Suppose that the energy of a particle can be represented by...

Suppose that the energy of a particle can be represented by the expression $E(z)=a z^2$ where $z$ is a coordinate or momentum and can take on all values from $-\infty$ to $+\infty$. (a) Show that the average energy per particle for a system of such particles subject to Boltzmann statistics will be $\bar{E}=k T / 2$. (b) State the principle of equipartition of energy and discuss briefly its relation to the above calculation.

Suppose that the energy of a particle can be represented by the expression $E(z)=a z^2$ where $z$ is a coordinate or momentum and can take on all values from $-\infty$ to $+\infty$.
(a) Show that the average energy per particle for a system of such particles subject to Boltzmann statistics will be $\bar{E}=k T / 2$.
(b) State the principle of equipartition of energy and discuss briefly its relation to the above calculation.

Key Concepts

Boltzmann Distribution

The Boltzmann distribution is a fundamental principle in statistical mechanics that provides the probability for a system to occupy a state with a certain energy. Under this distribution, the probability of a state having energy E is proportional to exp(-E/kT), where k is Boltzmann's constant and T the absolute temperature. This concept is crucial for determining ensemble averages (like average energy) by weighting each state's contribution appropriately, as exemplified when evaluating integrals over energy expressions.

Partition Function

The partition function is the normalization factor in the Boltzmann distribution, summing over all possible states of the system. It is integral to the calculation of statistical averages (such as energy or other observables) as it ensures the probabilities sum to one. In systems where energy depends quadratically on a variable (as in the example), the partition function is obtained by performing Gaussian integrals, leading to results that allow one to extract physical quantities like the average energy.

Equipartition Theorem

The equipartition theorem states that energy is distributed equally among all available quadratic degrees of freedom in a system at thermal equilibrium, with each contributing an average energy of (1/2)kT. This theorem provides a quick way to determine average energies without detailed integration, directly linking the calculation performed using Boltzmann statistics to the broader principle that each quadratic term in the energy adds (1/2)kT to the total energy per particle.

Gaussian Integrals

Gaussian integrals are mathematical tools used to evaluate integrals of the exponential of a quadratic function. In statistical mechanics, they are particularly useful when the energy expression is quadratic, as the integrations needed to compute the partition function or average quantities reduce to Gaussian integrals. Their well-known solutions simplify the calculation of statistical averages, and they underpin the derivation of results such as the kT/2 average energy per quadratic degree of freedom.

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