Simple liquids' quasiuniversality and the hard-sphere paradigm

Depending on the temperature T and pressure p, ordinary matter is found in one of three phases: solid, liquid, or gas [15–17]. A generic temperature-pressure phase diagram is shown in figure 1(a). The gas phase is found at high temperatures and low-to-moderate pressures, the solid (crystalline) phase at low temperatures and/or high pressures. The liquid phase is located in between. It is possible to move continuously from the liquid to the gas phase by circumventing the critical point (red) [2]. Consistent with this the liquid and gas phases have the same symmetries, namely translational invariance and isotropy, i.e. all positions and directions in space are equivalent. The solid phase is crystalline and breaks both these symmetries.

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It is often convenient to use instead a $\rho \,T$ phase diagram where $\rho \equiv N/V$ is the density of particles, i.e. their number N per volume V (figure 1(b)). In this diagram the liquid/solid and gas/liquid phase transitions give rise to regions of coexistence, which in the $Tp$ phase diagram collapse to the melting and vaporization lines respectively.

There are two special state points in a thermodynamic phase diagram: the triple point where all three phases coexist and the critical point (blue and red in figure 1). Below the triple-point temperature the liquid phase does not exist; above the critical-point temperature the gas and liquid phases merge. Close to the triple point the repulsive forces dominate the interactions, close to the critical point the attractive forces dominate [5]. Sometimes all states with temperature above the critical temperature are referred to as 'fluid' states. We do not make this distinction, however [18], because it is incompatible with the observation of invariant structure and dynamics along the freezing line [19]. Only the term 'liquid' is used below, by which is implied the condensed phase far from the gas phase in the traditional sense of this term (the colored regions in figure 1).

From daily life we are used to water and many other liquids, but throughout the universe the liquid state is actually quite rare. This reflects the fact that a liquid's existence depends on a delicate balance between attractive intermolecular interactions causing condensation and entropic forces preventing crystallization [20].

While van der Waals and followers emphasized the liquid–gas analogy, in his seminal book written during World War II the Russian physicist Frenkel [21] emphasized the solid–liquid resemblance. In honor of this it has been suggested to term the (blurred) line delimiting the genuine liquid phase from the more gas-like phase the 'Frenkel line' [13, 14, 22] (see, however, also [23]). This Topical Review focuses on the genuine liquid phase between the Frenkel line and the freezing line (colored in figure 1), defining the 'ordinary' liquid phase for which properties like density, energy, specific heat, compressibility, heat conductivity, etc, are closer to their solid than to their gas values [14, 22, 24].

In order to put simple liquid's quasiuniversality into perspective it is useful to recall a few well-known universalities.

The first example is the ideal-gas equation, $pV=N{{k}_{\text{B}}}T$, which applies for all dilute gases (p is the pressure, ${{k}_{\text{B}}}$ Boltzmann's constant, and T the temperature). This equation is cherished by physicists for its mathematical simplicity and general validity. In contrast, chemists typically regard it as useful, but not very informative—precisely because it is universal and does not relate to molecular details. Universality is also found close to the critical point, at which many properties conform to critical power-law scaling [25, 26]. For instance, at the critical density and close to the critical temperature T_c the compressibility diverges as $|T-{{T}_{\text{c}}}{{|}^{-\gamma}}$ with the universal exponent $\gamma =1.24$ [27]. A third example is that low-temperature crystals are well described by the harmonic approximation, leading to the well-known Debye universality of the specific heat's temperature dependence ($\propto {{T}^{\,3}}$) [28–30].

Universalities arise when things simplify in some limit, a limit that is usually quantified by a small dimensionless number λ. For a gas λ is the ratio between molecule size and average interparticle distance, and the ideal-gas equation becomes exact as $\lambda \to 0$. At the critical point $\lambda =\,|T-{{T}_{\text{c}}}|/{{T}_{\text{c}}}$, and the scaling laws become exact as $\lambda \to 0$. For a crystal λ is the ratio between nearest-neighbor distance and phonon mean-free path, and the harmonic description is exact in the $\lambda \to 0$ limit.

The 'ordinary' liquid phase excludes the critical point and exhibits no obvious asymptotically exact universalities. This reflects the fact that liquids, like atomic nuclei [31], are so-called strongly-coupled systems, i.e. with no obvious small dimensionless parameter λ to expand in. There are nevertheless intriguing, but approximate universalities in the liquid phase of many systems. Thus liquids with quite different interparticle interactions have often virtually the same structure and dynamics. In the literature the term 'quasiuniversality' has no clear definition, and until section 4.5 we follow the tradition of implying by a 'quasiuniversal liquid' a member of the large class of simple model systems with virtually the same structure and dynamics, which includes the HS and Lennard-Jones systems. It is important to keep in mind, though, that simple liquids' quasiuniversality is not generally valid, it is not exact in some obvious limit, and it applies only in part of the thermodynamic phase diagram.

Insights into the physics of liquids have often come from computer simulations [1]. Actually, some of the very first scientific computer simulations back in the 1950s studied the hard-sphere (HS) liquid (section 3). An early surprise was that the HS system has a first-order freezing transition [32].

The simplest liquid models are defined by pair potentials, v (r). If ${{r}_{ij}}=\,|{{\mathbf{r}}_{i}}-{{\mathbf{r}}_{j}}|$ is the distance between particles i and j, the potential energy U as a function of all particle coordinates is given by

$$\begin{eqnarray}&&U\left({{\mathbf{r}}_{1}},...,{{\mathbf{r}}_{N}}\right)=\underset{i<j}{\sum}\,v\,\left({{r}_{ij}}\right).\end{eqnarray} \tag{ 1 }$$

Such systems are traditionally referred to as 'simple liquids' [1, 9–12] because they describe point particles with isotropic interactions, i.e. with a mathematically simple Hamiltonian.

Recent developments have put simple liquids' structure and dynamics into focus again, and supplementing computer simulations different pair potentials can now also be studied experimentally, e.g. in carefully designed colloids [33, 34]. While most simple liquids conform to quasiuniversality (section 4), some pair-potential systems have unusual behavior. For instance, some such systems have a diffusion constant that increases upon isothermal compression and some of them melt instead of freeze upon compression [35–37]. Both features are anomalous and found, e.g. for water that is perhaps the most complex of all liquids. These findings call into question the name 'simple' for all pair-potential systems [38], but in order to avoid confusion we shall here nevertheless stick to the traditional definition of a simple liquid.

In many simulations one uses ad hoc pair potentials. The most famous one is the Lennard-Jones (LJ) [39] pair potential from 1924 defined by

$$\begin{eqnarray}&&{{v}_{\text{LJ}}}(r)\,=\,4\varepsilon \left[{{\left(\frac{r}{\sigma}\right)}^{-12}}-{{\left(\frac{r}{\sigma}\right)}^{-6}}\right].\end{eqnarray} \tag{ 2 }$$

The minimum value of ${{v}_{\text{LJ}}}(r)$ is $-\varepsilon$, which is found at $r={{2}^{1/6}}\sigma$ (compare figure 2). Commonly simulated simple liquid models are: the LJ pair potential (sometimes cut off at the minimum [40], sometimes with other exponents than 12 and 6), the purely repulsive inverse-power law (IPL) pair potentials $v(r)\propto {{(r/\sigma )}^{-n}}$ [41, 42], the Yukawa 'screened Coulomb' pair potential $v(r)\propto \exp (-r/\sigma )/r$ [43–45], the Morse pair potential that is a difference of two exponentials [46, 47], and the HS pair potential (section 3).

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The standard computer simulation method termed 'molecular dynamics' solves Newton's equations of motion numerically. If ${{\mathbf{F}}_{i}}=-{{\nabla}_{i}}U$ is the force on particle i with mass m_i, molecular dynamics traces out particle trajectories by solving a time-discretized version of Newton's second law, ${{m}_{i}}{{\ddot{{\mathbf{r}}}\,}_{i}}={{\mathbf{F}}_{i}}$ [1, 48–50]. Typical time steps are of order one femtosecond (10⁻¹⁵ s) in real units, so a sizable amount of computing power is required for getting accurate results (in particular for highly viscous liquids). For instance, simulating 1000 LJ particles for the equivalent of one millisecond requires about 10¹² time steps, which takes more than a year on a state-of-the art GPU-based computer [51].

Newtonian dynamics conserves the energy E. When considered at constant volume V and particle number N, this is referred to as NVE dynamics. Nowadays it is more common to use NVT dynamics, which maintains a predefined temperature by modifying Newton's second law in a clever way [48, 49, 52]. For most quantities, including spatial and time autocorrelation functions, this method gives the same results as for NVE simulations. It is also possible to use NVU dynamics, which traces out a geodesic curve on the constant-potential-energy hypersurface [53], leading to the same results as NVE or NVT dynamics [54]. Constant-pressure (NPT) dynamics is an option that is popular, in particular, among chemists and material scientists because experiments usually take place at ambient pressure.

The above dynamics are all deterministic, i.e. once set into motion the system's path is in principle uniquely determined. An alternative is the so-called Brownian or Langevin stochastic dynamics. For details about such dynamics the reader is referred to the literature [48]; we here just note that deterministic and stochastic dynamics give the same static—and in most cases also very similar dynamic—properties [54–56].

Two isolated, uncharged atoms or molecules do not interact when they are far from each other. Approaching one another, a regime of weak attractions is entered caused by the so-called van der Waals forces [57]. These basically reflect the fact that electronic wave functions can lower their energy by spreading over a larger volume. When the distance becomes comparable to the atomic/molecular size, strong repulsions appear deriving ultimately from the Fermi statistics of the electrons. This is why it is almost impossible to compress a liquid or a solid.

The idea of classifying forces into repulsive and attractive goes back in time at least to 1755 in which the famous philosopher Kant proposed that, fundamentally, there are only these two kinds of forces with which one point of matter can impress motion on another [58]. In 1873 van der Waals transformed this picture into a quantitative theory by basing his equation of state upon it [2, 16]. The modern van der Waals picture may be summarized as follows [1–8, 59]. The strongly repulsive forces imply that a liquid's atoms or molecules cannot approach one another below a certain distance, a distance that depends slightly on the temperature because at high temperature more kinetic energy is available for overcoming the repulsive forces. These forces thus give rise to severe constraints on the particles' motion. The attractive forces, on the other hand, do not really have any effect on structure and dynamics because they merely contribute a negative, density-dependent but spatially virtually constant potential. As stated eloquently by Hansen and McDonald [1]: '...the structure of most simple liquids, at least at high density, is largely determined by the way in which the molecular hard cores pack together. By contrast, the attractive interactions may, in a first approximation, be regarded as giving rise to a uniform background potential that provides the cohesive energy of the liquid but has little effect on its structure. A further plausible approximation consists in modeling the short-range forces by the infinitely steep repulsion of the hard-sphere potential'. The van der Waals picture works best for liquids of atoms or roughly spherical molecules that are not bonded to each other by covalent or hydrogen bonds [1], the so-called nonassociated liquids. In the words of Chandler [60], the intermolecular structure of a nonassociated liquid 'can be understood in terms of packing. There are no highly specific interactions in these systems'. In contrast, water is an example of an associated liquid, and its 'linear hydrogen bonding tends to produce a local tetrahedral ordering that is distinct from what would be predicted by only considering the size and shape of the molecule'.

What is henceforth referred to as the hard-sphere paradigm is the assumption that, to a good approximation, any simple liquid with strongly repulsive forces may be modeled as a system of hard spheres. This idea has dominated liquid-state research for many years, in particular after the 1960s when it first got convincing support from computer simulations. Extensive research since then has demonstrated that, based on the HS system as the zeroth-order approximation, it is possible to construct highly successful perturbation theories for, e.g. a liquid's thermodynamics and pair distribution function (the probability to find two particles a certain distance apart relative to that for a system of random particles at the same density) [1, 7, 40, 61–65].

Figure 3 shows coexisting liquid and solid phases of the LJ system (equation (2)). Each LJ particle is represented by a sphere, i.e. as a HS particle defined by having infinite pair potential when the pair distance r obeys $r<2{{\sigma}_{\text{HS}}}$ (${{\sigma}_{\text{HS}}}$ is the particle radius) and zero potential otherwise. This representation makes good sense, because the quasiuniversality of structure discussed below implies that the structures of the LJ and HS systems are virtually identical.

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HS particles have no rotational degrees of freedom and move in straight lines with constant velocity until they collide elastically, i.e. by conserving momentum and (kinetic) energy. At low density the HS system is gas like, at higher density it is liquid like, and at the highest densities it is crystalline [67, 68]. Using instead Brownian (stochastic) dynamics [48], which neither conserves momentum nor energy, leads to identical results for the static distribution functions and to very similar time-autocorrelation functions [56].

For a system of N HS particles in volume V one defines the packing fraction ϕ as the occupied fraction of the total volume, i.e.

$$\begin{eqnarray}&&\phi \,\equiv \,N\,\frac{4\pi \sigma _{\text{HS}}^{3}}{3V}.\end{eqnarray} \tag{ 3 }$$

The largest possible packing fraction is that of the face-centered cubic crystal in which $\phi =\pi \sqrt{2}/6=0.740$, a result known as 'Kepler's theorem' [69]. For many years this was 'known by all physicists and greengrocers, and conjectured to be true by most mathematicians...', but eventually Kepler's theorem was proven rigorously [70].

Above the packing fraction 0.492 the equilibrium HS system is partly crystalline, and for $\phi >0.545$ it is fully crystallized [33, 71, 72]. By applying a finite rate of compression it is possible to overcompress the liquid HS system to higher packing fractions than 0.545, just as real-life liquids may be supercooled below their freezing point [73–78]. In the metastable, overcompressed state the HS system's viscosity increases dramatically as a function of density, and the system eventually jams into a glass phase in which only localized vibrational-type motion survives [69], [79–81].

Like the ideal-gas model or the Ising model for magnetism, the HS model is widely regarded as the idealized model liquid [1, 68]. In mathematics, the HS system has been studied for centuries. More recently, the problem of packing spheres densely in d dimensions became of interest to computer scientists because of its importance for coding theory in communication [82]. In physics, the HS model is receiving renewed interest these years [69, 83, 84]; for instance it was recently solved exactly in infinite dimensions in a tour-de-force replica calculation [81]. The HS model is also the generic model for studies of jamming [69, 79, 83, 85], a field in which the last decade has brought significant advances based, in particular, on the notions of isostaticity and marginal stability [80], [86–90].

According to the HS paradigm any simple liquid with strongly repulsive forces corresponds to a HS system. This implies that any two such liquids must have very similar properties at two thermodynamic state points that map onto HS systems with the same packing fraction. We refer to this as the HS explanation of quasiuniversality. But what is meant by 'very similar properties'? In the present context this means very similar structure and dynamics when these are made dimensionless using proper units; in contrast, the thermodynamic equations of state connecting temperature, density, and pressure generally differ significantly among quasiuniversal systems.

Which unit system should be used for characterizing quasiuniversality? Generalizing equation (2) any potential energy function $U\left({{\mathbf{r}}_{1}},...,{{\mathbf{r}}_{N}}\right)\equiv U\left(\mathbf{R}\right)$ can be expressed in terms of an energy ε, a distance σ, and a dimensionless function $\rm{\tilde \Psi}$, as follows

$$\begin{eqnarray}&&U\left(\mathbf{R}\right)\,=\,\varepsilon \,\rm{\tilde \Psi}\left(\mathbf{R}/\sigma \right).\end{eqnarray} \tag{ 4 }$$

A 'microscopic' unit system may be defined based on the energy unit ε, length unit σ, and time unit $\sigma \sqrt{m/\varepsilon}$ (m is the particle mass). Is this the right unit system to use in tests for quasiuniversality? Perhaps surprisingly, even though results of computer simulations are usually reported in these microscopic units, the answer is no.

Consider a simple liquid for which the two thermodynamic state points $\left({{\rho}_{1}},{{T}_{1}}\right)$ and $\left({{\rho}_{2}},{{T}_{2}}\right)$ are well modeled by HS systems with the same packing fraction. According to the HS paradigm and the van der Waals picture these two state points have the same reduced pair distribution functions $g\left(\tilde{r}\right)$. In particular, these function's maxima must be at the same value of $\tilde{r}$. At a state point with density ρ the maximum nearest-neighbor distance scales with density as $\propto {{\rho}^{-1/3}}$. Thus the proper length unit to use is that defined by the density: ${{l}_{0}}={{\rho}^{-1/3}}$. Likewise, if the reduced velocity time-autocorrelation function $\langle \tilde{v}(0)\tilde{v}\left(\tilde{t}\right)\rangle$ is to be invariant, this must apply in particular to its zero-time value, $\langle {{\tilde{v}}^{2}}\rangle$. Since the kinetic energy scales in proportion to v² and since $\langle {{v}^{2}}\rangle \propto {{k}_{\text{B}}}T$ by the equipartition theorem, the proper energy unit must be the thermal energy ${{k}_{\text{B}}}T$. In turn, via the particle mass and thermal average velocity this leads to the time unit ${{\rho}^{-1/3}}\sqrt{m/{{k}_{\text{B}}}T}$. In summary, the HS paradigm implies that the correct units to use in relation to quasiuniversality are the following 'macroscopic' energy, length, and time units:

$$\begin{eqnarray}&&{{e}_{0}}={{k}_{\text{B}}}T,\,{{l}_{0}}={{\rho}^{-1/3}},\,{{t}_{0}}={{\rho}^{-1/3}}\sqrt{m/{{k}_{\text{B}}}T}.\end{eqnarray} \tag{ 5 }$$

Note that these units are experimentally assessable, but state-point dependent. In contrast, the microscopic units ε and σ are state-point independent, but not experimentally assessable—in fact, by redefining the function $\rm{\tilde \Psi}$ in equation (4) they can be based on any fixed energy and length.

Henceforth, whenever a quantity is referred to as 'reduced', it has been made dimensionless using the macroscopic units of equation (5). Reduced quantities are denoted by a tilde. For instance, if D is the diffusion constant, $\tilde{{D}}\,\equiv D/\left(l_{0}^{2}/{{t}_{0}}\right)=D{{\rho}^{1/3}}\sqrt{m/{{k}_{\text{B}}}T}$.

The simplest structural characteristic is the density spatial autocorrelation function. By neutron or x-ray scattering experiments one measures its Fourier transform, the so-called coherent scattering function $S\left(\mathbf{k}\right)$. If $\mathbf{k}$ is the scattering vector and the system's N particles are located at the positions ${{\mathbf{r}}_{1}},...,{{\mathbf{r}}_{N}}$, the coherent scattering function is defined by $S\left(\mathbf{k}\right)=\langle |{\sum}_{j}\exp \left(\text{i}\mathbf{k}\centerdot {{\mathbf{r}}_{j}}\right){{|}^{2}}\rangle /N$ in which the sharp brackets denote a thermal average [1]. A crystal's $S\left(\mathbf{k}\right)$ has pronounced peaks at the reciprocal lattice vectors [29, 30], whereas for a liquid the peaks are broader and $S\left(\mathbf{k}\right)$ is independent of the direction of $\mathbf{k}$.

Figure 4(a) shows static structure factors of the HS (dots) and LJ (full curve) liquids based on almost 50 years old computer simulations [92]. The striking finding already at that time was that these quite different systems have very similar structure. Figure 4(b) shows the structure factor of IPL systems defined by $v(r)\propto {{r}^{-n}}$ with exponents n ranging from 10 to 36, as well as that of the HS system (corresponding to the $n\to \infty$ limit), all simulated close to freezing. Again there is approximate identity. Experiments show that simple liquids like molten metals [94] and inactive gases [95] have structure factors close to those of the HS system.

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Experimental signatures of dynamic quasiuniversality have been obtained for molten metals and inactive gases. In the latter case, the half width of the coherent inelastic scattering factor follows the prediction of the HS model solved in the so-called Enskog theory [97]. The dynamics of liquid Gallium likewise follows the Enskog prediction [98]. Experimental data for the self-diffusion constant and viscosity of certain small-molecule liquids like methane and ethane have also been interpreted successfully in terms of the HS model [99], thus establishing experimental quasiuniversality for the dynamics.

Computer simulations have documented dynamic quasiuniversality in greater detail [42, 56, 96], [100–102]. Figure 5(a) shows the inverse diffusion constant as a function of the HS packing fraction ${{\phi}_{\text{HS}}}$ determined by fitting the given system's static structure factor to that of a HS system. The figure shows data for different LJ-type systems simulated with both Newtonian and Brownian dynamics. Figure 5(b) shows similar results, including here also experimental data for a clearly non-quasiuniversal pair-potential system. Figure 5(c) shows the product of diffusion constant and viscosity for IPL systems ($v(r)\propto {{r}^{-n}}$) plotted as a function of 1/n; the lower data set refers to this product in reduced units, demonstrating quasiuniversality.

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One defines the so-called excess entropy of a system as the entropy minus that of an ideal gas at the same density and temperature (appendix). Note that since the ideal gas is maximally disordered, the excess entropy is negative. In 1977 Rosenfeld pointed out that the excess entropy of a simple liquid, S_ex, in simulations determines the reduced diffusion constant and the reduced viscosity [103]. He justified this by arguing as follows. A thermodynamic state point of the HS system is characterized by the packing fraction ϕ (temperature merely scales the particle velocities). In particular, ϕ determines S_ex. Thus if two systems at two state points are well described by HS systems with the same packing fraction, they have the same excess entropy. For such systems the excess entropy determines the reduced-unit dynamics in a quasiuniversal way. By reference to Enskog theory Rosenfeld in 1999 interestingly managed to extend excess entropy scaling to the gas phase [104], but our focus below remains on the condensed 'ordinary' liquid phase.

Excess-entropy scaling did not receive a great deal of attention until about the year 2000, although a closely related two-particle entropy scaling was discussed briefly in 1996 by Dzugutov [108]. By now many systems, primarily in computer simulations, have been shown to conform to excess-entropy scaling [109–112]. Examples are given in figure 6, which in (a) shows numerical data for several systems. This figure also presents data for three exotic systems not obeying excess-entropy scaling revealed as an absence of data collapse; the two left-hand figures demonstrate quasiuniversal excess-entropy scaling for IPL systems. Other simple liquids like the LJ or the Yukawa (screened Coulomb) systems have reduced diffusion constants that fall virtually on top of the IPL data when plotted as functions of the excess entropy [103]. Figure 6(b) shows the reduced viscosity and diffusion constants for IPL systems as a function of the two-particle entropy, a quantity that gives the dominant contribution to S_ex. Figure 6(c) presents experimental data for nitrogen at various temperatures and pressures.

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Important features of simple liquids' thermodynamics are not quasiuniversal, for instance the equation of state connecting temperature, pressure, and density. There are some thermodynamic quasiuniversalities, however. One of these relates to the constant-volume specific heat C_V's temperature variation. As noted by Rosenfeld and Tarazona [103, 114], simple liquids obey ${{C}_{V}}\propto {{T}^{-2/5}}$ to a good approximation at constant density. Figure 7 gives computer simulation data confirming this. While the figure also includes simulation data for non-simple liquids like water, inspection reveals that the Rosenfeld–Tarazona relation works best for the LJ and IPL systems.

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Interesting thermodynamic quasiuniversalities have been reported going beyond the condensed liquid phase in focus here. These involve, e.g. the liquid-vapor coexistence region [115, 116], the so-called Zeno line [117], the behavior of supercritical fluids [118], and the existence of generalized corresponding-states equations [116, 119].

Quasiuniversality applies also for crystallization and melting [120–127]. Figures 8(a) and (b) show data for the structure factor of iron at ambient and high pressure along the freezing line. Invariance of the structure along the freezing line is demonstrated in (b) reporting reduced-unit structure factor data for iron at freezing up to a pressure of 58 GPa. This case is important because the Earth's core consists largely of molten iron [128–130]. Figure 8(c) gives the reduced viscosity of fifteen metallic elements plotted as a function of melting temperature over temperature. Clearly these metals behave very similarly. In particular, the reduced viscosity is quasiuniversal at freezing [131–133].

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Freezing and melting of simple systems are characterized by several quasiuniversal properties [123, 127], for instance the following:

• 1.  
  A crystal melts when the thermal vibrational amplitude is about 10% of the nearest-neighbor distance [120, 121, 123, 136, 137] (the Lindemann melting criterion from 1910).
• 2.  
  A liquid freezes when the maximum of its structure factor exceeds a quasiuniversal value close to, but slightly below 3 [138] (the Hansen–Verlet criterion).
• 3.  
  At freezing the ratio between the minimum and the maximum of the liquid's radial distribution function is $=0.20\pm 0.02$ [139] (the Raveche–Mountain–Streett criterion). Likewise, the ratio between the magnitudes of the structure factor's first and second peak is quasiuniversal at melting of crystals [140].
• 4.  
  For a liquid of particles subject to Brownian motion the ratio between the long- and short-time diffusion constants is about 0.1 at freezing [141, 142] (the Löwen–Palberg–Simon criterion).
• 5.  
  At freezing a liquid's C_V is close to 3k_B per particle [14, 24, 143].
• 6.  
  The constant-volume melting entropy is close to $0.8{{k}_{\text{B}}}$ per particle [123, 144].

The HS paradigm makes it possible to give a precise definition of quasiuniversality. Consider two systems at two thermodynamic state points. If the systems are here described by HS systems with the same packing fraction, all reduced-unit quantities referring to structure and dynamics of the two systems are identical to a good approximation. In practice one does not know the HS packing fraction, but this inspires to the following characterization of quasiuniversality:

• If two systems have in common a single reduced-unit quantity characterizing structure or dynamics at two—possibly different—thermodynamic state points, all other reduced-unit structural or dynamical quantities at these state points are also identical to a good approximation.

The above property defines an equivalence relation among liquids. There could be more than a single equivalence class (which we in section 8 suggest is the case for mixtures), but the focus below is on the large equivalence class of single-component systems that includes the HS system, the LJ and related systems, the IPL systems, and the Yukawa system. As we shall see, this class also includes the purely repulsive exponential pair potential, as the most prominent member in a certain sense.

The $\text{EXP}$ potential is the purely repulsive pair potential defined by

$$\begin{eqnarray}&&{{v}_{\text{EXP}}}(r)\,=\,\varepsilon \,{{\text{e}}^{-r/\sigma}}.\end{eqnarray} \tag{ 6 }$$

Born and Meyer already in 1932 used an exponentially repulsive term in a pair potential [161], justifying it from the fact—at that time recently established—that electronic bound-state wavefunctions decay exponentially in space. There are only few studies of the pure $\text{EXP}$ system in the literature; usually an $\text{EXP}$ term appears in tandem with an r⁻⁶ attractive term [161, 162] or multiplied by a 1/r term as in the Yukawa 'screened Coulomb' pair potential [43, 163, 164]. Kac and coworkers used a HS pair potential minus a long-ranged $\text{EXP}$ term for rigorously deriving the van der Waals equation of state in one dimension [165].

Like any other purely repulsive system, the $\text{EXP}$ system has a well-defined freezing transition, but no liquid-vapor transition—there is just a single fluid phase. The density-temperature phase diagram of the $\text{EXP}$ system is shown in figure 9(a), in which the full black line covers the solid–liquid coexistence region. The colors reflect the virial potential-energy (Pearson) correlation coefficient R defined by

$$\begin{eqnarray}&&R=\frac{\langle \Delta W\Delta U\rangle}{\sqrt{\langle {{(\Delta U)}^{2}}\rangle \langle {{(\Delta W)}^{2}}\rangle}}.\end{eqnarray} \tag{ 7 }$$

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Here $\Delta$ denotes deviation from the equilibrium value and the sharp brackets are constant-volume canonical averages; recalling that $\mathbf{R}\equiv \left({{\mathbf{r}}_{1}},...,{{\mathbf{r}}_{N}}\right)$ is the configuration vector and $U\left(\mathbf{R}\right)$ the potential energy, the virial function is defined by $W\left(\mathbf{R}\right)=-(1/3)\mathbf{R}\centerdot \nabla U\left(\mathbf{R}\right)$ (the average virial W determines the pressure via the identity $pV=N{{k}_{\text{B}}}T+W$ [1, 48, 167]).

We see from figure 9(a) that the $\text{EXP}$ system has strong WU correlations in the low-temperature part of its phase diagram. This is where ${{v}_{\text{EXP}}}(0)=\varepsilon \gg {{k}_{\text{B}}}T$, implying that the finite value of ${{v}_{\text{EXP}}}(0)$ is of no significance because zero particle separation is highly unlikely. Systems with R  >  0.9 were initially referred to as 'strongly correlating' [168], but that term was repeatedly confused with strongly correlated quantum systems and instead the term R (Roskilde) simple is now sometimes used [107], [169–176]. As shown in the next section, R systems are simple because in regard to structure and dynamics they have, just like the HS system, an essentially one-dimensional thermodynamic phase diagram.

The '$\text{EXP}$ explanation' of quasiuniversality given in section 6.3 below makes use of the theory of isomorphs [38, 166]. Independent of quasiuniversality, isomorphs are lines in the thermodynamic phase diagram of certain systems along which structure and dynamics in reduced units are invariant to a good approximation. Real-world liquids and solids with isomorphs are believed to include most metals and van der Waals bonded molecular systems, but exclude most covalently or hydrogen-bonded systems [19, 177].

An isomorph is defined as a configurational adiabat, i.e. a curve with ${{S}_{\text{ex}}}=\text{Const}.$, for any system characterized by the following scaling property [178]

$$\begin{eqnarray}&&U\left({{\mathbf{R}}_{\text{a}}}\right)<U\left({{\mathbf{R}}_{\text{b}}}\right)\,\,\Rightarrow \,\,U\left(\lambda {{\mathbf{R}}_{\text{a}}}\right)<U\left(\lambda {{\mathbf{R}}_{\text{b}}}\right).\end{eqnarray} \tag{ 8 }$$

Only for systems with a potential-energy that is a constant plus an Euler-homogeneous function is the above 'hidden-scale-invariance' identity rigorously obeyed. For the isomorph theory to work to a good approximation it is enough, however, that equation (8) applies for most configurations ${{\mathbf{R}}_{\text{a}}}$ and ${{\mathbf{R}}_{\text{b}}}$, i.e. for most of the typical configurations.

Equation (8) is mathematically equivalent to the conformal-invariance condition $U\left({{\mathbf{R}}_{\text{a}}}\right)=U\left({{\mathbf{R}}_{\text{b}}}\right)\Leftrightarrow U\left(\lambda {{\mathbf{R}}_{\text{a}}}\right)=U\left(\lambda {{\mathbf{R}}_{\text{b}}}\right)$, which by differentiation with respect to λ implies for the virial function $W\left({{\mathbf{R}}_{\text{a}}}\right)=W\left({{\mathbf{R}}_{\text{b}}}\right)$. Thus same potential energy implies same virial to a good approximation, so equation (8) implies strong correlations between the equilibrium fluctuations of W and U and that the correlation coefficient R of equation (7) is close to unity. Thus only R simple systems have isomorphs.

Figure 9(b) validates equation (8) for the $\text{EXP}$ system by showing how properly normalized potential energies of 20 selected configurations change under uniform scaling—the figure is constructed such that whenever the curves do not cross, equation (8) applies. Figure 9(c) shows how the virial and potential energy of the $\text{EXP}$ system fluctuate strongly correlated over the course of time. In summary, figure 9 shows that $\text{EXP}$ system is R simple in the low-temperature part of the phase diagram [160].

What are the characteristic properties of R systems? Following the Landau and Lifshitz theory of entropy fluctuations [179], one can define a microscopic excess entropy function by ${{S}_{\text{ex}}}\left(\mathbf{R}\right)\equiv {{S}_{\text{ex}}}(\rho,U){{|}_{U=U\left(\mathbf{R}\right)}}$ in which ${{S}_{\text{ex}}}(\rho,U)$ is the thermodynamic excess entropy of the state point with density ρ and average potential energy U [178]. By inversion, the potential-energy function obeys $U\left(\mathbf{R}\right)=U\left(\rho,{{S}_{\text{ex}}}\left(\mathbf{R}\right)\right)$ in which $U\left(\rho,{{S}_{\text{ex}}}\right)$ is the thermodynamic average potential energy as a function of density and excess entropy.

The above definition is completely general. Consider now a system obeying equation (8) and suppose that ${{\mathbf{R}}_{1}}$ is a configuration of density ${{\rho}_{1}}$ with the same reduced coordinate as ${{\mathbf{R}}_{2}}$, a configuration of density ${{\rho}_{2}}$, i.e. $\rho _{1}^{1/3}{{\mathbf{R}}_{1}}=\rho _{2}^{1/3}{{\mathbf{R}}_{2}}$ (compare the definition of reduced units equation (5)). It follows from the microcanonical expression for the excess entropy (appendix) that if 'Vol' is the reduced-coordinate configuration-space volume, one has ${{S}_{\text{ex}}}\left({{\mathbf{R}}_{1}}\right)/{{k}_{\text{B}}}=-N\ln N+\ln \left(\text{Vol}\left\{{{\mathbf{\tilde{R}}}^{\prime}}\,|\,U\left(\rho _{1}^{-1/3}{{\mathbf{\tilde{R}}}^{\prime}}\right)<U\left({{\mathbf{R}}_{1}}\right)\right\}\right)$ and ${{S}_{\text{ex}}}\left({{\mathbf{R}}_{2}}\right)/{{k}_{\text{B}}}=-N\ln N+\ln \left(\text{Vol}\left\{{{\tilde{{\mathbf{R}}}\,}^{\prime}}\,|\,U\left(\rho _{2}^{-1/3}{{\tilde{{\mathbf{R}}}\,}^{\prime}}\right)<U\left({{\mathbf{R}}_{2}}\right)\right\}\right)$. Applying $\lambda =\rho _{2}^{-1/3}\rho _{1}^{1/3}$ to the inequality of the first set, we see that the two sets are identical since equation (8) works 'both ways'. This means that ${{S}_{\text{ex}}}\left(\mathbf{R}\right)$ depends only on the configuration's reduced coordinate $\mathbf{\tilde{R}}\equiv {{\rho}^{1/3}}\mathbf{R}$, implying that

$$\begin{eqnarray}&&U\left(\mathbf{R}\right)=U\left(\rho,{{S}_{\text{ex}}}\left(\mathbf{\tilde{R}}\right)\right).\end{eqnarray} \tag{ 9 }$$

Since the thermodynamic excess entropy S_ex at any given state point is the average of the microscopic excess entropy function, ${{S}_{\text{ex}}}=\langle {{S}_{\text{ex}}}\left(\mathbf{\tilde{R}}\right)\rangle$, equation (9) implies invariant structure and dynamics along the isomorphs [178]. To show this in detail, note first that in the reduced unit system defined by equation (5) Newton's second law is ${{d}^{2}}\mathbf{\tilde{R}}/d{{\tilde{t}}^{2}}=\mathbf{\tilde{F}}$ in which the reduced force $\mathbf{\tilde{F}}$ is defined from the full 3N-dimensional force vector $\mathbf{F}$ by $\mathbf{\tilde{F}}\equiv \mathbf{F}{{\rho}^{-1/3}}/{{k}_{\text{B}}}T$. Since $\nabla ={{\rho}^{1/3}}\tilde{\nabla}$, equation (9) implies $\mathbf{F}=-\nabla U=-{{\left(\partial U/\partial {{S}_{\text{ex}}}\right)}_{\rho}}\,{{\rho}^{1/3}}\tilde{\nabla}{{S}_{\text{ex}}}\left(\mathbf{\tilde{R}}\right)$. Because ${{\left(\partial U/\partial {{S}_{\text{ex}}}\right)}_{\rho}}=T$ (appendix) this means that

$$\begin{eqnarray}&&\frac{{{\text{d}}^{2}}}{\text{d}{{{\tilde{t}}}^{2}}}\mathbf{\tilde{R}}\,=\,\mathbf{\tilde{F}}\,=\,-\tilde{\nabla}{{S}_{\text{ex}}}\left(\mathbf{\tilde{R}}\right)/{{k}_{\text{B}}}.\end{eqnarray} \tag{ 10 }$$

Equation (10) implies that the reduced-unit dynamics is invariant along the isomorphs. Since structure is found by averaging over the equilibrium time development, the reduced-unit structure is also isomorph invariant.

How to map out the isomorphs in the phase diagram? Calculating S_ex at a given state point is possible, but tedious. One can avoid this and identify the curves of constant S_ex by utilizing the fluctuation identity ${{(\partial \ln T/\partial \ln \rho )}_{{{S}_{\text{ex}}}}}=\langle \Delta W\Delta U\rangle /\langle {{(\Delta U)}^{2}}\rangle$ [166]. For instance, if the right-hand side is five at a certain state point, for a one percent density increase one stays on the same isomorph by increasing temperature five percent. The fluctuation identity was used in this way to step-by-step identify the isomorphs of figures 9(a) and 10. Another method for tracing out isomorphs utilizes equation (12) derived below, which for any two configurations at different densities with the same reduced coordinates, ${{\mathbf{R}}_{1}}$ and ${{\mathbf{R}}_{2}}$, implies

$$\begin{eqnarray}&&\frac{U\left({{\mathbf{R}}_{1}}\right)}{{{T}_{1}}}\,=\,\frac{U\left({{\mathbf{R}}_{2}}\right)}{{{T}_{2}}}+\text{Const}.\end{eqnarray} \tag{ 11 }$$

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Here T₁ and T₂ are the temperatures of the state points at which the two configurations are typical equilibrium configurations. Equation (11) implies the same canonical-ensemble probability of the configurations ${{\mathbf{R}}_{1}}$ and ${{\mathbf{R}}_{2}}$, which was the original isomorph definition [166]. In experiments, for any R simple system isomorphs may be identified with the lines of constant reduced viscosity or diffusion constant [19]; for highly viscous liquids the isomorphs are basically the lines of constant relaxation time, the so-called isochrones [76, 170], [183–185].

In the next section the isomorph theory is used to derive quasiuniversality for a large class of simple liquids. The theory is not limited to such systems, however. To illustrate this we show in figures 10(a) and (b) the incoherent intermediate scattering function as a function of reduced time for a molecular model along an isotherm and an isomorph respectively, demonstrating isomorph invariance of the dynamics. Figure 10(c) shows results for the pair correlation function of a crystal evaluated along an isomorph, an isochore, and an isotherm [181]. Figures 10(d) and (e) show results for a simple polymer model, again demonstrating isomorph invariance of the dynamics.

Other applications of the isomorph theory beyond the realm of simple liquids include, for instance, rationalizing simulations of non-linear shear flows and zero-temperature plastic flows of glasses [186, 187]. Several well-known melting line invariants like the Lindemann ratio may also be understood in terms of the isomorph theory, because the melting and freezing lines are both isomorphs to a good approximation [166].

Figure 9 demonstrated that the $\text{EXP}$ system obeys equation (9) well in the low-temperature part of the phase diagram. Expanding equation (9) to first order at a state point with excess entropy S_ex and average potential energy U leads to $U\left(\mathbf{R}\right)\cong U+T\left({{S}_{\text{ex}}}\left(\mathbf{\tilde{R}}\right)-{{S}_{\text{ex}}}\right)$ (recall that ${{\left(\partial U/\partial {{S}_{\text{ex}}}\right)}_{\rho}}=T$). In terms of the excess Helmholtz free energy ${{F}_{\text{ex}}}\equiv U-T{{S}_{\text{ex}}}$, if the $\text{EXP}$ system's microscopic excess entropy function is denoted by $S_{\text{ex}}^{\text{EXP}}\left(\mathbf{\tilde{R}}\right)$, one thus has

$$\begin{eqnarray}&&U\left(\mathbf{R}\right)\,\cong T\,S_{\text{ex}}^{\text{EXP}}\left(\mathbf{\tilde{R}}\right)+{{F}_{\text{ex}}}.\end{eqnarray} \tag{ 12 }$$

The $\text{EXP}$ system's reduced-unit dynamics is given by equation (10),

$$\begin{eqnarray}&&\frac{{{\text{d}}^{2}}}{\text{d}{{{\tilde{t}}}^{2}}}\mathbf{\tilde{R}}\,=\,-\tilde{\nabla}S_{\text{ex}}^{\text{EXP}}\left(\mathbf{\tilde{R}}\right)/{{k}_{\text{B}}}.\end{eqnarray} \tag{ 13 }$$

We proceed to show that a sum of two $\text{EXP}$ systems from the low-temperature part of the phase diagram defines a system that also obeys equation (13). This fact is the basis for defining below the $\text{EXP}$ 'family' of quasiuniversal pair potentials. Consider two systems, ${{v}_{1}}(r)={{\varepsilon}_{1}}\exp \left(-r/{{\sigma}_{1}}\right)$ and ${{v}_{2}}(r)={{\varepsilon}_{2}}\exp \left(-r/{{\sigma}_{2}}\right)$, obeying ${{k}_{\text{B}}}{{T}_{1}}\ll {{\varepsilon}_{1}}$ and ${{k}_{\text{B}}}{{T}_{2}}\ll {{\varepsilon}_{2}}$ at the respective state points of some configuration $\mathbf{R}$. The systems' dynamics are both described by equation (13), and equation (12) implies ${{U}_{1}}\left(\mathbf{R}\right)\cong {{T}_{1}}S_{\text{ex}}^{\text{EXP}}\left(\mathbf{\tilde{R}}\right)+{{F}_{\text{ex}}}_{,1}$ as well as ${{U}_{2}}\left(\mathbf{R}\right)\cong {{T}_{2}}S_{\text{ex}}^{\text{EXP}}\left(\mathbf{\tilde{R}}\right)+{{F}_{\text{ex}}}_{,2}$. For the 'sum' system defined by the pair potential ${{v}_{1}}(r)+{{v}_{2}}(r)$ one has $U\left(\mathbf{R}\right)={{U}_{1}}\left(\mathbf{R}\right)+{{U}_{2}}\left(\mathbf{R}\right)$, i.e.

$$\begin{eqnarray}&&U\left(\mathbf{R}\right)\cong \left({{T}_{1}}+{{T}_{2}}\right)S_{\text{ex}}^{\text{EXP}}\left(\mathbf{\tilde{R}}\right)+{{F}_{\text{ex}}}\end{eqnarray} \tag{ 14 }$$

in which ${{F}_{\text{ex}}}={{F}_{\text{ex}}}_{,1}+{{F}_{\text{ex}}}_{,2}$. This implies for the force vector

$$\begin{eqnarray}&&\mathbf{F}\,=\,-\left({{T}_{1}}+{{T}_{2}}\right)\,\nabla S_{\text{ex}}^{\text{EXP}}.\end{eqnarray} \tag{ 15 }$$

To show that the dynamics of the sum system is also described by equation (13) we first note that $T={{T}_{1}}+{{T}_{2}}$, in which T is the temperature of the sum system state point at which the configuration $\mathbf{R}$ is a typical equilibrium configuration. This is shown by making use of the configurational temperature identity ${{k}_{\text{B}}}T=\langle {{(\nabla U)}^{2}}/{{\nabla}^{2}}U\rangle$ [179, 188], which via equation (14) implies

$$\begin{eqnarray}&&{{k}_{\text{B}}}T\,=\,\left({{T}_{1}}+{{T}_{2}}\right)\,\langle \frac{{{\left(\nabla S_{\text{ex}}^{\text{EXP}}\right)}^{2}}}{{{\nabla}^{2}}S_{\text{ex}}^{\text{EXP}}}\rangle.\end{eqnarray} \tag{ 16 }$$

On the other hand, substituting ${{U}_{1}}\left(\mathbf{R}\right)\cong {{T}_{1}}S_{\text{ex}}^{\text{EXP}}\left(\tilde{{\mathbf{R}}}\,\right)+{{F}_{\text{ex}}}_{,1}$ into the configurational-temperature identity leads to ${{k}_{\text{B}}}{{T}_{1}}={{T}_{1}}\langle {{\left(\nabla S_{\text{ex}}^{\text{EXP}}\right)}^{2}}/{{\nabla}^{2}}S_{\text{ex}}^{\text{EXP}}\rangle$, i.e. ${{k}_{\text{B}}}=\langle {{\left(\nabla S_{\text{ex}}^{\text{EXP}}\right)}^{2}}/{{\nabla}^{2}}S_{\text{ex}}^{\text{EXP}}\rangle$. Thus equation (16) implies the required $T={{T}_{1}}+{{T}_{2}}$. Since the reduced force is defined by $\mathbf{\tilde{F}}\equiv \mathbf{F}{{\rho}^{-1/3}}/{{k}_{\text{B}}}T$, we conclude from equation (15) that $\mathbf{\tilde{F}}=-\tilde{\nabla}S_{\text{ex}}^{\text{EXP}}/{{k}_{\text{B}}}$. This shows that the sum system's equation of motion is also equation (13) (compare equation (10)).

The above generalizes to an arbitrary sum of $\text{EXP}$ terms and, in fact, also to a difference of such systems if thermodynamic stability is maintained [189, 190]. In order to state a precise (sufficient) condition for quasiuniversality of a system with pair potential $v(r)={\sum}_{j}{{\varepsilon}_{j}}\exp \left(-r/{{\sigma}_{j}}\right)$, we switch to reduced units. Defining the dimensionless numbers ${{\Lambda}_{j}}\equiv {{\varepsilon}_{j}}/{{k}_{\text{B}}}T$ and ${{u}_{j}}\equiv {{\rho}^{-1/3}}/{{\sigma}_{j}}$ and noting that the condition ${{k}_{\text{B}}}T\ll |{{\varepsilon}_{j}}|$ translates into $|{{\Lambda}_{j}}|\gg 1$, if the reduced pair potential of a system at the state points of interest is given by

$$\begin{eqnarray}&&\tilde{{v}}\,\left(\tilde{{r}}\,\right)\,=\,\underset{j}{\sum}\,{{\Lambda}_{j}}\exp \,\left(-{{u}_{j}}\tilde{{r}}\,\right)\,,\,\,|{{\Lambda}_{j}}|\gg 1,\end{eqnarray} \tag{ 17 }$$

the system obeys equation (13). In particular, any sum or product of two pair potentials that can be written as in equation (17) with all ${{\Lambda}_{j}}>0$ gives a pair potential that also obeys equation (13). Note that for equation (13) to apply it is not enough that the pair potential can be written as a superposition of exponentials with large coefficients—the 'wavevectors' u_j cannot be so closely spaced that large positive and negative terms almost cancel one another, because that would effectively result in a term that does not have a numerically large prefactor [160].

Thus any system for which $\tilde{v}\left(\tilde{r}\right)$ at the state point in question can be approximated as in equation (17) with u_js that are not too closely spaced has the same dynamics as the $\text{EXP}$ system to a good approximation. Such systems are quasiuniversal in the sense of section 4.5, because knowledge of a single number characterizing the reduced-unit structure or dynamics is enough to identify S_ex and thereby, via equation (13), all other reduced-unit structure and dynamics.

Having in mind the definition of quasiuniversality of section 4.5, the class containing the $\text{EXP}$ pair-potential system will be referred to as the '$\text{EXP}$ quasiuniversality class'. It follows from the above that whenever equation (17) applies to a good approximation for a given pair-potential system (at the state points of interest), this system is in the $\text{EXP}$ quasiuniversality class. It is an obvious conjecture that the $\text{EXP}$ quasiuniversality class does not include other regular pair-potential systems, but at this point there is no proof of this. It is likely that the $\text{EXP}$ quasiuniversality class contains also non-pair-potential systems, however (section 8 proposes that this is the case). In view of the above we shall refer to pair-potential systems obeying equation (17) (with not too closely spaced u_js) as members of the '$\text{EXP}$ family'.

In summary, any member of the $\text{EXP}$ family obeys equation (13) and is consequently in the $\text{EXP}$ quasiuniversality class. The HS system is in the $\text{EXP}$ quasiuniversality class, but not a member of the $\text{EXP}$ family because it does not obey equation (13). Finally, there may very well exist also non-pair potential members of the $\text{EXP}$ quasiuniversality class but these are by definition not in the $\text{EXP}$ family.

The numerical data presented in section 4 show that the IPL systems, ${{v}_{n}}\,(r)\equiv \varepsilon \,{{(r/\sigma )}^{-n}}$, are quasiuniversal. How would one justify this from the above [160]? Write first the reduced IPL pair potential in terms of the reduced radius $\tilde{r}$ as ${{\tilde{{v}}\,}_{n}}\,\left(\tilde{{r}}\,\right)={{\Gamma}_{n}}{{\tilde{{r}}\,}^{-n}}$ with ${{\Gamma}_{n}}\equiv {{\left(\rho {{\sigma}^{3}}\right)}^{n/3}}\,\varepsilon /{{k}_{\text{B}}}\,T$. The well-known identity ${\int}_{0}^{\infty}{{x}^{n-1}}\exp (-x)\text{d}x=(n-1)!$ implies ${{\tilde{{v}}\,}_{n}}\left(\tilde{{r}}\,\right)=\left[{{\Gamma}_{n}}/(n-1)!\right]{\int}_{0}^{\infty}{{u}^{n-1}}\exp \left(-u\tilde{{r}}\,\right)\text{d}u$. Discretizing the integral leads to ${{\tilde{{v}}\,}_{n}}\left(\tilde{{r}}\,\right)\cong \left[{{\Gamma}_{n}}/(n-1)!\right]\,\Delta u\,{\sum}_{j=0}^{\infty}{{\left((\,j+1/2)\Delta u\right)}^{n-1}}$ $\,\exp \left(-(\,j+1/2)\Delta u\,\tilde{{r}}\,\right)$. If one writes ${{\left((\,j+1/2)\Delta u\right)}^{n-1}}$ as $\exp \left[(n-1)\ln \left((\,j+1/2)\Delta u\right)\right]$, differentiation with respect to j shows that the dominant contributions to the sum come from the terms with $(n-1)/(\,j+1/2)\simeq \Delta u\,\tilde{r}$. For typical nearest-neighbor distances one has $\tilde{r}\simeq 1$, so the terms with $(\,j+1/2)\Delta u\simeq (n-1)$ are the most important ones. For these j's the prefactor of the exponential in the above sum is roughly ${{\Gamma}_{n}}\,\Delta u\,{{(n-1)}^{n-1}}/(n-1)!$. The largest realistic discretization step obeys $\Delta u\sim 1$. Thus for values of n larger than about 3  −  4, equation (17) is obeyed unless ${{\Gamma}_{n}}$ is very small, a condition that applies for the state points that have typically been studied [98], [191–194]. It follows that the IPL pair potentials are generally in the $\text{EXP}$ family, but that quasiuniversality is expected to gradually break down when n decreases and, in particular, goes below 3–4 [160] (note that for n  <  3 there is no thermodynamic limit because the integral ${\int}^{}v(r)\text{d}\mathbf{r}$ diverges). The gradual breakdown of quasiuniversality as the exponent is lowered is consistent with the finding that the crystal structure changes from face-centered cubic to body-centered cubic as n goes below 7 [195]. The one-component plasma [196], the n  =  1 case of a Coulomb system of identical particles in a charge-compensating background, is not in the $\text{EXP}$ family (although it interestingly does obey excess-entropy scaling [103]).

Adding or subtracting two pair potentials in the $\text{EXP}$ family defines a new system in the EXP family as long as the condition $|{{\Lambda}_{j}}|\gg 1$ is obeyed for all j and the resulting u_js are not close to one another. An example is the LJ pair potential equation (2) at the liquid state point defined by $\rho {{\sigma}^{3}}=1$ and ${{k}_{\text{B}}}T=2\varepsilon$. In terms of the reduced IPL functions introduced above, since ${{\Gamma}_{12}}={{\Gamma}_{6}}=1/2$ at this state point, we have here ${{\tilde{v}}_{\text{LJ}}}\left(\tilde{r}\right)=2\left({{\tilde{v}}_{12}}\left(\tilde{r}\right)-{{\tilde{v}}_{6}}\left(\tilde{r}\right)\right)$. This is of the form equation (17), implying quasiuniversality of the LJ liquid at that state point. Similar arguments may be applied for LJ-type systems with exponents different from 6 and 12, but if the two exponents are close, a more detailed investigation is required because one may run into the problem of having close u_js. The Yukawa pair-potential system is quasiuniversal in much of its phase diagram [45].

We finally note an alternative connection to the HS system. As shown by Brito and Wyart the HS system's interactions are described in a mean-field, i.e. time-averaged, sense by the continuous pair potential ${{v}_{\text{BW}}}(r)=-{{k}_{\text{B}}}\,T\ln \,(r/d-1)$ in which $d=2\sigma$ is the sphere diameter ($v(r)=\infty$ for r  <  d) [88]. This may be rewritten ${{v}_{\text{BW}}}(r)/{{k}_{\text{B}}}T=-\ln (r/d)+{\sum}_{n=1}^{\infty}{{(r/d)}^{-n}}/n$, which at high packing fractions is a sum of quasiuniversal IPL terms because here $r\cong d$ and consequently the large n terms dominate.