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Internal Energy and Pressure of Relativistic Gasses

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For zero mass particles, like neutrinos, the relativistic relationship between energy E and momentum p is

E(p) = c|p|,

where c is the speed of light. Calculate the pressure and energy of a classical relativistic zero mass gas as a function of density and termperature and show that pV = gE, where g is a numerical constant. What is g? Is it different from the non-relativistic case?

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Solution Preview

Before we start to work on this problem, let's first study the case of an ordinary nonrelativistic ideal gas consisting of particles of mass m. Although we're interested in the classical regime, it is still worthwhile to describe the gas quantum mechanically and then make suitable approximations. If you ignore quantum mechanics you'll miss some subtle things that are important even for a classical gas (like e.g. the factor 1/N! in formula 1.6, which is important if you want to compute the entropy)

Section 1: Partition function of a nonrelativistic gas

Section 2: Energy and Pressure of a dilute nonrelativistic ideal gas

Section 3: Energy and Pressure of a dilute relativistic ideal gas

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1 Partition function of a nonrelativistic gas
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The partition function is in general given by:

Z = Sum over r of Exp(- beta E_r) (1.1)

Here r enumerates the energy eigenstates of the system, E_r is the energy eigenvalue, and beta = 1/(k T), with k the Boltzmann constant. So, if you have a system with only two energy eigenstates with energies E1 and E2, the partition function will be:

Z = Exp(- beta E1) + Exp(- beta E2). (1.2)

To compute the partition function for a dilute ideal gas, we can make use of the fact that the partition function of a system that consists of non-interacting subsystems factorizes into the product of the partition functions of the individual systems (we can then take system to be the gas and the subsystems to be the individual particles in the gas). This is because the energy of such a system is given by the sum of the energies of the subsystems. From equation (1.1) you see that:

Z = Sum over r of Exp(- beta E_r) = Sum over r1,r2,r3...etc of Exp[- beta (E_r1 + E_r2 +E_r3 +...)] (1.3)

Here the r1, r2, r3,...etc enumerate the states of the subsystems. Clearly summing over the states of the entire system corresponds to summing over the states of the subsystems (but there is a catch here, I'll return to that after this derivation). In (1.3) you write summation of the terms in the exponential as a product:

Z = Sum over r1,r2,r3...etc of Exp[- beta (E_r1 + E_r2 +E_r3 +...)] = Sum over r1,r2,r3...etc of Exp(-beta E_r1)
Exp(-beta E_r2) Exp(-beta E_r3)...

When we sum over a particular ri, all the factors except the Exp(-beta Eri) stay constant, so you can bring all those other factors outside that summation. If you do this for all the ri you get:

Z = Product over i of Sum over ri of Exp(- beta E_ri) (1.4)

Now, Sum over ri of Exp(- beta E_ri) is, by definition, the partition function of subsystem i, so we see that Z is the product of the partition functions of ...

Solution Summary

After explaining some basic statistical mechanics, this solution explains the partition function as well as the calculations for the pressure of a dilute nonrelativistic ideal gas and a dilute relativistic gas in 1864 words.

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