A magnetic system of spins is at thermodynamic equilibrium...

A magnetic system of spins is at thermodynamic equilibrium at temperature $T$. Let $\mu$ be the magnetic moment of each spin; and let $M$ be the mean magnetization per spin, so $-\mu<M<\mu$. The free energy per spin, for specified magnetization $M$, is $F(M)$. (1) Compute the magnetization $M$ as a function of external magnetic field strength $B$, given that $$ F(M)=\lambda \begin{cases}0, & |M / \mu| \leq 1 / 2, \\ (|M / \mu|-1 / 2)^2, & 1 \geq|M / \mu| \geq 1 / 2,\end{cases} $$ where $\lambda$ is a constant. (2) Suppose, instead, that someone gives you $$ F(M)=\lambda\left[(M / \mu)^4-(M / \mu)^2\right], $$ you should respond that this is unacceptable - this expression violates a fundamental convexity principle of thermodynamics. (a) State the principle. (b) Check it against the above expression. (c) Discuss, by at least one example, what would go wrong with thermodynamics if the principle is not satisfied.

A magnetic system of spins is at thermodynamic equilibrium at temperature $T$. Let $\mu$ be the magnetic moment of each spin; and let $M$ be the mean magnetization per spin, so $-\mu<M<\mu$. The free energy per spin, for specified magnetization $M$, is $F(M)$.
(1) Compute the magnetization $M$ as a function of external magnetic field strength $B$, given that
$$
F(M)=\lambda \begin{cases}0, & |M / \mu| \leq 1 / 2, \\ (|M / \mu|-1 / 2)^2, & 1 \geq|M / \mu| \geq 1 / 2,\end{cases}
$$
where $\lambda$ is a constant.
(2) Suppose, instead, that someone gives you
$$
F(M)=\lambda\left[(M / \mu)^4-(M / \mu)^2\right],
$$
you should respond that this is unacceptable - this expression violates a fundamental convexity principle of thermodynamics. (a) State the principle. (b) Check it against the above expression. (c) Discuss, by at least one example, what would go wrong with thermodynamics if the principle is not satisfied.

Key Concepts

Thermodynamic Equilibrium

Thermodynamic equilibrium refers to the state of a system where macroscopic observables remain constant over time, meaning that the system's free energy is at a minimum. In this state, the responses of the system to small perturbations are governed by the convexity of the free energy, ensuring a unique, stable equilibrium configuration.

Free Energy

Free energy is a thermodynamic potential that measures the useful work obtainable from a system, particularly under conditions of constant temperature and volume (Helmholtz free energy) or constant temperature and pressure (Gibbs free energy). In equilibrium, a system minimizes its free energy, and this minimization condition is used to determine the equilibrium state, such as the relationship between an external field and an order parameter.

Magnetization

Magnetization is the measure of the degree to which a material becomes magnetized in response to an external magnetic field. It represents the average magnetic moment per spin or per unit volume, and its equilibrium value is determined by how the free energy of the system varies with changes in the magnetization, reflecting the balance between the external field and intrinsic interactions.

Convexity Principle in Thermodynamics

The convexity principle in thermodynamics states that thermodynamic potentials, such as the free energy, must be convex functions of their natural variables to guarantee stability. This means, for instance, that the second derivative (or curvature) of the free energy with respect to an extensive variable (such as magnetization) must be non-negative. If this condition is violated, the system would exhibit unstable behavior as small fluctuations could spontaneously grow, leading to non-physical outcomes and inconsistencies in the thermodynamic description.

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