulants in the time $t$}
The infinite number of terms in the time expansion $\Delta r_i$ in
\Eq{newexpansion} means that we would have to combine in
\Eq{kappaintermsofmu} an infinite number of terms to get the power
expansion in $t$ of the cumulants. Obviously if we are interested in
the cumulants up to $\order{t^{2n-1}}$ we should not have to retain
all these terms in \Eq{newexpansion}, but only those up to
$\order{t^{2n-1}}$. As a matter of fact, we need even less terms,
namely only terms up to $\order{t^n}$ in \Eq{newexpansion}, as the
following reasoning shows. The moments $\mu_\ell$ which occur in the
expression in \Eq{kappaintermsofmu} for the cumulants $\kappa_n$, can
be worked out by taking the terms up to \textorder{t}{n} in
\Eq{newexpansion}, which gives
\begin{align}
\mu_\ell &=
\average{
\big[u_it+X_1 t +\ldots +X_n t^n\big]^\ell
+\order{t^{\ell+n}}
}
.
\end{align}
A product of two $\mu_\ell$'s can then be written as
\begin{eqnarray}
\mu_{\ell_1}\mu_{\ell_2} &=&
\average{
\big[u_it+X_1 t
+\ldots
+X_n t^n
\big]^{\ell_1}}
\average{
\big[ u_it+X_1 t
+\ldots
+X_n t^n
\big]^{\ell_2}}
\nonumber\\&&
+\order{t^{\ell_1+\ell_2+n}}.
\end{eqnarray}
Applying this recursively to a product of $\mu_{\ell_\gamma}$
($\gamma=1\ldots \alpha$), the remainder term in replacing the full
expression for $\Delta r_i(t)$ by the first $n$ terms is
$\order{t^{\sum_{\gamma=1}^\alpha\ell_\gamma+n}}$. If some of the
$\ell_\gamma$ values occur more than once in this product, this can be
rewritten as $\order{t^{\sum_{\ell=1}^\infty \ell p_\ell +n}}$, where
$p_\ell$ is the frequency of occurrence of $\ell$. According to
\Eq{kappaintermsofmu}, the $n$th cumulant is a sum of terms which are
products of $\mu_{\ell}$ with frequencies $p_\ell$ such that the
$\ell p_{\ell}$ add up to $n$, \emph{i.e.}\ $\sum_\ell \ell p_\ell=n$, so for each of those terms the remainder in replacing the
full expression for $\Delta r_i(t)$ by the first $n$ terms, is
$\order{t^{2n}}$ and
\begin{align}
\kappa_n &=
\bcumulant{\Big[u_it+\sum_{\gamma=1}^{n}X_\gamma t^{\gamma}\Big]^n}
+ \order{t^{2n}}
.
\label{lemmaA3}
\end{align}
We remark that the first term on the right hand side (\rhs) of this
equation gives all powers $t^n$ up to $t^{2n-1}$, which we are
interested in, as well as some of the higher powers of $t^{\alpha\geq
2n}$ which we are not interested in. The second term, \emph{i.e.},
$\order{t^{2n}}$ only contains higher powers of $t^{\alpha\geq 2n}$.
So with \Eq{lemmaA3} we have established that for the lower powers
only $n$ terms in the time expansion of $\Delta r_i(t)$ are needed,
but we have not separated the powers of $t$ lower and higher than $2n$
completely yet.
For that purpose we continue from \Eq{lemmaA3} and use first the
translation invariance property of cumulants, which states that
$\cumulant{(X+C)^n}=\cumulant{X^n}$ if $n>1$ and $C$ a constant while
for $n=1$ one has $\cumulant{X+C}=\cumulant{X}+C$,\cite{Cramer46,VanKampen}
to obtain from \Eq{lemmaA3} (with $C=u_it$)
\begin{align}
\kappa_n &=
\bcumulant{\Big[\sum_{\gamma=1}^{n} X_\gamma t^{\gamma}\Big]^n}
+ u_it\,\delta_{n1}+
\order{t^{2n}}.
\label{an29}
\end{align}
Furthermore, we can write the first term on the \rhs\ of in this
expression, using \Eqs{GXdef} and \eq{defcumvar} with $\alpha=1$ and
$k_1=k$, as
\begin{align}
\bcumulant{\Big[\sum_{\gamma=1}^{n} X_\gamma t^{\gamma}\Big]^n}
&=
\dd[n]{\ }{(\mathrm{i} k)}
\log\baverage{\exp \sum_{\gamma=1}^n \mathrm i k X_\gamma t^\gamma}
\bigg|_{k=0}
\nonumber\\
&=
\dd[n]{\ }{(\mathrm{i} k)}
\left[
\log\baverage{\exp \sum_{\gamma=1}^n \mathrm ik_\gamma X_\gamma t^\gamma}
\bigg|_{k_{1\ldots n}=k}\right]
\Bigg|_{k=0}.
\label{a29}
\end{align}
The introduction of the $k_\gamma$ ($\gamma=1\ldots n$) here may seem
unnecessary, as they are all set equal to $k$ and subsequently to zero
at the end, but now one can use the chain rule and a
multinomial expansion to find from \Eq{a29}
\begin{align}
\bcumulant{\Big[\sum_{\gamma=1}^{n} X_\gamma t^{\gamma}\Big]^n}
&=
\left(\sum_{\gamma=1}^n\dd{\ }{(\mathrm{i} k_\gamma)} \right)^{\!n}
\log\baverage{\exp \sum_{\gamma=1}^n \mathrm ik_\gamma X_\gamma t^\gamma}
\bigg|_{k_{1\ldots n}=0}
\nonumber\\
&=
\mathop{\sum_{n_\gamma\geq 0}}_{\sum_{\gamma=1}^n
n_\gamma=n}
\!\!
\frac{n!}{n_1! \cdots n_n!}
\nonumber\\&\qquad\times
\prod_{\gamma=1}^n\dd[n_\gamma]{\ }{(\mathrm{i} k_\gamma)}
\log\baverage{\exp \sum_{\gamma=1}^n \mathrm ik_\gamma X_\gamma t^\gamma}
\bigg|_{k_{1\ldots n}=0},
\label{a30}
\end{align}
where on the \rhs\ of \Eq{a30} we now recognize the multi-variable cumulant
defined in~\Eqs{GXdef} and \eq{defcumvar} with $\alpha=n$, whence
\begin{align}
\bcumulant{\Big[\sum_{\gamma=1}^{n} X_\gamma t^{\gamma}\Big]^n}
&= \mathop{\sum_{n_\gamma\geq 0}}_{\sum_{\gamma=1}^n
n_\gamma=n}
\!\!
\frac{n!}{n_1! \cdots n_n!}
\cumulant{X_1^{n_1}X_2^{n_2}\ldots X_n^{n_n}}
t^{\sum_{\gamma=1}^nn_\gamma\gamma}.
\label{a32}
\end{align}
For $\kappa_n$ in \Eq{an29}, we need this quantity only explicitly up
to $\order{t^{2n-1}}$, so powers of $t$ higher than $2n-1$ may be
discarded (\emph{i.e.}, combined with the $\order{t^{2n}}$ term) and
only powers $t^{\alpha<2n}$ need to be kept. Hence, in the exponent on
the \rhs\ of \Eq{a32}, we only need terms with
$\sum_{\gamma=1}^nn_\gamma\gamma<2n$. Since also
$n=\sum_{\gamma=1}^nn_\gamma$, we obtain
$\sum_{\gamma=1}^nn_\gamma\gamma < 2 \sum_{\gamma=1}^nn_\gamma$, or
$n_1>\sum_{\gamma=2}^n n_\gamma(\gamma-2)$. Combining this condition
with \Eqs{an29} and \eq{a32}, and using $X_1=V_i$ (cf.\ \Eq{wdef}), we
find that the expansion of $\kappa_n$ in time $t$ up to
$\order{t^{2n-1}}$ is given by
\begin{multline}
\kappa_n =
\!\!\!
\mathop{\mathop{\sum_{n_\gamma\geq 0}}_{\sum_{\gamma=1}^n
n_\gamma=n}}_{n_1>\sum_{\gamma=2}^n n_\gamma(\gamma-2)}
\!\!\!
\frac{n!}{n_1! \cdots n_n!}
\cumulant{V_i^{n_1} X_2^{n_2} \ldots
X_n^{n_n}}
t^{\sum_{\gamma=1}^n n_\gamma\gamma}
+u_it\,\delta_{n1}
+\order{t^{2n}}.
\label{expanded}
\end{multline}
We note that in \Eq{expanded} only cumulants appear, instead of
moments, and, more importantly, that each power of $t$ is easily
identified and we have separated the powers of $t$ lower and higher
than $2n$, something that in the straightforward moment-based
approach mentioned above only happens after a lengthy calculation.
\subsection*{Second part: Writing cumulants in terms of velocities}
The dependence of $\kappa_n$ in \Eq{expanded} on the peculiar
velocities $V^N$ follows from the dependence of the $X_\gamma$ on the
$V^N$. We note that for $\gamma\geq2$, $X_\gamma$ is a polynomial in
the peculiar velocities $V^N$ of degree $\gamma-2$, with coefficients
that can depend on the positions of the particles and the time, as can
be seen inductively as follows. For $\gamma=2$, \Eqs{wdef}, \eq{Vdef}
and \eq{motion2} show that $X_2=(dV_i/dt)/2=F_i(r^N,t)/(2m_i)$, which is
independent of $V^N$ so it is a polynomial in the peculiar velocities
of degree zero. Using the recursion relation \Eq{newerrecursion} one
now sees that if $X_\gamma$ is a polynomial in $V^N$ of degree
$\gamma-2$ then on the \rhs\ of \Eq{newerrecursion}, the term
$(u_j+V_j)\partial{X_\gamma}/\partial{r_j}$ is a polynomial of degree
$\gamma-1$, while $(F_j/m_j)\partial{X_\gamma}/\partial{V_j}$ has a
degree of $\gamma-3$, and $\partial{X_\gamma}/\partial{t}$ has a
degree $\gamma-2$. The highest power of $V^N$ in $X_{\gamma+1}$ is
thus indeed $\gamma-1$.
Therefore we can write for the dependence of $X_\gamma$ on $V^N$
\begin{eqnarray}
X_{\gamma} &=& \sum_{p=0}^{\gamma-2} \sum_{\sum_j p_j=p}
a_{\{p_j\}}(r^N,t) V_1^{p_1}\cdots V_N^{p_N}
\end{eqnarray}
or, focusing on powers of $V_i$,
\begin{eqnarray}
X_{\gamma}
&=& \sum_{p_i=0}^{\gamma-2}
b_{\{p_j\}}(r^N,V^N_{j\neq i},t) \,V_i^{p_i}
\label{Xpoly}
\end{eqnarray}
As it turns out, in the third part of the proof we will only need this
polynomial nature, while the precise and explicit forms of
$a_{\{p_j\}}(r^N,t)$ or $b_{\{p_j\}}(r^N,V^N_{j\neq i},t)$ are not
required.
\subsection*{Third part: Using the Gaussian nature of velocities}
Given the polynomial nature of the $X_\gamma$ as a function of $V_i$
in \Eq{Xpoly}, we see that in the second condition under the summation
sign in \Eq{expanded}, \emph{i.e.}, $n_1>\sum_{\gamma=2}^n
n_\gamma(\gamma-2)$, the expression $n_\gamma(\gamma-2)$ is also the
degree in $V_i$ of the polynomial $X_\gamma^{n_\gamma}$ which occurs
on the \rhs\ of \Eq{Xpoly}. Then,
$\sum_{\gamma=2}^nn_\gamma(\gamma-2)$, which we will denote by $d$, is
also the combined degree of the product of $X_2^{n_2}$ up to $X_n^{n_n}$
that occurs in \Eq{expanded}. Thus the sum in \Eq{expanded} is over
cumulants for which the power $n_1$ of the peculiar velocity $V_i$ is
higher than the combined degree $d$ of $X_2^{n_2}\cdots X_n^{n_n}$.
The auxiliary \Theorem{cumulant with v power} in the Appendix can then
be applied to cumulants of this form
$\cumulant{V_i^{n_1}X_2^{n_2}\ldots X_n^{n_n}}$. \Theorem{cumulant
with v power} states that if $X_\gamma$ ($\gamma=2\ldots n$) are
polynomials of degree $d_\gamma$ in a zero-mean Gaussian distributed
$V$, and $n_\gamma$ are nonnegative integers, then $\cumulant{V^{n_1}
X_2^{n_2} \ldots X_n^{n_n}} =0$ if $n_1> d
\equiv\sum_{\gamma=2}^{n}n_\gamma d_\gamma$, except when all
$n_{\gamma\geq 2}=0$ and $n_1=2$, in which case
$\cumulant{V^{n_1}X_2^{n_2}\ldots X_n^{n_n}} = \cumulant{V^2}$.
We can therefore complete the proof of the main Theorem by applying
the auxiliary \Theorem{cumulant with v power} to each term in the
summation on the \rhs\ of \Eq{expanded}, with $V=V_i$ and
$d_\gamma=\gamma-2$ (see \Eq{Xpoly}). Note that the variables
$V_{j\neq i}$ and $r_i$ are independent of $V_i$, so that even though
\Theorem{cumulant with v power} is only formulated for the average of
$V=V_i$, we may afterwards also average over the other degrees of freedom and
still get the properties of \Theorem{cumulant with v power}.
The degree $d$ mentioned in \Theorem{cumulant with v power} is for our
case given by $d=\sum_{\gamma=2}^n n_\gamma d_\gamma =
\sum_{\gamma=2}^{n} n_\gamma (\gamma-2)$. As the restriction on the
summation in \Eq{expanded} shows, for each explicit term in the sum in
\Eq{expanded}, $n_1$ is larger than this $d$. Then \Theorem{cumulant
with v power} tells us that the cumulant occurring in each term is zero
except when $n_2=n_3=\cdots n_n=0$ and $n_1=2$. This exception means,
since also $\sum_{\gamma=1}^n n_\gamma=n$, that $n=n_1=2$. So the only
possible nonzero term in {the sum in} \Eq{expanded} occurs for $n=2$.
Furthermore, for $n=1$, only the last term $u_it\,\delta_{n1}$ in
\Eq{expanded} is left.
Thus, using \Theorem{cumulant with v power}, we have shown that each
term in \Eq{expanded} is zero separately except for $n=1$ and $n=2$,
so that
\begin{equation}
\kappa_n = \begin{cases}
\order{t^{2n}} & \quad \text{if }\: n> 2
\\
{\order{t^n}} & \quad \text{if }\: n\leq2
\end{cases}
\label{thisone}
\end{equation}
remains on the \rhs\ of \Eq{expanded}. Given that $\kappa_n$ was
expanded here as a power series in $t$, \Eq{thisone} coincides with the
formulation \Eq{Theorem} of the main Theorem, which is therefore now
proved (with the proviso that Theorem A is proved in the
Appendix).\qed
\subsection*{An expression for the coefficient $c_n$ in the Theorem}
We will now determine the coefficient of $t^{2n}$ in $\kappa_n$,
\emph{i.e.}, the $c_n$ of \Eq{Theorem} of the Theorem. For $n=1$ and
$n=2$ it is straightforward to show that
\begin{subequations}
\label{cn}
\begin{eqnarray}
c_1 &=& \average{v_i} \:=\: u_i\\
c_2 &=& \average{V_i} \:=\: 1/(\beta_im_i).
\end{eqnarray}
To find $c_n$ for $n>2$ one can use the same calculation of $\kappa_n$
as used above, but one has to take one additional term
$X_{n+1}t^{n+1}$ in the time expansion of $\Delta r_i(t)$ in
\Eq{lemmaA3}, \emph{i.e.}, one writes $\kappa_n
=\cumulant{[u_it+\sum_{\gamma=1}^{n+1}X_\gamma t^{\gamma}]^n} +
\order{t^{2n+1}}$. Performing then the same kind of manipulations as
in the proof above one arrives at
\begin{equation}
c_{n>2} =
\mathop{
\mathop{
\sum_{n_\gamma\geq0}
}_{\sum_{\gamma=1}^{n+1}n_\gamma=n}
}_{\sum_{\gamma=1}^{n+1}\gamma n_\gamma=2n}
\frac{n!}{n_1! \cdots n_{n+1}!}
\cumulant{V_i^{n_1} X_2^{n_2} \ldots
X_{n+1}^{n_{n+1}}}
.
\end{equation}
\end{subequations}
Examples of this for $n=3$ and $n=4$ are
\begin{eqnarray}
c_3 &=& 3\cumulant{V_i^2X_4}+6\cumulant{V_i X_2 X_3}
+\cumulant{X_2^3}
\\
c_4 &=& 4\cumulant{V_i^2X_5} +
6\cumulant{V_i^2X_3^2}+12\cumulant{V_i^2X_2X_4}
\nonumber\\
&&+12\cumulant{V_iX_2^2X_3}+\cumulant{V_2^4}.
\end{eqnarray}
Although the $X_\gamma$ are useful to derive these expressions for
$c_n$, to evaluate the $c_n$ in practice requires additional work. One
would first need to write the $X_\gamma$ out as
$d^{\gamma-1}V_i/t^{\gamma-1}$ (cf.~\Eq{wdef}) and work out the
derivatives of $V_i$ using the recursion relation
\eq{recursion}. Furthermore, the cumulants would have to be worked out
in terms of averages. The values of these averages will depend on the
system and their evaluation will in general require a numerical
approach.
\section{Applications and discussion}
\label{discussion}
In this paper we have presented and proved a theorem that states that
$n$th order cumulant of the displacement of a single particle scales
as $t^{2n}$ for small $t$ and all $n>2$ for a large class of systems for
which the initial velocities are Gaussian and independent.
While it is true that only displacements in one direction are
considered, for isotropic systems, the result may be translated to
displacements in more than one dimension. Work on the solution for
non-isotropic systems is underway.
We will now discuss a number of applications of the Theorem just
proved.
\emph{1) The equilibrium Van Hove self-correlation function}. a) In
incoherent neutron scattering on an equilibrium fluid, one essentially
measures the equilibrium Van Hove self-correlation function
$G_s(r,t)$, which is the Fourier inverse of the incoherent scattering
function
$F_s(k,t)$.\cite{VanHove54,BoonYip80,HansenMcDonald86,Squires,Arbeetal02,Arbeetal03}
If the wave vector $k$ in $F_s(k,t)$ is small, then according to
\Eq{Fsrelation} $F_s(k,t)\approx\exp[-\kappa_2k^2/2]$, \emph{i.e.},
nearly Gaussian, and so its inverse Fourier transform $G_s(r,t)$ is
also approximately Gaussian. Corrections to this Gaussian behavior can
be found by resumming $F_s(k,t)=\exp\sum_n \kappa_n(\mathrm i k)^n/n!$
in \Eq{Fsrelation} to the form
\begin{equation}
F_s(k,t)=e^{-\frac12\kappa_2 k^2}\left[1+\sum_{n=2}^\infty (-1)^nb_{2n}
k^{2n}\right]
\end{equation}
using that odd cumulants are zero. The coefficients $b_{2n}$ are given by
\begin{equation}
b_{2n} \:=\:
\mathop{\sum_{p_\ell\geq 0}}_{\sum_{\ell=2}^\infty \ell p_\ell = n}
\prod_{\ell=2}^\infty\:\:
\left[
\frac{1}{p_\ell!}\left(\frac{\kappa_{2\ell}}{(2\ell)!}\right)^{p_\ell}
\right]
.
\label{equilbnintermsofkappan}
\end{equation}
Fourier inverting the resummed form of $F_s(k,t)$ leads to
\begin{equation}
G_s(r,t) = \frac{e^{-w^2}}{\sqrt{2\pi\kappa_2}}\left[
1+
\sum_{n=2}^\infty
\frac{b_{2n}}{(2\kappa_2)^n}H_{2n}(w)\right],
\label{equilGs}
\end{equation}
where $H_n$ is the $n$th Hermite polynomial and the dimensionless
$w\equiv r/\sqrt{2\kappa_2}$. We note that the series in \Eq{equilGs}
appears to have a fairly rapid convergence.\cite{LevesqueVerlet70}
Taking just the first few terms would give
\begin{equation}
G_s(r,t) = \frac{e^{-w^2}}{\sqrt{2\pi\kappa_2}}\left[
1+\frac{\kappa_4}{4!4\kappa_2^2}H_4(w)+\frac{\kappa_6}{6!8\kappa_2^3}H_6(w)+\ldots
\right].
\label{afewgoodterms}
\end{equation}
The current Theorem provides a justification for the expansion in
\Eq{equilGs} of the self part of the Van Hove function $G_s$ for short
times $t$. As \Eqs{equilbnintermsofkappan} and \eq{equilGs} show, the
cumulants $\kappa_{n\geq4}$ give, via the $b_{2n}$, corrections to a
Gaussian behavior of $G_s(r,t)$. The Gaussian factor $e^{-w^2}$
suggests that typical values of $w$ are $\order{1}$ in \Eq{equilGs},
so also $H_{2n}(w)=\order{1}$. Its prefactor in \Eq{equilGs} is,
however, $t$-dependent through $b_{2n}/(2\kappa_2)^{n}$. Given the
relation between $b_n$ and $\kappa_n$ in \Eq{equilbnintermsofkappan}
it is easy to see that they scale similarly, \emph{i.e.}, if the
conditions of the Theorem are satisfied so that
$\kappa_{n>2}=\order{t^{2n}}$ then also $b_{n}=\order{t^{2n}}$. Since
$\kappa_2=\order{t^2}$, we obtain
\begin{equation}
\frac{b_{2n}}{(2\kappa_2)^{n}} = \order{t^{2n}}.
\end{equation}
\emph{This means that the series in \Eq{equilGs} is well-behaved for
small times $t$, in that each next term is smaller than the previous
one, and that by truncating the series one obtains for small $t$
approximations which can be systematically improved by taking more
terms into account. Note that in contrast if $\kappa_n$ had been
$\order{t^n}$, each term in the series in \Eq{GFexpand} would have
been of the same order.}
b) An expansion of a similar form as \Eq{equilGs} was found by
Rahman\cite{Rahman64} for $G_s(r,t)$, and by Nijboer and
Rahman\cite{NijboerRahman66} for $F_s(k,t)$. Their expressions are in
terms of the so-called \emph{non-Gaussian parameters}
$\alpha_n$. These non-Gaussian parameters have recently also found an
application in the context of supercooled liquids and glasses, where
they have been proposed as a kind of order parameter for the glass
transition\cite{OdagakiHiwatari91,Arbeetal02,Arbeetal03} and as
measures of `dynamical heterogeneities' in supercooled liquids and
glasses.\cite{KobAndersen95,HurleyHarrowell96,Kobetal97,ZangiRice04}
Given the interest in these non-Gaussian parameters $\alpha_n$, we will now
compare the cumulants $\kappa_n$ with the
$\alpha_n$. The non-Gaussian parameters $\alpha_n$ are defined in
terms of the distance $r=\sqrt{\Delta x_i^2(t)+\Delta y_i^2(t)+\Delta
z_i^2(t)}$ traveled by a particle in time $t$ in a three dimensional
fluid, as\cite{Rahman64}
\begin{eqnarray}
\alpha_n&\equiv&
\frac{\saverage{r^{2n}}}{\saverage{r^2}^n(2n+1)!!/3^n}-1.
\label{alphandef}
\end{eqnarray}
For isotropic fluids,
$\average{r^{2n}}=(2n+1)\average{\Delta x_i^{2n}(t)}=(2n+1)\mu_{2n}$,
so that \Eq{alphandef} can be written as
\begin{eqnarray}
\alpha_n
&=&
\frac{\mu_{2n}-(2n-1)!!\mu_2^n}{(2n-1)!!\mu_2^n}.
\label{alphan}
\end{eqnarray}
We now see that even though both the $\alpha_n$ and the cumulants
$\kappa_{n>2}$ are, by construction, zero for Gaussian distributed
variables, in \Eq{alphan} the $\alpha_n$ are $2n$-th moments
$\mu_{2n}$ with only the most factored term, $\mu_2^n$, subtracted,
while the cumulants $\kappa_{2n}$ in \Eq{kappaintermsofmu} have all
possible factored terms subtracted. Using \Eq{alphan} and the inverse
of the relation between $\kappa_n$ and $\mu_n$ in
\Eq{kappaintermsofmu}, it is possible to express the $\alpha_n$ in terms of
$\kappa_n$ as
\begin{equation}
\alpha_n \:=\: n!
\mathop{\sum_{0\leq p_\ell < n}}_{\sum_{\ell=1}^\infty \ell p_\ell = n}
\prod_{\ell=1}^\infty\:\:
\left[
\frac{1}{p_\ell!}\left(\frac{2^\ell\kappa_{2\ell}}{(2\ell)!\kappa_2^\ell}\right)^{p_\ell}
\right]
.
\label{alphanintermsofkappan}
\end{equation}
According to this formal relation, the first few $\alpha_n$ are given by
\begin{subequations}
\label{43}
\begin{eqnarray}
\alpha_2 &=&\frac13 \frac{\kappa_4}{\kappa_2^2}
\label{Rahmansalpha2}
\\
\alpha_3 &=&\frac{\kappa_4}{\kappa_2^2}+\frac1{15}\frac{\kappa_6}{\kappa_2^3}
\end{eqnarray}
\end{subequations}
The Theorem says that for small times $t$, $\kappa_2=\order{t^2}$ and
$\kappa_{n>2}=\order{t^{2n}}$, so
$\kappa_{2n}/\kappa_2^n=\order{t^{2n}}$. According to
\Eq{alphanintermsofkappan}, all $\alpha_n$ have a contribution from
$\kappa_{4}/\kappa_2^2$, so that all $\alpha_n$ are of $\order{t^4}$,
in contrast to the cumulants $\kappa_{2n}$ which are of increasing
order in $t$ with increasing $n$. In fact, using
\Eq{alphanintermsofkappan} and the Theorem, one can derive
straightforwardly that the dominant term for small $t$ in
\Eq{alphanintermsofkappan} is the one with $p_1=n-2$, $p_2=1$ and
$p_{\ell>2}=0$, which leads to
\begin{equation}
\alpha_n \sim \frac{n(n-1)}{2}\,\alpha_2
\label{alphanrel}
\end{equation}
plus a correction of $\order{t^6}$. Thus, for small $t$, $\alpha_3$ is
approximately three times $\alpha_2$, $\alpha_4$ six times
$\alpha_2$ \emph{etc.} Such approximate relations are indeed borne out by
Rahman's data on $\alpha_2$, $\alpha_3$ and
$\alpha_4$.\cite{Rahman64,BoonYip80}
In terms of the $\alpha_n$, the expansion of $G_s$ in
\Eq{afewgoodterms} becomes, with the help of \Eq{43},
\begin{equation}
G_s(r,t) = \frac{e^{-w^2}}{\sqrt{2\pi\kappa_2}}\left[
1+\frac{3\alpha_2}{4!4}H_4(w)+\frac{15(\alpha_3-3\alpha_2)}{6!8}H_6(w)+\ldots
\right].
\label{afewgoodalphaterms}
\end{equation}
Although formally equivalent to \Eq{afewgoodterms}, for small $t$
\Eq{afewgoodalphaterms} is somewhat less convenient, because one
cannot see right away that the last term is $\order{t^6}$ rather than
$\order{t^4}$, as one may naively suspect from $\alpha_2=\order{t^4}$
and $\alpha_3=\order{t^4}$. In \Eq{afewgoodterms}, this time ordering
is clear because the Theorem says that $\kappa_6=\order{t^{12}}$ and
$\kappa_2=\order{t^2}$.
We note that surprisingly, the data of Rahman also show that the
relation between $\alpha_n$ and $\alpha_2$ in \Eq{alphanrel} is still
approximately satisfied for larger times.\cite{Rahman64} This
seems even true for hard spheres.\cite{Desai66} Thus the higher order
non-Gaussian parameters $\alpha_{n>2}$ are apparently dominated by
$\alpha_2$ for larger times just as they are for smaller times
$t$. This dominance of $\alpha_2$ makes it hard to extract from these
higher order non-Gaussian parameters any information that was not
already contained in $\alpha_2$. The cumulants $\kappa_n$, or perhaps
the $b_n$, may contain additional information about correlations in
(supercooled) fluids in a more accessible form (compared to the
$\alpha_n$), and may therefore be a more suitable choice to
investigate such correlations for all times $t$.
c) Returning to the series in \Eq{equilGs}, although it may be
well-behaved for small enough $t$, it is not known up to what time
that remains so. Since $b_n=\order{t^{2n}}$ only to leading order in
$t$, the point at which the series in \Eq{equilGs} is no longer
guaranteed to be useful may be related to where the $\order{t^{2n}}$
scaling of the $\kappa_n$ breaks down. Rahman\cite{Rahman64} (see also
ref.~\cite{BoonYip80} for a broader overview) investigated this
and related questions numerically for a model of liquid argon
(temperature 94~K, density $1.4\cdot 10^{3}$ kg/m$^{3}$). Figure 1
shows a sketch of the non-Gaussian parameter $\alpha_2$ as a function
of $t$ (based on fig.~7 in ref.~\cite{Rahman64} and fig.~4.11 in
ref.~\cite{BoonYip80}). One sees that $\alpha_2$ is a very flat
function near $t=0$, which persists only up to roughly $t\approx
0.1$--$0.2$~ps. At that point the curve shoots up rapidly, giving a
large `hump' which last up to about 10 ps, and after which it starts
to decrease to zero.
\renewcommand{\figurename}{\small Figure}
\begin{figure}[bt]
\centerline{\includegraphics[width=0.83\textwidth]{figure1}}
\caption{Sketch of the behavior of the non-Gaussian parameter
$\alpha_2=\kappa_4/(3\kappa_2^2)$ for regular liquids in equilibrium
in different time regimes (based on fig.~7 in
ref.~\cite{Rahman64} and fig.~4.11 in
ref.~\cite{BoonYip80}), with our physical interpretation for
each regime.}
\end{figure}
Although somewhat outside the scope of this paper, we would like to
give an interpretation of the numerical results sketched in Figure
1. \emph{a)}~The flat behavior of $\alpha_2$ near $t=0$ is the
$\order{t^4}$ behavior as given by the Theorem proved in this
paper. \emph{b)}~Because hard spheres can be seen as a limit of a
smooth interparticle potential in which the steepness goes to
infinity,\cite{Sears72} and the potential used by Rahman is rather
steep, the shoot-up phenomenon at $\approx 0.1$~ps is probably related
to the hard-spheres result of De Schepper \emph{et
al.}\cite{DeSchepperetal81} that $\kappa_{n}=\order{|t|^{n+1}}$, or
$\alpha_2=\order{|t|^5/t^4}=\order{|t|}$, as follows. The steep but
smooth potential of Rahman will resemble a hard sphere fluid on time
scales $t_s$ on which a collision has been
completed.\cite{DeSchepperCohen76} Thus at $t=t_s$ the scaling
$\order{t^4}$ for $\alpha_2$ ought to go over to $\order{|t|}$ which
would require the kind of sharp increase observed by Rahman at
$\approx 0.1$ ps. \emph{c)}~The persistence of the non-Gaussianity
occurs because the particle is trapped in a `cage' formed by its
neighboring particles, with which it has repeated and correlated
collisions. \emph{d)}~The decay of $\alpha_2$ to zero indicates that
the motion becomes Gaussian and presumably sets in where the particle
manages to escape its cage. After escaping it finds itself in a new
cage environment consisting largely of particles with which it has not
interacted before. This motion from cage to cage is called cage
diffusion.\cite{CohenDeSchepper91} \emph{e)}~From a central limit
theorem argument using that successively visited cages after many cage
escapes have little correlation with each other, one would then expect
Gaussian (and presumably but not necessarily diffusive) behavior.
Also in simulations of a supercooled argon-like
mixture,\cite{KobAndersen95} $\alpha_2$ plotted as a function of time
shows a flat curve for short times and a sharp increase around $0.1$
ps, while $\alpha_3(t)$ shows similar behavior. The interesting part
from the perspective of supercooled liquids and glasses, however, is
in how far that increase continues and on what time scale and how
$\alpha_2$ decays back to zero, which takes a very long time for
supercooled liquids and is related to the time scale at which the
particle escapes its cage. But the Theorem has nothing to say about
$\alpha_2$ on that time scale.
\emph{2) Local equilibrium systems.} In local equilibrium a fluid has
roughly an equilibrium distribution except that the temperature,
fluid velocity and density are spatially dependent. The class of
initial distributions in \Eq{distribution} does not seem to be of that form,
and indeed, if $\beta_i$ and $u_i$ are allowed to vary with $r_i$, then
the proof as given here runs into difficulties. Nonetheless, one can
construct distributions of the form \eq{distribution} which physically
describe precisely the local equilibrium situation. Imagine
dividing the physical volume $\mathcal V$ up into $A$ cells and assigning to
each cell $a$ a temperature $\beta_a$, a fluid velocity $u_a$ and a
density $n_a$. Divide the particles of the system up as well, putting
$N_a=n_a \mathcal V/A$ particles in each cell, such that particles $1$ through
$N_1$ are in cell $1$, $N_1+1$ through $N_1+N_2$ are in cell $2$
etc. This can be accomplished by choosing $f(r^N)$ in
\Eq{distribution} such that the chance for these particles to be
outside their cell is zero. Next, we set all the $\beta_i$ and $u_i$
of the particles in cell $a$ to $\beta_a$ and $u_a$. If the cells are
big enough so that fluctuations in the number of particle may be
neglected, this situation describes local equilibrium just as well as
spatially dependent $\beta(r_i)$ and $u(r_i)$ can, and for this
constructed local equilibrium, the Theorem holds.
\emph{3) Far-from-equilibrium phenomena on the picosecond time scale.}
The cumulants whose short time scaling was obtained here also occur
naturally in the Green's function theory which was developed for
far-from-equilibrium phenomena on the picosecond and nanometer
scales.\cite{Kincaid95,KincaidCohen02a,KincaidCohen02b,CohenKincaid02,VanZonCohen05b}
Considering for example a mixture of two components, one can write
the density of component $\lambda$ ($\lambda=1$ or $2$) at position
$x$ and for simplicity in one dimension as:\cite{VanZonCohen05b}
\begin{equation}
n_\lambda(x, t) = \int\! dx'\,G_\lambda(x,x',t)n_\lambda(x',0),
\end{equation}
where the non-equilibrium \emph{Green's function} $G_\lambda(x,x',t)$
is the probability that a particle of component $\lambda$ is at
position $x$ at time $t$ given that it was at position $x'$ at time
zero. By an expansion detailed in a future
publication\cite{VanZonCohen05b} $G_\lambda$ can be written similarly as
$G_s$ in \Eq{equilGs}, as
\begin{equation}
\label{GFexpand}
G_\lambda(x,x',t)=
\frac{e^{-w^2}}{\sqrt{2\pi \kappa_2}}.
\left[
1 + \sum_{n=3}^{\infty} \frac{b_n}{(2\kappa_2)^{n/2}}
H_n(w)
\right]
\end{equation}
Here, the dimensionless $w\equiv(x-x'-\kappa_1)/\sqrt{2\kappa_2}$ and
\begin{equation}
b_n \:=\:
\mathop{\sum_{p_\ell\geq 0}}_{\sum_{\ell=3}^\infty \ell p_\ell = n}
\prod_{\ell=3}^\infty\:\:
\left[
\frac{1}{p_\ell!}\left(\frac{\kappa_\ell}{\ell!}\right)^{p_\ell}
\right]
.
\label{bnintermsofkappan}
\end{equation}
Furthermore, in the Green's function theory, the $\kappa_n$, and thus
the $b_n$ through \Eq{bnintermsofkappan}, depend on $x'$ because in
that theory the single particle $i$ of component $\lambda$ is required to
have been at the position $x'$ at time zero. This requirement can be
imposed by multiplying the probability distribution function in
\Eq{distribution} by $\delta(x_i-x')$ (times a proper
normalization). The resulting distribution describes the subensemble
of the original ensemble for which particle $i$ is at $x'$ at time
zero. Note also that it is still of the same form as \Eq{distribution},
with $f(r^N)\rightarrow f'(r^N)=\delta(x_i-x')f(r^N)$, so the Theorem
still applies. Note that the $f(r^N)$ may describe any non-equilibrium
distribution of the positions of the particles.
The Gaussian factor in \Eq{GFexpand} suggests as before that typical values of $w$ are $\order{1}$, so $H_n(w)=\order{1}$. Its prefactor
in \Eq{GFexpand} is $b_n/(2\kappa_2)^{n/2}$. Using the Theorem that
$\kappa_n=\order{t^{2n}}$ and the relation between $b_n$ and $\kappa_n$
in \Eq{bnintermsofkappan} it is easy to see that
$b_{n}=\order{t^{2n}}$. Since $\kappa_2=\order{t^2}$, we obtain
\begin{equation}
\frac{b_n}{(2\kappa_2)^{n/2}} = \order{t^n}.
\end{equation}
\emph{This means that the series for the non-equilibrium Green's
functions $G_\lambda$ in \Eq{GFexpand} are well-behaved for small
times $t$ (just as the equilibrium Van Hove self-correlation function
was) and that by truncating the series one obtains for small $t$
approximations which can be systematically improved by taking more
terms into account.}
We note that in non-equilibrium situations,
$\alpha_2=\kappa_4/(3\kappa_2^2)$ may have a similar behavior as
sketched in Figure 1 for the equilibrium $\alpha_2$. Although a proper
numerical test is yet to be performed, this expectation is roughly
consistent with numerical results of the Green's function for heat
transport:\cite{KincaidCohen02a,KincaidCohen02b,CohenKincaid02} In
that case the contribution of the non-Gaussian corrections in
\Eq{GFexpand} were most significant on the sub-picosecond time scale,
whereas extrapolation indicated that hydrodynamic-like results may
occur for times as short as 2 ps.\cite{KincaidCohen02b}
In view of the possible application to nano-technology, it is
important to understand the behavior on the (sub)picosecond time scale
and the related (sub) nanometer length scale over which a particle
typically moves at such time scales. The Green's function theory can
potentially describe a system on all time scales. The current theorem
assures that this theory can at least consistently describe the short
time scales, by showing that the expansion of the Green's function is
well-behaved. For practical applications, and to know how short the
time scales must be for the theorem to apply, it is still necessary to
determine the coefficient of the $\order{t^{2n}}$ of $\kappa_n$,
\emph{i.e.} the $c_n$ in \Eq{cn}. This will require a numerical
evaluation of the moments of derivatives of the forces, which we plan
to do in the future. The behavior of $\kappa_n$ at longer time scales
will also be investigated in the future.\cite{VanZonCohen05b}
\section*{Acknowledgments}
This work was supported by the Office of Basic Energy Sciences of
the US Department of Energy under grant number DE-FG-02-88-ER13847.
\appendix
\section*{Appendix}
\renewcommand{\theequation}{A.\arabic{equation}}
\setcounter{equation}{0}
In the text, we needed the following auxiliary theorem to prove the main Theorem.
\begin{theorem}
\label{cumulant with v power}
Let $V$ be a zero mean Gaussian variable and let
$\{X_\gamma\}_{\gamma=2\ldots\alpha}$ be a set of functions of $V$
which are polynomials of degree $d_\gamma$. Let
$\{n_\gamma\}_{\gamma=1\ldots\alpha}$ be a set of nonnegative
integers, and let $d =\sum_{\gamma=2}^{\alpha}n_\gamma d_\gamma$. Then
if $n_1>d$ and at least one $n_{\gamma>1}\neq 0$, the following
cumulant vanishes:
\begin{equation}
\cumulant{V^{n_1} X_2^{n_2} \ldots X_\alpha^{n_\alpha}} =0.
\label{vanishing}
\end{equation}
while if all $n_{\gamma>1}$ are zero, one has
\begin{equation}
\cumulant{V^{n_1}} = \cumulant{V^2}\delta_{n_12}.
\label{nonvanishing}
\end{equation}
\end{theorem}
Since we have not found a proof of this theorem in the literature, we
will give it here, but before we can prove \Theorem{cumulant with v
power}, we need four lemmas and the definition of a $\theta$-modified
average. This definition will serve, in conjunction with
\Lemma{theta and normal cumulants} below, to construct a convenient
generating function (which takes the form of a $\theta$-modified
cumulant) for the quantities $\cumulant{V^{n_1} X_2^{n_2} \ldots
X_\alpha^{n_\alpha}}$ that occur in \Theorem{cumulant with v power}.
Working with this generating function will be more convenient than
trying to calculate each $\cumulant{V^{n_1} X_2^{n_2} \ldots
X_\alpha^{n_\alpha}}$ individually. After this generating function
has been introduced, \Lemma{average with v power} and
\Lemma{theta average polynomial} are presented and proved, which
use the Gaussian nature of $V$ to give properties of averages and
$\theta$-modified averages of powers of $V$ which allow one to prove a
central polynomial property (\Lemma{theta cumulant polynomial}) of
the generating function of the quantities $\cumulant{V^{n_1} X_2^{n_2}
\ldots X_\alpha^{n_\alpha}}$. This central polynomial property is the
main ingredient in the proof of \Theorem{cumulant with v power} at the
end of this Appendix.
The definition of $\theta$-modified averages and cumulants is:
\begin{definition}
Given the zero-mean Gaussian variable $V$, the $\theta$-modified average
of a variable $X$ is defined as
\begin{equation}
\average{X}_\theta \equiv
\frac{\saverage{e^{\theta V}
X}}{\average{e^{\theta V}}}
=
e^{-\frac12\theta^2\langle{V^2}\rangle}\average{e^{\theta V} X}
.
\label{modavdef}
\end{equation}
Similarly, $\theta$-modified moments are defined as the
$\theta$-modified averages of powers of variables (\emph{i.e.},
functions of $V$), and $\theta$-modified cumulants are defined as
having the same relation to $\theta$-modified moments as normal
cumulants have to normal moments. $\theta$-modified cumulants are
therefore also given through the $\theta$-modified cumulant generating
function as:
\begin{equation}
\cumulant{X_1^{n_1} \ldots X_\alpha^{n_\alpha}}\big._\theta
\equiv
\prod_{\gamma=1}^\alpha \dd[n_\gamma]{\ }{(\mathrm{i} k_\gamma)}
\log\baverage{\exp\sum_{\gamma=1}^\alpha \mathrm{i} k_\gamma
X_\gamma}_{\!\!\theta}\bigg|_{k_{1\ldots\alpha}=0}.
\label{modcumdef}
\end{equation}
\end{definition}
\noindent
The generating function nature of the $\theta$-modified cumulants
follows from:
\begin{lemma}{\emph{(relation between cumulants and $\theta$-modified cumulants)$\,$}}\footnote{For
$\alpha=n_1=1$ this lemma coincides with the last exercise of section
XVI.3 in Van Kampen.\cite{VanKampen}}\
\label{theta and normal cumulants}
The $\theta$-modified cumulants of a set of variables $X_1$, \ldots
$X_\alpha$ are related to the normal cumulants by
\begin{equation}
\cumulant{X_1^{n_1} \ldots X_\alpha^{n_\alpha}}\big._\theta
=
\begin{cases}\displaystyle
\sum_{n=0}^\infty \frac{
\cumulant{V^n X_1^{n_1} \ldots X_\alpha^{n_\alpha}}
}{n!}
\,\theta^n
&\quad\text{if at least one $n_\gamma\neq0$,}\\
0&\quad\text{otherwise.}
\end{cases}
\label{modcum}
\end{equation}
\end{lemma}
\begin{proof}
The case $n_1=\cdots=n_\alpha=0$ is trivial: $\cumulant{X_1^{n_1}
\ldots X_\alpha^{n_\alpha}}\big._\theta =\cumulant{1}\big._\theta=0$
since a zeroth cumulant is always zero. For the other cases, we start
with the \rhs\ of \Eq{modcum} and use the expression for the cumulants
in \Eq{defcumvar}:
\begin{multline}
\sum_{n=0}^\infty \frac{
\cumulant{V^n X_1^{n_1} \ldots X_\alpha^{n_\alpha}}
}{n!}
\,\theta^n
\\
=
\sum_{n=0}^\infty \frac{\theta^n}{n!}
\dd[n]{\ }{(\mathrm{i} k_0)}
\prod_{\gamma=1}^\alpha \dd[n_\gamma]{\ }{(\mathrm{i} k_\gamma)}
\log\baverage{\exp\Big[\mathrm{i} k_0 V+\sum_{\gamma=1}^\alpha \mathrm{i} k_\gamma
X_\gamma\Big]}
\Bigg|_{k_{0\ldots\alpha}=0}
\end{multline}
We recognize the Taylor series, \emph{i.e.}, that
$\sum_{n=1}^{\infty}([-i\theta]^n/n!)\, \partial^n f(k_0)/\partial
{k_0^n}\Big|_{k_0=0}=f(-i\theta)$ to write this as
\begin{multline}
\sum_{n=0}^\infty \frac{
\cumulant{V^n X_1^{n_1} \ldots X_\alpha^{n_\alpha}}
}{n!}
\,\theta^n
\\
=
\prod_{\gamma=1}^\alpha \dd[n_\gamma]{\ }{(\mathrm{i} k_\gamma)}
\log\baverage{\exp\Big[\theta V+\sum_{\gamma=1}^\alpha \mathrm{i} k_\gamma
X_\gamma\Big]}\bigg|_{k_{1\ldots\alpha}=0}
\end{multline}
Using definition \eq{modavdef}, this becomes
\begin{multline}
\sum_{n=0}^\infty \frac{
\cumulant{V^n X_1^{n_1} \ldots X_\alpha^{n_\alpha}}
}{n!}
\,\theta^n
\\
=
\prod_{\gamma=1}^\alpha \dd[n_\gamma]{\ }{(\mathrm{i} k_\gamma)}
\bigg[
\log\baverage{\exp\sum_{\gamma=1}^\alpha \mathrm{i} k_\gamma
X_\gamma}_{\!\!\theta}+ \frac12\theta^2\saverage{V^2}
\bigg]\Bigg|_{k_{1\ldots\alpha}=0}
\end{multline}
Since we are considering the case that at least one $k_\gamma$ is
nonzero, the contribution from the $k_\gamma$-independent
$\frac12\theta^2\saverage{V^2}$ vanishes when
$\partial/\partial({\mathrm i k_\gamma})$ acts on it. The remainder is
by definition \eq{modcumdef} equal to the \lhs\ of \Eq{modcum}.
\end{proof}
A consequence of this lemma, \emph{i.e.}, of \Eq{modcum}, is that
repeated derivatives with respect to $\theta$ of $\cumulant{X_1^{n_1}
\ldots X_\alpha^{n_\alpha}}_\theta$ taken at $\theta=0$ generate the
$\cumulant{V^{n} X_1^{n_1} \ldots X_\alpha^{n_\alpha}}$.\vspace{2mm}
The second lemma concerns the average of a product of powers of $V$
and a single polynomial function of $V$.
\begin{lemma}{\emph{(Averages of powers of $V$ times a
polynomial in $V\!$)}}
\label{average with v power}
Let $X$ be a variable which is a polynomial function of degree $d$ in
the zero mean Gaussian variable $V$ and which is even if $d$ is even
and odd if $d$ is odd. The quantities ($n\geq 0$)
\begin{equation}
A_n
=\average{V^{n} X}
\label{Andef}
\end{equation}
are a) zero in cases where $n$ is odd but $d$ is even and in cases
where $n$ is even but $d$ is odd, b) or otherwise given in terms of
$A_n$ with lower $n$ as follows. For even $n=2s$ and $d=2s^*$, set
$\delta=0$, while for odd $n=2s+1$ and $d=2d^*+1$, set $\delta=1$,
respectively. Then
\begin{eqnarray}
A_{2s+\delta} &=&
\sum_{s'=0}^{s^*} \frac{(2s-1+2\delta)!!}{
(2s'-1+2\delta)!!}
\frac{
\saverage{V^2}^{s-s'}A_{2s'+\delta}
}{s'!(s^*-s')!}
\mathop{\prod_{s''=0}^{s^*}}_{s''\neq s'}
(s-s'').
\label{odd}
\end{eqnarray}
\end{lemma}
\begin{proof}
a) Note first that $d$ should be even if $n$ is even and odd if $n$ is
odd, respectively, in order to get a non-zero result for $A_n$,
because otherwise, with the even nature of the zero mean Gaussian
distribution, the integrand in the average in \Eq{Andef} would be odd
in $V$ and yield zero.
b) For even $n=2s$ and even $d=2s^*$, \emph{i.e.}, $\delta=0$, one can
write for $X$
\begin{equation}
X = \sum_{s'=0}^{s^*} C_{s'}\, V^{2s'}
\end{equation}
with general coefficients $C_s$. Using
$\saverage{V^{2p}}=(2p-1)!!\saverage{V^2}^p$,\cite{Cramer46} the
result for $A_{2s}$ in terms of the $C_{s'}$ becomes
\begin{eqnarray}
A_{2s}
&=& \sum_{s'=0}^{s^*}
C_{s'}\saverage{V^{2s'+2s}}
\:=\:
\sum_{s'=0}^{s^*}
(2s+2s'-1)!! C_{s'}\saverage{V^2}^{s'+s}.
\label{temp}
\end{eqnarray}
Now let
\begin{equation}
a_s \equiv \frac{A_{2s}}{\saverage{V^2}^{s}(2s-1)!!},
\label{asdef}
\end{equation}
which is given, according to \Eq{temp}, by
\begin{eqnarray}
a_s &=&
\sum_{s'=0}^{s^*}
(2s+2s'-1)\cdots(2s+1)C_{s'}\saverage{V^2}^{s'}.
\end{eqnarray}
The maximum number of factors $(2s+2s'-1)$ in this expression is
$s^*$. Since this is the only way in which $s$ enters on the \rhs,
this implies that $a_s$ is a polynomial in $s$ of degree $s^*$. The
classic Lagrange interpolation formula applied to a
polynomial\cite{NumericalRecipes} then gives the general expression
for the polynomial $a_s$ in terms of its values $a_{s'}$ on the
$s^*+1$ points $s'=0 \ldots s^*$:\footnote{The Lagrange formula is
commonly used as an interpolation formula using polynomials for
functions whose values are known at a finite number of
points. However, in the present case in which the function is known to
be a polynomial, the formula is just an exact representation of the
function in terms of the values it takes on the $s^*+1$ points.}
\begin{equation}
a_s = \sum_{s'=0}^{s^*} \frac{a_{s'}}{s'!(s^*-s')!}
\mathop{\prod_{s''=0}^{s^*}}_{s''\neq s'} (s-s'').
\label{Lagrange}
\end{equation}
Re-substituting \Eq{asdef} in this equation and multiplying by
$\saverage{V^2}^{s}(2s-1)!!$ yields \Eq{odd} for $\delta=0$.
The same procedure can be followed for odd values for $n=2s+1$ and
$d=2s^*+1$, in which case $\delta=1$. Writing $X=\sum_{s'=0}^{s^*}
C_{s'}\, V^{2s'+1}$ gives $\saverage{V^{2s+1} X} = \sum_{s'=0}^{s^*}
(2s+2s'+1)!!C_{s'}\saverage{V^2}^{s+s'+1}$. The quantities
\begin{equation}
a_s \equiv \frac{A_{2s+1}}{\saverage{V^2}^{s}(2s+1)!!}
\label{astdef}
\end{equation}
are then seen to be once more polynomials in $s$ of degree $s^*$, and
can therefore be expressed by the Lagrange formula in
\Eq{Lagrange}. Re-substituting \Eq{astdef} in \Eq{Lagrange} then leads
to \Eq{odd} for $\delta=1$.
\end{proof}
The third lemma concerns $\theta$-modified averages of polynomials $X$
in $V$ in terms of averages of powers of $V$ times $X$.
\begin{lemma}{\emph{($\theta$-modified average of polynomials
in $V$)$\,$}}
\label{theta average polynomial}
Let $X$ be a variable which is a polynomial of degree $d$ in the
zero mean Gaussian variable $V$. Then the $\theta$-modified average
of $X$ is a polynomial in $\theta$ of the same degree $d$ and given by
\begin{equation}
\average{X}_\theta =
\mathop{\sum_{k,s'\geq0}}_{k+2s'\leq d}
\frac{[-\frac12\average{V^2}]^{s'}\saverage{V^kX}}{s'!k!}\,
\theta^{k+2s'}.
\label{generalpoly}
\end{equation}
\end{lemma}
\begin{proof}
The proof will be in three parts: a, b and c.
a) Consider first the case that $X$ is an even polynomial in $V$ and
write its degree as $d=2s^*$. Using \Eq{modavdef}, expanding
$e^{\theta V}$ and using that only even functions of $V$ have a
nonzero average, one gets
\begin{eqnarray}
\average{X}_\theta &=&
e^{-\frac12\theta^2\langle{V^2}\rangle}
\sum_{s=0}^\infty \frac{\average{V^{2s}X}}{(2s)!}\theta^{2s}
\end{eqnarray}
where part a) of \Lemma{average with v power} was used which states
that $\average{V^nX}=0$ if $n$ is odd and $d$ is even. Using part b)
of \Lemma{average with v power}, \emph{i.e.}\ \Eq{odd} with
$\delta=0$ as well, one gets
\begin{equation}
\average{X}_\theta =
e^{-\frac12\theta^2\langle{V^2}\rangle}
\sum_{s=0}^\infty
\sum_{s'=0}^{s^*}
\frac{(2s-1)!!}{(2s)!}\theta^{2s}
\frac{\saverage{V^2}^{s-s'}
\saverage{V^{2s'}X}}{(2s'-1)!!s'!(s^*-s')!}
\mathop{\prod_{s''=0}^{s^*}}_{s''\neq s'}(s-s'')
\end{equation}
Noting that ${(2s-1)!!}/{(2s)!}= {1}/({2^s s!})$, this becomes
\begin{equation}
\average{X}_\theta =
e^{-\frac12\saverage{V^2}\theta^2}
\sum_{s=0}^\infty
\sum_{s'=0}^{s^*}
\frac{[\frac12\saverage{V^2}\theta^2]^s}{s!}
\frac{\saverage{V^{2s'}X}}{\saverage{V^2}^{s'}(2s'-1)!!s'!(s^*-s')!}
\mathop{\prod_{s''=0}^{s^*}}_{s''\neq s'}(s-s'')
\label{temp6}
\end{equation}
In order to get \Eq{generalpoly}, \Eq{temp6} can be rewritten by using
for $s'\leq s^*$ [as in the second summation in \Eq{temp6}] the
algebraic equality\footnote{The proof of the algebraic relation
\eq{algebraiclemma} for $s'\leq s^*$ goes via induction for both $s^*$ and
$s'$. The start of the induction is the case $s^*=0$, $s'=0$, for
which \Eq{algebraiclemma} is easily verified. Next, one assumes that
\Eq{algebraiclemma} is satisfied for specific vales $s^*=u^*$ and $s'=u'$
with $u'\leq u^*$. By writing the \lhs\ of \Eq{algebraiclemma} as
$\exp[-\saverage{V^2}\theta^2/2]
\prod_{s''=0;s''\neq s'}^{s^*}[\theta^2 d/d(\theta^2) - s'']
\exp[\saverage{V^2}\theta^2/2]$, it can be shown that \eq{algebraiclemma}
then also holds for $s^*=u^*+1$ and $s'=u'$. Thus \Eq{algebraiclemma} is
proved for values $s'=u'\leq u^*=s^*-1$ but not yet for $s'=s^*$. The
latter case can be proved directly in \Eq{algebraiclemma} by substituting
$s'=s^*$. This completes the proof.}
\begin{multline}
e^{-\frac12\saverage{V^2}\theta^2}
\sum_{s=0}^\infty
\frac{[\frac12\saverage{V^2}\theta^2]^s}{s!}\mathop{\prod_{s''=0}^{s^*}}_{s''\neq s'}
(s-s'')
\\
=
\sum_{s''=0}^{s^*-s'}(-1)^{s''}\Big[\frac12\saverage{V^2}\theta^2\Big]^{s^*-s''}\frac{(s^*-s')!}{(s^*-s'-s'')!}.
\label{algebraiclemma}
\end{multline}
Using \Eqs{temp6} and
\eq{algebraiclemma} and that ${(2s-1)!!}/{(2s)!}= {1}/({2^s s!})$, one finds
\begin{equation}
\average{X}_\theta =
\sum_{s'=0}^{s^*}\sum_{s''=0}^{s^*-s'}
\frac{\saverage{V^{2s'}X}}{\average{\frac12 V^2}^{s'}(2s')!}
\frac{(-1)^{s^*+s'+s''}[\frac12\saverage{V^2}\theta^2]^{s^*-s''}}{(s^*-s'-s'')!}.
\end{equation}
Changing the summation variables from ($s'$, $s''$) to ($s$, $s'$) by
replacing $s'$ by $s$ and $s''$ by $s^*-s-s'$, respectively, gives
\begin{equation}
\average{X}_\theta =
\mathop{\sum_{s,s'\geq0}}_{s+s'\leq s^*}
\frac{[-\frac12\saverage{V^2}]^{s'}\saverage{V^{2s}X}}{s'!(2s)!}\,
\theta^{2s+2s'},
\label{moreeven}
\end{equation}
which is of the form \Eq{generalpoly} for an even
polynomial $X$, so the lemma is now proved for this case.
b) Consider next the case that $X$ is an odd polynomial in $V$ of
degree $d=2s^*+1$. Write, using \Eq{modavdef} and the part a) of
\Lemma{average with v power},
\begin{equation}
\average{X}_\theta =
e^{-\frac12\theta^2\langle{V^2}\rangle}
\sum_{s=0}^\infty \frac{\average{V^{2s+1}X}}{(2s+1)!}\theta^{2s+1}
\end{equation}
Using part b) of \Lemma{average with v power}, \emph{i.e}\ \Eq{odd}
with $\delta=1$ as well, one gets
\begin{equation}
\average{X}_\theta =
e^{-\frac12\theta^2\langle{V^2}\rangle}
\sum_{s=0}^\infty
\sum_{s'=0}^{s^*}
\frac{(2s+1)!!}{(2s+1)!}\theta^{2s+1}
\frac{\saverage{V^2}^{s-s'}
\saverage{V^{2s'+1}X}}{(2s'+1)!!s'!(s^*-s')!}
\mathop{\prod_{s''=0}^{s^*}}_{s''\neq s'}(s-s'')
\end{equation}
Noting that $(2s+1)!!/(2s+1)!=1/(2^s s!)$ and performing the same
manipulations as in a), one gets
\begin{equation}
\average{X}_\theta =
\mathop{\sum_{s,s'\geq0}}_{s+s'\leq s^*}
\frac{[-\frac12\saverage{V^2}]^{s'}\saverage{V^{2s+1}X}}{s'!(2s+1)!}\,
\theta^{2s+2s'+1},
\label{moreodd}
\end{equation}
which is of the form \Eq{generalpoly} for an
odd polynomial $X$, so the lemma is now proved for this case too.
c) To derive the form in \Eq{generalpoly} for a general polynomial,
one splits the general polynomial $X$ up into an even part $Y$ and an
odd part $Z$ in $V$. Using \Eqs{moreeven} and \eq{moreodd} and using
that $\saverage{YV^{2s+1}}=\saverage{ZV^{2s}}=0$, one obtains
straightforwardly \Eq{generalpoly}.
\end{proof}
The next and final lemma we need before we can prove \Theorem{cumulant
with v power} concerns $\theta$-modified cumulants of several
polynomials in $V$.
\begin{lemma}{\emph{($\theta$-modified cumulants of polynomials in $V$)$\,$}}
\label{theta cumulant polynomial}
Let $\{X_\gamma\}_{\gamma=1\ldots\alpha}$ be a set of polynomials of
degree $d_\gamma$ in the zero mean Gaussian variable $V$. Then the
$\theta$-modified cumulant $\cumulant{X_1^{n_1} \ldots
X_\alpha^{n_\alpha}}\big._\theta$ is a polynomial in $\theta$ of the
combined degree $d =\sum_{\gamma=1}^{\alpha}n_\gamma d_\gamma$.
\end{lemma}
\begin{proof}
Because the $\theta$-modified cumulants are defined formally in
precisely the same way as normal cumulants [\emph{i.e.},
\Eq{modcumdef} vs.\ \Eq{defcumvar}], the relation \Eq{cumintermsofmom}
between cumulants and moments applies to the $\theta$-modified
cumulants and moments as well. According to that relation, the
$\theta$-modified cumulant $\cumulant{X_1^{n_1} \ldots
X_\alpha^{n_\alpha}}\big._\theta$ can be expressed as a sum of terms
each of which contains a product of the moments
$\saverage{X_1^{\ell_{1}}\cdots X_\alpha^{\ell_{\alpha}}}_\theta$
raised to the power $p_{\{\ell\}}$ (where the product is over all
possible sets $\{\ell\}=\{l_1,\ldots, l_\alpha\}$ with the
restrictions stated in \Eq{cumintermsofmom}). Because $X_\gamma$ is
required by the conditions of the lemma to be a polynomial in $V$ of
degree $d_\gamma$, $X_\gamma^{\ell_{\gamma}}$ is a polynomial in $V$
of degree $\ell_{\gamma}d_\gamma$, and the product $X_1^{\ell_{1}}
\cdots X_\alpha^{\ell_{\alpha}}$ is a polynomial in $V$ of degree
$\sum_{\gamma=1}^\alpha \ell_\gamma d_\gamma$. According to
\Lemma{theta average polynomial}, each $\theta$-modified moment
$\saverage{X_1^{\ell_{1}} \cdots X_\alpha^{\ell_{\alpha}}}_\theta$
is then a polynomial in $\theta$ of degree $\sum_{\gamma=1}^\alpha
\ell_\gamma d_\gamma$. Its $p_{\{\ell\}}$-th power in
\Eq{cumintermsofmom} is then a polynomial in $\theta$ of degree
$p_{\{\ell\}}\sum_{\gamma=1}^\alpha \ell_\gamma d_\gamma$ , and the
product (over $\{\ell\}$) that occurs on the \rhs\ of
\Eq{cumintermsofmom} is of degree
$d\equiv\sum_{\{\ell\}}p_{\{\ell\}}\sum_{\gamma=1}^\alpha \ell_\gamma
d_\gamma $. Since $p_{\{\ell\}}$ is summed over in
\Eq{cumintermsofmom} with the restriction that
$\sum_{\{\ell\}}p_{\{\ell\}}\ell_\gamma=n_\gamma$, the degree $d$ of
the expressions $\prod_{\{\ell\}} \saverage{X_1^{\ell_{1}}\cdots
X_\alpha^{\ell_{\alpha}}}_\theta^{p_{\{\ell\}}}$ can be rewritten as
$d=\sum_{\gamma=1}^\alpha n_\gamma d_\gamma$. Each term in the sum
over $p_{\{\ell\}}$ on the \rhs\ of \Eq{cumintermsofmom} is of
this degree $d$, so the full expression, \emph{i.e.} $\cumulant{X_1^{n_1}
\ldots X_\alpha^{n_\alpha}}\big._\theta$, is also a polynomial in
$\theta$ of degree~$d$.
\end{proof}
\subsubsection*{Proof of \Theorem{cumulant with v power}.}
We can now give the proof of \Theorem{cumulant with v power}. To
consider $\cumulant{V^{n_1}X_2^{n_2}\ldots X_\alpha^{n_\alpha}}$ as on
the \lhs\ of \Eq{vanishing} in the case that at least one
$n_{\gamma>1}\neq 0$, we look at its generating function, \emph{i.e.},
$\cumulant{X_2^{n_2}\ldots X_\alpha^{n_\alpha}}_\theta$. According to
\Eq{modcum} of \Lemma{theta and normal cumulants}, this generating
function $\cumulant{X_2^{n_2} \ldots X_\alpha^{n_\alpha}}_\theta$
admits a power series in $\theta$. At the same time, according to
\Lemma{theta cumulant polynomial} (whose proof required Lemmas
\ref{average with v power} and \ref{theta average polynomial}),
$\cumulant{X_2^{n_2} \ldots X_\alpha^{n_\alpha}}_\theta$ is given by a
polynomial in $\theta$ of degree $d =\sum_{\gamma=2}^{\alpha}n_\gamma
d_\gamma$, so its power series in $\theta$ terminates after
$\theta^d$. Since these lemmas should be valid for all $\theta$, each
coefficient of the $\theta^{n>d}$ terms must be zero, \emph{i.e.}, using
\Eq{modcum} of \Lemma{theta and normal cumulants},
$\cumulant{V^{n_1} X_2^{n_2} \ldots X_\alpha^{n_\alpha}}/n_1!=0$ if
$n_1>d$. Since $n_1!\neq0$, it follows that $\cumulant{V^{n_1}
X_2^{n_2} \ldots X_\alpha^{n_\alpha}} =0$, which is \Eq{vanishing} of
\Theorem{cumulant with v power}. Finally, for the case $n_2=\cdots=n_\alpha=0$,
\Eq{nonvanishing} of \Theorem{cumulant with v power} follows from the
Gaussian nature of $V$. \qed
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\end{document}
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8#334 /Udieresis 8#335 /Yacute 8#336 /Thorn 8#337 /germandbls 8#340 /agrave
8#341 /aacute 8#342 /acircumflex 8#343 /atilde 8#344 /adieresis 8#345 /aring
8#346 /ae 8#347 /ccedilla 8#350 /egrave 8#351 /eacute
8#352 /ecircumflex 8#353 /edieresis 8#354 /igrave 8#355 /iacute
8#356 /icircumflex 8#357 /idieresis 8#360 /eth 8#361 /ntilde 8#362 /ograve
8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide
8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex
8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def
/Times-Roman /Times-Roman-iso isovec ReEncode
/Times-Italic /Times-Italic-iso isovec ReEncode
/Times-Roman /Times-Roman-iso isovec ReEncode
/$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def
/$F2psEnd {$F2psEnteredState restore end} def
$F2psBegin
10 setmiterlimit
0.06000 0.06000 sc
%
% Fig objects follow
%
/Times-Roman-iso ff 240.00 scf sf
2550 1950 m
gs 1 -1 sc (a\) smooth) col0 sh gr
/Times-Roman-iso ff 240.00 scf sf
2550 2235 m
gs 1 -1 sc ( potential) col0 sh gr
/Times-Roman-iso ff 240.00 scf sf
3450 1425 m
gs 1 -1 sc (b\) hard sphere-like) col0 sh gr
/Times-Roman-iso ff 240.00 scf sf
3450 1710 m
gs 1 -1 sc ( collisions) col0 sh gr
% Polyline
15.000 slw
n 2400 1575 m 2400 4800 l
10125 4800 l gs col0 s gr
% Polyline
7.500 slw
gs clippath
1980 1560 m 1920 1560 l 1920 1712 l 1950 1592 l 1980 1712 l cp
eoclip
n 1950 2400 m
1950 1575 l gs col0 s gr gr
% arrowhead
n 1980 1712 m 1950 1592 l 1920 1712 l col0 s
% Polyline
45.000 slw
[90 45 15 45] 0 sd
n 3075 4875 m
5250 1425 l gs col0 s gr [] 0 sd
% Polyline
7.500 slw
gs clippath
2865 4590 m 2985 4590 l 2985 4302 l 2925 4542 l 2865 4302 l cp
eoclip
n 2925 2325 m
2925 4575 l gs col0 s gr gr
% arrowhead
15.000 slw
n 2865 4302 m 2925 4542 l 2985 4302 l col0 s
% Polyline
7.500 slw
gs clippath
7665 5355 m 7665 5295 l 7513 5295 l 7633 5325 l 7513 5355 l cp
eoclip
n 6975 5325 m
7650 5325 l gs col0 s gr gr
% arrowhead
n 7513 5355 m 7633 5325 l 7513 5295 l col0 s
% Polyline
2 slj
45.000 slw
n 2400 4800 m 2401 4800 l 2404 4800 l 2412 4800 l 2426 4800 l 2447 4800 l
2472 4799 l 2501 4799 l 2532 4798 l 2563 4798 l 2594 4797 l
2622 4796 l 2648 4796 l 2673 4795 l 2695 4794 l 2716 4792 l
2736 4791 l 2756 4789 l 2775 4788 l 2794 4785 l 2814 4783 l
2833 4781 l 2854 4777 l 2874 4774 l 2896 4770 l 2917 4765 l
2939 4760 l 2961 4754 l 2983 4747 l 3004 4740 l 3025 4733 l
3045 4724 l 3065 4716 l 3084 4706 l 3102 4696 l 3120 4686 l
3138 4675 l 3150 4666 l 3163 4657 l 3176 4647 l 3189 4636 l
3203 4624 l 3217 4611 l 3231 4597 l 3245 4582 l 3260 4566 l
3275 4549 l 3291 4531 l 3307 4512 l 3323 4491 l 3340 4470 l
3356 4448 l 3373 4425 l 3390 4401 l 3407 4376 l 3424 4351 l
3441 4325 l 3459 4298 l 3476 4270 l 3494 4242 l 3513 4213 l
3526 4191 l 3539 4169 l 3553 4146 l 3568 4122 l 3583 4097 l
3598 4071 l 3614 4045 l 3631 4018 l 3648 3989 l 3665 3960 l
3683 3930 l 3701 3900 l 3720 3868 l 3740 3837 l 3759 3804 l
3779 3771 l 3800 3738 l 3820 3704 l 3841 3671 l 3862 3637 l
3883 3603 l 3904 3570 l 3925 3537 l 3946 3504 l 3967 3471 l
3987 3439 l 4008 3408 l 4029 3377 l 4049 3346 l 4069 3316 l
4090 3286 l 4110 3257 l 4130 3228 l 4150 3200 l 4170 3172 l
4191 3144 l 4212 3116 l 4233 3088 l 4254 3060 l 4276 3031 l
4298 3003 l 4321 2975 l 4344 2946 l 4368 2917 l 4392 2889 l
4417 2860 l 4442 2832 l 4467 2803 l 4493 2775 l 4518 2747 l
4544 2719 l 4571 2692 l 4597 2665 l 4623 2638 l 4649 2612 l
4676 2587 l 4702 2563 l 4727 2539 l 4753 2515 l 4778 2493 l
4804 2471 l 4829 2450 l 4853 2430 l 4878 2410 l 4902 2391 l
4927 2373 l 4951 2355 l 4975 2338 l 5001 2319 l 5027 2302 l
5053 2285 l 5080 2268 l 5108 2251 l 5136 2235 l 5165 2219 l
5194 2203 l 5224 2187 l 5255 2173 l 5286 2158 l 5318 2144 l
5351 2131 l 5384 2118 l 5417 2105 l 5451 2094 l 5486 2083 l
5520 2072 l 5555 2063 l 5589 2054 l 5624 2046 l 5659 2039 l
5693 2033 l 5728 2028 l 5762 2023 l 5796 2019 l 5830 2016 l
5864 2014 l 5898 2013 l 5932 2012 l 5966 2012 l 6000 2013 l
6033 2014 l 6066 2016 l 6099 2018 l 6133 2022 l 6168 2026 l
6203 2030 l 6239 2036 l 6276 2042 l 6314 2049 l 6352 2056 l
6390 2065 l 6429 2074 l 6468 2084 l 6508 2094 l 6548 2105 l
6587 2117 l 6627 2129 l 6667 2142 l 6706 2156 l 6744 2170 l
6783 2185 l 6820 2199 l 6857 2215 l 6893 2230 l 6928 2246 l
6962 2262 l 6995 2279 l 7027 2296 l 7058 2312 l 7088 2329 l
7118 2347 l 7146 2364 l 7173 2382 l 7200 2400 l 7228 2420 l
7254 2440 l 7281 2460 l 7307 2481 l 7333 2503 l 7358 2526 l
7383 2549 l 7408 2574 l 7433 2599 l 7457 2625 l 7482 2651 l
7506 2678 l 7530 2706 l 7554 2735 l 7577 2764 l 7600 2794 l
7623 2824 l 7646 2855 l 7669 2885 l 7691 2916 l 7713 2947 l
7734 2979 l 7755 3010 l 7776 3041 l 7797 3072 l 7817 3103 l
7837 3134 l 7857 3164 l 7877 3195 l 7897 3226 l 7917 3256 l
7938 3288 l 7956 3315 l 7974 3343 l 7993 3372 l 8013 3401 l
8032 3431 l 8053 3460 l 8073 3491 l 8095 3522 l 8116 3553 l
8138 3585 l 8161 3617 l 8184 3650 l 8207 3682 l 8230 3715 l
8254 3748 l 8278 3781 l 8302 3814 l 8327 3847 l 8351 3879 l
8375 3911 l 8400 3943 l 8424 3974 l 8447 4005 l 8471 4034 l
8494 4064 l 8517 4092 l 8540 4119 l 8562 4146 l 8584 4171 l
8605 4196 l 8626 4219 l 8646 4242 l 8667 4264 l 8686 4285 l
8706 4306 l 8725 4325 l 8751 4351 l 8777 4375 l 8803 4399 l
8829 4421 l 8856 4443 l 8882 4464 l 8909 4483 l 8935 4502 l
8962 4520 l 8990 4537 l 9017 4554 l 9044 4569 l 9072 4583 l
9099 4596 l 9126 4608 l 9152 4620 l 9179 4630 l 9205 4639 l
9230 4648 l 9256 4655 l 9280 4662 l 9305 4668 l 9329 4674 l
9353 4679 l 9376 4683 l 9400 4688 l 9424 4691 l 9448 4695 l
9473 4698 l 9499 4701 l 9525 4704 l 9554 4706 l 9583 4708 l
9615 4710 l 9648 4712 l 9684 4714 l 9722 4716 l 9761 4717 l
9802 4719 l 9845 4720 l 9887 4721 l 9929 4722 l 9970 4723 l
10007 4723 l 10040 4724 l 10068 4724 l 10090 4725 l 10106 4725 l
10117 4725 l 10122 4725 l
10125 4725 l gs col0 s gr
% Polyline
[90] 0 sd
n 2400 4800 m 2403 4800 l 2409 4799 l 2419 4798 l 2436 4796 l 2458 4793 l
2485 4790 l 2517 4786 l 2552 4781 l 2589 4777 l 2628 4772 l
2666 4767 l 2703 4763 l 2739 4758 l 2774 4754 l 2806 4750 l
2837 4747 l 2867 4743 l 2895 4740 l 2922 4737 l 2948 4734 l
2974 4731 l 2999 4728 l 3025 4725 l 3049 4722 l 3073 4720 l
3098 4717 l 3123 4714 l 3149 4711 l 3176 4708 l 3203 4705 l
3230 4701 l 3259 4698 l 3287 4694 l 3316 4690 l 3346 4685 l
3375 4681 l 3404 4676 l 3434 4671 l 3463 4665 l 3491 4660 l
3520 4654 l 3547 4648 l 3574 4642 l 3601 4636 l 3627 4629 l
3652 4622 l 3677 4615 l 3701 4608 l 3725 4600 l 3749 4592 l
3772 4583 l 3796 4574 l 3820 4565 l 3844 4555 l 3869 4544 l
3893 4533 l 3918 4521 l 3943 4509 l 3968 4496 l 3993 4483 l
4018 4469 l 4043 4456 l 4068 4442 l 4092 4427 l 4116 4413 l
4140 4399 l 4163 4384 l 4185 4370 l 4207 4356 l 4228 4342 l
4248 4329 l 4268 4315 l 4287 4302 l 4306 4288 l 4325 4275 l
4345 4261 l 4365 4246 l 4385 4232 l 4405 4216 l 4426 4201 l
4447 4184 l 4469 4167 l 4493 4148 l 4518 4129 l 4544 4108 l
4571 4086 l 4600 4063 l 4628 4040 l 4657 4017 l 4686 3994 l
4712 3972 l 4736 3952 l 4757 3936 l 4773 3922 l 4785 3912 l
4794 3905 l 4798 3902 l
4800 3900 l gs col0 s gr [] 0 sd
% Polyline
7.500 slw
gs clippath
3793 3436 m 3879 3332 l 3711 3193 l 3807 3360 l 3625 3297 l cp
eoclip
n 3900 1800 m 3899 1803 l 3897 1808 l 3893 1819 l 3886 1835 l 3878 1858 l
3867 1886 l 3855 1919 l 3840 1956 l 3825 1997 l 3809 2039 l
3794 2082 l 3778 2125 l 3763 2166 l 3749 2206 l 3735 2245 l
3723 2281 l 3711 2315 l 3701 2347 l 3692 2378 l 3683 2407 l
3675 2434 l 3668 2461 l 3661 2487 l 3655 2512 l 3650 2538 l
3644 2565 l 3639 2592 l 3635 2619 l 3631 2647 l 3627 2675 l
3623 2703 l 3620 2732 l 3618 2760 l 3616 2789 l 3614 2817 l
3613 2846 l 3612 2874 l 3612 2901 l 3612 2928 l 3613 2954 l
3614 2978 l 3616 3002 l 3618 3025 l 3620 3046 l 3623 3067 l
3626 3086 l 3630 3104 l 3633 3121 l 3638 3138 l 3644 3160 l
3652 3181 l 3661 3201 l 3672 3220 l 3685 3239 l 3699 3257 l
3715 3276 l 3733 3295 l 3753 3313 l 3772 3331 l 3790 3346 l
3825 3375 l gs col0 s gr gr
% arrowhead
0 slj
15.000 slw
n 3625 3297 m 3807 3360 l 3711 3193 l col0 s
/Symbol ff 210.00 scf sf
1800 2400 m
gs 1 -1 sc (2) col0 sh gr
/Symbol ff 300.00 scf sf
1575 2325 m
gs 1 -1 sc (a) col0 sh gr
/Times-Roman-iso ff 270.00 scf sf
5250 1800 m
gs 1 -1 sc (t) col0 sh gr
/Times-Roman-iso ff 270.00 scf sf
4800 4200 m
gs 1 -1 sc (t) col0 sh gr
/Symbol ff 210.00 scf sf
4875 4050 m
gs 1 -1 sc (4) col0 sh gr
/Times-Roman-iso ff 270.00 scf sf
8925 5175 m
gs 1 -1 sc (~10 ps) col0 sh gr
/Times-Roman-iso ff 270.00 scf sf
5175 5175 m
gs 1 -1 sc (~1 ps) col0 sh gr
/Times-Roman-iso ff 270.00 scf sf
2700 5175 m
gs 1 -1 sc (~0.1 ps) col0 sh gr
/Times-Italic-iso ff 300.00 scf sf
6675 5400 m
gs 1 -1 sc (t) col0 sh gr
/Times-Roman-iso ff 240.00 scf sf
8775 4425 m
gs 1 -1 sc ( diffusion) col0 sh gr
/Times-Roman-iso ff 240.00 scf sf
8775 4125 m
gs 1 -1 sc (e\) cage) col0 sh gr
/Times-Roman-iso ff 240.00 scf sf
5625 2400 m
gs 1 -1 sc (c\) caging) col0 sh gr
/Times-Roman-iso ff 240.00 scf sf
7650 2700 m
gs 1 -1 sc (d\) escape from cage) col0 sh gr
$F2psEnd
rs
---------------0505301454874--