The Virial Theorem & Thermal Equilibrium

The Virial Theorem & Thermal Equilibrium

This section covers two distinct but fundamental topics in statistical mechanics. First, the Virial Theorem, which serves as a tool to understand how real gases deviate from ideal behavior by relating kinetic energy to interaction forces. Second, we derive the conditions for Thermal Equilibrium using the microcanonical ensemble, providing the rigorous statistical definition of temperature.

1. The Virial Theorem

The Virial Theorem links the long-term average of the kinetic energy of a stable system to the forces acting within it. It is particularly useful for deriving the Equation of State for interacting systems (real gases).

Statement and Formula

The theorem states that the average kinetic energy \( \langle T \rangle \) is related to the "Virial" of the forces:

\[ \langle T \rangle = -\frac{1}{2} \sum_{i} \langle \mathbf{r}_i \cdot \mathbf{F}_i \rangle \]

In terms of Generalized Coordinates (\( q_n \)) and the Hamiltonian (\( H \)), we define the Virial (\( \mathcal{V} \)) as:

\[ \mathcal{V} = \sum_{n} q_n \frac{\partial H}{\partial q_n} \]

Connection to Equipartition

From our previous study of the microcanonical ensemble, we know that for a system in equilibrium:

\[ \left\langle q_n \frac{\partial H}{\partial q_n} \right\rangle = kT \]

Summing this over all \( 3N \) degrees of freedom gives the total Virial of the system:

\[ \langle \mathcal{V} \rangle = 3NkT \]

Significance: This relationship is what allows us to study the equation of state for interacting systems. If there are no interactions (ideal gas), the virial comes purely from wall collisions ($PV$). If there are interactions, this term captures the deviations from the ideal gas law.

2. Thermal Equilibrium (Microcanonical Derivation)

How do we define temperature statistically? We consider two systems, \( A \) and \( A' \), which are in thermal contact but isolated from the rest of the universe.

System Setup

• Total System (\( A^0 \)): Isolated, composed of \( A + A' \).
• Constraint: Total energy is conserved, though it fluctuates between the subsystems. \[ E + E' = E^0 \quad (\text{Constant}) \] \[ E' = E^0 - E \]

Probability Analysis

The number of accessible microstates for system \( A \) at energy \( E \) is \( \Gamma(E) \). For system \( A' \), it is \( \Gamma'(E') \). The probability \( P(E) \) of finding the composite system in a configuration where system \( A \) has energy \( E \) is:

\[ P(E) = C \cdot \Gamma(E) \cdot \Gamma'(E^0 - E) \]

The Physics of the Peak: \( \Gamma(E) \) is a rapidly increasing function of \( E \). However, since \( E' = E^0 - E \), as \( E \) increases, \( E' \) decreases, making \( \Gamma'(E') \) a rapidly decreasing function. The product of these two opposing functions creates a sharply pronounced maximum at the equilibrium energy.

The Equilibrium Condition

The most probable state corresponds to the maximum of \( P(E) \). It is mathematically easier to maximize the logarithm, \( \ln P(E) \):

\[ \ln P(E) = \ln C + \ln \Gamma(E) + \ln \Gamma'(E^0 - E) \]

Differentiating with respect to \( E \) and setting to zero:

\[ \frac{\partial \ln \Gamma(E)}{\partial E} + \frac{\partial \ln \Gamma'(E')}{\partial E} = 0 \]

Using the chain rule, where \( \frac{\partial E'}{\partial E} = -1 \):

\[ \frac{\partial \ln \Gamma(E)}{\partial E} - \frac{\partial \ln \Gamma'(E')}{\partial E'} = 0 \]

This yields the fundamental condition for equilibrium:

\[ \beta(E) = \beta'(E') \]

Defining Temperature and Entropy

From this equality, we identify the parameter \( \beta \) with temperature:

• Temperature: \( \beta \equiv \frac{1}{kT} \)
• Entropy (Boltzmann): \( S = k \ln \Gamma \)

The condition of maximum probability is therefore equivalent to the Maximization of Total Entropy: \[ S_{total} = S + S' \rightarrow \text{Maximum} \]