Microcanonical Ensemble and Boltzmann Entropy

In our previous lesson, we established the postulate of equal a priori probabilities (PEAPP) as the fundamental rule for describing an isolated system in equilibrium. This principle, justified by both dynamical reasoning and information theory, asserts that every accessible microstate is equally likely. This provides the statistical foundation we need to move from microscopic laws to macroscopic thermodynamic behavior.

This lesson will formalize this foundation by defining the microcanonical ensemble, the statistical ensemble representing an isolated system. We will then take a monumental step: deriving one of the most important equations in all of physics, Boltzmann's entropy formula, $S = k_{B} ln Ω$ . This derivation will forge the definitive link between the microscopic world of countable states and the macroscopic, measurable world of thermodynamics.

1. The Microcanonical Ensemble (N, V, E)

The concepts from our last discussion—an isolated system with fixed particle number $N$ , volume $V$ , and energy $E$ , governed by the postulate of equal a priori probabilities—are formally collected under the name microcanonical ensemble.

An ensemble, in statistical mechanics, is a conceptual collection of a vast number of systems, each representing a possible microstate that the real system could be in. For the microcanonical ensemble, all systems in this collection are constrained to have the same $N$ , $V$ , and $E$ .

The state of the ensemble is described by a probability distribution over the phase space. Based on PEAPP, this distribution is uniform over all accessible microstates and zero everywhere else.

For a classical system with Hamiltonian $H (P, Q)$ , where $(P, Q)$ represents the set of all momenta and positions, the probability density $ρ (P, Q)$ can be written formally.

Lecture Notes on Statistical Mechanics and Thermodynamics

These notes from the University of Leipzig provide a concise and formal definition of the microcanonical ensemble, which aligns with your preference for mathematical rigor.

Please read the subsection "Micro-Canonical Ensemble in Classical Mechanics" on pages 31-32. Focus on how the probability density is constructed. Pay attention to the two equivalent ways of defining it: Using a Dirac delta function, Equation (4.5), which enforces a perfectly sharp energy E. Using a narrow energy shell, Equation (4.6), between E-\Delta E and E, which is often more physically realistic. You will notice that for systems with a large number of particles, the results obtained from both definitions are practically identical. The number of states in this shell is denoted W(E), which is our familiar \Omega(N,V,E).

For a quantum system, the picture is conceptually simpler because energy levels are often discrete. We don't need to construct phase-space volumes; we can simply count the quantum states that have the required energy. The microcanonical ensemble is then an equal-weighted mixture of all energy eigenstates within the specified narrow energy range $[E, E + δ E]$ .

This diagram illustrates the quantum mechanical view of the microcanonical ensemble. On the left, a vast number of unique quantum microstates exist. Those with the same macroscopic properties (fixed total energy U, volume V, and particle number N) are grouped together to form the ensemble. On the right, these states, labeled by $\eta_i$, are shown as the accessible states that define the ensemble. The number of these states is $\Omega(N,V,E)$.

2. The Statistical Definition of Entropy

With the microcanonical ensemble defined, our task is to connect the number of accessible microstates, $Ω (N, V, E)$ , to a thermodynamic function. The function that emerges is entropy. The derivation is one of the most elegant arguments in physics and relies on considering two systems in thermal contact.

Imagine two large systems, A and B, which are initially isolated. System A has energy $E_{A}$ and $Ω_{A} (E_{A})$ microstates. System B has energy $E_{B}$ and $Ω_{B} (E_{B})$ microstates. We then bring them into thermal contact, allowing them to exchange energy but not particles or volume. The combined system A+B is itself isolated, with a constant total energy $E_{t o t a l} = E_{A} + E_{B}$ .

Since the two subsystems are weakly interacting, for any given microstate of A, system B can be in any of its accessible microstates. Therefore, the total number of microstates for the combined system, when system A has energy $E_{A}$ , is the product:

Ω_{t o t a l} (E_{A}) = Ω_{A} (E_{A}) \times Ω_{B} (E_{t o t a l} - E_{A})

According to the fundamental postulate applied to the total isolated system, every microstate of A+B is equally likely. This means the probability of finding the combined system in a macrostate where system A has energy $E_{A}$ is proportional to the number of microstates corresponding to that macrostate:

P (E_{A}) \propto Ω_{t o t a l} (E_{A}) = Ω_{A} (E_{A}) Ω_{B} (E_{t o t a l} - E_{A})

When the systems reach thermal equilibrium, they will settle into the most probable macrostate—the one with the maximum number of available microstates. To find this maximum, we can maximize $ln P (E_{A})$ with respect to $E_{A}$ , as the logarithm is a monotonic function.

The following text provides a step-by-step derivation of this maximization, leading directly to the definition of entropy.

[PDF] Introduction to Statistical Mechanics - Salvatore Assenza

This resource by Salvatore Assenza provides a very clear and self-contained derivation of Boltzmann's entropy formula.

Please read Section 3.1, "Entropy as the states-counting variable" (pages 8-10). Follow the argument carefully, starting from the two systems in thermal contact. The key steps are: Writing the total number of states as a product, Equation (20). Maximizing this quantity with respect to energy exchange, which leads to the equilibrium condition in Equation (24). Recognizing that this condition defines thermal equilibrium and identifying the quantity (\partial \ln \Omega / \partial E)_{N,V} with the inverse temperature from thermodynamics, 1/T. This forces the identification of entropy S with k_B \ln \Omega, as shown in Equation (25).

This derivation is the heart of statistical mechanics. It shows that the thermodynamic condition for thermal equilibrium (equal temperatures) is equivalent to the statistical condition of maximum probability (maximum number of microstates). The quantity that connects them is entropy, defined as:

S (N, V, E) = k_{B} ln Ω (N, V, E)

This is Boltzmann's entropy formula, famously engraved on his tombstone in Vienna. It gives a precise, microscopic meaning to entropy: it is a measure (on a logarithmic scale) of the number of ways the microscopic constituents of a system can be arranged to produce the same macroscopic state.

The following video lecture provides a complementary perspective on this derivation.

Mod-01 Lec-22 The microcanonical emsemble

This NPTEL lecture walks through the same logical steps, reinforcing the key ideas.

First, watch the recap of the two-system setup from the start of the lecture. This reinforces the factorization of \Omega_{total}. Then, skip to the maximization step. The lecturer finds the equilibrium condition by maximizing the log of the probability from this timestamp. This leads to the equality of \frac{\partial \ln \Omega}{\partial E} for both systems. Finally, see the formal definition of entropy being introduced at this point, where the lecturer explicitly states S = k_B \ln \Omega and explains why \ln \Omega is the quantity of interest.

3. Properties and Interpretations of Boltzmann's Entropy

This statistical definition of entropy is powerful because it is consistent with the properties we expect from thermodynamics and provides a deeper, information-theoretic interpretation.

Additivity and Extensivity

Entropy is an extensive thermodynamic property, meaning that the entropy of a composite system is the sum of the entropies of its parts. Our statistical definition beautifully captures this. If we have two independent systems A and B, the total number of microstates is $Ω_{t o t a l} = Ω_{A} \times Ω_{B}$ . The total entropy is then:

S_{t o t a l} = k_{B} ln (Ω_{A} Ω_{B}) = k_{B} (ln Ω_{A} + ln Ω_{B}) = S_{A} + S_{B}

The logarithm in Boltzmann's formula turns the multiplicative nature of state-counting into the additive nature of entropy.

Connection to Information Entropy

In our first module, we introduced the Gibbs-Shannon formula for information entropy, $S_{G} = - k_{B} \sum_{i} p_{i} ln p_{i}$ . The Boltzmann entropy is precisely the Gibbs-Shannon entropy evaluated for the microcanonical ensemble.

In the microcanonical ensemble, each of the $Ω$ accessible states has a probability $p_{i} = 1/Ω$ , and all other states have $p_{i} = 0$ . Plugging this into the Gibbs-Shannon formula gives:

S_{G} = - k_{B} i = 1 \sum Ω \frac{1}{Ω} ln (\frac{1}{Ω}) = - k_{B} \cdot Ω \cdot \frac{1}{Ω} (- ln Ω) = k_{B} ln Ω = S_{B o l t z mann}

This shows a deep consistency in our framework. The Boltzmann entropy is not a new, ad-hoc definition but rather the logical consequence of applying the principle of maximum entropy to an isolated system. It represents the amount of microscopic uncertainty that remains after all macroscopic constraints ( $N, V, E$ ) have been specified.

Conclusion

In this lesson, we have formally defined the microcanonical ensemble and used it to derive the statistical basis for entropy. This is a cornerstone of our course, bridging the microscopic and macroscopic worlds.

Key Takeaways:

The microcanonical ensemble is the statistical description of an isolated system with fixed $N$ , $V$ , and $E$ . Its fundamental characteristic is the uniform probability distribution over all $Ω (N, V, E)$ accessible microstates.
Boltzmann's entropy formula, $S = k_{B} ln Ω$ , connects the number of accessible microstates to the thermodynamic entropy.
This formula is derived by finding the condition for equilibrium between two systems in thermal contact: equilibrium corresponds to the configuration that maximizes the total number of microstates.
The statistical entropy $S = k_{B} ln Ω$ is consistent with the thermodynamic property of extensivity and is a specific instance of the more general Gibbs-Shannon information entropy.

We have now defined entropy statistically. In our next lesson, we will complete the bridge to thermodynamics by using this definition. We will derive expressions for temperature, pressure, and chemical potential as partial derivatives of the entropy function $S (N, V, E)$ , demonstrating that our statistical framework can fully reproduce the fundamental relations of thermodynamics.

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