H o=

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f d plo log 10 =

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/2l - ~} n{IOg [nC T~kT f

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Using the equation of state we can rewrite this as

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-kVHo = iNk log (PV SI 3 ) + constant

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(4.34)

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We recognize that the right-hand side is the entropy of an ideal gas in thermodynamics. It follows from (4.34), (4.33), and (4.31) that dS = dQ/T. Thus we have derived all of classical thermodynamics for a dilute gas; and moreover, we were able to calculate the equation of state and the specific heat. The third law of thermodynamics cannot be derived here because we have used classical mechanics and thus are obliged to confine our considerations to high temperatures.

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4.3 THE METHOD OF THE MOST PROBABLE DISTRIBUTION

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We have noted the interesting fact that the Maxwell-Boltzmann distribution is independent of the detailed form of molecular interactions, as long as they exist. This fact endows the Maxwell-Boltzmann distribution with universality. We might suspect that as long as we are interested only in the equilibrium behavior of

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THERMODYNAMICS AND KINETIC THEORY

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Fig. 4.3 The microcanonical ensemble corresponding to a gas contained in a finite volume with energy between E and E + ~.

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a gas there is a way to derive the Maxwell-Boltzmann distribution without explicitly mentioning molecular interactions. Such a derivation is now supplied. Through it we shall understand better the meaning of the Maxwell-Boltzmann distribution. The conclusion we reach will be the following. If we choose a state of the gas at random from among all its possible states consistent with certain macroscopic conditions, the probability that we shall choose a MaxwellBoltzmann distribution is overwhelmingly greater than that for any other distribution. We shall use the approach of the Gibbsian ensemble described in Sec. 3.4. We assume that in equilibrium the system is equally likely to be found in any state consistent with the macroscopic conditions. That is, the density function is a constant over the accessible portion of r space. Specifically we consider a gas of N molecules enclosed in a box of volume V with perfectly reflecting walls. Let the energy of the gas lie between E and E + IJ., with IJ. E. The ensemble then consists of a uniform distribution of points in a region of r space bounded by the energy surfaces of energies E and E + IJ., and the surfaces corresponding to the boundaries of the containing box, as illustrated schematically in Fig. 4.3. Since the walls are perfectly reflecting, energy is conserved, and a representative point never leaves this region. By Liouville's theorem the distribution of representative points moves likes an incompressible fluid, and hence maintains a constant density at all times. This ensemble is called a microcanonical ensemble. Next consider an arbitrary distribution function of a gas. A molecule in the gas is confined to a finite region of p. space because the values of p and q are restricted by the macroscopic conditions. Cover this finite region of p. space with volume elements of volume w = d 3p d 3q, and number them from 1 to K, where K is a very large number which eventually will be made to approach infinity. We refer to these volume elements as cells. An arbitrary distribution function is defined if we specify the number of molecules n i found in the i th cell. These are called occupation numbers, and they satisfy the conditions

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L ni = N

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(4.35) (4.36)

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THE EQUILIBRIUM STATE OF A DILUTE GAS

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is the energy of a molecule in the ith cell:

f..=I

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where p; is the momentum of the ith cell. It is in (4.36), and only in (4.36), that the assumption of a dilute gas enters. An arbitrary set of integers {n;} satisfying (4.35) and (4.36) defines an arbitrary distribution function. The value of the distribution function in the ith cell, denoted by /;, is

/; =

n --.!.w

(4.37)

This is the distribution function for one member in the ensemble. The equilibrium distribution function is the above averaged over the microcanonical ensemble, which assigns equal weight to all systems satisfying (4.35) and (4.36):