Configurational integrals, which sit at the heart of statistical mechanics and computational chemistry/physics, especially in molecular simulations (Monte Carlo, Molecular Dynamics).
Let’s carefully build this up from first principles, step by step, with all the mathematical formulations that appear in computing configurational integrals.
1. Partition Function as the Starting Point
In statistical mechanics, the canonical partition function for (N) particles at temperature (T) in volume (V) is:
[ Q_N(V, T) = \frac{1}{N! h^{3N}} \int d^{3N}p , d^{3N}q , e^{-\beta H(p, q)} ]
- (N!) : indistinguishability correction
- (h) : Planck’s constant (ensures correct units in phase space)
- (d^{3N}p) : integration over momentum coordinates of all (N) particles
- (d^{3N}q) : integration over spatial coordinates of all (N) particles
- (H(p, q)) : Hamiltonian of the system
- (\beta = \frac{1}{k_B T}), where (k_B) is Boltzmann’s constant
2. Separation into Momentum and Configurational Parts
The Hamiltonian separates into kinetic and potential energy:
[ H(p, q) = K(p) + U(q) ]
where
- (K(p) = \sum_{i=1}^N \frac{p_i^2}{2m_i})
- (U(q)) is the potential energy depending on positions only
So:
[ Q_N(V, T) = \frac{1}{N! h^{3N}} \int d^{3N}p , e^{-\beta K(p)} \int d^{3N}q , e^{-\beta U(q)} ]
3. Momentum Integral (Gaussian Integral)
The momentum part is separable and Gaussian:
[ \int d^{3N}p , e^{-\beta \sum_i \frac{p_i^2}{2m_i}} = \prod_{i=1}^N \left( \int d^3p_i , e^{-\beta \frac{p_i^2}{2m_i}} \right) ]
Each integral is a Gaussian in 3D:
[ \int d^3p , e^{-\beta \frac{p^2}{2m}} = \left( 2\pi m k_B T \right)^{3/2} ]
So the total kinetic integral is:
[ \int d^{3N}p , e^{-\beta K(p)} = (2\pi k_B T)^{3N/2} \prod_{i=1}^N m_i^{3/2} ]
4. Configurational Integral
What remains is the configurational integral:
[ Z_N(V, T) = \int d^{3N}q , e^{-\beta U(q)} ]
This is the central object in computing molecular systems:
- (d^{3N}q) means integrate over all possible positions of (N) particles.
- (U(q)) encodes the interactions (pair potentials, many-body forces, etc).
Thus:
[ Q_N(V, T) = \frac{1}{N! h^{3N}} (2\pi k_B T)^{3N/2} \left( \prod_{i=1}^N m_i^{3/2} \right) \cdot Z_N(V, T) ]
5. Potential Energy Decomposition
For pairwise additive interactions:
[ U(q) = \sum_{i<j} u(r_{ij}) ]
| where (r_{ij} = | q_i - q_j | ). |
So:
[ Z_N(V, T) = \int_V \cdots \int_V \exp!\left( -\beta \sum_{i<j} u(r_{ij}) \right) d^3q_1 \cdots d^3q_N ]
This is intractable analytically for large (N). That’s why Monte Carlo and Molecular Dynamics are used.
6. Thermodynamic Connections
The configurational integral connects directly to thermodynamic observables:
-
Helmholtz free energy: [ A = -k_B T \ln Q_N(V,T) ]
-
Pressure (virial route): [ P = k_B T \frac{\partial \ln Z_N}{\partial V} ]
-
Internal energy: [ U = -\frac{\partial \ln Z_N}{\partial \beta} ]
7. Approximations and Series Expansions
In practice, we approximate (Z_N).
Mayer Cluster Expansion:
Introduce the Mayer f-function:
[ f_{ij}(r_{ij}) = e^{-\beta u(r_{ij})} - 1 ]
Then:
[ Z_N = \int \cdots \int \prod_{i<j} \left(1 + f_{ij}\right) d^3q_1 \cdots d^3q_N ]
This expansion is used to derive virial coefficients for real gases.
8. Computing (Z_N) in Simulation
-
Monte Carlo: Sample configurations ({q}) with probability (P(q) \propto e^{-\beta U(q)}), then compute averages.
-
Molecular Dynamics: Generate trajectories consistent with (e^{-\beta U(q)}), and time averages approximate ensemble averages.
9. Example: Ideal Gas Case
For (U(q) = 0):
[ Z_N(V, T) = \int_V \cdots \int_V 1 , d^3q_1 \cdots d^3q_N = V^N ]
So:
[ Q_N^{\text{ideal}} = \frac{1}{N! h^{3N}} (2\pi k_B T)^{3N/2} \left( \prod_{i=1}^N m_i^{3/2} \right) V^N ]
✅ Summary of Key Mathematical Formulas in configurational integrals:
-
Partition function: [ Q_N = \frac{1}{N! h^{3N}} \int d^{3N}p , d^{3N}q , e^{-\beta H} ]
-
Configurational integral: [ Z_N = \int d^{3N}q , e^{-\beta U(q)} ]
-
Pairwise potential: [ Z_N = \int \cdots \int \exp!\Big(-\beta \sum_{i<j} u(r_{ij})\Big) d^{3N}q ]
-
Mayer expansion: [ Z_N = \int \cdots \int \prod_{i<j} (1 + f_{ij}) d^{3N}q ]
Would you like me to derive the computational Monte Carlo estimator for (Z_N) (how we numerically compute configurational integrals in practice), or should I keep it purely analytical expansions like cluster/Mayer series?