ChaosBook.org chapter: {Weak noise dynamics} 
					\hfill Feb 19 2004

\input{defs}


\noindent
\tangent
When a deterministic trajectory is smeared out under influence
of 
{Gaussian white} noise of strength $\DiffC$,
the deterministic dynamics 
is recovered in the weak noise limit $\DiffC\rightarrow 0$. 

\section{Weak noise}

Consider a deterministic flow
\beq
\dot{x}=v(x)+\xi(t)
\,,
\ee{LangEq}
perturbed by a weak stochastic term $\xi(t)$.
The best studied 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 kicks $\xi(t)$ which tend to
smear out a particle trajectory, and a damping term 
which tries to drive the velocity to zero. Assuming that the distribution
of kicks is Gaussian (white noise), and the kicks so frequent that
they can be assumed 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.
\index{Langevin equation}

Here we shall consider more general
stochastic differential equations in the weak noise limit. 
The equation \refeq{LangEq} is meaningless unless we specify 
the probability distribution for $\xi(t)$'s. If the noise is
weak, $|v(x)| > |\xi|$, and uncorrelated in time
\beq
\spaceAver{\xi(t)\xi(t')} = 2\DiffC \delta(t-t')
\,,
\ee{TimeCorel}
\index{white noise}
\index{noise!white}
it is common to assume that the noise is
{\em white}, {\ie}, specified by
a Gaussian distribution after an infinitesimal time step
$\timeStep$,
\beq
p(\xi, \timeStep)
=\sqrt{\frac{\timeStep}{4\pi \DiffC}}e^{-\frac{\xi^2}{4\DiffC} \timeStep }
\ee{WhiteInfStep}
with boundary condition that as $\DiffC \to 0$, the distribution
tends to the (deterministic) Dirac delta function.
The time evolution of  $\msr(x,t)$, the probability distribution of $x$,
 is governed by the (generalized)
Fokker-Planck % partial differential
equation
\index{Fokker-Planck equation}
\beq
\partial_t \msr + \nabla(\msr v(x))=\DiffC\nabla^2 \msr.
\ee{FokkPlanck}
\PC{eliminate symbol nabla $\nabla$}
 (The symbol nabla $\nabla$ denotes the divergence when it acts on vectors
 and the gradient, when it acts on scalars.)
We say that the system is bound if
for large values of $x$
the probability distribution vanishes
\[
\msr(x,t)\rightarrow 0
	\,,\qquad \mbox{as} \;\;
|x|\rightarrow \infty
\,.
\]
In this case the total probability is conserved
\begin{equation}
\int dx \, \msr
\msr(x,t)=e^{-\Phi(x,t)/\DiffC}.
\end{equation} 
The time evolution of $\Phi$ is given by 
\[
\partial_t \Phi + \nabla\Phi v+(\nabla\Phi)^2
	-\DiffC\nabla v+\DiffC\nabla^2\Phi=0
\,.
\]
Consider now the weak noise limit
and drop the terms proportional to $\DiffC$.
The remaining equation
\begin{equation}
\partial_t \Phi + H(x,\nabla \Phi)=0.
\end{equation}
is reminiscent of the Hamilton-Jacobi
equation of classical physics. The function $\Phi$ can be interpreted
as a noise ``action'' corresponding to the noise ``Hamiltonian''
\[
H(x,p)=pv(x)+p^2
\,.
\]
with the Hamilton equations of motion
\PC{replace boldface ${\bf D}$}
\begin{eqnarray}
\dot{x}&=&\partial_p H =v+2p,\\ 
\dot{p}&=&-\partial_x H =-\Mvar p
\,.
%\dot{p}&=&-\partial_x H =-{\bf D}v p,\\
\end{eqnarray}
\PC{link to mat of variations $\Mvar$ def}
and the ``noise''Lagrangian
\beq
L(x,\dot{x})=(\dot{x}-v(x))^2.
\ee{OnsagMachlup}
What is the meaning of this Hamiltonian, Lagrangian, action?
We can figure out this by taking two points $x$ and $x'$ and
asking which is the most probable noisy path that connects them in time 
$T$. The probability of a given path $x(t)$ is the same as the probability
of the noise sequence $\xi(t)$ which generates the path. This probability
is proportional with the product of the noise probability functions
along the path, or the stochastic Wiener path integral
\PC{normalize}
\begin{equation}
P(x,x',t)\sim \exp\left(-\frac{1}{\DiffC}\int_0^T dt \xi^2(t)\right).
\end{equation}
The most probable path is the one which maximizes the integral inside of
the exponential.  If we express the noise as 
\[
\xi(t)=\dot{x}(t)-v(x(t))
\]
the condition leads to the 
least action principle 
\index{least action principle}
\PC{of Fermat?}
\begin{equation}
\mbox{min} \int_0^Tdt(\dot{x}(t)-v(x(t)))^2=\mbox{min}\int_0^T 
L(x(t),\dot{x}(t))dt,
\end{equation}
like in the classical mechanics. Following the usual mechanics textbook
derivation of the path minimizing the action integral, we can
convince ourselves, that it is given by the 
solution of the Hamilton's equations. To a given $x$, $x'$ and $T$
we have to find the initial $p$ which realizes this path. 


\Remarks

\remark{Literature.}{
Theory of stochastic processes is a vast subject, spanning over centuries
and over disciplines ranging form pure mathematics to impure financial
dealings. 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 variational
methods with the Lagrangian \refeq{OnsagMachlup}, but for 
linear stochastic ODE's only ({\em ie}, no state space varying
velocity field $v(x)$).
\index{Langevin equation}
\index{Onsager-Machlup}
        % ``Fluctuations and Irreversible Processes",
}


\Resume

The effect of the noise can be taken into account by
adding noise corrections to the classical trace formula. 

\input{refsNoise10-1-7}

