Consider a system composed of a very large number N of...

Consider a system composed of a very large number $N$ of distinguishable atoms, non-moving and mutually non-interacting, each of which has only two (non-degenerate) energy levels: $0, \varepsilon>0$. Let $E / N$ be the mean energy per atom in the limit $N \rightarrow \infty$. (a) What is the maximum possible value of $E / N$ if the system is not necessarily in thermodynamic equilibrium? What is the maximum attainable value of $E / N$ if the system is in equilibrium (at positive temperature, of course)? (b) For thermodynamic equilibrium, compute the entropy per atom, $S / N$, as a function of $E / N$.

Consider a system composed of a very large number $N$ of distinguishable atoms, non-moving and mutually non-interacting, each of which has only two (non-degenerate) energy levels: $0, \varepsilon>0$. Let $E / N$ be the mean energy per atom in the limit $N \rightarrow \infty$.
(a) What is the maximum possible value of $E / N$ if the system is not necessarily in thermodynamic equilibrium? What is the maximum attainable value of $E / N$ if the system is in equilibrium (at positive temperature, of course)?
(b) For thermodynamic equilibrium, compute the entropy per atom, $S / N$, as a function of $E / N$.

Key Concepts

Boltzmann_Distribution

The Boltzmann distribution relates the probability of finding a particle in a specific energy state to the state’s energy and the system’s temperature. Under equilibrium, it ensures that states with lower energy are exponentially more populated, and this distribution is key for deriving thermodynamic quantities like the mean energy and entropy.

Entropy_and_Microstate_Counting

Entropy quantifies the degree of disorder or the number of microstates corresponding to a system's macroscopic state. For a two-level system, the entropy per particle can be derived from the probabilities of occupying the two energy levels, using an expression like S = -k_B(p ln p + (1-p) ln(1-p)), thereby linking microscopic configurations to macroscopic observables.

Non_Equilibrium_Maximum_Energy

Without the restrictions imposed by equilibrium, a system can be arranged in a way that maximizes the mean energy per particle by, for example, populating the highest energy state exclusively. This demonstrates that the energy bound in an arbitrary state can be different (usually higher) than the one allowed under equilibrium conditions.

Distinguishable_Particles

Distinguishable particles are those that can be individually labeled, eliminating the need for symmetrization or antisymmetrization in counting microstates. In the context of the problem, this simplifies the combinatorial analysis since each atom’s state is independent and uniquely identified.

Two_Level_System

A two-level system is one in which each particle can occupy only one of two possible energy states (often labeled as 0 and ?). This model is fundamental in statistical mechanics because it provides a clear, simple framework to study energy distribution, microstates, and thermodynamic properties such as entropy and heat capacity.

Thermodynamic_Equilibrium

Thermodynamic equilibrium is the state in which a system’s macroscopic properties remain constant in time, and the energy distribution among the constituents is governed by statistical laws like the Boltzmann distribution. Equilibrium imposes constraints on properties such as average energy per particle, with temperature mediating the balance between energy levels.

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