# Problems and solutions to statistical mechanics problems

List of solved problems:

Derive Strirling's approximation for N!.

Derive the Gaussian approximation of the binomial distribution.

Derive expressions for the mean and standard deviation for the binomial distribution.

Compute the pressure of an ideal Bose gas at absolute zero.

Compute the entropy of an ideal gas using the micro-canonical ensemble.

Compute the internal energy of N harmonic oscillators as a function of temperature using the micro-canonical ensemble.

Compute the susceptibility of a spin 1/2 particle .

Compute the entropy of a harmonic oscillator.

Compute the energy fluctuations of a system at constant temperature.

Compute the internal energy of a diatomic molecule.

Estimate the maximum possible height of trees.

Compute the temperature of the relic neutrino background of the universe.

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#### Solution Summary

In this study guide we give a basic overview of statistical mechanics and explain the tools needed to do computations. We have included a number of exercises to practice the theory. The main focus is on students who are preparing for exams. We give a brief overview of the theory in each chapter. The given overviews contain the essential points needed for the student to be able to start working on the problems. Students who have difficulty understanding a particular overview are recommended to study the relevant part of the theory from a textbook. Unlike typical textbooks, this study guide focusses mainly on the fundamentals. We don't e.g. derive the ideal gas law or study the ideal fermi gas in detail; such topics are discussed in most textbooks. We do discuss a number of problems that are useful to gain a deeper understanding of the theory and to gain practice with using the mathematical tools needed to perform computations.

Problems in Statistical Mechanics

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3. Consider again the lattice of N spin-1/2 particles in an external homogeneous magnetic field, where each particle has two possible states: spin "down" with energy e = 0 and spin "up" with energy e = 1/2". The microstate of the system is specified by the energy states of all the particles, i.e. the list (e1, e2...

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