\Chapter{noise}{6mar2004}{Noise}
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% old title:Quantum and noise trace formulas}
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% PC: o shucks, I like it anyway                19jun2003
% PC: most likely forget this chapter            4aug2000
% PC                                            14aug1998
% vattay@ipncls.in2p3.fr                         5jun1995


% \PublicPrivate{
\begin{bartlett}
\bauthor{M. %ichel
	 de Montaigne
	}
He who establishes his argument by noise and command shows 
that his reason is weak.
\end{bartlett}
%       }{ % switch \PublicPrivate{
% \begin{bartlett}
% In fact I'm Vattay G\'abor, just these Indo-Europeans
% mix up the right order.
% \end{bartlett}
%       }% end \PublicPrivate

\authorGVPC

\noindent
\tangent
This chapter (which reader can safely skip on the first reading)
is about noise, how it affects classical dynamics, and the ways it mimicks
quantum dynamics.

\fastTrackChap{c-traceSemicl}

Why - in a study of deterministic and quantum chaos - 
start discussing noise? 
First, in physical settings any dynamics 
takes place against a noisy background, and whatever prediction we
might have, we have to
check its robustness to noise.
Second, as we show in this chapter,
to the leading order in noise strength
the semiclassical Hamilton-Jacobi formalism carries over to
weakly stochastic flows in totto. As classical noisy dynamics is 
more intuitive than quantum dynamics, this exercise helps demistify
some of the formal machinery of semiclassical quantization.
Surprisingly, symplectic structure emerges here not as a deep principle
of mechanics, but an artifact of the leading approximation to
quantum/noisy dynamics, not respected by higher order corrections.
Third, the variational principle derived here will be refashioned into
a powerful tool for determining periodic orbits
in \refchap{c-relax}.

We start by deriving the continuity
equation for purely deterministic, noiseless flow, and 
then incorporate noise in stages:
diffusion equation, Langevin equation,
Fokker-Planck equation,
Hamilton-Jacobi formulation,
stochastic path integrals.

\section{Deterministic transport}
\label{s_mat_inv}
%\section{Conservation laws}
% Predrag moved back to dense.tex    28aug97
% PC, EAS moved this from dense.tex  29may97
%       Predrag			 18/4-97
%       Predrag edits chug3.tex  19/3-95
%       EAS                      14/3-95

\hfill 		(E.A. Spiegel and P. Cvitanovi\'c)

% Predrag				15feb2004
% moved here from \Chapter{appendMeasure}{15nov2002}{Implementing evolution}
% extracted from measure.tex, version of  12dec99
\noindent
% The large body of accrued wisdom
% on the subject of flows called
Fluid dynamics is about physical
flows of media with %(typically)
continuous densities.  On the
other hand, the flows in state spaces of dynamical systems
% complicated systems
% are not only abstract, but may lead to fractal or other forms
% of complication in the densities. The latter
frequently
require more abstract tools.
% and formal methods that will be usefull in our discussions.
% that we develop in this text.
To sharpen our intuition about those, it is helpful to outline the
more tangible fluid dynamical vision. 

Consider first the simplest property of a fluid flow called {\em material
invariant}.
% by fluid dynamicists.
% who have no choice but to ponder such matters.
A \indx{material invariant} $\mInv(x)$ is a property attached to each point
$x$ that is preserved by the flow, $\mInv(x)=\mInv(\flow{t}{x})$;
for example, at this point a green particle 
(more formally: a {\em passive scalar})
is embedded into the fluid.
\index{passive scalar}
% The flow is a transformation of
% coordinates under which $x\mapsto f^t(x)$.  A material invariant
% is invariant under this transformation: 
As $\mInv(x)$
% =\mInv(\flow{t}{x})$ 
is invariant,
its total time derivative vanishes, $\dot{\mInv}(x)=0$. 
Written in
terms of partial derivatives this is 
the {\em conservation equation}
for the material invariant
\index{conservation!equation}
\beq
%{d\over dt} \mInv := 
\partial_t \mInv + v \cdot \partial \mInv = 0\,.
\ee{3.13}
Let the {\em density}\index{density} of representative points be $\msr(x,t)$.
% where $x$ stands for the coordinates of a general point.  
% To each
% representative point $x$ we assign the value $\mInv(x)$ of some
% physical property of a system. 
The manner in which the
flow redistributes $\mInv(x)$ is governed by a partial differential
equation whose form is relatively simple because the representative
points are neither created nor destroyed.  This conservation property is
expressed in the integral statement
\[
\partial_t \int_{ V} dx \,\msr \mInv = - \int_{\partial{ V}}
d\sigma \, \hat{n}_i v_i \msr \mInv \,, 
\] %\label{intS}
where ${ V}$ is an arbitrary volume in the state space $\pS$,
$\partial{ V}$
is its surface, $\hat{n}$ is its outward normal, and
repeated indices are summed over throughout.
% and $\partial_t$ is the partial derivative with respect to time.
The divergence theorem turns the surface integral
into a volume integral,
\[
\int_{ V} \left[ \partial_t (\msr \mInv) + \partial_i(v_i \msr \mInv)
              \right] dx = 0\,, 
%\label{MintS}
\]
where $\partial_i$ is the partial derivative operator
with respect to $x_i$.  Since the integration is over an
arbitrary volume, we conclude that
\beq
\partial_t (\msr \mInv) + \partial_i(\msr \mInv v_i) = 0\,.
\label{Cons} \eeq
The choice $\mInv\equiv 1$ 
% is especially important since it 
yields 
% the equation for $\msr$ itself,
the {\em continuity equation}\index{continuity equation}
for the density:
\beq
\partial_t \msr  + \partial_i(\msr v_i) = 0
\,.
\label{Cont} 
\eeq
% The choice $\mInv(\pSpace) = 1$ if fluid is green at $\pSpace$,
% and zero otherwise
% yields the continuity equation for the passive scalar density.
% % transported by the fluid. 

We have used here the language of fluid mechanics to ease the visualization,
but, as we already saw in the discussion of
infinitesimal action of the \FPoper\ \refeq{3.Cont},
% \refchap{c-measure},
continuity equation
applies to any deterministic state space flow.

\section{Brownian difussion}
\label{s-TracerDiff}

Consider tracer molecules, let us say green molecules,
embedded in a denser gas of transparent molecules.
Assume that the density of tracer molecules 
$\msr$ compared to the background gas density is low, so we can neglect 
green-green collisions. 
Each green molecule,
jostled by frequent collisions with the background gas,  executes 
its own Brownian motion. 
The molecules 
are neither created nor destroyed,
so their number within an arbitrary volume ${ V}$ changes with
time only by the current density $j_i$ flow through its
surface $\partial{ V}$
(with $\hat{n}$ its outward normal):
\beq
\partial_t \int_{ V} d\pSpace \,\msr = - \int_{\partial{ V}}
d\sigma \, \hat{n}_i j_i  \,. 
\ee{intScurrJ}
The divergence theorem turns this into the conservation law
for tracer density:
\beq
\partial_t \msr + \partial_i j_i = 0
\,.
\ee{DiffCons}
The tracer density $\msr$ is defined as 
the average density of a ``material particle'', 
averaged over a subvolume
large enough to contain many green
(and still many more background) molecules, 
but small compared to the macroscopic observational scales. What is $j$?
If the density is constant, on the average as many molecules leave the
material particle volume as they enter it, so a reasonable phenomenological
assumption is that the {\em average} current density
({\em not} the individual particle
current density $\msr v_i$  in  \refeq{Cont})
is driven by the density gradient 
\index{Fick law}
\beq
j_i =  -\DiffC {\partial \msr \over \partial \pSpace_i}
\,.
\ee{FickLaw}
This is the {\em Fick law},
with the diffusion constant $\DiffC$ a phenomenological parameter.
For simplicity here we assume that $\DiffC$ is a scalar;
in general $\DiffC \to \DiffC_{ij}(\pSpace,t)$
is a space- and time-dependent tensor.
Substituting this $j$ into \refeq{DiffCons}
yields the {\em diffusion equation}
\index{diffusion!equation}
\beq
{\partial~ \over \partial t} \msr(\pSpace,t)
	= \DiffC {\partial^2~ \over \partial \pSpace^2} \msr(\pSpace,t)
\,.
\ee{DiffEq}
This linear equation has an exact solution in terms of
an initial Dirac delta density distribution,
$ \msr(\pSpace,0)  = \delta(\pSpace-\pSpace_0)$,
\index{Dirac delta function}
\beq
\msr(\pSpace,t)  = 
		{1 \over \left(4 \pi \DiffC t \right)^{d/2}}
%                  \exp\left\{- {(\pSpace-\xInit)^2
%				 \over
%				4 \DiffC t }
%		      \right\}
                  e^{- {(\pSpace-\xInit)^2
				 \over
				4 \DiffC t }
		      }
		= {1 \over \left(4 \pi \DiffC t \right)^{d/2}}
                  e^{- {\dot{\pSpace}^2
				 \over
				4 \DiffC} t 
		      }
\,,
\ee{freeDiffusion}
in the spirit of the quantum free particle propagation
of \refsect{free-prop}.
The average distance covered in time $t$ obeys the 
Einstein diffusion formula
\index{Einstein!diffusion formula}
\beq
\spaceAver{(\pSpace-\xInit)^2}_t
= \int d\pSpace \,
	\msr(\pSpace,t) 
	(\pSpace-\xInit)^2
= 2 \DiffC t
\,.
\ee{EinstDiffusion}


\section{Weak noise}
\label{s-WeakNoise}

So far we have considered tracer molecule dynamics which is purely
Brownian, with no deterministic ``drift''. 
Consider next a deterministic flow
$ \dot{\pSpace}=v(\pSpace)$
perturbed by a stochastic term $\xi(t)$,
\beq
\dot{\pSpace}=v(\pSpace)+\xi(t)
\,.
\ee{LangEq}
Assume that $\xi(t)$'s fluctuate around 
$
[\dot{\pSpace} - v(\pSpace)]
$
with a Gaussian 
probability density 
\beq
p(\xi,\delta t) 
%               =   \exp\left\{- {(\pSpace-\xInit)^2
%				 \over
%				4 \DiffC t }
%		      \right\}
		= \left({\delta t \over 4 \pi \DiffC} \right)^{d/2}
%                  e^{- {(\dot{\pSpace} -v)^2
                  e^{- {\xi^2
				 \over
				4 \DiffC} \delta t 
		      }
\,,
\ee{velocDiff}
and are uncorrelated in time
(white noise)
\index{white noise}
\index{noise!white}
\PC{explain what average?}
\beq
\spaceAver{\xi(t)\xi(t')} = 2\DiffC \delta(t-t')
\,.
\ee{TimeCorel}
The normalization factors in 
\refeq{freeDiffusion} and \refeq{velocDiff}
differ, as $p(\xi,\delta t)$ is a probability density for velocity
$\xi$, and $\msr(\pSpace,t)$ is a probability density for position $\pSpace$.
The material particle now drifts along the trajectory $\pSpace(t)$, 
so the
velocity diffusion follows \refeq{freeDiffusion} for infinitesimal time
$\delta t$ only.
As $\DiffC \to 0$, the distribution
tends to the (noiseless, deterministic) Dirac delta function.

An example is 
the Langevin equation for a Brownian
particle, in which one replaces the Newton's equation
for force
by two counter-balancing forces: random 
accelerations $\xi(t)$ which tend to
smear out a particle trajectory, and a damping term 
which drives the velocity to zero. 
% Assuming that the distribution
% of kicks is Gaussian (white noise), and the kicks so frequent that
% they can be considered uncorrelated, one derives a 
% fluctuation-dissipation theorem that relates the temperature of the
% background gas to the damping constant 
% and the mass of the Brownian particle.
% Here we shall consider more general
% stochastic differential equations in the weak noise limit. 
\index{Langevin equation}
\PC{explain what $|v(\pSpace)| > |\xi|$ means}

The phenomenological Fick law current \refeq{FickLaw} is now a sum of two components,
the material particle center-of-mass 
deterministic drift $v(\pSpace)$ and the weak noise term
\beq
j_i = \msr v_i
	 -\DiffC {\partial \msr \over \partial \pSpace_i}
\,,
\ee{FickDrift}
Substituting this $j$ into \refeq{DiffCons} yields the  %(generalized)
{\em Fokker-Planck equation}
% which governs 
% he time evolution of  $\msr(\pSpace,t)$,
% the probability distribution of $\pSpace$,
\index{Fokker-Planck equation}
\beq
\partial_t \msr + \partial_i(\msr v_i)=\DiffC \, \partial^2 \msr.
\ee{FokkPlanck}
%  (The symbol nabla $\nabla$ denotes the divergence when it acts on vectors
%  and the gradient, when it acts on scalars.)
The left hand side,
$d\msr/dt = \partial_t \msr + \partial \cdot (\msr v) $,
is deterministic, with
the 
{continuity equation} \refeq{Cont}
recovered in the weak noise limit $\DiffC\rightarrow 0$. 
The right hand side describes the diffusive transport 
in or out of the material particle volume.
If the density is lower than in
the immediate neighborhood, the local curvature is
positive, $\partial^2 \msr > 0$, and the density grows.
Conversely, for negative curvature 
% (top of a hill)
diffusion lowers the
local density, thus smoothing the variability of $\msr$.
Where is the density going globally? 
\index{continuity equation}

If the system is bound,
the probability density vanishes
sufficiently fast outside the central region,
$
\msr(\pSpace,t)\rightarrow 0
$ 
as
%	\,,\qquad \mbox{as} \;\;
$
|\pSpace|\rightarrow \infty
$,
and the total probability is conserved
\[
\int d\pSpace \, \msr(\pSpace,t) = 1
\,.
\]
{\em Any} initial density
$ \msr(\pSpace,0)$
is smoothed by diffusion
and with time tends to
the invariant density
\beq
\msr_0(\pSpace) = \lim_{t\to\infty} \msr(\pSpace,t)
\,,
\ee{natMdefStoch}
an eigenfunction 
$\msr(\pSpace,t)= e^{\eigenvL t}\, \msr_0(\pSpace)$ 
of the time-independent  Fokker-Planck equation 
\beq
\left(\partial_i v_i -\DiffC \, \partial^2 + \eigenvL_\alpha \right) \msr_\alpha = 0
\,,
\ee{ti-FokkPlanck}
with vanishing eigenvalue $\eigenvL_0=0$.
Provided the noiseless classical flow is hyperbolic,
in the vanishing noise limit the leading eigenfunction of the
Fokker-Planck equation tends to 
natural measure \refeq{natMdefined}
of the corresponding deterministic flow,
the leading eigenvector of the  \FPoper. 

If the system is open, 
there is a continuous outflow of probability from
the region under study, the leading eigenvalue is contracting,
$\eigenvL_0<0$,
and the density of the system tends to zero.
In this case the leading eigenvalue $\eigenvL_0$ of
the time-independent  Fokker-Planck equation
\refeq{ti-FokkPlanck}
can be interpreted by saying that
a finite density can be maintained by pumping back probability
into the system at a constant rate $\gamma= -\eigenvL_0$.
% This yields an extra term, $-\gamma\msr$, in the Fokker-Planck equation.
The value of
$\gamma$ for which any initial probability density converges to a
finite equilibrium density is called the {\em escape rate}.
In the noiseless limit this coincides with
the deterministic escape rate \refeq{shrinkM}.

% For finite but small noise strength $\DiffC$ the material particle
% is transported by the flow, but for varying $\msr$ particles
% are diffusing in or out of the material particle volume,
% $d\msr/dt = \DiffC \partial^2 \msr$.
We have introduced noise phenomenologically, and used 
the weak noise assumption in retaining only 
the first derivative of $\msr$ in formulating the Fick law \refeq{FickLaw}
and including noise additively in \refeq{FickDrift}.
A full theory of stochastic ODEs is much subtler, but this will do for our
purposes.


\section{%WKB%
Weak noise approximation}
\label{s-HamiltNoise}

In the spirit of the WKB approximation, we shall now
study the evolution of the probability distribution by
rewriting it as
\beq
\msr(\pSpace,t)=e^{-R(\pSpace,t)/ \DiffC}
\,.
\ee{noiseWKB} 
The time evolution of $R$ is given by 
\[
\partial_t R +  v\partial R+(\partial R)^2 =
	  \DiffC\partial v + \DiffC\partial^2 R
\,.
\]
Consider now the weak noise limit
and drop the terms proportional to $\DiffC$.
The remaining equation
\[
\partial_t R + H(\pSpace,\partial R)=0
\]
is the Hamilton-Jacobi
equation \refeq{ham_jac}. The function $R$ can be interpreted
as the {Hamilton's principal function},
corresponding to the Hamiltonian
\index{Hamilton!principal function}
\[
H(\pSpace,p)= p \, v(\pSpace)+p^2
\,,
\]
with the Hamilton's equations of motion
\bea
\dot{\pSpace} &=& ~~\partial_p H =\, v+2p
	\continue 
\dot{p}       &=& -\partial_\pSpace H =-\Mvar \, p
\,,
%\dot{p}&=&-\partial_x H =-{\bf D}v p,\\
\label{HamEqsNoise}
\eea
where $\Mvar$ is the matrix of variations
\refeq{DerMatrix}
\index{matrix!of variations}
\[
{\Mvar}_{ij}(\pSpace) ={\pde v_i(\pSpace)\over \pde \pSpace_j  }
\,.
\]
The noise Lagrangian \refeq{Lgrngn} is then
\beq
L(\pSpace,\dot{\pSpace})
	= \dot{\pSpace} \cdot p - H
	= {1 \over 4} \left[\dot{\pSpace}-v(\pSpace)\right]^2
\,.
\ee{OnsagMachlup}
We have come the full circle - the Lagrangian is the exponent 
of our assumed Gaussian distribution \refeq{velocDiff} for noise 
$\xi^2=[\dot{\pSpace}-v(\pSpace)]^2$.
What is the meaning of this Hamiltonian, Lagrangian?
Consider two points $\xInit$ and $\pSpace$.
Which noisy path is the most probable path that connects them in time 
$t$? The probability of a given path ${\cal P}$ is given by the probability
of the noise sequence $\xi(t)$ which generates the path. This probability
is proportional to the product of the noise probability functions
\refeq{velocDiff}
along the path, and the total probabbility for reaching $\pSpace$
from $\xInit$ in time $t$ is given by the sum over
all paths, or the stochastic path integral (Wiener integral)
\index{Wiener integral}
\bea
P(\pSpace,\xInit,t)
	& \sim &
	\sum_{\cal P}
	\prod_j p(\xi(\tau_j),\delta \tau_j) 
	= \int \prod_j 
	  d\xi_j
	  \left({\delta \tau_j \over 4 \pi \DiffC }\right)^{d/2}
%                  e^{- {(\dot{\pSpace} -v)^2
                  e^{- {\xi(\tau_j)^2
				 \over
				4 \DiffC} \delta \tau_i 
		      }
	\continue
	& \to &
	{1 \over Z} 
	\sum_{\cal P}
	\exp\left(-\frac{1}{4 \DiffC}\int_0^t d\tau \, \xi^2(\tau)\right)
\,,
\label{WiPathInt}
\eea
where $\delta \tau_i = \tau_i - \tau_i$, and the normalization constant is
\[
	{1 \over Z} 
	= \lim \prod_i 
	  \left({\delta \tau_i \over 4 \pi \DiffC}\right)^{d/2}
%                  e^{- {(\dot{\pSpace} -v)^2
\,.
\]
The most probable path is the one maximizing the integral inside 
the exponential.  If we express the noise
\refeq{LangEq} as 
\[
\xi(t)=\dot{\pSpace}(t)-v(\pSpace(t))
\,,
\]
the probability is maximized by the 
variational principle 
\index{variational principle}
\[
\mbox{min} \int_0^t d\tau[\dot{\pSpace}(\tau)-v(\pSpace(\tau))]^2
	=\mbox{min}\int_0^t 
	 L(\pSpace(\tau),\dot{\pSpace}(\tau))d\tau
\,.
\]
By the standard arguments, 
for a given $\pSpace$, $\pSpace'$ and $t$ the the probability is maximized by 
a solution of Hamilton's equations \refeq{HamEqsNoise} that connects the 
two points $\xInit \to \pSpace'$  in time $t$. 


\Remarks

\remark{Literature.}{
The theory of stochastic processes is a vast subject, spanning over centuries
and over disciplines ranging from pure mathematics to impure finance.
We enjoyed reading van Kampen classic\rf{vKampen81}, 
especially his railings against those who blunder carelessly into
nonlinear landscapes (with
this chapter we join the list of van Kampen's sinners).
A more specialized monograph like Risken's\rf{Rysken89} will do just as well.
The ``Langevin equation'' introduces noise and damping only into the
acceleration of Newton's equations; here we are considering more general
stochastic differential equations in the weak noise limit. 
Onsager-Machlup seminal paper\rf{Onsager53} was the 
first to introduce a variational
method - the ``principle of least dissipation'' -
based on the Lagrangian of form
\refeq{OnsagMachlup}. This paper deals only with a
finite set of linearly damped thermodynamic variables.
Here the setting is much more general: we study fluctuations
over a state space varying
velocity field $v(\pSpace)$.
\index{Langevin equation}
\index{Onsager-Machlup}
\index{van Kampen, N. G.}
        % ``Fluctuations and Irreversible Processes",
}

\Preliminary{

\remark{Operator ordering.}{
The It{\'o} and Stratanovich operator ordering issues arise in
the order beyond the leading approximation considered here.
\PC{not true? rethink}
        } %end \remark{Operator ordering.}{

	     } %end \Preliminary{

\Resume

When a deterministic trajectory is smeared out under the influence
of 
{Gaussian} noise of strength $\DiffC$,
the deterministic dynamics 
is recovered in the weak noise limit $\DiffC\rightarrow 0$. 
The effect of the noise can be taken into account by
adding noise corrections to the classical trace formula. 
\MAP{expand}

\PC{recheck whether something can be used from chapter/refsQmnoise}
\input{chapter/refsNoise}

