How to Derive the Equilibrium Velocity Distribution Two Ways (Neither of Which is What You are Expecting) Cosmology Summer School Santa Cruz, July 26, 2013 Tim Maudlin, NYU
The Target Statistical Mechanics uses certain statistical techniques to � derive “laws” for the behavior of systems with many degrees of freedom. The prototype such system is a box of monatomic gas with something like a mole of atoms. Of particular interest are the “equilibrium” states of such a system. Many “laws”, e.g. the “ideal gas law” PV=nRT, describe relations between various (macroscopic) parameters of systems at equilibrium . What we are interested in here is a derivation of the velocity distribution of the atoms in our gas at a given temperature T at equilibrium. But what does “at equilibrium” mean ? �
Equilibrium Some macrostates (e.g. the state of a gas that had been � confined to half of a box and then had the partition removed, or of two boxes at different temperatures put in thermal contact) spontaneously change with respect to their macroscopic parameters. And in an isolated system, those parameters often settle, after a little time, into a static state in which they persist for very long times. If there is a state into which the system so settles, it is called “equilibrium”. Note that what counts as “equilibrium” may be sensitive to � the definition of “a little time” and “very long times”.
Note � The very definition just given of an “equilibrium” state already has a time asymmetry built into it: “settling into” and “remaining in” denote a temporal progress from past to future. So a certain time-directedness is already built into the very description of the phenomenon we are trying to explain.
The Velocity Distribution � Isolated boxes of gas tend to have equilibrium states into which they settle. In these states, the temperatures at different locations are uniform, pressures are uniform, densities are uniform (ignoring gravitational effects), the ideal gas law relates these, etc. � “Tend to” here means: every known isolated box of gas through all history has behaved this way. � At equilibrium, we expect the pressure, temperature and density to be uniform. But what about the distribution of velocities of the atoms?
A Note About “Uniform” � The sense in which the density of the gas is “uniform” is this: subdivide the volume V of the box into many, many parts with volume P i , but each part large enough to contain many, many atoms. At any moment, each such part has a well-defined density: N i /P i . The density if “uniform” if these densities are all equal to within some small epsilon. Similarly for temperature, pressure, etc. � Note that this requires there to be many, many atoms. A “single particle gas” makes no sense at all in this context.
Boltzmann 1872
Boltzmann con’t
The Velocity Distribution � Our target, then, is in this sense the empirical velocity distribution of a gas at temperature T at equilibrium. Take the set of all of velocities (or speeds) and divide it into ranges such that every range will have many, many atoms. We want to derive the (stable to within epsilon) distribution of velocities in the gas in these ranges at equilibrium. � That is, we want the distribution that any initial state will tend to go to and remain in.
Note on Terminology � The velocity distribution is, in the technical mathematical sense, a “probability distribution” or a “probability measure”. That is, the proportion of atoms in the union of two disjoint velocity ranges is the sum of the proportions in each range, and the sum of all the proportions over all ranges is unity. But the meaning of the distribution has nothing at all to do with “probability”. There is a definite, exact, empirical fact about what the distribution is at all times, and we want to derive the distribution that the system settles down to.
Temperature � One might take the given temperature of the system as a clue to the velocity distribution. Since temperature is a measure of mean kinetic energy, T = <1/2m v 2 >, one might guess that most of the atoms should have a speed near √ (2T/m). But how near, and with what sort of spread?
Simplifying Assumptions To make our problem easier (using assumptions we can � ultimately go back and derive), we will assume that at equilibrium the velocity distribution is isotropic (in physical space) and the spatial distribution of atoms is uniform (in physical space). This means that all we need is a speed distribution: the � velocity distribution follows from the speed distribution and the isotropy. Note: the notions of isotropy and uniformity here require a � measure to be defined , but the relevant measure is just given by the metric of space itself.
Any Guesses?
Three Derivations � We are going to look at three derivations of the equilibrium velocity distribution: � Maxwell 1860: “Illustrations of the dynamical theory of gases” � Maxwell 1867: “On the dynamical theory of gases” � Boltzmann 1872: “Further studies on the thermal equilibrium of gas molecules”
What These Derivations Are Not � David has outlined or mentioned two possible approaches to a question like this. Let’s recall them. � Approach 1) choose (how?) some measure over the phase space of the system. For each possible velocity distribution, calculate the measure of the volume of phase space that has that distribution. If one such distribution absolutely dominates in terms of the measure, say that if the dynamics is “random” (ergodic? something else?), then no matter where the system begins, it will “probably” “soon” wander into that distribution and stay there “for a long time”.
What They Are Not (Con’t) � Approach 2) Choose a measure over phase space (somehow). Choose a macrostate that specifies a velocity distribution. Use the exact microdynamics to time-evolve the measure restricted to the macrostate in time. If after some period of time, almost all of the time-evolved measure now occupies a macrostate characteristic of a certain velocity distribution, and then stays there for a long time, declare that the equilibrium state for the initial macrostate. Repeat for all possible initial macrostates.
Problems For Approach 1, we must somehow justify the use of the � measure on phase space and explain what we mean by a “random” dynamics. It is not clear that one could derive anything like relaxation times from this approach. Note that the exact microdynamics plays no role. Given a tractable definition of “random dynamics”, perhaps this can be done. For Approach 2, we must justify the choice of measure on � phase space. The calculation would give relaxation times, based in the microdynamics, which have to be recalculated for each initial state. The relevant calculations are practically impossible.
In Any Case � None of the three papers we are going to look at proceeds in anything like either of these fashions. Nonetheless, they both derive an equilibrium velocity distribution, and indeed the same equilibrium velocity distribution. Presumably, it is the same one that would be derived using Approach 2, but we can’t directly verify this because we can’t actually calculate using Approach 2. � So how do they work?
What to Watch For � In his 1872 paper, Boltzmann makes an incredibly strong claim about what he has achieved. So strong, in fact, that it is provably false. Both the reversibility and recurrence objections demonstrate it is false. When this was pointed out, Boltzmann quickly saw how to explain what he had done and add the appropriate caveats. But the fact that he made the claim shows that it was not entirely clear even to him what he was doing.
The Quote
Keep Your Eyes Peeled! � So here are some questions: � At what point, and in what way, does some time- asymmetric assumption appear in the reasoning? � At what point, and in what way, does any “probabilistic” or “randomness” assumption appear in the reasoning? How is it justified?
Maxwell 1860 � Maxwell’s 1860 derivation of the equilibrium velocity distribution is amazingly short, and amazingly puzzling. � All that Maxwell assumes is that the equilibrium velocity distribution has two formal properties: � 1) It is isotropic. � 2) The velocities of a particle in orthogonal directions are uncorrelated.
Clarification of Terms It is clear what “isotropic” means here: for any given speed, � the number of particles moving in any particular (coarse- grained) direction is the same, within epsilon. (In a one- particle gas, this cannot possibly obtain!) The condition of the velocities in orthogonal directions � being uncorrelated also has an exact meaning. Take, for example, the overall distribution of X-velocities. Now restrict attention to all the particles with any given Y-velocity, and look a the distribution of X-velocities in that subgroup. It should be the same (within epsilon) for all Y-velocities. So knowing the Y-velocity of a particle gives no information about its X-velocity (or Z-velocity), for any orthogonal directions.
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