Related Experiment Video
Updated: Dec 7, 2025
05:51
Isotopic Effect in Double Proton Transfer Process of Porphycene Investigated by Enhanced QM/MM Method
Published on: July 19, 2019
6.5K
Sensitivity of pair statistics on pair potentials in many-body systems
Haina Wang1, Frank H Stillinger1, Salvatore Torquato2
1Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA.
The Journal of Chemical Physics
|October 2, 2020
Summary
Henderson's theorem in statistical mechanics, crucial for inverse problems, has practical limits. Distinct potentials can yield similar pair correlations, creating ambiguity in determining unique interactions.
Area of Science:
- Classical statistical mechanics
- Condensed matter physics
- Computational physics
Background:
- Henderson's theorem posits a unique pair potential for a given pair correlation function.
- The theorem is foundational for inverse-problem studies in statistical mechanics.
- Quantitative scrutiny of the theorem's practical utility is limited.
Purpose of the Study:
- To quantitatively assess the sensitivity and practicality of Henderson's theorem.
- To identify conditions where the theorem's uniqueness claim may fail.
- To explore the implications for inverse-problem solutions in statistical mechanics.
Main Methods:
- Development and application of sensitivity metrics.
- Analysis of pair correlation functions and structure factors.
- Theoretical extension of Henderson's theorem.
Main Results:
- Henderson's theorem exhibits practical shortcomings for certain densities and temperatures in both disordered and ordered phases.
- Distinct potential functions can produce nearly identical pair correlation functions and structure factors within their correlation lengths.
- Ambiguity arises in inverse problems relying solely on pair information due to limited range and precision.
- Systems with similar pair statistics can exhibit different higher-order correlations and dynamics.
Conclusions:
- Caution is advised when claiming uniqueness for effective pair potentials derived from inverse problems.
- The theorem's practical applicability is constrained by data limitations.
- A generalized Henderson's theorem is proven, extending uniqueness to multi-body interactions.
Related Concept Videos
03:06The Pauli Exclusion Principle
57.9K
The arrangement of electrons in the orbitals of an atom is called its electron configuration. We describe an electron configuration with a symbol that contains three pieces of information:
57.9K
04:01VSEPR Theory and the Effect of Lone Pairs
51.0K
Effect of Lone Pairs of Electrons on Molecule Geometry
51.0K
01:02¹H NMR: Interpreting Distorted and Overlapping Signals
1.3K
Spin systems where the difference in chemical shifts of the coupled nuclei is greater than ten times J are called first-order spin systems. These nuclei are weakly coupled, and their chemical shifts and coupling constant can generally be estimated from the well-separated signals in the spectrum.
As Δν decreases and the signals move closer, the doublets appear increasingly distorted. The intensities of the inner lines increase at the cost of those of the outer lines as the signals are...
As Δν decreases and the signals move closer, the doublets appear increasingly distorted. The intensities of the inner lines increase at the cost of those of the outer lines as the signals are...
1.3K
01:20Spin–Spin Coupling: Two-Bond Coupling (Geminal Coupling)
1.4K
Two NMR-active nuclei bonded to a central atom can be involved in geminal or two-bond coupling. Geminal coupling is commonly seen between diastereotopic protons in chiral molecules and unsymmetrical alkenes, among others.
The central atom need not be NMR-active because its electrons are affected by the electron polarization of the spin-active atoms. However, spin information is transmitted less effectively than in one-bond coupling, and 2J values are usually weaker than 1J values. The energy of...
The central atom need not be NMR-active because its electrons are affected by the electron polarization of the spin-active atoms. However, spin information is transmitted less effectively than in one-bond coupling, and 2J values are usually weaker than 1J values. The energy of...
1.4K
01:14Atomic Nuclei: Nuclear Spin State Population Distribution
2.2K
Near absolute zero temperatures, in the presence of a magnetic field, the majority of nuclei prefer the lower energy spin-up state to the higher energy spin-down state. As temperatures increase, the energy from thermal collisions distributes the spins more equally between the two states. The Boltzmann distribution equation gives the ratio of the number of spins predicted in the spin −½ (N−) and spin +½ (N+) states.
2.2K
01:08NMR Spectroscopy: Spin–Spin Coupling
2.7K
The spin state of an NMR-active nucleus can have a slight effect on its immediate electronic environment. This effect propagates through the intervening bonds and affects the electronic environments of NMR-active nuclei up to three bonds away; occasionally, even farther. This phenomenon is called spin–spin coupling or J-coupling. Coupling interactions are mutual and result in small changes in the absorption frequencies of both nuclei involved. While nuclei of the same element are involved...
2.7K