Comparing isotropic solids and fluids at either imposed volume or pressure, we investigate various correlations of the instantaneous pressure and its ideal and excess contributions. Focusing on the compression modulus *K*, it is emphasized that the stress fluctuation representation of the elastic moduli may be obtained directly (without a microscopic displacement field) by comparing the stress fluctuations in conjugated ensembles. This is made manifest by computing the Rowlinson stress fluctuation expression *K*_{row} of the compression modulus for NPT-ensembles. It is shown theoretically and numerically that *K*_{row}|_{P} = *P*_{id}(2 − *P*_{id}/*K*) with *P*_{id} being the ideal pressure contribution.

## INTRODUCTION

Among the fundamental properties of any equilibrium system are its elastic moduli characterizing the fluctuations of its extensive and/or conjugated intensive variables.^{1–7} The isothermal compression modulus *K* of an isotropic solid or fluid may thus be obtained in the NPT-ensemble at imposed particle number *N*, pressure *P*, and temperature *T* from the fluctuations

*strain fluctuation*relation

^{2}

with *k*_{B} being Boltzmann's constant. Equivalently, *K* may be obtained in a canonical NVT-ensemble using Rowlinson's *stress fluctuation* relation^{4,8,9}

with β = 1/*k*_{B}*T* being the inverse temperature,

_{B}a Born-Lamé coefficient

^{9}which for pairwise additive potentials becomes a simple sum of moments of derivatives of the potential with respect to the particle distance.

^{8}In this Communication, we emphasize that the stress fluctuation representation of the elastic moduli

^{4–6}may be obtained directly from the well-known transformation rules between conjugated ensembles.

^{10}Focusing on the compression modulus

*K*this is made manifest by computing Rowlinson's expression

*K*

_{row}deliberately for NPT-ensembles where the volume is allowed to freely fluctuate. We show that

with *P*_{id} being the ideal pressure contribution. We demonstrate first Eq. (3) by considering theoretically the fluctuations of the instantaneous normal pressure

## BACKGROUND

As discussed in the literature,^{2,8,10} a simple average

*V*→ ∞). A correlation function

*V*or

*P*are imposed. As shown by Lebowitz, Percus, and Verlet,

^{10}one verifies that

where *K* = −*V*∂*P*/∂*V*^{2} has been used. For

i.e., the compression modulus *K* may be obtained from the difference of the pressure fluctuations in both ensembles. Interestingly, the numerically more convenient Rowlinson expression *K*_{row} for NVT-ensembles can be derived from Eq. (5)^{9} without using a microscopic displacement field (only possible for solids)^{5} and avoiding the volume rescaling trick used originally for liquids.^{4}

## MC-GAUGE

There is a considerable freedom for defining the instantaneous pressure

*P*=

*P*

_{id}+

*P*

_{ex}does not change.

^{8}It is convenient for the subsequent derivations and the presented MC simulations to define the instantaneous ideal pressure

and the instantaneous excess pressure

^{8,9}Within this “MC-gauge” the thermal momentum fluctuations are assumed to be integrated out and the (effective) Hamiltonian

*s*of the system may be written

with

## NON-AFFINE CONTRIBUTION

In the following, the concise notation

## AFFINE (BORN) CONTRIBUTION

The second moment of any intensive variable computed in an ensemble, where its mean value is imposed, is obtained readily by integration by parts. Using Eq. (7), this shows that

where a prime denotes a derivative with respect to the indicated variable. Albeit the indicated averages are taken over all states *s* and all volumes

*P*, being

*simple averages*they can also be evaluated for sufficiently large systems in the NVT-ensemble yielding identical results. Denoting the last term in Eq. (9) by η

_{A, ex}, one can show that it is equivalent for pair interaction potentials to the already mentioned Born-Lamé coefficient:

^{9}η

_{A, ex}= η

_{B}+

*P*

_{ex}. Substituting Eqs. (8) and (9) into the Legendre transform Eq. (5), this confirms Eq. (2).

## CORRELATIONS AT CONSTANT *P*

We focus now on stress fluctuations in the NPT-ensemble. By comparing with Eq. (5), one sees that if the Rowlinson formula *K*_{row} is applied at imposed *P*, this must yield

Interestingly, Eq. (10) does not completely vanish for finite *T* as does the corresponding stress fluctuation expression for the shear modulus *G* at imposed shear stress τ.^{9} As a next step, we demonstrate the relations

from which Eq. (3) is then directly obtained by substitution into Eq. (10). Returning to the general transformation relation Eq. (4), we first note that the *l.h.s.* must vanish if at least one of the observables is a function of

making the steepest-descent approximation

*V*/

*K*= −∂

*V*/∂

*P*.

^{11}Remembering Eq. (6), this implies Eq. (11). With

to leading order for *V* → ∞. This relation implies finally the claimed correlation between ideal and excess pressure fluctuations, Eq. (12), using

^{12}Please note that in Ref. 9, ideal and excess pressure fluctuations have incorrectly been assumed to be uncorrelated.

## SOME ALGORITHMIC DETAILS

The numerical results reported here to check our predictions have been obtained by MC simulation of (i) one-dimensional (1D) nets with permanent cross-links and (ii) two-dimensional (2D) glass-forming liquids. Periodic boundary conditions are used and the pressure *P* is first imposed using a standard MC barostat.^{8,9} After equilibrating and sampling in the NPT-ensemble, the volume is fixed,

^{13}For the 1D nets, we assume ideal harmonic springs,

*U*= ∑

_{l}

*k*

_{l}(

*x*

_{l}−

*R*

_{l})

^{2}/2, with

*x*

_{l}being the distance between the connected particles, the reference length

*R*

_{l}of the springs being set to unity and the spring constants

*k*

_{l}being taken randomly from a uniform distribution of half-width δ

*k*centered around a mean value also set to unity. Only simple networks are presented here where two particles

*i*− 1 and

*i*along the chain are connected by

*one*spring

*l*=

*i*, i.e., at zero temperatures all forces

*f*

_{l}along the chain become identical. This implies

*K*∼ 1/⟨1/

*k*

_{l}⟩. The compression modulus decreases thus strongly with δ

*k*as indicated by the bold line in the inset of Fig. 1. Our 2D systems are polydisperse Lennard-Jones (pLJ) beads

^{7}kept at a constant pressure

*P*= 2 as described in Ref. 9.

## COMPUTATIONAL RESULTS

As shown in Fig. 1, the compression modulus *K* may be determined using the volume fluctuations in the NPT-ensemble, Eq. (1), or using Rowlinson's stress fluctuation formula, Eq. (2), for the NVT-ensemble. The same values of *K* are obtained from the Legendre transform for the pressure fluctuations, Eq. (5), and from the ideal pressure fluctuations η_{id}|_{P}, Eq. (11), which thus confirms both relations. As seen in the inset of Fig. 1, the compression modulus of the 1D nets decreases with δ*k*. Also indicated is the “affine” contribution η|_{P} to *K*, measuring the mean spring constant ⟨*k*_{l}⟩ = 1, and the “non-affine” contribution η|_{V} which is seen to increase with δ*k*. The decrease of *K* is thus due to the increase of the non-affine contribution.

As shown in the inset of Fig. 2, we have also checked the correlations between the ideal and the excess pressure fluctuations η_{mix}|_{P}. To make both models comparable, the reduced correlation function *y* = η_{mix}|_{P}/*P*_{id} is traced as a function of the reduced ideal pressure *x* = *P*_{id}/*K* with *K* as determined independently above. A perfect data collapse on the prediction *y* = 1 − *x* (bold line) is observed for all systems. The main panel of Fig. 2 shows finally the scaling of the Rowlinson formula computed in the NPT-ensemble. As before a scaling collapse of the data is achieved by plotting *y* = *K*_{row}|_{P}/*P*_{id} *vs.* *x*. The bold line indicates our key prediction, Eq. (3). Interestingly, the latter result does not depend on the MC-gauge which has been used above to simplify the derivation of Eq. (3). Please note that it is not possible to increase *x* beyond unity for our liquid systems (*K* ⩾ *P*_{id}) and the deviations from the low-temperature plateau *y* = 2 are thus necessarily small. The additional *P*_{id}/*K* correction has thus been overlooked in our previous publication.^{9}

## CONCLUSION

Emphasizing the underlying Legendre transform, Eq. (5), of the stress fluctuation formalism, we have investigated here the well-known Rowlinson stress fluctuation expression *K*_{row} for the compression modulus, Eq. (2), using deliberately the NPT-ensemble. Correcting several statements made in Ref. 9, it has been demonstrated theoretically and numerically that Eq. (3) holds. The latter result, as the other correlation relations indicated in the paper, may allow to readily calibrate (correctness, convergence, and precision) various barostats commonly used.^{8}

## ACKNOWLEDGMENTS

P.P. thanks the Région Alsace and the IRTG Soft Matter and F.W. the DAAD for funding. We are indebted to A. Blumen (Freiburg) and A. Johner (Strasbourg) for helpful discussions.

## REFERENCES

*V*→ ∞ can be also seen by using that the distribution of

*d*= 1 “volume” corresponds to the linear length of the system and in

*d*= 2 to its surface. Pressure and elastic moduli take units of energy per

*d*-dimensional volume.