Physics 301 3-Oct-2005 8-1ReadingThis week, we’ll concentrate on the material in K&K chapter 4. This might be calledthe t hermodynamics of oscillators.Classical Statistical MechanicsRecall that statistical mechanics was developed before quantum mechanics. In ourdiscussions, we’ve made use of the fact that quantum mechanics allows us to speak ofdiscrete states (sometimes we have to put our system in a box of volume V ), so it makessense to talk about the number of states available to a system, to define the entropy asthe lo garithm of t he number of states, and to speak of maximizing the number of states(entropy) available to the system. If one didn’t know about quantum mechanics and didn’tknow about discrete states, how would one do statisti cal mechanics?Answer: in classical statistical mechanics, the phase space volume plays the role ofthe number of states. We’ve mentioned phase space briefly. Here’s a slightly more detail eddescription. In classical mechanics, one has the Lagrangian, L(q, ˙q, t) which is a functionof generalized coordinates q, their velocities, ˙q, and possibly the time, t. The equations ofmotion are Lagrange’s eq uat ionsddt∂L∂ ˙q−∂L∂q= 0 .Note that q might be a single variable or it might stand for a vector of coordinates. Inthe la tter case, there is one equation of motion for each coordinate. The Hamiltonian isdefined by a Legendre transformation,H(q, p, t) = p ˙q − L(q, ˙q, t) ,wherep =∂L∂ ˙q,is called the momentum conjugate to q (or the canonical momentum). The equatio ns ofmotion become (assuming neither L nor H is an explicit function of time)˙q =∂H∂p, ˙p = −∂H∂q,so that each second order equation of motion has been replaced by a pair of first orderequations of motio n.If p and q are given for a particle at some initial time, then the time development of pand q are determined by the eq uat ions of motion. If we consider a single pai r of conjugateCopyrightc 2005, Princeton University Physics Department, Edward J. G rothPhysics 301 3-Oct-2005 8-2coordinates, p and q (i. e., a one-dimensional system), and we consider a space with axes qand p, then a point in this space represents a state of the system. The equat ions of motiondetermine a trajectory (or orbit) in t his space that the system follows. The q-p space iscalled phase space. If we consider a 3-dimensional particle, then three coordinates andthree momenta are required to describe the particle. Phase space becomes 6-dimensionaland is a challenge to draw. If we consider N 3-dimensional particles, then phase spacebecomes 6N -dimensional. Or, one might draw N trajectories in a 6-dimensional space.As an example of a phase space that we might actually be able to draw, consider two1-dimensional particles moving along a common line. Suppose they are essentially freeparticles. The phase space coordinates are q1, p1, q2, and p2. (Subscripts refer to particles1 and 2.) The figure shows an attempt at drawing a t rajectory in the 4-dimensional phasespace. Since we have free particles, p1and p2are constants and q1and q2are linearfunctions of time, for example, q1= p1t/m1. The figure shows a trajectory for q1and forq2. As shown, q1has a po si tive momentum, so its trajectory is from left to right, whileq2has a negative momentum, so its trajectory is from right t o left. Each point on thetrajectory of q1correspo nds to exactly one point o n the trajectory of q2—the points arelabeled by time and points at the same time are corresponding points. If we could drawin four dimensions, there would be a singl e l ine representing both particles and we wouldnot have to point out this correspondence.Note that at some time, both particles are at the same physical place in space andsimply pass through each other as we’ve drawn the trajectories above. Instead of passingthrough each other, suppose they have a collision and “bounce backwards.” This might berepresented by the diagram shown in the next figure. This has been drawn assuming equalCopyrightc 2005, Princeton University Physics Department, Edward J. G rothPhysics 301 3-Oct-2005 8-3masses, equal and opposite momenta, and an elastic collision. I’m sure you can work outdiagrams for other cases.Suppose we are considering a low density gas (again!) . We certainly would not wantto try to draw a phase space for all the particles in the gas and we certainly wouldn’twant to try to draw all the trajectories including collisions. In the two particle case we’vebeen considering, suppose we blinked while the collision occurred. What would we see?The answer (for a suitable bli nk ) is shown in the next figure. We’d see particles 1 and 2moving along as free particles before we blinked and again a fter we blinked, but while weblinked, they changed their momenta. We’ve already mentioned that in a low density gas,the mol ecules t ravel several molecular diameters between collisions while collisions occuronly when molecules are within a few molecular diameters o f each other. One way to treata low density gas is to treat the molecules as free particles and to try to add in somethingto account for the coll isions. By looking at the drawing of the col lision (where we blinked),we can see that one way is to say that the particles fol low phase space trajectories for afree particle, except every now and then a trajectory ends and reappears—at random—somewhere else. The disappearance and reappearance of phase space trajectories does notreally happen; it’s an approximate way to treat collisions.All this is motivation for the idea that collisions randomize the distribution of particlesin phase space. Of course the randomization must be consistent with whatever constraintsare placed on the sy stem (such as fixed total energy, etc.) In general, if a system isin thermal contact with another system, we would expect that the exchanges of energy,Copyrightc 2005, Princeton University Physics Department, Edward J. G rothPhysics 301 3-Oct-2005 8-4required for thermal equilibrium, would result in randomization of the phase space location.The classical statistical mechanics analog of our postulate that all accessible statesare equally probable is the postulate that all accessible regions of phase space are equal lyprobable. In other words, a point in phase space plays the role of a state. The levelingof t he probabilities i s of course accomplished by the collisions and energy t ransfers we’vejust b een discussing.