Ammm:Md erg

The Liouville equation

http://en.wikipedia.org/wiki/Liouville%27s_theorem_%28Hamiltonian%29#Liouville_equation

Phase space distribution conserves over time as with the density of matter (mass conservation)

Liousville eq describes the time evolution of phase space distribution function (hamiltonian describes individual member of the ensemble). Consider a dynamical system with canonical coordinates $q_{i}$ and conjugate momenta $p_{i}$, where $i=1,\dots ,d$. Then the phase space distribution $\rho (p,q)$ determines the probability $\rho (p,q)\,d^{d}q\,d^{d}p$ that the system will be found in the infinitesimal phase space volume $d^{d}q\,d^{d}p$. The Liouville equation governs the evolution of $\rho (p,q;t)$ in time $t$:

${\frac {d\rho }{dt}}={\frac {\partial \rho }{\partial t}}+\sum _{i=1}^{d}\left({\frac {\partial \rho }{\partial q_{i}}}{\dot {q}}_{i}+{\frac {\partial \rho }{\partial p_{i}}}{\dot {p}}_{i}\right)=0.$

Time derivatives are denoted by dots, and are evaluated according to Hamilton's equations for the system. This equation demonstrates the conservation of density in phase space (which was Gibbs's name for the theorem). Liouville's theorem states that

The distribution function is constant along any trajectory in phase space.

A simple proof of the theorem is to observe that the evolution of $\rho$ is defined by the continuity equation:

${\frac {\partial \rho }{\partial t}}+\sum _{i=1}^{d}\left({\frac {\partial (\rho {\dot {q}}_{i})}{\partial q_{i}}}+{\frac {\partial (\rho {\dot {p}}_{i})}{\partial p_{i}}}\right)=0.$

The differential form of the continuity equation is:
${\partial \rho  \over \partial t}+\nabla \cdot (\rho \mathbf {u} )=0$

That is, the tuplet $(\rho ,\rho {\dot {q}}_{i},\rho {\dot {p}}_{i})$ is a conserved current. Notice that the difference between this and Liouville's equation are the terms

$\rho \sum _{i=1}^{d}\left({\frac {\partial {\dot {q}}_{i}}{\partial q_{i}}}+{\frac {\partial {\dot {p}}_{i}}{\partial p_{i}}}\right)=\rho \sum _{i=1}^{d}\left({\frac {\partial ^{2}H}{\partial q_{i}\,\partial p_{i}}}-{\frac {\partial ^{2}H}{\partial p_{i}\partial q_{i}}}\right)=0,$

Recall the property of H discussed previously.

where $H$ is the Hamiltonian, and Hamilton's equations have been used. That is, viewing the motion through phase space as a 'fluid flow' of system points, the theorem says that the convective derivative of the density $d\rho /dt$ is zero follows from the equation of continuity by noting that the 'velocity field' $({\dot {p}},{\dot {q}})$ in phase space has zero divergence (which follows from Hamilton's relations).

Another illustration is to consider the trajectory of a cloud of points through phase space. It is straightforward to show that as the cloud stretches in one coordinate – $p_{i}$ say – it shrinks in the corresponding $q^{i}$ direction so that the product $\Delta p_{i}\,\Delta q^{i}$ remains constant.

Equivalently, the existence of a conserved current implies, via Noether's theorem, the existence of a symmetry. The symmetry is invariant under time translations, and the generator (or Noether charge) of the symmetry is the Hamiltonian.

Why do time average in MD?

http://en.wikipedia.org/wiki/Ergodic_hypothesis

In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a particle in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., that all microstates of the same energy will be accessed with equal oppurtunity over a long period of time.

For example, a double-well potential energy sruface has two microstates.
If the two wells have the same energy (e.g. two conformations of the same molecule with same energy), 
the system will spend the same amount of time in either well.
If one state has lower energy, the system will spend more time in the lower energy well than in the 
higer energy state, because the volume of the lower energy state is larger(the Boltzman factor)

The ergodic hypothesis in MD ismualtions assumes that the average of a process parameter over time and the average over the statistical ensemble are the same. The assumption is however tim is "long enough", otherwise the sampling is not proportional to the phase space volume. In the above example, one conformation could up more than the other even if the energy is the same.

Liouville's Theorem shows that, for conserved classical systems, the local density of microstates following a particle path through phase space is constant as viewed by an observer moving with the ensemble (i.e., the total or convective time derivative is zero). Thus, if the microstates are uniformly distributed in phase space initially, they will remain so at all times.

Liouville's theorem ensures that the notion of time average makes sense, 
but ergodicity does not follow from Liouville's theorem.