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Deriving the Rosenfeld functional from the virial expansion
1Institute of Technical Thermodynamics, RWTH Aachen University, Schinkelstraße 8, 52062 Aachen, Germany. stephan.korden@rwth-aachen.de
Summary
This study systematically derives the Rosenfeld functional for hard convex particles using Mayer cluster calculations. The findings extend the virial expansion, offering a new framework for understanding particle interactions and free energy calculations.
Area of Science:
- Statistical Mechanics
- Thermodynamics
- Computational Physics
Background:
- The Rosenfeld functional provides a semiheuristic approach to hard convex particle systems.
- Existing methods lack systematic derivations for complex particle geometries.
Purpose of the Study:
- To replace the semiheuristic derivation of the Rosenfeld functional with a systematic Mayer cluster calculation.
- To extend the virial expansion of free energy by introducing loop order.
- To derive Rosenfeld's weight functions from intersection probability.
Main Methods:
- Systematic calculation of Mayer clusters.
- Decomposition of cluster integrals into intersection pattern diagrams classified by loop number.
- Generalization of the Blaschke, Santalo, and Chern equation for intersection probability.
Main Results:
- The virial expansion is extended by an expansion in loop order, with the Rosenfeld functional as the leading contribution.
- Rosenfeld's weight functions are derived from generalized intersection probability.
- The 0-loop order is exactly derived, reproducing the Rosenfeld functional.
Conclusions:
- A systematic, loop-expansion-based framework is established for hard convex particle systems.
- The derivation provides a rigorous foundation for the Rosenfeld functional.
- The study highlights the impact of particle dimensions, topologies, and geometries on the mathematical structure.
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