Statistical Definitions of Temperature, Pressure, and Chemical Potential

Welcome back. In our last lesson, we forged the crucial link between the microscopic world of quantum states and the macroscopic world of thermodynamics by deriving Boltzmann's entropy formula, $S = k_{B} ln Ω$ . We established that for an isolated system, equilibrium corresponds to the macrostate with the largest number of accessible microstates, and therefore, the maximum entropy.

This lesson will build directly upon that foundation. We will use the statistical definition of entropy to derive the meanings of three fundamental thermodynamic quantities: temperature, pressure, and chemical potential. You will see how these familiar concepts emerge naturally as partial derivatives of the entropy function $S (N, V, E)$ . This process completes the bridge from statistical first principles to the core relations of equilibrium thermodynamics, demonstrating the profound predictive power of the microcanonical ensemble.

1. Temperature: The Measure of Thermal Equilibrium

Let's begin by revisiting the central argument from our previous lesson. We considered two systems, 1 and 2, brought into thermal contact. The total system is isolated, so their total energy $E_{t o t a l} = E_{1} + E_{2}$ is constant. Equilibrium is reached when the total number of microstates, $Ω_{t o t a l} = Ω_{1} (E_{1}) Ω_{2} (E_{t o t a l} - E_{1})$ , is maximized. This led to the condition:

(\frac{\partial ln Ω _{1}}{\partial E _{1}})_{N_{1}, V_{1}} = (\frac{\partial ln Ω _{2}}{\partial E _{2}})_{N_{2}, V_{2}}

Using our new definition of entropy, $S = k_{B} ln Ω$ , this equilibrium condition becomes:

(\frac{\partial S _{1}}{\partial E _{1}})_{N_{1}, V_{1}} = (\frac{\partial S _{2}}{\partial E _{2}})_{N_{2}, V_{2}}

This equation tells us that when two systems are in thermal equilibrium, this specific partial derivative has the same value for both. This implies that the quantity $(\partial S / \partial E)_{N, V}$ must be related to the thermodynamic property that is equalized at thermal equilibrium: temperature.

From classical thermodynamics, the fundamental relation for a simple system undergoing a reversible process is $d E = T d S - P d V$ . If we consider a process at constant volume ( $d V = 0$ ), this simplifies to $d E = T d S$ , which can be rearranged to $1/ T = (\partial S / \partial E)_{V}$ . This perfect correspondence allows us to define temperature within our statistical framework.

Definition: Temperature
In the microcanonical ensemble, the absolute temperature $T$ is defined by the relation:

\frac{1}{T} \equiv (\frac{\partial S}{\partial E})_{N, V}

This definition gives temperature a precise microscopic meaning: it is inversely proportional to how rapidly the system's entropy increases with energy.

This figure illustrates the relationship between entropy and energy. Panel (a) shows that for a typical system, the entropy curve S(E) is concave. Since temperature is the inverse of the slope ($1/T = \partial S/\partial E$), a steeper slope at lower energy (point A) corresponds to a lower temperature, while a flatter slope at higher energy (point B) corresponds to a higher temperature. Panel (c) shows how two systems initially at different temperatures (different slopes) exchange energy to reach a common equilibrium state 'e' with a single, intermediate slope and temperature.

The following reading provides a formal argument confirming that this definition aligns with our intuition that energy flows from a hotter body to a colder one.

Part II - Statistical Physics

These lecture notes by Dexter Chua formally define temperature and rigorously prove that it behaves as expected.

Please read the subsection titled "Temperature" on page 7. Focus on the formal definition and the proof in the subsequent proposition that energy flows from the system with higher T to the one with lower T, as dictated by the second law (\delta S \ge 0). This confirms our statistical definition is consistent with the zeroth and second laws of thermodynamics.

This graph visualizes the process of reaching thermal equilibrium between a system and its environment. Both start in initial states (i and j). They exchange an amount of energy $|\Delta E|$ until they reach their respective equilibrium states (s and e), where the slopes of their S(E) curves are equal. This equalization of slopes signifies that $1/T_{sys} = 1/T_{env}$, or simply $T_{sys} = T_{env}$.

2. Pressure and Chemical Potential

We can extend the same logic to define pressure and chemical potential. We consider two systems separated by a partition that can move and is also permeable to particles. The total system is isolated, so the total volume $V_{t o t a l} = V_{1} + V_{2}$ , total particle number $N_{t o t a l} = N_{1} + N_{2}$ , and total energy $E_{t o t a l} = E_{1} + E_{2}$ are all constant.

The total entropy is $S_{t o t a l} (E_{1}, V_{1}, N_{1}) = S_{1} (E_{1}, V_{1}, N_{1}) + S_{2} (E_{t o t a l} - E_{1}, V_{t o t a l} - V_{1}, N_{t o t a l} - N_{1})$ . For the combined system to be in equilibrium, $S_{t o t a l}$ must be at a maximum. This requires its derivatives with respect to all free variables ( $E_{1}, V_{1}, N_{1}$ ) to be zero. We have already seen that $\partial S_{t o t a l} / \partial E_{1} = 0$ leads to $T_{1} = T_{2}$ . Now we examine the other conditions.

Mechanical Equilibrium and Pressure
For mechanical equilibrium (the partition stops moving), we require $\partial S_{t o t a l} / \partial V_{1} = 0$ . This leads to:

(\frac{\partial S _{1}}{\partial V _{1}})_{E_{1}, N_{1}} = (\frac{\partial S _{2}}{\partial V _{2}})_{E_{2}, N_{2}}

Again, a quantity is equalized at equilibrium. From thermodynamics, we know this is pressure. The fundamental relation $d E = T d S - P d V + μ d N$ can be rewritten as $d S = \frac{1}{T} d E + \frac{P}{T} d V - \frac{μ}{T} d N$ . This shows that the partial derivative $(\partial S / \partial V)_{E, N}$ must be $P / T$ .

Definition: Pressure

\frac{P}{T} \equiv (\frac{\partial S}{\partial V})_{E, N} ⟹ P \equiv T (\frac{\partial S}{\partial V})_{E, N}

This defines pressure as the entropic response to a change in volume at constant energy and particle number.

Diffusive Equilibrium and Chemical Potential
For diffusive equilibrium (net particle flow ceases), we require $\partial S_{t o t a l} / \partial N_{1} = 0$ , which yields:

(\frac{\partial S _{1}}{\partial N _{1}})_{E_{1}, V_{1}} = (\frac{\partial S _{2}}{\partial N _{2}})_{E_{2}, V_{2}}

This identifies the chemical potential, $μ$ . From the rearranged thermodynamic relation, we see that $(\partial S / \partial N)_{E, V}$ corresponds to $- μ / T$ .

Definition: Chemical Potential

- \frac{μ}{T} \equiv (\frac{\partial S}{\partial N})_{E, V} ⟹ μ \equiv - T (\frac{\partial S}{\partial N})_{E, V}

The chemical potential measures the change in entropy with respect to particle number. The negative sign is a convention ensuring that particles flow from a region of higher chemical potential to one of lower chemical potential, a property you can verify with an argument identical to the one for temperature.

The following resource formally derives these definitions and discusses their physical meaning.

Lecture 7: Ensembles

This lecture by Matthew Schwartz uses the elegant method of considering systems in equilibrium to derive the definitions of pressure and chemical potential.

First, read Section 5, "Gibbs ensemble," on pages 8-9. Although the name "Gibbs ensemble" is used, the logic is purely microcanonical. It formalizes the argument we just made for two systems exchanging volume, leading to the definition of pressure in Equation (43). Next, read Section 6, "Grand canonical ensemble," on page 10. Again, focus on the initial argument. The reasoning about two systems exchanging particles leads to the definition of chemical potential in Equation (49). Finally, read the subsection "To see how chemical potential is embedded in the microcanonical ensemble" on page 10. This part explicitly connects the Lagrange multipliers from our earlier MaxEnt discussion (which you may recall from Module 1) to the definition of chemical potential, confirming the result within the microcanonical framework.

3. The Fundamental Thermodynamic Relation

We have now defined $T$ , $P$ , and $μ$ entirely from the properties of the function $S (E, V, N)$ . We can consolidate these results by considering the total differential of the entropy:

d S (E, V, N) = (\frac{\partial S}{\partial E})_{V, N} d E + (\frac{\partial S}{\partial V})_{E, N} d V + (\frac{\partial S}{\partial N})_{E, V} d N

Substituting our new definitions into this expression gives:

d S = \frac{1}{T} d E + \frac{P}{T} d V - \frac{μ}{T} d N

Multiplying by $T$ , we can rearrange this into the energy representation, which is perhaps the most famous equation in thermodynamics:

d E = T d S - P d V + μ d N

This is the fundamental thermodynamic relation (or Gibbs-Duhem equation). Its derivation from purely statistical arguments is a paramount achievement of statistical mechanics. It shows that the entire structure of equilibrium thermodynamics is a direct consequence of counting microstates and applying the postulate of equal a priori probabilities.

The following video lecture walks through this entire sequence of derivations, from defining the intensive variables to arriving at the fundamental relation.

Mod-01 Lec-22 The microcanonical emsemble

This lecture from NPTEL provides an excellent summary of the logic we've developed in this lesson.

First, watch the derivation of the inverse temperature, \beta, from this segment. This recaps our first step. Next, see how the same principle of maximizing \ln \Omega is extended to define pressure and chemical potential at this point. Finally, watch from this timestamp as the lecturer introduces S = k_B \ln \Omega and shows how all these partial derivatives combine to form the fundamental thermodynamic relation, dE = TdS - PdV + \mu dN, and connects it to the first and second laws of thermodynamics. This provides a powerful synthesis of the lesson's content.

Conclusion

In this lesson, we have completed the construction of the microcanonical framework by giving precise statistical definitions to the key intensive variables of thermodynamics. We have shown that the abstract counting of states in an isolated system logically implies the entire structure of thermodynamic relationships.

Key Takeaways:

Temperature, pressure, and chemical potential are not arbitrary parameters but emerge as derivatives of the entropy function $S (N, V, E)$ with respect to its natural variables.
The statistical definitions are:
- Temperature: $\frac{1}{T} = (\frac{\partial S}{\partial E})_{N, V}$
- Pressure: $\frac{P}{T} = (\frac{\partial S}{\partial V})_{E, N}$
- Chemical Potential: $- \frac{μ}{T} = (\frac{\partial S}{\partial N})_{E, V}$
These definitions naturally lead to the fundamental thermodynamic relation, $d E = T d S - P d V + μ d N$ , demonstrating a perfect consistency between statistical mechanics and classical thermodynamics.

The microcanonical ensemble provides the absolute foundation for statistical mechanics. However, the constraint of fixed energy and the need to count $Ω (E)$ can be mathematically cumbersome. In our next lesson, we will relax the constraint of fixed energy and consider a system in contact with a large heat reservoir at a fixed temperature $T$ . This will lead us to the canonical ensemble, a much more practical and widely used tool for calculating thermodynamic properties.

Recommended resources

1. Temperature: The Measure of Thermal Equilibrium

2. Pressure and Chemical Potential

3. The Fundamental Thermodynamic Relation

Conclusion