Radial Distribution FunctionEdit
Radial Distribution Function
The radial distribution function, often denoted g(r), is a fundamental descriptor in statistical mechanics and condensed-matter physics that characterizes how particle density varies as a function of distance from a reference particle. In homogeneous, isotropic systems such as many liquids and disordered solids, g(r) provides a direct measure of the local structure and the extent of spatial correlations between particles. It is intimately related to the concept of the pair correlation function and serves as a bridge between microscopic configurations and macroscopic observable properties.
In practical terms, g(r) compares the local density at a distance r from a reference particle to the average density ρ of the system. If the system were perfectly uniform, g(r) would be identically equal to 1 for all r. Deviations from unity reveal the presence of short-range order (e.g., nearest-neighbor shells in a liquid) or long-range correlations (as in a crystal). The radial distribution function is thus a probabilistic measure: it quantifies how likely it is to find a particle at a distance r from a given particle, relative to an ideal gas at the same density.
For a finite system, g(r) can be defined rigorously as g(r) = [1/(N ρ 4π r^2 Δr)] ⟨number of pairs with separation in [r, r+Δr]⟩, where N is the number of particles, ρ is the number density, and Δr is a thin distance bin. In the limit of small Δr, this becomes a smooth function that encodes structural information. In mixtures, one often uses partial radial distribution functions g_AB(r) that describe correlations between species A and B.
Definition and mathematical formulation
The radial distribution function is most naturally discussed alongside related correlation concepts. The pair correlation function h(r) is defined by h(r) = g(r) − 1, so that g(r) − 1 measures deviations from ideal-gas behavior. The Fourier transform of h(r) is related to the static structure factor S(k), a quantity directly accessible in scattering experiments: S(k) − 1 = 4πρ ∫_0^∞ [g(r) − 1] r^2 (sin(kr)/(kr)) dr. This relationship underpins how structural information inferred from real-space distributions translates into reciprocal-space signals measured in X-ray, neutron, or electron scattering.
In the classic theory of liquids, the Ornstein–Zernike equation provides a formal link between the total correlation function h(r) and the direct correlation function c(r). In its integral form, h(r) is the convolution of c(r) with h(r) itself, reflecting how local packing and longer-range ordering propagate through the fluid. Solving the Ornstein–Zernike equation with appropriate closure relations yields g(r) for model systems, and these closures (e.g., Percus–Yevick, Hypernetted-Chain) remain central to liquid-state theory. See Ornstein–Zernike equation for more details.
Partial radial distribution functions, g_AB(r), extend the concept to multi-component systems. In mixtures or solutions, g_AA(r), g_BB(r), and g_AB(r) capture how the presence of one species influences the probability of finding another at separation r. These functions are essential for understanding solvent structure around solutes, ionic liquids, and colloidal suspensions.
Calculation and computation
Computing g(r) from microscopic configurations can be done in several ways, with molecular simulations being a prominent approach. In molecular dynamics (MD) or Monte Carlo (MC) simulations, one generates an ensemble of configurations consistent with the underlying Hamiltonian and thermodynamic conditions. For each configuration, one counts pair separations and builds a histogram in radial bins, then averages over the ensemble and normalizes by the ideal gas reference: g(r) ≈ [V/(N^2 4π r^2 Δr)] ⟨∑_{i≠j} δ(r − |ri − rj|)⟩, where V is the system volume, and the brackets denote ensemble averaging. In practice, one uses finite bins and appropriate finite-size corrections, along with corrections for periodic boundary conditions to account for the finite simulation box.
Experimentally, g(r) is not measured directly in real space for most systems. Instead, one infers it from structure-factor measurements S(k) obtained via diffraction techniques. The inverse Fourier transform of S(k) − 1 yields h(r), and from h(r) one recovers g(r) = 1 + h(r). In many liquids and simple solids, the agreement between g(r) computed from simulations and that inferred from scattering data is a primary test of the underlying interaction model and thermodynamic state.
Applications and interpretation
Liquids: In liquids, g(r) often shows a pronounced first peak near the contact distance, reflecting nearest-neighbor coordination. The height and position of this peak reveal the degree of short-range order and packing efficiency. The subsequent minima and shells indicate longer-range correlations and the extent of structuring in the liquid. The integral of g(r) up to a distance r gives the average number of neighbors within that radius (the coordination number).
Solids and crystals: In crystalline solids, g(r) exhibits sharp, well-defined peaks at the lattice spacings corresponding to the crystal structure. The presence and spacing of these peaks encode the symmetry and periodicity of the lattice, while their widths reflect thermal motion and defects. Contrast with liquids highlights how g(r) distinguishes order at long range in crystals from the more limited or decaying order in liquids.
Complex fluids and mixtures: For solutions, polymers, and colloidal suspensions, partial RDFs reveal specific interactions and solvent-mediated effects. For example, in polymer melts, g_AB(r) can illuminate how solvent quality and chain architecture influence local packing around monomers.
Quantum and quantum-statistical systems: In quantum fluids and dense electron-nuclear systems, quantum effects modify g(r), and a quantum radial distribution function or pair distribution function may be defined within appropriate formalisms. This is relevant for dense hydrogen, electron gas models, and certain metallic systems at low temperatures.
Limitations and debates
Completeness of g(r): While g(r) captures two-body spatial correlations, it does not fully characterize all many-body structural information or dynamics. Different microscopic states can share similar g(r) profiles, and higher-order correlations (three-body and beyond) can influence properties such as transport coefficients and phase behavior. Researchers often supplement g(r) with additional metrics, such as angular distribution functions, three-body correlation functions, or time-correlation functions.
Non-equilibrium and anisotropy: g(r) assumes isotropy and, in its simplest form, equilibrium conditions. In driven or anisotropic systems (e.g., under shear, strong external fields, or confinement), one uses generalized or directional correlation measures, such as angle-resolved or time-dependent distribution functions, to capture the full structure.
Finite-size effects and sampling: In simulations, the finite size of the system and the finite sampling time can influence g(r), especially at long distances or in systems with slow dynamics. Careful convergence checks, finite-size extrapolation, and validation against experimental data are standard practices.
Multi-component interpretation: In mixtures, interpreting g_AB(r) requires careful consideration of concentration, interaction asymmetry, and possible preferential binding. The extraction of meaningful thermodynamic quantities from g(r) often relies on combining it with theories of solution structure and equation-of-state models.
See also