A More Systematic View =========================== Ensemble ---------------- A standard procedure of solving mechanics problems is Initial condition / Description of states -> Time evolution -> Extraction of observables States ~~~~~~~~~~~~~~~~~~~~~~~ **Density of states in phase space** Continuity equation .. math:: \partial _ t \rho + \nabla \cdot (\rho \vec u) =0 This conservation law can be more simpler if dropped the term :math:`\nabla\cdot \vec u = 0` for incompressibility. Or more generally, .. math:: \partial _ t \rho + \nabla \cdot \vec j = 0 and here :math:`\vec j` can take other definitions like :math:`\vec j = - D \partial_x \rho`. This second continuity equation can represent any conservation law provided the proper :math:`\vec j`. .. admonition:: From continuity equation to Liouville theorem :class: toggle From continuity equation to Liouville theorem: We start from .. math:: \frac{\partial}{\partial t} \rho + \vec \nabla \cdot (\rho \vec v) Divergence means .. math:: \vec \nabla \cdot = \sum_i \left( \frac{\partial}{\partial q_i} + \frac{\partial}{\partial p_i} \right) . Then we will have the initial expression written as .. math:: \frac{\partial}{\partial t} \rho + \sum_i \left( \frac{\partial}{\partial q_i} (\rho \dot q_i) + \frac{\partial}{\partial \dot p_i} \right) . Expand the derivatives, .. math:: \frac{\partial}{\partial t} \rho + \sum_i \left[ \left( \frac{\partial}{\partial q_i} \dot q_i + \frac{\partial}{\partial p_i} \dot p_i\right) \rho + \dot q_i \frac{\partial}{\partial q_i} \rho + \dot p_i \frac{\partial}{\partial p_i} \rho \right] . Recall that Hamiltonian equations .. math:: \dot q_i = \frac{\partial H}{\partial p_i} \dot p_i = - \frac{\partial H}{\partial q_i} Then .. math:: \left( \frac{\partial}{\partial q_i} \dot q_i + \frac{\partial}{\partial p_i} \dot p_i\right) \rho . Finally convective time derivative becomes zero because :math:`\rho` is not changing with time in a comoving frame like perfect fluid. .. math:: \frac{d}{d t} \rho \equiv \frac{\partial}{\partial t}\rho + \sum_i \left[ \dot q_i \frac{\partial}{\partial q_i} \rho + \dot p_i \frac{\partial}{\partial p_i} \rho \right] =0 Time evolution ~~~~~~~~~~~~~~~~~~~~~ Apply Hamiltonian dynamics to this continuity equation, we can get .. math:: \partial_t \rho = \{H, \rho\} which is very similar to quantum density matrix operator .. math:: \mathrm i \hbar \partial_t \hat \rho = [ \hat H, \hat \rho ] That is to say, the time evolution is solved if we can find out the Poisson bracket of Hamiltonian and probability density. Requirements for Liouville Density ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1. Liouville theorem; 2. Normalizable; .. hint:: What about a system with constant probability for each state all over the phase space? This is not normalizable. Such a system can not really pick out a value. It seems that the probability to be on states with a constant energy is zero. So no such system really exist. I guess? Like this? .. image:: images/sandiaPeaks.png :scale: 90% :align: center Someone have 50% probability each to stop on one of the two Sandia Peaks for a picnic. Can we do an average for such a system? **Example by Professor Kenkre.** And one more for equilibrium systems, :math:`\partial_t \rho =0`. Extraction of observables ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ It's simply done by calculating the ensemble average .. math:: \langle O \rangle = \int O(p_i; q_i;t) \rho(p_i;q_i;t) \sum_i dp_i dq_i dt where :math:`i=1,2,..., 3N`.