[Pathria 3.14.] Consider a system of N classical particles with mass m moving in a cubic box with volume V =
LxLxL.The particles interact via a short-range d pair potential u(rij) and each particle interacts with each wall with a short-ranged interaction uwall(z), where z is the perpendicular distance of a particle from the wall. Write down the Lagrangian for this model and use a Legendre transformation to determine the Hamiltonian H.
(a) Show that the quantity P = −∂H ∂V= − 1 3L2∂H ∂Lcan clearly be identied as the instantaneous pressure — that is, the force per unit area on the walls.
(b) Reconstruct the Lagrangian in terms of the relative locations of the particles inside the box ri = Lsi, where the variables si all lie inside a unit cube. Use a Legendre transformation to determine the Hamiltonian with this set of variables.
(c) Recalculate the pressure using the second version of the Hamiltonian. Show that the pressure now includes three contributions: i)a contribution proportional to the kinetic energy, ii) a contribution related to the forces between pairs of particles, and iii) a contribution related to the force on the wall. Show that in the thermodynamic limit the third contribution is negligible compared to the other two. Interpret contributions i) and ii) and compare to the virial equation of state (3.7.15)in the text.
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