Phase space in classical physics

Transcription

1 Pase space in classical pysics Quantum mecanically, we can actually COU te number of microstates consistent wit a given macrostate, specified (for example) by te total energy. In general, eac microstate will be specified by one or more quantum numbers. One migt, for example, ask ow many microstates are consistent wit a given energy (macrostate). is number of states is wat we ave been calling te multiplicity of a system. Classically, it s a bit trickier. A microstate of a gas, for example, is specified by te positions and momenta of eac of te molecules tat make up te gas; tat is, molecule one is specified by p and r, or if you prefer, by te 6 coordinates px, py, p z, x, y, z. e difficulty, of course, is tat tere are an infinite number of suc states, since te variables are continuous. Let s finesse tis problem for just a moment. For large, it will turn out tat OE macroscopic state will be associated wit virtually ALL of te microstates. Here is a place were large numbers lead to intuitively surprising results. Suppose tis room were divided down te middle by a partition, and all te gas were on one side. Clearly, tere are a HUGE number of states corresponding to te equilibrium state. ow remove te partition. e gas expands and fills te entire room. As we ave seen, te probability tat purely by cance, te gas will all come over to one side again is vanisingly small. us, uge as te first number is, it is altogeter negligible compared wit te number of states tat is available after te partition is removed. Moreover, for large, te peak is enormously sarp, so tat te likeliood of even small deviations is tiny. Let s return to te initial question: How do we estimate te number of states of a system for a classical system? Here is one common approac: First, we define te notion of pase space eac momentum and position coordinate counts as one dimension of pase space. us, for a single particle, tere are tree degrees of freedom (x, y, and z); and te state of te particle is defined by tose tree coordinates, and also by te tree coordinates of te momentum. us, one would ave a 6 dimensional pase space. Similarly, for a gas of particles confined to a volume V, one would speak of a degrees of freedom, and a 6 dimensional pase space. ow, we can calculate te volume of te pase space: Γ dp dp dp dp dx dx dx dx were te integral is over eiter all states wit energy less tan E 0, or possibly all states wit te energy in some small range, as in E0 < E < E0 + if you will, te volume of a sell of constant energy. Curiously enoug, it won t matter wic see below! If te latter, te usual notation is to set

2 pase space argument page of 5 Ω dp dp dp dp dx dx dx dx Before te advent of quantum mecanics, one approac was to look for ways of categorizing states into bins, and ten take te limit as te size of eac bin 0. (Boltzmann). It actually works pretty well, but we won t do it discuss reasons. Instead, we look at G& s trick, wic you will find in many oter books: Consider a quantum mecanical version of a one-dimensional, one particle system, and count te number of states. e classical version of suc a system will ave a two-dimensional pase space (one dimension corresponding to position, te oter to momentum). ote tat a two dimensional pase space volume as dimensions [ px ] wit units joulesec; note tat Planck s constant as te same units. Calculate te volume of pase space (in tis case, te area of te twodimensional pase space), and compare tis classical system to te corresponding QM system, in a semi-classical limit. One finds tat QM number of states classical volume of pase space or in oter words, eac quantum state corresponds to a classical volume of pase space! Examples: particle in a box: See G& Sections 4.., 4..5, 4..6 and class notes. simple armonic oscillator: See G& Section 4..4 and class notes. us, for a collection of point masses, free to move in tree dimensions, one would ave QM number of states classical volume of pase space We refer to tis collection of states as an ensemble. And at some point, we need to state te Postulate of Equal Apriori Probability all microstates in te ensemble ave equal probabilities. After Gibbs, we call suc an ensemble te microcanonical ensemble.

3 pase space argument page of 5 Connection to Baierlein It turns out to be straigtforward to calculate te volume of pase space for an ideal gas. ote tat for a single particle, we ave dx dx dx V dxdydz ow since it s an ideal gas, te particles don t know te oters are tere, and so we may treat eac one independently. e first step is easy. Ω V dp dp dp dp dx dx dx dx dp dp dp dp p px + py + pz For a single particle, te energy E is given by E ; tus, for a m m collection of particles wit total energy E in te range E0 < E < E0 +, we ave me pi α i α ow tis double sum can be tougt of as te square of te radius R( E ) of a dimensional space: me R ( E), or R me e volume of an dimensional pase space is just a constant te total volume of pase space Γ for all energies E < E0 is R ; ence, we ave B B Γ V R V me ( ) / were B is just a geometry-dependent constant. (If you re interested, G& work it out in an Appendix. But it is never needed in calculations.) Consequently, a surface of constant energy is given by B B Ω V R V me ( ) But of course, for large, we can neglect te, and ence we ave

4 pase space argument page 4 of 5 B Ω V ( ) me / us, as advertised above, te volume of a sell of constant energy in pase space is virtually te same as te volume of te entire space for E < E0. It s anoter surprising consequence of working wit large numbers! For mysterious reasons of my own (it as to do wit te nature of identical particles in quantum mecanics), I will put in an additional factor of! in te denominator. us, for an ideal gas, te quantity Ω, wic you will see variously described as te volume of classical pase space, te number of states, or peraps, te multiplicity, is given by Ω B V me! ( ) / ote, by te way, tat Ω depends on only tree macroscopic variables, EV,, and. At tis point, Baierlein s mysterious statement becomes obvious. For an ideal gas, te energy depends only on te temperature. But on pages, Baierlein is considering a constant temperature process. Hence te ratio of multiplicities depends only on te ratio of te volumes raised to te t power, as we see from te above equation! You migt be interested to see were we go from ere. We will be getting a little aead of ourselves, but it s interesting to see anyway. We sall sortly define te entropy as S kln Ω were k is Boltzmann s constant, so-called because it was introduced into pysics by Planck and Einstein. We sall also define te termodynamic temperature as S E V, Hence, putting everyting togeter, S B k ln + k lnv + ln E E E! V, ote tat te first term is just an constant, and ence its derivative is zero. We ave k E E k or 4

5 pase space argument page 5 of 5 But tis result is te same one we got from our simple kinetic teory argument. Here we ave obtained it in a far more general way. It gets even better: It s easy enoug to sow (tink about an adiabatic, quasi-static process) tat E p V But using a teorem tat I ope you remember from multivariable calculus, it follows tat E V S, V S E S E V S p S V E or in oter words, p S B k ln + k lnv + ln E V V! E, or p pv k, or, written anoter way, V k us, te defining equations for an ideal gas, pv E k k follow in a natural way from our pase space argument!! 5