Apparent density fluctuations in N-constant ensemble simulations
In computer simulations performed in constant number of particles ensembles, although the total number of particles N contained in the simulation box does not fluctuate, hence giving a zero apparent compressibility, there are still local fluctuations in the number of particles. It is shown herein that these apparent fluctuations produce a compressibility that can be computed from the calculated radial distribution function, and which matches to a great accuracy the compressibility of the fluid for the open system.
This statement implies that the radial distribution function evaluated in simulation of constant number of particles is identical to that evaluated in the grand canonical ensemble, for the entire distance range within half-box width. This is illustrated for the hard sphere and Lennard-Jones fluids and for molecular models of water. The origin of this apparent fluctuation is that the bulk of the remaining particles, outside the range over which the distribution function is calculated, act as a reservoir of particles for those within this range, thanks to the periodic boundary conditions. The implications on the calculation of the Kirkwood-Buff integrals are discussed.
Laboratoire de Physique Théorique des Liquides (UMR CNRS 1600), Université Pierre et Marie Curie, 4 Place Jussieu, F75252, Paris cedex 05, France.
Laboratoire de Spectrochimie Infrarouge et Raman (UMR CNRS 8516), Centre d’Etudes et de Recherches Lasers et Applications, Université des Sciences et Technologies de Lille, F59655 Villeneuve d’Ascq Cedex, France.
The statistical theory of liquids is now mostly textbook knowledge. Within such a theory one shows that the isothermal compressibility of a liquid is related to the fluctuation in number of particles of the system, through a fluctuation-dissipation type relation:
where is the ideal gas compressibility, is the Boltzmann factor and the number density. The first equality in (1) is a thermodynamic definition, while the two last equalities are derived in the grand canonical ensemble, where the fluctuation in the number of particle is allowed. in this equation is the integral of the radial distribution function (RDF)
From Eq.(1) it is seen that the compressibility is related to the fluctuation in number of particles through the second equality. Therefore, we do not expect these relations to hold in any constant N ensemble such as the micro-canonical (constant N,V,E), canonical (constant N,V,T) or isobaric (constant N,P,T) ensembles. In such ensembles, the last equality is expected to give . Many textbooks show that this is indeed the case, when is evaluated in N-constant ensemble. One such demonstration is provided in the next section of this report. Since the RDF can be evaluated by simulations in different ensembles, one expects the above constraint to be verified in N-constant ensembles. We show here that this is not the case, and that, in fact, the effective compressibility evaluated through Eq.(1) using the RDF, as well as the RDF itself, evaluated in the canonical ensemble, for example, match the expected ones with great accuracy. This contradiction is lifted by considering contributions from the periodic boundary conditions, as discussed later. In fact, this result should not surprise us, since it is well known that the chemical potential can be evaluated from simulations in N-constant ensemble. The procedure consist in evaluating the insertion free energy of an additional particle, which implicitly supports the existence of local density fluctuations. It is these local fluctuations that give rise to the apparent global density fluctuation, in the very absence of any such macroscopic feature.
Such behaviour is observed equally in mixtures, where, in addition to density fluctuation, concentration fluctuation also play an important role. Hence, it gives some additional support for the evaluation, through computer simulations, of the so-called Kirkwood-Buff integrals (KBIs), that have attracted recent interest in the modeling of the force field of aqueous mixtures[4, 5]. Indeed, such mixtures tend to show appreciable micro-segregation, that can be detected to some extent through the comparison with the experimental KBIs[4, 6, 7]. Since these quantities are simply the integrals of the various site-site radial distribution functions, the evaluation of the latter by computer simulations has an undeniable interest. However, the theory behind the KBIs is strictly a grand canonical approach. Then, one is interested to know what are the limitations of the RDF computed in constant-N ensemble simulations
It is worthwhile mentioning that, the problematic of comparing the RDF evaluated in different ensembles, have been addressed in the past by few authors[8, 9]. To our knowledge, however, the fact that the isothermal compressibility and the RDF match that evaluated from grand canonical ensemble has not been addressed in present terms.
The paper is organized as follows; the next section presents the theoretical material that is needed to clarify the issue raised here . The results section contains an illustration on the hard sphere, Lennard-Jones and some water models. Finally the discussions and conclusions are given in section 4.
2 Theoretical details
As stated in the introduction, there is a connection between the fluctuation of the number of particles N and the isothermal compressibility , which is found in many textbooks of the liquid state theory. We recall here briefly the main steps for completeness sake.
The RDF is a key quantity in the analysis of liquids since it provides a direct information about the microstructure of liquids. This function is related to the second member of the whole hierarchy of density correlation functions, that can be constructed as statistical moments from one single function, the microscopic density in a N-particles system
with being the Dirac delta function and with the convention that represents the position and the orientation of molecule labeled . This definition is equivalent to a snapshot of the instantaneous position and orientation of the particles, and thus represents one microstate of the system. It is interesting to note that this quantity is the basic observable that can be perused by computer simulations. In practice, however, we are essentially interested in averages of this quantity and its various correlations. These averages can be performed in different ensembles. For a given ensemble, one can define, for example, the one-body function which for the translationally and rotationally invariant systems (homogeneous and isotropic) that we consider in this report is simply
that is the number density divided by the solid angle (where or depending on the symmetry of the molecule).
Going one step further, one can equally define the second moment, namely the two particles density as
The pair correlation function is then defined by
This definition implies that when the particles are uncorrelated, that is for example when they are infinitely far apart, then one has exactly , which means:
It is clear that this relationship is trivially valid for an infinite system, but must be revised for a finite system with N particles in a volume V. This is, for example, the case of the micro-canonical (constant NVE), canonical (constant NVT) or isobaric (constant NPT) statistical ensembles, in which most simulations are performed. In the following discussion, we take as example the canonical ensemble, but the reasoning equally holds for the other two ensembles.
where is the canonical ensemble partition function, with being the total interaction energy between the N particles. The RDF is then defined as the angle average of as:
It is perhaps worthwhile noting here that the definition (8) contains the factor , where the numerator arises from the number of pairs in an ensemble of particles, while the denominator is coming from the factor in (6). It is this factor that is responsible for the asymptotic limit of the RDF in finite systems.
Let us examine the asymptotic behaviour from Eqs.(8,9) when . From Eq.(8), when , that is when the two particles are far apart, one can neglect the interaction between them in , hence one gets directly from Eqs.(8,9), for the canonical ensemble
This relation hold equally in the micro-canonical ensemble. In the isobaric, constant NPT ensemble, the corresponding limit is , where is a formal function whose form is given in the appendix. This size dependence is expected to be observed in the canonical ensemble, and one can see from Eq.(1) that the integral should trivially satisfy
since N does not fluctuate.
The RDF can also be directly obtained from computer simulations from Eq.(5). This equation tells us that we only need to compute the histogram which counts the pairs of particles at a distance between and . The resulting expression is:
where is the volume of the spherical element considered, reduced by the total volume of the simulation cell. The above expression is the equivalent of the second equality in Eq.(5). In practice, this calculation is performed and averaged over several configurations or microstates. It is very important to note that Eq.(12) is valid for any statistical ensemble. It is the nature of fluctuations of a particular ensemble that will affect the form the histogram and determine the corresponding asymptotic behaviour of . Hence, it is not appropriate to introduce additional factors into this expression, in order to enforce expected behaviour. This point is worth mentioning, since incorrect additional factors are often found in some algorithms.
An important remark must be made at this point. The computer simulations of the various N constant ensembles are not strictly concerned by arguments about finite or closed systems, since the periodic boundary conditions allow to mimic an infinite system. This is, of course, size dependent. It turns out that this fact permits some kind of pseudo-fluctuations in the number of particles within specific subvolumes, that we consider to be local fluctuations, while the total number of particles in the simulation cell does not fluctuate. In order to grasp what this feature could induce, let us imagine a very large system, that is constrained to be finite (a fluid in a huge box). Then, we consider computing the g(r) in a very small sub-system inside this large system. Clearly, the outer part of the system acts as a particle reservoir, and we naturally expect the resulting g(r) to contain the fluctuation of particles correctly described, that is the correct asymptotic limit, i.e. unity. Since the g(r) of small systems are computed within a sub cell of the system box, usually in a spherical shell with radius about L/2, where L is the simulation box size, we expect some influence on the g(r) of the fluctuations of the number of particles of this subsystem. This is what we examine in the next section. It should be clear, at this point, that our considerations about apparent fluctuations hold only for subvolumes smaller than that of the largest sphere inscribed within the cubical box, and that fluctuations beyond this range will inevitably show that the total number of particles is fixed.
These definitions can be equally extended to mixtures. In particular, the Kirkwood-Buff integrals are defined between species labeled and as:
where is the corresponding RDF.
Nothing, in the general formalism outlined above, prepare for the actual results we have observed in our simulations. We have evaluated the RDF and the resulting compressibility (through (1)) for the hard sphere fluid and two water models. Constant NVT Monte Carlo (MC) simulations have been performed for the hard sphere and Lennard-Jones fluids, for various system sizes (N=500, 2048, 4000). These two models are the archetypal models for simple liquids, and thus serve as a perfect basis to test the claims. Molecular Dynamics simulations in the constant NPT ensemble have been performed for the SPC/E and TIP5P water models, with N=864, 2048 and N=10976. The calculations show in an unbiased manner that the RDF have the correct asymptotic limit, i.e. unity.
3.1 The hard sphere fluid
One component hard sphere (HS) fluid has been equilibrated using a constant NVT Monte Carlo code, for various system sizes, and for the fixed reduced number density , where is the hard sphere diameter. This density can be considered as dense enough for a liquid phase. The RDF for systems with N=500, 2048 and 10976 have been computed with statistics performed over 100 million particle moves. This exceeds by far the usually encountered numbers, but it is necessary in order to have accurate evaluation of the g(r) at large distances.
Fig.1 shows the RDFs in the entire range, and the inset shows the integrand . The expected asymptotes are also plotted, each for the range over which the RDFs have been evaluated. The inset indicates clearly that oscillates around zero, rather than the expected asymptotes. The RDF calculated from the Percus-Yevick (PY) theory is equally shown. Since this RDF is, by definition, calculated in the grand canonical ensemble, it serves as a test of the asymptotic limit, despite being an approximate theory. One notices in particular the very good agreement at large distances between the oscillatory structure from the PY theory and the simulations with N=10976, which is clearly seen in the inset.
3.2 The Lennard-Jones fluid
The one component Lennard-Jones (LJ) fluid has been also studied by an NVT constant MC algorithm, for reduced density and reduced temperature (where is the energy depth of the LJ interaction). This choice corresponds to a point in the dense and high temperature state of the LJ fluid. Systems sizes of N=500, 2048 and 10976 have been studied, with statistics similar to that of the hard sphere fluid.
Fig.2 is the analogous of Fig.1 but for the LJ fluid. Once again, it is observed that the RDF tends to unity, despite the fact that we are in an constant N ensemble. The PY results are also plotted and serve as a reference, despite the approximate nature of this theory. In the high temperature region, this theory is expected to be more appropriate than the hypernetted chain (HNC) theory, since in the high temperature regime the correlations are expected to be dominated by excluded volume effects, and be closer to the HS regime. It is observed that the long range oscillatory structures, between the simulations and the theory, are again in very good agreement. The large oscillations for the N=10976 system are due to the high temperature, and are difficult to reduce, despite statistics collected over 100 million moves. Again it is observed that none of the data follow the expected 1/N asymptotes that are plotted in the inset.
Fig.3 shows the running reduced compressibility for the same system. The numerical value obtained from the PY theory is , shown as an horizontal line, and serves as a comparison point for the unknown exact compressibility for this state point. The agreement between the asymptotic limit evaluated through Eqs.(1,2) indicates that the pseudo macroscopic density fluctuation in this system where neither N nor V are changing, tend to a non zero value, which is quite nicely seen for the N=10976 system. This value of the compressibility, estimated to be about , is below both that of the approximate PY and HNC(0.07564) theories, which tends to overestimate the correct compressibility. The virial pressures (compressibility factor ) obtained from the HNC and PY theories are 6.398 and 4.90, respectively, while that obtained from the simulations is 5.140. The reduced energies per particle are -1.167 (HNC), -1.453(PY) and -1.395 (MC). The simulated values are seen to be in between those of the two theories. This reflects the trend generally observed that the two theories bracket the exact results.
As a side remark, it is rather worthy of note that the convergence of the value of the compressiblity should need systems sizes as large as N=10976 for systems as simple as the HS or LJ fluids. Properties such as the pressure or the internal energy, converge faster, that is within a smaller range of distance. The typical example of this is the pressure for hard spheres that requires only the contact value of the g(r). The principal reason for these properties to converge faster is that the integrand is weighted by the interaction (or the derivative), which screens the long range oscillatory behaviour of the RDF.
3.3 Water models
Molecular Dynamics simulations in constant NPT ensemble, using the DL-POLY2 code, have been performed for two water models, namely the SPC/E and TIP5P models, under ambient conditions. The oxygen-oxygen RDF have been computed for several system sizes, which is N=864, 2048 and 10976 for the SPC/E model, and N=2048 for the TIP5P model. Averages have been performed over several hundreds of thousand steps.
Fig.4 shows the RDF, and the inset shows the integrand . Again, the asymptotic limit is near perfectly unity, in contradiction with the fact that this calculation is done in the N,P,T constant ensemble, where the strict application of the arguments developed in Section 2, give the following expectation, (see the appendix).
Fig.5 shows the experimental and calculated KB integrals from simulations. The experimental compressibility at room temperature is . From this value we estimate the experimental G to be , which is shown as an horizontal line. The agreement is very good, indicating again that the true and apparent compressibilities are in good correspondence. One can note the fact that both water models reproduce well the experimental compressibility, which is an indication of their accuracy with that respect (TIP4P is equally in the same accuracy region, although not shown here).
4 Discussion and conclusion
The results shown in the preceding section demonstrate numerically that the apparent density fluctuations in N constant ensemble simulations match near perfectly those of a corresponding system where N would be allowed to fluctuate. In addition, the RDF evaluated in such ensemble tend to the correct asymptotic behaviour, i.e. unity. These findings are purely empirical, and nothing in the theoretical considerations of Section 2 allows us to anticipate or justify these findings.
Examining the conditions of the realization of the various N-constant ensemble in computer simulations, we notice that these ensembles are not “closed”, since they are made infinite through the periodic boundary conditions. In fact, a finite constant N system cannot be “closed” in the sense of having a fixed boundary, since this boundary will always influence the distribution of the particles by making the system inhomogeneous, and thus rendering useless the formalism exposed in the section 2. In this, the term “closed” used in Ref., and various subsequent reports by other authors, is totally inappropriate. The absence of boundaries permits an effective fluctuation of the number of particles inside any subvolume, and the numerical evidence here simply indicates that this fluctuation is enough to induce an apparent macroscopic fluctuation through the RDF, that is near exactly that of the open system. The fact that a small system of few hundreds to few thousands particles can reproduce a macroscopic quantity is not new; many macroscopic thermodynamic quantities are evaluated within such small microscopic subsystems that are the simulation cells. In fact, as stated in the introduction, the very fact that one can use an N-constant cell to compute the chemical potential, indicates that the fluctuations in the system are the appropriate response to local variations in pressure and volume.
One of the interesting conclusions of this study is the perspective of its application to mixtures, and particularly to aqueous mixtures. Indeed, in such systems, concentration fluctuations are very important, due to specific self-clustering tendencies induced by the ability of water molecules to link together through hydrogen bonds. One of the questions that have aroused in recent works [6, 7] is the ability of the constant N simulations to reproduce properly the correct long range behaviour of the various RDFs, and in particular that between water molecules. The present study gives some confidence in the constant N simulation studies of such systems, once care is taken to consider sizes large enough to support local immiscibility without leading to macroscopic phase separation.
The computation of the formal limit of g(r) in the NPT ensemble follows a logic similar to that exposed in Section 2. We start from the equivalent of (8) which is now
where is the partition function of the (N,P,T) constant isobaric ensemble. The number density is now defined as , where is the effective average volume of the system under constant pressure . When , that is when the interaction between particles 1 and 2 can be neglected, Eq.(14) reduces to
with . The indicates that the volume average is taken for (N-3) particles only. Eq.(15) can be rewriten as
The second fraction involving the volumes can be evaluated easily in the ideal gas limit, but is not so for the case the fully interacting system. Nevertheless, one sees that the asymptotic limit of the isotropic part of g(1,2) is not trivially unity, as would be expected in an open system.
- [*] Department of Physics, Faculty of Natural Sciences, Mathematics and Kinesiology, University of Split, Nikole Tesle 12, 21000, Split, Croatia.
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Fig.1. (color online). RDF of the hard sphere fluid at . MC simulation results for N=500 (green), N=2048 (blue) and N=10976 (red). PY result in black. Inset: plots of with same color conventions.
Fig.2. (color online). RDF of the Lennard-Jones fluid at and . MC simulation results for N=500 (green), N=2048 (blue) and N=10976 (red). HNC result in black. Inset: plots of with same color conventions.
Fig.3. (color online). Running (reduced) compressibility integral (see text) for the system shown in Fig.2, with same color convention. The horizontal line is the corresponding HNC compressibility.
Fig.4. (color online). Water oxygen-oxygen RDF from MD simulations for pure water at ambient conditions. SPC/E with N=864 (magenta); SPC/E with N=2048 (blue); SPC/E with N=10976 (black); TIP5P with N=2048 (cyan). The inset shows the corresponding .
Fig.5. (color online). The running KB integral for the systems shown in Fig.4, with same conventions. The horizontal line is the experimental result (see text).