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On statistical mechanics of smallsystems: accurate analytical equationof state for confined fluidsMehrdad Khanpourab, Luis A. Rivera-Riverab & Thomas D. Sewellba Department of Chemistry, Ayatollah Amoli Branch, Islamic AzadUniversity, Amol, Iranb Department of Chemistry, University of Missouri-Columbia,Columbia, MO 65211-7600, USAPublished online: 23 Feb 2015.

To cite this article: Mehrdad Khanpour, Luis A. Rivera-Rivera & Thomas D. Sewell (2015): Onstatistical mechanics of small systems: accurate analytical equation of state for confined fluids,Physics and Chemistry of Liquids: An International Journal, DOI: 10.1080/00319104.2015.1006631

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On statistical mechanics of small systems: accurate analytical equationof state for confined fluids

Mehrdad Khanpoura,b*, Luis A. Rivera-Riverab and Thomas D. Sewellb*

aDepartment of Chemistry, Ayatollah Amoli Branch, Islamic Azad University, Amol, Iran;bDepartment of Chemistry, University of Missouri-Columbia, Columbia, MO 65211-7600, USA

(Received 9 October 2014; accepted 8 January 2015)

Some relations are derived using statistical mechanics to describe the effects ofsurroundings on the properties of systems for sizes below the thermodynamic limit.A general expression for the free energy of closed, small systems is derived and thenused to obtain the dependence of the thermal properties on density and temperature,including general expressions for equations of state and internal energies.Comparisons between predictions of the current theory and the results of moleculardynamics (MD) simulations are made for 3D hard-sphere and Lennard-Jones fluidsfor which the surroundings are modelled as reflecting hard walls that confine thesystem along one direction. The analytical predictions are in excellent agreementwith the MD results.

Keywords: small systems; confined fluid; equation of state; analytic theory; statisticalmechanics

1. Introduction

The thermodynamics of small systems has been studied extensively (see, for example,Refs. [1–17]) due to their relevance for understanding myriad small systems that occur innature; atomic and molecular clusters, aerosols, micelles, biological macromolecules,crystal nuclei in supersaturated solution, and colloids, to name a few. However, we stilldo not have an established, accepted theory to explain satisfactorily their structural andthermal behaviour. For example, the phase diagram of confined water, which is found invarious geological materials on Earth, differs from that of the bulk.[16] The geometry ofthe confinement and the potential it exerts on the water molecules alter significantly boththe structure and thermodynamics. More fundamentally, it appears that energy andentropy of small or confined systems are not strictly extensive.[13,14]

In an attempt to account for the non-extensive energy and entropy, a definition ofentropy different from Boltzmann’s has been suggested upon which a non-extensivestatistical mechanics can be constructed [13,14] by incorporating this definition withinotherwise normal statistical mechanics. The framework requires the introduction of anempirical parameter q that to our knowledge has no a priori meaning and therefore cannotbe computed theoretically from first principles. Rather, the value of q must be determined

*Corresponding authors. Email: mtankhanpour@yahoo.com (Mehrdad Khanpour);sewellt@missouri.edu (Thomas D. Sewell)Present address of Luis A. Rivera-Rivera: Department of Chemistry, Texas A&M University,College Station, TX 77843-3325, USA

Physics and Chemistry of Liquids, 2015http://dx.doi.org/10.1080/00319104.2015.1006631

© 2015 Taylor & Francis

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empirically for each problem/system studied. Furthermore, the theory is unable to explainwhy large systems always have q = 1, whereas small systems usually have values of qdifferent from unity. For these reasons we think that the non-extensive thermodynamicsdoes not fundamentally explain non-extensiveness of small systems.

The mathematical framework of classical thermodynamics, based on four fundamentalpostulates, can describe the behaviour of large systems. When statistical mechanics wasconstructed it became clear that it would reduce to thermodynamics in the so-calledthermodynamic limit (N→∞, V→∞, N/V = finite). For systems below the thermodynamiclimit there are finite-size effects.[17] These are important for small systems but becomenegligible for large systems. Hence, a system can be regarded as small when it consists ofa finite number of particles confined to a restricted volume and has non-negligibleinteractions with the surroundings. The structural and thermal properties of confinedsystems are commonly treated by density functional theory (DFT),[18–22] which isbased on the interaction of a system with an external potential. The one-particle densityfunction varies with respect to the external potential and therefore position. The funda-mental theorem of DFT states that the one-particle density function of a system isuniquely determined by the external potential.[19] There are multiple versions of DFTbased on this theorem, but all of them involve some approximation to the free energydensity functional. The one-particle density function is obtained by minimisation of theproposed free energy density functional with respect to it. Once the one-particle densityfunction is obtained, one can calculate structural and thermal properties of the system. Toour knowledge, however, none of the existing DFT methods can provide analyticalexpressions for thermodynamic properties.

Density functional theory is much different in appearance from ordinary statisticalthermodynamics. Thus, the question arises: can one write down a global theory, as withthermodynamics for infinite systems, that is able to describe small systems? We assert thatsmall systems can be studied by using statistical mechanics amended to account explicitlyfor the effects of the surroundings on the system. The way to do so is to work out andapply the statistical mechanics as completely as possible prior to taking the thermody-namic limit, at which point the effects of the surroundings on the system should disappearimplicitly from all calculated properties. This approach allows us to determine the extentto which the number of particles or the spatial extent of the system, both of which areassumed finite prior to the thermodynamic limit, influences the thermodynamic properties.In other words, it allows us to quantify the differences between statistical properties of thesystem before and after applying the thermodynamic limit.

The approach we have taken is to study a system with and without the thermodynamiclimit starting from the general framework. We assume that the thermodynamics of thesystem absent the effects of the surroundings is completely known, and then introduce thesurroundings to evaluate the effect on the thermodynamic functions describing the system.Thus, the procedure is to develop a kind of thermodynamic perturbation theory from thebulk properties to the small-system ones. In general, the pre- and post-limit systems willnot have the same thermodynamic properties; for example, they should have differentequations of state and internal energies due to the effects of the surroundings.

Here we develop analytical expressions for the thermodynamics of small systemsbased on the approach just described. The effects of the surroundings, treated as anexternal source of potential energy, are included in a way such that, for sufficientlylarge systems, they become negligible and the normal thermodynamic description isrecovered. We invoke statistical mechanics without assuming the thermodynamic limit;that is, both the number of particles and the volume of the system are considered without

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any restriction. Within this framework even the apparent non-extensiveness of, forexample, the free energy can be satisfactorily explained. Indeed, we will show that, withinthe developed framework, the non-extensiveness of the entropy and all energy functionsoriginates from the non-negligible influences of the surroundings on the small systems.For large systems the effects due to the surroundings become negligible and we recoverthe extensiveness of energy and entropy functions.

The remainder of the manuscript is organised as follows. In Section 2 we present thegeneral theory and derive the specific expressions for the free energy and hence allthermodynamic functions. In Section 3 we apply the results from Section 2 to twomodel systems, namely 3D atomic fluids confined between hard-wall potentials alongone direction with periodic boundaries in the transverse directions, with inter-particlepotentials described either as hard spheres or using the 12–6 Lennard-Jones potential. Weconsider each system in the small- and large-system limits to demonstrate the validity ofthe general theory for predicting the free energy. Concluding remarks are given inSection 4.

2. Theory

Consider an isolated system consisting of N particles contained in a volume V andcharacterised by quantum mechanical energy states Ej(N,V). If we now introduce thesurroundings, which are assumed to be in thermal equilibrium with the system, then theenergy states of the resulting closed system must reflect this interaction, leading to newenergy states E0

j(N,V). Assuming a one-to-one correspondence between the energy statesin the isolated and closed systems (which is valid when the effect of the surroundings isweak), we have E0

j ¼ Ej þ αj where αj stems from the surroundings and encompasses

completely the effects of the surroundings on the energy states of the closed systemrelative to those of the isolated one. Because the formal structure of statistical mechanicsis independent of the specific nature of the interactions, the volume of the system, and thenumber of particles, we are still allowed to use it but now must take into account explicitlythe effects of the surroundings on the energy states of the previously isolated system.

The next step is to consider the partition function of the system under the effects of the

surroundings. Working in the canonical ensemble, we have Q0 ¼ Pje�βE0

j ¼ Pje�β Ejþαjð Þ

where β = (kT)−1, k is Boltzmann constant and T is temperature. This temperature is thetemperature of the surroundings and, because the system is in equilibrium with thesurroundings, it is the temperature of the system irrespective of the size of the system.Having the partition function gives us, in principle, access to all system thermodynamicproperties, in this case including the effects of the surroundings. It seems natural tocompare the thermodynamic properties of the system with and without the effects ofsurroundings. We will therefore write

Q0 ¼Xj

e�β Ejþαjð Þ ¼Xj

e�βEje�βαj ¼ QXj

e�βEj

Qe�βαj ¼ Q e�βαjib;

�(1)

where Q ¼ Pj e

�βEj is the partition function of the system free from the effects of thesurroundings and the subscript b in the last equality emphasises that the average is takenover the bulk probability distribution (i.e. independent of the surroundings). Note that at

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this stage of the theory there is no explicit reference to the volume or number of particlesof the system. By defining the thermodynamic quantity α through the relation

e�βα ¼ e�βαj� �

b (2)

and using the relation A ¼ � 1=βð Þ lnQ, where A represents the Helmholtz free energy, weobtain

A0 ¼ Aþ α: (3)

Let us call α the outside free energy. From the macroscopic point of view, this outside freeenergy takes into account all effects of the surroundings on the system. Based on Equation(3), we obtain the following relations for the internal energy and entropy of the systemwith and without the effect of the surroundings:

E0 ¼ E þ @ βαð Þ@β

(4)

S0 ¼ S � @α@T

: (5)

Clearly, computation of thermodynamic properties in the general case is contingent onknowing α.

Now consider two systems in the same thermal environment but so far apart as to bepractically non-interacting, each with definite values of N, V, E and S. If we consider thetwo systems as a single entity, then obviously N ¼ N1 þ N2, V ¼ V1 þ V2, E ¼ E1 þ E2

and S ¼ S1 þ S2. If the two systems are brought sufficiently close together then onceagain we will have N ¼ N1 þ N2 and V ¼ V1 þ V2 but in this case we will not have E ¼E1 þ E2 or S ¼ S1 þ S2 due to interactions between particles of the two subsystems.Rather, because the energy states of each subsystem are affected by the presence of theother subsystem, we must use Equation (3) to describe the situation correctly; that is, wehave

E ¼ E01 þ E0

2 ¼ E1 þ E2 þ @ βα1ð Þ@β

þ @ βα2ð Þ@β

(6a)

S ¼ S1 þ S2 � @α1@T

� @α2@T

: (6b)

There are no ‘cross terms’ in Equation (6) because we regard system two as being part ofthe surroundings for system one, and vice versa. Thus, in general the internal energy andentropy of the interacting system are non-additive due to the appearance of @ βα1ð Þ=@βand @ βα2ð Þ=@β, and @α1=@T and @α2=@T , respectively, in Equation (6). However, as willbe seen, when the outside effects are properly included, we recover the additivity ofextensive properties.

It is of interest to calculate the size of system fluctuations in the presence or absence ofthe effects of the surroundings. The internal energy fluctuation is defined by

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σE ¼ E2� �� Eh i2

� �1=2. It can be found immediately using σ2E ¼ �@E=@β, which can be

combined with E0 ¼ E þ @ βαð Þ=@β under isothermal conditions to yieldσ2E0 ¼ σ2E � @2 βαð Þ=@β2 ¼ σ2E 1� 1=σ2E

� �@2 βαð Þ=@β2� �

, which is exact. Taking thesquare root, expanding, and retaining only the first two terms (valid when the effect ofthe surroundings is small) yields σE0 ¼ σE 1� 1=2σ2E

� �@2 βαð Þ=@β2� �

. Furthermore, we

know that σ2E ¼ NkT 2Cv, where Cv is the per-particle heat capacity of the system. Hencethe energy fluctuation for a system, including the weak effect of the surroundings, is givenapproximately by

σE0 ¼ σE 1� 1

2NkT 2Cv

@2 βαð Þ@β2

� : (7)

The second term in square brackets accounts for the effects of the surroundings on thefluctuation of the system energy. Later (specifically, following Equation (13)), we willemploy this formula to distinguish between a small system, for which the influence of thesurroundings is appreciable, and the bulk, for which such effects are negligible.

It is also of interest to study the change in the Gibbs–Boltzmann probability functionof the system due to the effects of the surroundings. From the general framework ofstatistical mechanics, we have Pj ¼ e�βEj=Q for a bulk system and P0

j ¼ e�βE0j=Q0 for a

small one. The latter equation can be written as

P0j ¼ Pje

�β αj�αð Þ: (8)

Thus, the Gibbs–Boltzmann probability for small systems is influenced by the effects ofthe surroundings and differs in general from that for a bulk system, again due to thepresence of the αs whose evaluation requires knowledge of the effects of the surroundingson the energy states.

Strictly speaking, it is necessary to compute the full interaction between the particlesof the system and the surroundings. Because this is practically impossible, we usuallyassume a model potential exerted on the system particles by the surroundings. We denotethis potential by W ¼ PN

i¼1 w rið Þ, where w rið Þ is the external potential exerted on the ithparticle of the system located at ri. At this point we invoke classical statistical mechanicsand write Equation (1) in integral form (discarding the ideal gas part):

Q0 ¼ðe�βU�βWdr3N ¼

ðe�βUe�β

PN

i¼1w rið Þdr3N ; (9)

where U represents the inter-particle potential energy. Let us define f rið Þ ¼ e�βw rið Þ � 1,by which we have

e�βPN

i¼1w rið Þ ¼

YNi¼1

1þ f rið Þð Þ ¼ 1þXNi¼1

f rið ÞþXNi¼1

XNj¼1;j�i

f rið Þf rj� �þ . . .

which, in combination with Equation (9), gives

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Q0 ¼ðe�βU�βWdr3N ¼ Q

ðe�βU

Q

YNi¼1

1þ f rið Þð Þdr3N :

Making use of this, in conjunction with the n-particle distribution function

ρ nð Þb r1; r2; . . . ; rnð Þ of the corresponding bulk system, defined by

ρ nð Þb r1; r2; . . . ; rnð Þ ¼ N !

N � nð Þ!ðe�βU

Qdrnþ1 . . . drN ;

where Q ¼ðe�βUdr3N is the partition function of the bulk system, gives

Q0 ¼ Q 1þXn¼1

1

n!

ðρ nð Þb r1; r2; . . . ; rnð Þ

Yn1

f r1ð Þ . . . f rnð Þdr1 . . . drn" #

; (10)

and hence the free energy of small systems is found to be

As ¼ Ab � 1

βln 1þ

ðρ 1ð Þb r1ð Þf r1ð Þdr1 þ 1

2

ðρ 2ð Þb r1; r2ð Þf r1ð Þf r2ð Þdr1 dr2 þ . . .

� ; (11)

where subscripts s and b denote the small and bulk systems, respectively. Equation (11) isthe main result required to calculate all thermodynamic properties of small systems.Applying it, the outside free energy α is found explicitly as

α ¼ � 1

βln 1þ a1 þ a2 þ . . .ð Þ; (12)

where

a1 ¼ðρ 1ð Þb r1ð Þf r1ð Þdr1; a2 ¼ 1

2

ðρ 2ð Þb r1; r2ð Þf r1ð Þf r2ð Þdr1 dr2; and so on: (13)

At this point we are able to identify more precisely the conditions under which asystem should be considered small. We do so by examining the behaviour of thefluctuations using Equation (7) wherein, based on our adopted picture, a system issmall when, in addition to finiteness of both N and V, the effects of surroundings aresignificant and therefore the second term in Equation (7) cannot be neglected. Because wehave the explicit equation for the outside free energy, Equation (12), we can compute thefluctuations. Noting that the as are dependent on temperature, we have

@2 βαð Þ@β2

¼ �d2a1dβ2

þ d2a2dβ2

þ . . .

1þ a1 þ a2 þ . . .þ

da1dβ þ da2

dβ þ . . .� �2

1þ a1 þ a2 þ . . .ð Þ2 :

Also, noting from Equation (13) that all as are directly proportional to the average bulkdensity of the substance (ρb ¼ N=V ), and therefore also to their partial derivatives with

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respect to β, we write the second partial derivative as @2 βαð Þ=@β2 ¼ D� N=Vð Þ, where Dstands for the proportionality factor. Inserting this into Equation (7) gives

σE0 ¼ σE 1� D

2VkT 2Cv

� �: (14)

Aside from temperature, which is supposed to be well defined and the same for small andbulk systems, and the heat capacity, it is mainly the volume of the system that determineswhether a system can be considered as bulk or must be treated as small. Anotherinteresting conclusion is that, at the points for which critical phenomena occur (i.e. attemperatures for which CV ! 1), the second term vanishes and the effects of thesurroundings disappear completely regardless of the system size. That is, at the criticalpoints small systems behave like the corresponding bulk ones.

3. Results and discussion

To test the developed theory, we consider a 3D particle fluid confined between tworeflecting hard walls oriented normal to one Cartesian direction, with periodic boundaryconditions in the transverse directions. The primary simulation cell used is cuboid, withthe hard walls located at x = 0 and Lx and with square cross-sectional area Ayz normal to x.The potential between particles and the confining walls is defined as

w xð Þ ¼1 x < σ

20 σ

2 � x � Lx � σ21 x > Lx � σ

2

8<: (15)

where σ is the hard sphere diameter. We note that w xð Þ ¼ w Lx � xð Þ, which means thepotential is symmetric. Our main goals are to find the equation of state for this system andcompare it to the corresponding equation of state in the thermodynamic limit, and tocompare the analytical predictions to the results obtained using molecular dynamics (MD)simulations for closely related systems. We do this for two different particle–particlepotentials, namely hard spheres and the 12–6 Lennard-Jones (L-J) function.

We begin the calculation of the outside free energy α by computing the ais. Accordingto the general scheme, we should compute the integrals in Equation (13) to find anapproximation for the confined systems:

a1 ¼ðρ 1ð Þb rð Þ e�βw xð Þ � 1

h id3r;

a2 ¼ 1

2

ððρ 2ð Þb r1; r2ð Þ e�βw xð Þ � 1

h ie�βw x0ð Þ � 1h i

d3r d3r0; etc:

The a1 integral is easy to calculate if we assume that the one-particle distribution function

of a bulk fluid system is constant, that is, ρ 1ð Þb rð Þ ¼ ρb. The result is

a1 ¼ðρ 1ð Þb rð Þ e�βw xð Þ � 1

h id3r ¼ ρb

ðððe�βw xð Þ � 1h i

dxdydz ¼ ρbAyz

ðL0e�βw xð Þ � 1h i

dx

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a1 ¼ ρbAyz

ðL0e�βw xð Þ � 1h i

dx ¼ ρbAyz

ðσ=20

�1ð Þdxþ 0þðLxLx�σ=2

�1ð Þdx" #

¼ �ρbAyzσ:

The higher-order integrals are difficult to compute because of the correlations betweenparticle pairs, triples, etc. As the simplest approximation, we shall assume that allcorrelation functions involving two or more particles are unity. This is a reasonableapproximation for short-range particle-wall potentials because the relevant integrandsare non-zero only when two or more particles are in contact with the walls. Under the

stated assumption, one finds a2 ¼ ρ2bA2yzσ

2� �

=2, and more generally an ¼ ρnbAnyzσ

n� �

=n!.

Inserting all as into Equations (12) and (13) gives

α ¼ �kTa1 ¼ kTρbAyzσ: (16)

From this point onward we set σ = 1 and scale all distances by σ. Note that the densityof N hard spheres with diameter σ confined in a cuboid with volume AyzLx is~ρs ¼ N= Ayz Lx � 1ð Þ�

, whereas for the corresponding bulk fluid it is ~ρb ¼ N=AyzLx.Thus, if the density of a confined fluid is defined to be ρs ¼ N=AyzLx, then the densityfor the corresponding bulk hard spheres is ρb ¼ N= Ayz Lx þ 1ð Þ� ¼ ρs= 1þ L�1

x

� �.[23]

Using this we obtain our final expression for the free energy of confined hard spheresinteracting with hard walls:

As ρs;Tð Þ ¼ Ab ρb; Tð Þ þ kTN

Lx þ 1: (17)

Note that the second term is directly proportional to the number of particles but isinversely proportional to the system size (i.e. Lx). This demonstrates clearly our claimthat as Lx goes to infinity, the effect of the surroundings vanishes and the thermalproperties, including the internal energy and entropy, become extensive.

Using Equation (17) one can compute all thermodynamic quantities for the system.For example, the internal energy has not been changed in this approximation:Es ρs; Tð Þ ¼ Eb ρb; Tð Þ, whereas the equation of state of the confined hard spheres willbe altered considerably by the presence of the hard walls at x = 0 and Lx. At constanttemperature and number of particles dAs ¼ �PdV ¼ �Pd AyzLx

� � ¼ �PAyzdLx�PLxdAyz. Because for our problem the surface area Ayz is constant, we have Px ¼�dAs=AyzdLx for the pressure in the x-direction. We thus find

Px ρs; Tð Þ ¼ Pb ρb; Tð Þ þ NT

Ayz Lx þ 1ð Þ2 : (18)

Thus, if we know the equation of state for the bulk system, then we can calculate thestress in the small system along the direction of confinement. For example, combiningEquation (18) with the Carnahan–Starling equation of state for bulk hard spheres [24]leads to the following equation for the x-component of the pressure tensor:

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Px ¼1216

π3N3

A3yz Lxþ1ð Þ3 � 1

36π2N2

A2yz Lxþ1ð Þ2 � 1

6πN

Ayz Lxþ1ð Þ � 1

� NT

16

πNAyz Lxþ1ð Þ � 1

h i3Ayz Lx þ 1ð Þ

þ NT

Ayz Lx þ 1ð Þ2 : (19)

It is interesting to rewrite Px in terms of the small system density. The result is

Px ¼1

216π3ρ3s1þ 1

Lxð Þ3 �136

π2ρ2s1þ 1

Lxð Þ2 �16

πρs1þ 1

Lxð Þ � 1

� ρsT

16

πρs1þ 1

Lxð Þ � 1

� 31þ 1

Lx

� � þ ρsT

Lx þ 1ð Þ 1þ 1Lx

� � : (20)

This equation shows transparently that Px ! PCS when Lx ! 1 (noting that in this limitρs ! ρb), where PCS is the Carnahan–Starling equation for bulk hard spheres. We see thatthe pressure exerted on the walls depends not only on the density and temperature of thefluid but also on the length of the container in the x-direction (i.e. the finite volume).

To assess the reliability of Equations (19) and (20), we performed MD simulations for(nearly) hard spheres and L-J fluids confined between reflecting walls. For the ‘hardsphere’ simulations, we used the cut-and-shifted 50–49 Weeks–Chandler–Anderson(WCA) potential [25]

u50;49 rð Þ ¼ 50 5049

� �49ε σ

r

� �50 � σr

� �49h iþ ε r< 50

49

� �σ

0 r � 5049

� �σ

(; (21)

which at dimensionless temperature T = 1.5 yields a close approximation to hard spherebehaviour. Henceforth, for simplicity we refer to the 50–49 WCA potential as a hardsphere potential. All simulations were performed using the Large-scale Atomic/MolecularMassively Parallel Simulator (LAMMPS) [26] code. We set σ = 1 and scaled all distancesby σ; all particle masses were set to unity; and ε = 1. Initial particle positions were chosensuch that all particles were located within the primary simulation cell, with particlesseparated by at least σ/2 (edge-to-edge). Initial velocities were sampled from the Maxwelldistribution for the reduced temperature T = 1.5. Isochoric–isothermal (NVT) simulationswere performed using the Nosé–Hoover [27] thermostat with the thermostat couplingparameter set to 0.1 in reduced time units. Trajectory integration was performed for 108

time steps using the velocity Verlet integrator with reduced time step Δt = 0.001. The final5.0 × 107 steps of each trajectory were used to calculate the average values and samplestandard deviations for the x-component of the pressure tensor Px and the internal energyE. The quantity Px corresponds to the Pxx component of the pressure tensor, calculated inLAMMPS as the sum of the components of the kinetic energy and virial tensors.

3.1. Hard sphere particle–particle potential

In the first of three sets of simulations to be discussed for the case of hard sphere particle–particle interaction, the transverse lengths were set to Ly = Lz = 20, the number of particleswas set to N = 12,500 and Lx was varied between 34.7222 and 625.0000. Figure 1contains the predictions for Px obtained from MD (circles) and the present analyticaltheory (specifically Equation (20), solid curve). The Carnahan–Starling equation of state

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for bulk hard spheres is also included (dashed curve) to show clearly the differencebetween the isothermal equations of state of the confined and bulk hard spheres. Errorbars for the MD results correspond to one sample standard deviation. The good overallagreement between the MD results and Equation (20) at all densities is apparent.

In Section 2 we showed, based on analysis of fluctuations, that what largely distin-guishes a small system from its bulk counterpart is the volume relative to the number ofparticles. Based on Equation (20), Px depends on temperature, density and Lx (i.e. thevolume under the constraint that LyLz = Ayz = constant). To test this general assertionquantitatively for the confined hard spheres, we performed MD simulations for systemswith fixed surface area Ayz = LyLz = 202 = 400 and N and Lx varied in tandem such that theratio ρ = N/(LxAyz) = 0.5. Table 1 contains values of Px obtained from the simulations andusing Equation (20) for seven values of N between 1250 and 125,000 and also for the 3Dperiodic case with N = 12,500. The agreement between MD and Equation (20) is good,especially for larger values of Lx. This confirms our assertion that at constant density, it islargely the volume of the system that determines whether the system should be consideredsmall or bulk. The discrepancies between MD and theory are larger for smaller values ofLx because the approximations made to obtain Equations (19) and (20), in particularneglecting the two-particle, three-particle, etc. correlations, are less valid for smallersystems.

The third set of MD simulations for the confined hard spheres was performed forN = 12,500 with Lx and Ayz varied in tandem such that the density is ρ = 0.5 (i.e. variablesurface-to-volume ratio of the confinement). Table 2 contains results from MD and usingEquation (20) for three cases of confined fluid plus the bulk case in which 3D periodic

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

Px

Carnahan–Starling EquationPresent TheoryMD Results

Figure 1. Normal pressure component Px versus density for hard spheres (with N = 12,500 andAyz = 400) confined between reflecting hard walls along x, obtained using MD (circles), from thepresent theory (solid curve, Equation (20)), and from the Carnahan–Starling equation for bulk hardspheres (dashed curve). Error bars for the MD results correspond to one sample standard deviation.

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boundaries were applied. The MD results together with the predictions of theory showthat, for constant density, at higher Lx the confined system behaves more like the bulk one,whereas at lower Lx the difference in their behaviour becomes more significant.

3.2. Lennard-Jones 12–6 particle–particle potential

The second system studied is the L-J fluid confined between hard walls. The dimension-less L-J parameters are σ = 1 and ε = 1. Because Equation (16) is unaffected by the changefrom hard spheres to L-J for the particle–particle interaction, the internal energy is notchanged and therefore Es ρs; Tð Þ ¼ Eb ρb; Tð Þ and the equation of state is still given byEquation (18). Indeed, the only difference that arises in the theoretical treatment is that theCarnahan–Starling equation for bulk hard spheres must be replaced by the equation ofstate for the bulk L-J system, which we calculated using MD for the specific densitiesstudied. Comparisons between MD results and the present theory for this case are shownin Figures 2 and 3. Figure 2 contains Px versus ρ and Figure 3 U versus ρ. In both figuresthe MD results are shown as circles, and predictions from the present theory are shown assolid curves. The results for the bulk L-J fluid are also added as dashed curves for

Table 2. Values of Px obtained from MD and using Equation (20) for the confined hard spheresystem at constant density ρ = 0.5 and number of particles N = 12,500.

Ayz LxPx

MDPx

Equation (20) Percentage errora

40 × 40 (PBC)b 15.625 2.45 (8)c 2.4494 −0.0210 × 10 250 2.44 (8) 2.4272 −0.5220 × 20 62.5 2.40 (8) 2.3702 −1.2440 × 40 15.625 2.27 (8) 2.1669 −4.54

Notes: aDefined as Px% ¼ Px;theory�Px;MD

Px;MD� 100.

bResult for bulk fluid.cNumbers in parentheses indicate the 1σ uncertainty in the digit preceding them, based on the sample standarddeviation from the MD simulations.

Table 1. Values of Px obtained from MD and using Equation (20) for the confined hard spheresystem at constant density ρ = 0.5 and square cross-sectional area Ayz = (20)2 = 400.

N LxPx

MDPx

Equation (20) Percentage errora

12,500 (PBC)b 62.5 2.45 (8)c 2.4494 −0.02125,000 625 2.44 (2) 2.4389 −0.0562,500 312.5 2.44 (4) 2.4311 −0.3625,000 125 2.43 (6) 2.4079 −0.9112,500 62.5 2.40 (8) 2.3702 −1.246250 31.25 2.4 (1) 2.2983 −4.242500 12.5 2.2 (2) 2.1068 −4.241250 6.25 2.1 (2) 1.8511 −11.85

Notes: aDefined as Px% ¼ Px;theory�Px;MD

Px;MD� 100.

bResult for bulk fluid.cNumbers in parentheses indicate the 1σ uncertainty in the digit preceding them, based on the sample standarddeviation from the MD simulations.

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0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

Px

Bulk Lennard-Jones Equation of State Present TheoryMD Results

Figure 2. Normal pressure component Px versus density (with N = 12,500 and Ayz = 400) for the L-J particles confined between reflecting hard walls along x, obtained using MD (circles), from thepresent theory (solid curve, Equation (18)), and from the bulk L-J equation of state (dashed curve).Error bars for the MD results correspond to one sample standard deviation.

0 0.2 0.4 0.6 0.8 1–4

–3

–2

–1

0

1

2

E/N

Bulk Lennard-Jones Present WorkMD Data

Figure 3. Reduced internal energy per particle E/Nɛ versus density (with N = 12,500 andAyz = 400) for the L-J particles confined between reflecting hard walls along x, obtained usingMD (circles), from the present theory (solid curve), and from the bulk L-J internal energy (dashedcurve). Error bars for the MD results correspond to one sample standard deviation.

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comparison. As before, the agreement among all predictions for Px is quite good at lowdensity (see Figure 2), with deviations evident and increasing for ρ > 0.5. In contrast tothe hard sphere results discussed in Section 3.1, here the analytical model predicts anequation of state softer than obtained from MD, which indicates that for better results weshould find and employ better approximations (which we do not pursue in the presentstudy). The results for the internal energy versus density (see Figure 3) exhibit goodagreement for all densities studied. The present theory agrees with the MD results towithin 1σ uncertainty for all but the highest density studied.

It is interesting to note that all the analytic results shown are obtained based on thesimplest level of approximation, that is, the assumption that ρ nð Þ r1; r2; . . . ; rnð Þ ¼ ρnb for alln. Obviously, better approximations for the correlation functions should yield ‘improved’results, especially if the methods are extended to the case of long-range particle-wallpotentials.

4. Conclusions

Small systems have been described using statistical mechanics by evaluating thermalproperties as completely as possible prior to taking the thermodynamic limit. The effectsof the surroundings on the system are considered explicitly but disappear naturally as thethermodynamic limit is approached. We derived some general relations within thistheoretical framework that can be applied to closed, small systems. We applied theserelations to predict the variation of the macroscopic properties of hard sphere and 12–6Lennard-Jones fluids periodic in two directions but confined between reflecting hard wallpotentials along the third. The approximate analytical free energies of these systems wereobtained first, from which the equations of state and internal energies were derived.Comparisons between the present theory and MD results for bulk hard spheres andLennard-Jones systems demonstrated good overall agreement.

AcknowledgementsMK thanks the Islamic Azad University, Ayatollah Amoli Branch, for its generous financial supportduring the research project. The authors thank Prof. Donald L. Thompson for useful discussions.

Disclosure statementNo potential conflict of interest was reported by the authors.

FundingThis research was supported by the US Army Research Office [grant number W911NF 12-1-0146].

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