Question
Derivation of the Boltzmann distributionDerive the Boltzmann distribution using the counting argument from Section $6.1 .4 .$ Modify the argument given there such that every particle has at least an energy of $\varepsilon$
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Each energy level $i$ has energy $\varepsilon_i$ and contains $n_i$ particles. The total energy of the system is $E$ and the total number of particles is $N$. Show more…
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For a system obeying Boltzmann statistics, we know what $\mu$ is from Chapter $6 .$ Suppose, though, that you knew the distribution function (equation 7.31 ) but didn't know $\mu .$ You could still determine $\mu$ by requiring that the total number of particles, summed over all single-particle states, equal $N$ Carry out this calculation, to rederive the formula $\mu=-k T \ln \left(Z_{1} / N\right) .$ (This is normally how $\mu$ is determined in quantum statistics, although the math is usually more difficult.)
Quantum Statistics
Bosons and Fermions
Boltzmann Distribution Generalize Exercise 55 to $n$ variables: Show that there is a constant $\mu$ such that the maximum of $$S=x_{1} \ln x_{1}+\cdots+x_{n} \ln x_{n}$$ subject to the constraints $$x_{1}+\cdots+x_{n}=N, \quad E_{1} x_{1}+\cdots+E_{n} x_{n}=E$$ occurs for $$x_{i}=A^{-1} e^{\mu E_{i}},$$ where $$\quad A=N^{-1}\left(e^{\mu E_{1}}+\cdots+e^{\mu E_{n}}\right)$$ This result lies at the heart of statistical mechanics. It is used to deter- mine the distribution of velocities of gas molecules at temperature $T$ ; $x_{i}$ is the number of molecules with kinetic energy $E_{i} ; \mu=-(k T)^{-1}$ , where $k$ is Boltzmann's constant. The quantity $S$ is called the entropy.
DIFFERENTIATION IN SEVERAL VARIABLES
Lagrange Multipliers: Optimizing with a Constraint
According to Maxwell-Boltzmann statistical count, the total number of arrangements of the particles is given by: W = N! ̧̣̀̑̐̓_{r=1}^s [g_r^{n_r} / n_r!] a) Derive the general form of Maxwell Boltzmann energy distribution law. b) By using the continuous variation of free particles in an ideal gas, show that the expression of the multiplier ̲ is: ̲ = log [V/N (2̰mkT / h^2)^(3/2)] Given that, e^-̲ = N / ∑ g_r e^(-E_r / kT) c) Deduce the Maxwell Boltzmann energy distribution law for particles in an ideal gas. d) Show that the entropy: S = kN log Z + (U/T), where Z is the partition function Z = ∑ g_r e^-̱ E_r. e) Find the most probable and the root mean square speed of N₂ molecule at 27
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