A system consists of N very weakly interacting particles at...

A system consists of $N$ very weakly interacting particles at a temperature sufficiently high such that classical statistics are applicable. Each particle has mass $m$ and oscillates in one direction about its equilibrium position. Calculate the heat capacity at temperature $T$ in each of the following cases: (a) The restoring force is proportional to the displacement $x$ from the equilibrium position. (b) The restoring force is proportional to $x^3$. The results may be obtained without explicitly evaluating integrals.

A system consists of $N$ very weakly interacting particles at a temperature sufficiently high such that classical statistics are applicable. Each particle has mass $m$ and oscillates in one direction about its equilibrium position. Calculate the heat capacity at temperature $T$ in each of the following cases:
(a) The restoring force is proportional to the displacement $x$ from the equilibrium position.
(b) The restoring force is proportional to $x^3$.

The results may be obtained without explicitly evaluating integrals.

Key Concepts

Power?Law Potentials and Scaling Arguments

When the potential energy of a system can be expressed as a power law, V(x) ? x^n, scaling arguments can be applied to determine how the average energy scales with temperature. This approach leverages dimensional analysis of the partition function and energy integrals, allowing one to deduce that, in one dimension, the average potential energy contributes an amount proportional to kBT divided by the exponent of the power law (e.g. 1/n times kBT) while the kinetic part still contributes ½kBT, thereby influencing the overall heat capacity.

Equipartition Theorem

The equipartition theorem states that for a system in thermal equilibrium, each quadratic degree of freedom contributes an amount equal to one?half the thermal energy (½kBT) to the average energy. In the context of oscillatory systems, this theorem is used to assign energy contributions both from the kinetic part (which is quadratic in momentum) and from the quadratic part of the potential energy arising from the restoring force, thereby allowing for the calculation of heat capacities without explicit integration.

Classical Partition Function

The classical partition function is a key concept in statistical mechanics that encapsulates the statistical properties of a system in thermal equilibrium at high temperatures. It is computed as an integral over phase space of the Boltzmann factor exp(–?H), where H is the Hamiltonian. The partition function enables the derivation of thermodynamic quantities such as the internal energy and heat capacity, often through scaling arguments when the Hamiltonian contains power-law potentials.

Relationship Between Force and Potential Energy

In classical mechanics, the restoring force is related to the form of the potential energy by F(x) = –dV/dx. This relation implies that if the restoring force is proportional to a power of the displacement (e.g., proportional to x for harmonic or x^3 for anharmonic systems), the corresponding potential energy will be a power-law function of displacement (quadratic for harmonic, quartic for certain anharmonic cases). Recognizing this correspondence is essential for applying techniques like the equipartition theorem or scaling arguments to compute thermal properties.

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