THERMODYNAMIC LIMIT FOR POLYDISPERSE FLUIDS

S. Banerjee^{∗}^{1,2}, R. B. Griffiths^{1}, and M. Widom^{1},

1Department of Physics, Carnegie Mellon University, Pittsburgh, Pa. 15213 2Institute for Physical Science and Technology, University of Maryland, College Park, Md.

20740 ^{†} (October 31, 2000)

Abstract

We examine the thermodynamic limit of fluids of hard core particles that are polydisperse in size and shape. In addition, particles may interact magnetically. Free energy of such systems is a random variable because it depends on the choice of particles. We prove that the thermodynamic limit exists with probability 1, and is independent of the choice of particles. Our proof applies to polydisperse hard-sphere fluids, colloids and ferrofluids. The existence of a thermodynamic limit implies system shape and size independence of thermodynamic properties of a system.

Keywords: Thermodynamic limit, Free energy, Colloids, Random, Polydisperse, hard-core, Ferrofluid, Polar fluid, Magnetic fluid

∗Electronic address: sb7w@andrew.cmu.edu †Present address

I. INTRODUCTION

The free energy of a system of particles, as defined in statistical mechanics, depends explicitly on the the size and shape of the container holding the particles (the “system shape”) through the dependence of the partition function on these quantities. Thermodynamics, in contrast, assumes a free energy density independent of the system shape and size. For typical systems, as the system size grows the contribution of the boundary to the statistical free energy becomes negligible compared with the bulk contribution. In the thermodynamic limit (system size going to infinity) the free energy density becomes independent of system shape and the boundary conditions.

Ruelle [1] and Fisher [2] proved the existence of thermodynamic limits for a large class of fluids and solids with interactions that fall off faster than r^{−3} at large separation. Interactions which fall off as r^{−3} or slower, complicate the thermodynamic limit. For neutral systems with coulombic interactions, Lieb [3] proved the existence of a thermodynamic limit using the screening of the interaction at large distances. For dipolar interactions, Griffiths [4] proved the existence of a thermodynamic limit for dipolar lattices using magnetization reversal in a domain. The present authors [5] extended Griffiths’ proof to several non-lattice models with dipolar interactions.

The systems discussed above contain identical particles with specified interactions. In this paper we study random systems, fluids consisting of hard core particles that are polydisperse in size and shape. A proof of the thermodynamic limit for polydisperse fluids is needed because they occur abundantly in nature and technology, and because their thermodynamic properties are of great interest [6–9].

For a system where the particles are chosen at random from a distribution of size and shape, the free energy is a random variable because it depends on the choice of particles. In general, proof of a thermodynamic limit depends upon the subadditivity of free energy as the system size grows. Thus we must exploit the theory of subadditive random variables [10]. Furthermore, depending on the size and shape distribution of particles, a particular choice of particles may or may not pack into the available volume. Due to these complications, the proof requires a restriction on the particle size and shape distribution in the form of a “packing condition”. Thermodynamic limits of random systems have been investigated previously [11–13]. Those results, however, apply only to lattice models.

For our calculations, we fix the choice of particles in a canonical ensemble. Thus, we study “quenched” systems. In contrast, in an “annealed” system the relative concentration of particles of various types are controlled using chemical potentials [8,14]. The quenched and annealed free energies have been studied extensively in the context of spin glass systems [13,15–17]. The quenched free energy is calculated by fixing the interaction between spins whereas the annealed free energy is calculated by averaging the partition function over the random interactions first. The quenched free energy is the desirable quantity since in a real system the interactions are fixed. However, it is often easier to calculate the annealed free energy. It is claimed that for polydisperse fluids the annealed free energy equals the quenched free energy [14].

Polydisperse fluids often phase separate into different liquid or even solid phases with different concentration of particle types in each phase [6–9]. In our terminology quenching fixes the overall concentration of various particle types and not the particle concentration in each phase. This terminology should not be confused with that often used in the literature [9] of polydisperse colloidal systems to describe systems that have reached equilibrium (annealed) in relative particle concentration between the two phases, and systems that have not reached equilibrium (quenched).

In this paper we prove that the free energy density F/V of any random choice of particles goes to some finite limiting value f with probability 1, independent of the choice of particles, as the system size goes to infinity. In Sec. II we carry out the proof for fluids of polydisperse hard core particles. In Sec. III we extend the proof to fluids of hard core particles with magnetic interactions. These systems may be liquids such as ferrofluids [19]. Finally, in Sec. IV we summarize our results and discuss some models that are not covered by our proof.

II. HARD CORE PARTICLES POLYDISPERSE IN SHAPE AND SIZE

In this section we show that, for a fluid of hard core particles polydisperse in size and shape contained within a box of any shape, the free energy density goes to a limit with probability 1 as we take the system size to infinity. Our proof involves three main steps. First, we consider systems of particles contained within cubical containers and show that the free energy density of systems shaped as cubes goes to a limit with probability 1 as the system size goes to infinity. In the second step, we show that this limiting free energy density is independent of the choice of particles. In the the final step, we prove system shape independence of the limit by showing that the limiting free energy density for a system of any arbitrary shape is the same as that of a system shaped as a cube.

In Sec. II A we define our model that includes the particle size and shape distribution, a choice of particles from the distribution, the Hamiltonian, the partition function, and the free energy. We impose an artificial upper bound on the free energy to deal with choices of particles that do not fit inside given containers without overlapping. This section also defines a packing condition on the particle size and shape distribution which guarantees that with high probability the particles fit within containers without overlapping.

In Sec. II B we consider systems shaped as cubes and show that the free energy density satisfies the conditions of a subadditive ergodic theorem [10] so that the free energy density of a cube-shaped system goes to a limit for almost every choice of particles as the cube size goes to infinity. To make use of the subadditive ergodic theorem we define a lattice and divide space into basic cubes defined on the lattice. A choice of particles identifies some specific particles associated with each basic cube. Any box defined on the lattice holds particles assigned to the basic cubes inside the box. The lattice, the basic cubes, and the boxes, are a device for assigning particles to containers in a random but well-defined fashion and employing the subadditive ergodic theorems in the literature.

In Sec. II C we show that the variance of the free energy density around its mean vanishes as the cube size goes to infinity, proving that the limit is in fact independent of the specific choice of particles. In Sec. II D we prove the system shape independence of the free energy density by filling up an arbitrary shaped system with cubes and showing that the free energy density of any arbitrary shape has the same limit as that of a cube.

A. Definitions and assumptions

Let X be a set of particle sizes and shapes (see Fig. 1) with some probability measure μ_{1} defined on it. Consider a container S, of volume V_{S} and a choice c_{N_{S}} of N_{S} hard core particles contained in S, independently chosen from X, one at a time. Denote the set of all such choices by C_{N_{S}}. The probability measure μ_{N_{S}} defined on C_{N_{S}} is the N_{S}-fold product of the probability measure μ_{1} defined on X. Let c denote a choice of an infinite sequence of particles and C denote the set of all possible choices. The probability measure μ defined on C is the infinite-fold product of μ_{1}.

The particles can be placed anywhere inside S with any orientation as long as the Hamiltonian H_{S} = H_{S}^{HC} is finite, where

H_{S}^{HC}(c_{N_{S}}) = \{ \begin{matrix}0  if no particles overlap each other or the boundaries of S \\ +∞ otherwise, \end{matrix} (1)

is the hard core repulsion between the particles. Although not indicated explicitly, H_{S}^{HC}(c_{N_{S}}) depends on the particle center of mass positions {r_{i}} and the particle orientations {Ω_{i}}. Define the partition function

Z_{S}(c_{N_{S}}) = \frac{1}{Ω^{N_{S}}N_{S}!} ∫_{S} ∏_{i=1}^{N_{S}} dr_{i}dΩ_{i}e^{−H_{S}(c_{N_{S}})/k_{B}T} ,  (2)

where Ω = 4π is the integral of dΩ over all possible orientations of a particle. The prefactor N_{S}! is chosen to make the free energy extensive. This factor may be modified depending on the details of the set X to account for the entropy of mixing (see appendix A). However, the factor N_{S}! suffices as long as we do not “unmix” the particles [6,24].

Because the particle sizes and shapes are drawn at random, sometimes a choice of particles may not fit into the box S without overlapping. Some choices that fit into S may have insufficient room to move. Such choices have large, physically unrealistic, free energies. We wish to remove these unphysical choices from consideration. However, their exclusion causes mathematical inconvenience, since the probability measure for choices would no longer be the product of the single particle probability measure on X. On the other hand, including such choices causes the partition function to vanish for some c_{N_{S}} ∈ C_{N_{S}}, leading to infinite free energies.

To handle these difficulties, we define an arbitrary threshold z_{0}^{N_{S}}, for the partition function of a container S. The constant z_{0} > 0 can be interpreted as a (geometric) mean free phase space volume per particle. We define the free energy in the following artificial manner,

F_{S}(c_{N_{S}}) = \{ \begin{matrix}−N_{S}k_{B}T ln z_{0}  if Z_{S}(c_{N_{S}}) < z_{0}^{N_{S}} \\ −k_{B}T ln Z_{S}(c_{N_{S}})  otherwise. \end{matrix}  (3)

Provided that the particle size and shape distribution X obeys the packing condition discussed below, choices with Z_{S}(c_{N_{S}}) < z_{0}^{N_{S}} occur sufficiently infrequently that the limiting free energy density is independent of the arbitrary constant z_{0} (see Sec. II C).

Consider a cube Γ of volume V_{Γ} containing N_{Γ} particles such that ρ_{Γ} = N_{Γ}/V_{Γ}. Let P_{Γ} be the set of choices c_{N_{Γ}} ∈ C_{N_{Γ}} for which Z_{Γ_{Γ}}(c_{N_{Γ}}) ≥ z_{0}^{N_{Γ}}, where N_{Γ} is the number of particles contained in Γ. Formally,

P_{Γ} ≡ {c_{N_{Γ}} ∈ C_{N_{Γ}} | Z_{Γ}(c_{N_{Γ}}) ≥ z_{0}^{N_{Γ}}}  (4)

is the set of choices that “pack” in Γ. Let {Γ} be a sequence of cubes. Our packing condition on the particle size and shape distribution X requires that for any z_{0} > 0 there exists a “critical packing density” ρ^{∗}(z_{0}) > 0, such that for any density ρ_{Γ} ≤ ρ < ρ^{∗},

lim_{V_{Γ}→∞} μ_{N_{Γ}}(P_{Γ}) = 1.  (5)

Here μ_{N_{Γ}} is the probability measure associated with the sample space C_{N_{Γ}}. See appendix A for examples of some particle size and shape distributions that satisfy the packing condition.

B. Thermodynamic limit for rectangular and cubical boxes

In this section we show that the free energy density of rectangular boxes goes to a limit with probability 1 (i.e. for almost every c ∈ C) as the system size goes to infinity. We assume that the particle distribution X obeys the packing condition Eq. (5) and that the particle density ρ < ρ^{∗}. For the proof we make use of a subadditive ergodic theorem (theorem 2.7 in Ref. [10]) by Akcoglu and Krengel.

Consider a tiling of non-negative real space ℝ_{+}^{3} by “basic” cubes with each side of length l_{0}. The vertices of the basic cubes define a lattice L such that any point in L can be written as (n_{1}, n_{2}, n_{3})l_{0}, where n_{i} ∈ ℤ_{+} (nonnegative integers). For vectors a = (a_{i}) and b = (b_{i}) in L, the half-open interval [a, b) denotes the set {u|u = (u_{i}) ∈ L, a_{i} ≤ u_{i} < b_{i}}. Physically, it represents a rectangular box in ℝ_{+}^{3}. We call a the “base” of the box. Let I be the class of all such rectangular boxes. Denote the cube [w, w + 2^{k}d) by Γ_{k}^{w}, where w is any vector in L, d is the vector (l_{0}, l_{0}, l_{0}) and k is any nonnegative integer. Any box I ∈ I can be completely filled with the “basic” cubes Γ_{0}^{w}, where w runs over the intersection of L with I.

Let c be a choice (infinite sequence) of particles. Break up c into successive sequences of N_{0} particles and put one such sequence into each basic cube Γ_{0}^{w} using a one-to-one map from a one-dimensional rectangular array (of sequences of particles) to a three dimensional rectangular array (of basic cubes). Denote the segment of particles in Γ_{0}^{w} by c_{N_{0}}^{w}. For any vector u ∈ L, define a transformation τ_{u} on C such that c′ = τ_{u}(c) is another choice in C with c_{N_{0}}^{w}′ = c_{N_{0}}^{w+u} for every w ∈ S. Since μ is a product measure, τ_{u} is a measure preserving transformation on C.

Consider a rectangular box I ∈ I, of volume V_{I}. Let the choice of particles assigned to I be the union of choices c_{N_{0}}^{w} assigned to basic cubes Γ_{0}^{w} contained in I (see Fig. 2). The number of particles in I is N_{I} = N_{0}V_{I}/V_{Γ_{0}}. The subadditive ergodic theorem requires that for each choice c ∈ C the free energy F_{I} satisfies the following three conditions:

(i) Translation invariance: For every box I ∈ I and every vector u ∈ L

F_{I} ∘ τ_{u} = F_{I+u}.  (6)

(ii) Subadditivity: For any box I ∈ I composed of nonoverlapping boxes I_{1}, ..., I_{n} ∈ I

F_{I} ≤ ∑_{i=1}^{n} F_{I_{i}}.  (7)

(iii) Lower bound: For some constant ω_{A} and for all I ∈ I

\frac{1}{V_{I}} F_{I} ≥ ω_{A},  (8)

where V_{I} is the volume of box I. Then, the theorem states that for a sufficiently regular sequence of rectangular boxes {I_{k}}, of increasing size, the free energy density F_{I_{k}}/V_{I_{k}} goes to a limit f(c) almost everywhere in C as k → ∞. Note that this limit may depend on the choice of the particles c. A sequence of rectangular boxes {I_{k}} of increasing size is sufficiently regular (as defined by Akcoglu and Krengel [10]) if there exists another sequence of rectangular boxes {I′_{k}} such that box I′_{k} fully covers the box I_{k}, the ratio V_{I_{k}}/V_{I′_{k}} is always greater than or equal to some fixed nonzero constant, and lim_{k→∞} I′_{k} = ℝ_{+}^{3}.

From our definition of the transformation τ_{u}, the free energy trivially satisfies (6). For any box I ∈ I composed of nonoverlapping boxes I_{i} ∈ I, the partition functions satisfy

Z_{I} ≥ ∏_{i} Z_{I_{i}},  (9)

because each particle has more room to move inside I than an individual box I_{i} (see Ref. [2] for the detailed argument). The free energy therefore satisfies subadditivity (7). The free energy also satisfies the lower bound (8) with ω_{A} = k_{B}Tρ ln(ρ) which is temperature times the ideal gas entropy per particle.

Hence, the free energy satisfies all the conditions of Akcoglu and Krengel’s subadditive ergodic theorem. To apply this to the special case of a cube, construct a sequence of cubes of increasing size {Γ_{k}} such that V_{Γ_{k}} → ∞ as k → ∞ and the distance of the bases of the cubes from the origin grows less rapidly than their size. This ensures that the sequence is sufficiently regular in the sense of Akcoglu and Krengel [10]. Then, according to the theorem the free energy density f_{Γ_{k}} ≡ F_{Γ_{k}}/V_{Γ_{k}} goes to some limit f_{Γ}(c), the free energy density in a cube, for almost every c ∈ C.

It is interesting to note that the sequence of cubes {Γ_{k}} can be chosen so that no cubes overlap with each other and thus have no particles in common. This suggests that the limiting free energy density f_{Γ}(c) should not depend on c. We demonstrate that fact in the following section.

C. Choice independence of the limiting free energy density

In this section we prove that the limiting free energy density f_{Γ}(c) is independent of the choice of particles c. We show that the free energy density averaged over all choices goes to a limit and its variance goes to zero as the cube size goes to infinity. This implies that the limit is same for any choice with probability 1. Furthermore, the limiting value and its variance are independent of z_{0}, the arbitrary constant introduced in defining free energy.

Let ⟨f_{Γ_{k}}⟩ denote the free energy density of a cube Γ_{k} averaged over all choices c_{N_{k}} ∈ C_{N_{k}} of N_{k} particles in Γ_{k}. From subadditivity (7) of the free energy it follows that

⟨f_{Γ_{k+1}}⟩ ≤ ⟨f_{Γ_{k}}⟩.  (10)

The sequence of average free energy densities {⟨f_{Γ_{k}}⟩} is monotonically decreasing and has a lower bound (8). The average free energy density therefore goes to some limit f̄_{Γ} as k → ∞.

To show that the variance of the free energy density of cubes vanishes as the cube size goes to infinity, we construct a cube Γ_{k+1} by putting together eight nonoverlapping cubes Γ_{k}. By using subadditivity (7), the lower bound (8) and the existence of a limiting average free energy density for cubes we show (see appendix B) that for any ϵ > 0 there exists a sufficiently large k_{0} such that for any k > k_{0},

⟨(f_{Γ_{k+1}} − ⟨f_{Γ_{k+1}}⟩)^{2}⟩ ≤ \frac{1}{8}⟨(f_{Γ_{k}} − ⟨f_{Γ_{k}}⟩)^{2}⟩ + ϵ.  (11)

Iterating this procedure n times we write

⟨(f_{Γ_{k+n}} − ⟨f_{Γ_{k+n}}⟩)^{2}⟩ ≤ \frac{1}{8^{n}} ⟨(f_{Γ_{k}} − ⟨f_{Γ_{k}}⟩)^{2}⟩ + ∑_{j=0}^{n−1} \frac{1}{8^{j}} ϵ.  (12)

For a fixed value of k there exists n_{0} such that, for all n > n_{0}, the first term on the right hand side in the inequality is less than ϵ. The second term on the right hand side approaches (8/7)ϵ from below. Hence, for any given ϵ, there exist k_{0} and n_{0} so that for any k > k_{0} and n > n_{0}, the variance in the free energy density satisfies the upper bound

⟨(f_{Γ_{k+n}} − ⟨f_{Γ_{k+n}}⟩)^{2}⟩ ≤ \frac{15}{7}ϵ.  (13)

For a sufficiently large system the variance of the free energy density becomes arbitrarily small. We previously proved (Sec. II B) that f_{Γ_{k}} goes to a limit f_{Γ}(c) for almost every choice c of particles. A vanishing variance implies (by using Chebyshev’s inequality [18]) that the probability of f_{Γ}(c) differing from f̄_{Γ} by more than some arbitrarily small fixed amount, also vanishes. The values of f_{Γ}(c) thus converge “in probability” to f̄_{Γ}. The existence of the limit f_{Γ}(c) with probability 1 together with convergence in probability to f̄_{Γ} implies that f_{Γ}(c) equals f̄_{Γ} with probability 1. In appendix C we show that f̄_{Γ} is a convex and continuous function of the density ρ.

Finally, we show that the limit f̄_{Γ} and the variance are independent of the arbitrary constant z_{0}. Let P_{Γ_{k}} be the set of choices that pack in a cube Γ_{k} as defined in Sec. II A. Let Q_{Γ_{k}} be the complement of P_{Γ_{k}} so that μ_{N_{k}}(P_{Γ_{k}}) + μ_{N_{k}}(Q_{Γ_{k}}) = 1. Write the free energy density for a cube Γ_{k} averaged over all choices as

f̄_{Γ_{k}} = ∫_{P_{Γ_{k}}} dμ_{N_{k}} f_{Γ_{k}} + ∫_{Q_{Γ_{k}}} dμ_{N_{k}} f_{Γ_{k}}.  (14)

From the definition of free energy (3) it follows that

f̄_{Γ_{k}} = ∫_{P_{Γ_{k}}} dμ_{N_{k}} f_{Γ_{k}} − k_{B}T ln z_{0} μ_{N_{k}}(Q_{Γ_{k}}).  (15)

By the packing condition, μ_{N_{k}}(P_{Γ_{k}}) → 1 and μ_{N_{k}}(Q_{Γ_{k}}) → 0, as k → ∞. Hence f̄_{Γ} is independent of z_{0}. Now write the variance of the free energy density as

⟨(f_{Γ_{k}} − f̄_{Γ_{k}})^{2}⟩ = ∫_{P_{Γ_{k}}} dμ_{N_{k}} (f_{Γ_{k}} − f̄_{Γ_{k}})^{2} + ∫_{Q_{Γ_{k}}} dμ_{N_{k}} (f_{Γ_{k}} − f̄_{Γ_{k}})^{2}.  (16)

From the definition of free energy (3) it follows that

⟨(f_{Γ_{k}} − f̄_{Γ_{k}})^{2}⟩ = ∫_{P_{Γ_{k}}} dμ_{N_{k}} (f_{Γ_{k}} − f̄_{Γ_{k}})^{2} + (k_{B}T ln z_{0} + f̄_{Γ_{k}})^{2} μ_{N_{k}}(Q_{Γ_{k}}).  (17)

Since μ_{N_{k}}(Q_{Γ_{k}}) → 0 as k → ∞, because of the packing condition (5), the contribution of the choices c_{N_{k}} ∈ Q_{Γ_{k}} to the variance vanishes. Hence the variance is independent of z_{0}. We can then remove the free energy threshold in Eq. (3) by taking the limit of z_{0} → 0.

D. System shape independence of the limiting free energy density

Now we prove that for a sufficiently regular (in the sense of Fisher [2]) sequence of shapes {S_{j}}, the free energy density goes to a limit as V_{S_{j}} → ∞, provided that the density of particles ρ = N_{S_{j}}/V_{S_{j}} remains fixed. Note that the volumes V_{S_{j}} must be adjusted so that N_{S_{j}} is an integer. To prove shape independence we show that the limiting free energy density satisfies the bounds

lim_{j→∞} sup f_{S_{j}} ≤ f̄_{Γ}  (18)

and

lim_{j→∞} inf f_{S_{j}} ≥ f̄_{Γ}  (19)

with probability 1. To show these bounds we use a technique similar to that of Fisher in his proof for identical particles.

Consider a maximal filling of the shape S_{j} by n_{jk} cubes Γ_{k}^{w} so that each of the cubes Γ_{k}^{w} is fully contained within S_{j} (see Fig. 3). The length l_{0} of the side of the basic cubes Γ_{0} is chosen such that there exists an integer N_{0} such that N_{0}/l_{0}^{3} = ρ. This ensures that it is possible to achieve density ρ in Γ_{k}^{w} exactly. The total number of particles in S_{j} must equal N_{S_{j}} = ρV_{S_{j}}. Therefore, in general, not all n_{jk} cubes contain equal numbers of particles. Let there be two types of cubes, A and B, such that type A cubes contain N_{jk} = [N_{S_{j}}/n_{jk}] particles and type B cubes contain N_{jk} + 1 particles independently chosen from the particle distribution X. Here [y] denotes the greatest integer less than or equal to y. Note that because of our choice of l_{0}, N_{jk} must equal 2^{k}N_{0} plus some nonnegative integer. To ensure that the number of particles in S_{j} is N_{S_{j}}, the numbers of type A and type B cubes (respectively n_{jk}^{A} and n_{jk}^{B}), obey

N_{S_{j}} = n_{jk}^{A}N_{jk} + n_{jk}^{B}(N_{jk} + 1)  (20)

and n_{jk} = n_{jk}^{A} + n_{jk}^{B}.

Let ρ_{jk}^{A} = N_{jk}/V_{Γ_{k}} and ρ_{jk}^{B} = (N_{jk} + 1)/V_{Γ_{k}} be the density of particles in cubes of type A and B respectively, and ϕ_{jk}^{A} = n_{jk}^{A}/n_{jk} and ϕ_{jk}^{B} = n_{jk}^{B}/n_{jk} be the fractions of cubes of type A and B respectively. Define n_{jk}V_{Γ_{k}}/V_{S_{j}} = 1 − ζ_{jk}, so that ζ_{jk} is the fraction of volume in S_{j} unfilled by the cubes Γ_{k}^{w}. The density ρ of particles in S_{j} is related to the densities of particles in the cubes by

ρ = (1 − ζ_{jk})(ϕ_{jk}^{A}ρ_{jk}^{A} + ϕ_{jk}^{B}ρ_{jk}^{B}).  (21)

The density ρ_{jk}^{A} must always be greater than or equal to ρ because if ρ_{jk}^{A} < ρ, then ρ_{jk}^{B} ≤ ρ (since type B cubes have only one more particle than type A cubes), and Eq. (21) cannot be satisfied for ζ_{jk} > 0. In the event that ζ_{jk} = 0 we have ϕ_{jk}^{A} = 1 and the equality ρ_{jk}^{A} = ρ holds.

Let c be a choice (infinite sequence) of particles. Put the first N_{S_{j}} particles of the sequence into S_{j} and denote them by c_{S_{j}}. Split c_{S_{j}} into n_{jk}^{A} successive sequences of N_{jk} particles followed by n_{jk}^{B} successive sequences of N_{jk} + 1 particles. Label these sequences as c_{N_{jk}}^{w} and c_{N_{jk}+1}^{w} (depending on their length) using a map from the one-dimensional array (of sequences of particles) to a three-dimensional array (of cubes Γ_{k}^{w}). Put particles sequences labeled by superscript w into cube Γ_{k}^{w}. From the subadditivity of free energy it follows that

F_{S_{j}}(ρ, c_{S_{j}}) ≤ ∑_{w∈A} F_{Γ_{k}^{w}}(ρ_{jk}^{A}, c_{N_{jk}}^{w}) + ∑_{w∈B} F_{Γ_{k}^{w}}(ρ_{jk}^{B}, c_{N_{jk}+1}^{w}),  (22)

where the sum over w ∈ A and w ∈ B includes all cubes Γ_{k}^{w} of type A and type B respectively. We include the density as an argument for free energy and free energy density for the sake of clarity where two or more different densities are present in the same equation. The density is assumed to be ρ if not indicated explicitly. The free energy densities satisfy

f_{S_{j}}(ρ, c_{S_{j}}) V_{S_{j}} ≤ V_{Γ_{k}}{∑_{w∈A} f_{Γ_{k}^{w}}(ρ_{jk}^{A}, c_{N_{jk}}^{w}) + ∑_{w∈B} f_{Γ_{k}^{w}}(ρ_{jk}^{B}, c_{N_{jk}+1}^{w})}.  (23)

Rearranging terms we get

f_{S_{j}}(ρ, c_{S_{j}}) ≤ (1 − ζ_{jk})\{\frac{ϕ_{jk}^{A}}{n_{jk}^{A}} ∑_{w∈A} f_{Γ_{k}^{w}}(ρ_{jk}^{A}, c_{N_{jk}}^{w}) + \frac{ϕ_{jk}^{B}}{n_{jk}^{B}} ∑_{w∈B} f_{Γ_{k}^{w}}(ρ_{jk}^{B}, c_{N_{jk}+1}^{w})\}.  (24)

If n_{jk}^{B} = 0, then the sum over type B cubes on the right hand side is not present because there are no type B cubes.

Take the limit j → ∞, holding k fixed. For sufficiently regular shapes (as defined by Fisher [2]) ζ_{jk} → 0. Since ρ_{jk}^{A} ≥ ρ can only take discrete values, it follows from Eq. (21) that there exists j_{0}, such that for all j ≥ j_{0}

ρ_{jk}^{A} = ρ.  (25)

Therefore, ϕ_{jk}^{A} → 1 and ϕ_{jk}^{B} → 0 as j → ∞.

The free energies f_{Γ_{k}^{w}}(ρ) are identically distributed, independent random variables, because the cubes Γ_{k}^{w} are identical in shape and the particles are independently chosen. By the Strong Law of Large Numbers [18]

lim_{j→∞} \frac{1}{n_{jk}^{A}} ∑_{w∈A} f_{Γ_{k}^{w}} → ⟨f_{Γ_{k}}⟩  (26)

with probability 1 . Therefore, the limit of Eq. (24) becomes

lim_{j→∞} sup f_{S_{j}} ≤ ⟨f_{Γ_{k}}⟩  (27)

with probability 1. Now take the limit k → ∞ to get the desired upper bound (18) with probability 1.

To show the lower bound on the limiting free energy (19) enclose the shape S_{j} completely inside the smallest possible cube Γ_{K} whose edge length is an integer multiple of 2^{k}l_{0} (see Fig 3). Fill the empty space between S_{j} and Γ_{K} with n_{jk} cubes Γ_{k}^{w} which lie completely outside S_{j}. Fix the density of particles in Γ_{K} at ρ by using two types of cubes Γ_{k}^{w}, type A cubes that contain N_{jk} = [ρ(V_{Γ_{K}} − V_{S_{j}})/n_{jk}] particles and type B cubes that contain N_{jk}^{B} = N_{jk} + 1 particles.

The number of particles in Γ_{K},

N_{Γ_{K}} = N_{S_{j}} + n_{jk}^{A}N_{jk} + n_{jk}^{B}(N_{jk} + 1),  (28)

where n_{jk}^{A} and n_{jk}^{B} are the number of type A and type B cubes respectively. Dividing by V_{Γ_{K}} on both sides gives

ρ = \frac{V_{S_{j}}}{V_{Γ_{K}}} ρ + (\frac{n_{jk}V_{Γ_{k}}}{V_{Γ_{K}}} )(ϕ_{jk}^{A}ρ_{jk}^{A} + ϕ_{jk}^{B}ρ_{jk}^{B}).  (29)

We rewrite this relation, introducing Φ_{j} ≡ V_{S_{j}}/V_{Γ_{K}} ≤ 1, and introducing ζ_{jk} ≥ 0 as the ratio of volume that is unfilled by the cubes Γ_{k}^{w} to that of S_{j}. Formally, (V_{Γ_{K}} − n_{jk}V_{Γ_{k}})/V_{S_{j}} = 1 + ζ_{jk}. Then Eq. (29) becomes

ρ(1 − Φ_{j}) = (1 − Φ_{j}(1 + ζ_{jk}))(ϕ_{jk}^{A}ρ_{jk}^{A} + ϕ_{jk}^{B}ρ_{jk}^{B}).  (30)

It follows from Eq. (30) that ρ_{jk}^{A} ≥ ρ, because if ρ_{jk}^{A} < ρ then ρ_{jk}^{B} ≤ ρ and Eq. (30) cannot be satisfied for ζ_{jk} > 0. In the event that ζ_{jk} = 0 then ϕ_{jk}^{A} = 1 and ρ_{jk}^{A} = ρ.

Let c be a choice (infinite sequence) of particles. Denote the first N_{Γ_{K}} particles of c by c_{Γ_{K}}. Split c_{Γ_{K}} into a subsequence of N_{S_{j}} particles followed by n_{jk}^{A} successive subsequences of N_{jk} particles and n_{jk}^{B} successive subsequences of N_{jk} + 1 particles. Denote the subsequence of N_{S_{j}} particles by c_{S_{j}}, and subsequences of N_{jk} and N_{jk} + 1 particles by c_{N_{jk}}^{w} and c_{N_{jk}+1}^{w} respectively, using a map from the one-dimensional array (of sequences of particles) to a three-dimensional array (of cubes Γ_{k}^{w}). Put particles denoted by c_{S_{j}} into S_{j} and particles labeled by superscripts w into cubes Γ_{k}^{w}. From the subadditivity of free energy it follows that

F_{Γ_{K}}(ρ, c_{Γ_{K}}) ≤ F_{S_{j}}(ρ, c_{S_{j}}) + ∑_{w∈A} F_{Γ_{k}^{w}}(ρ_{jk}^{A}, c_{N_{jk}}^{w}) + ∑_{w∈B} F_{Γ_{k}^{w}}(ρ_{jk}^{B}, c_{N_{jk}+1}^{w}),  (31)

Rewriting the above in terms of free energy densities and isolating f_{S_{j}} gives

f_{S_{j}}(ρ, c_{S_{j}}) ≥ \frac{V_{Γ_{K}}}{V_{S_{j}}} f_{Γ_{K}}(ρ, c_{Γ_{K}}) − \frac{V_{Γ_{k}}}{V_{S_{j}}} \{∑_{w∈A} f_{Γ_{k}^{w}}(ρ_{jk}^{A}, c_{N_{jk}}^{w}) + ∑_{w∈B} f_{Γ_{k}^{w}}(ρ_{jk}^{B}, c_{N_{jk}+1}^{w})\}. (32)

Rearranging terms further, write

f_{S_{j}}(ρ, c_{S_{j}}) ≥ (1 + ζ_{jk})\{\frac{ϕ_{jk}^{A}}{n_{jk}^{A}} ∑_{w∈A} f_{Γ_{k}^{w}}(ρ_{jk}^{A}, c_{N_{jk}}^{w}) + \frac{ϕ_{jk}^{B}}{n_{jk}^{B}} ∑_{w∈B} f_{Γ_{k}^{w}}(ρ_{jk}^{B}, c_{N_{jk}+1}^{w})\}  (33)

+ \frac{V_{Γ_{K}}}{V_{S_{j}}} \{f_{Γ_{K}}(ρ, c_{Γ_{K}}) − (\frac{ϕ_{jk}^{A}}{n_{jk}^{A}} ∑_{w∈A} f_{Γ_{k}^{w}}(ρ_{jk}^{A}, c_{N_{jk}}^{w}) + \frac{ϕ_{jk}^{B}}{n_{jk}^{B}} ∑_{w∈B} f_{Γ_{k}^{w}}(ρ_{jk}^{B}, c_{N_{jk}+1}^{w}))\}.

Take the limit j → ∞ so that ζ_{jk} → 0. Since ρ_{jk}^{A} ≥ ρ is discrete, it follows from Eq. (30) that there exists j_{0} such that ρ_{jk}^{A} = ρ for j ≥ j_{0}. Therefore, ϕ_{jk}^{A} → 1 and ϕ_{jk}^{B} → 0 as j → ∞. For the sequence {S_{j}} of sufficiently regular shapes [2], (V_{Γ_{K}}/V_{S_{j}}) ≥ δ_{S} for some δ_{S} > 0. Using the Strong Law of Large Numbers [18] we get that the inequality

lim_{j→∞} inf f_{S_{j}} ≥ ⟨f_{Γ_{k}}⟩ − δ_{S}|f_{Γ} − ⟨f_{Γ_{k}}⟩|  (34)

is satisfied with probability 1. Now take the limit k → ∞ so that ⟨f_{Γ_{k}}⟩ → f̄_{Γ}. Therefore, the lower bound (19) is satisfied with probability 1.

Combining the two bounds (18) and (19) we get

lim_{j→∞} f_{S_{j}} = f̄_{Γ}  (35)

with probability 1. Therefore, the limiting free energy density f exists with probability 1 for any shape and is independent of the shape of the system. Its value is equal to f̄_{Γ}.

III. HARD CORE MAGNETIC PARTICLES POLYDISPERSE IN SIZE AND SHAPE

In this section we extend our proof to fluids which consist of polydisperse hard core particles, as in Sec. II, but in addition, they interact magnetically with each other. One example of such a system is a ferrofluid [19], which is a colloidal suspension of ferromagnetic particles in a carrier liquid. The thermodynamic limit in this case is complicated by the 1/r^{3} fall-off of the magnetic interaction where the exponent of 1/r exactly matches the dimensionality of space.

We prove the existence of a thermodynamic limit with probability 1 for fluids of magnetic hard core particles using a strategy similar to that of hard core particles in Sec. II. We show that the free energy satisfies translation invariance (6), subadditivity (7), and the lower bound (8). Once these relations are established, the rest of the proof of a thermodynamic limit is identical to that of the hard core particles in Sec. II and is omitted to avoid repetition.

A. Definitions and assumptions

Let the particles be chosen from a distribution of size, shape and magnetization, X. The magnetization is uniform inside each particle and has a definite relation to the orientation of the particle. The magnitude of the magnetization is the same for all particles. We consider both superparamagnetic particles where the magnetization M_{i} can reverse independently of the particle orientation [20,21] Ω_{i}, and non-superparamagnetic particles with magnetization fixed relative to particle orientation. For the non-superparamagnetic particles, we require that the particle shape have a two-fold rotation symmetry about any axis perpendicular to its magnetization (see Fig. 4) [5].

Consider a container S and a choice of particles c_{N_{S}}. The magnetic interaction between any two non-overlapping particles i, j inside S is

U_{S}^{ij} = ∫_{v_{i}} d^{3}r ∫_{v_{j}} d^{3}r′\{\frac{M_{i}(r) · M_{j}(r′)}{|r − r′|^{3}} − \frac{3(M_{i}(r) · (r − r′)) (M_{j}(r′) · (r − r′))}{|r − r′|^{5}}\}, (36)

where v_{i} and v_{j} are the regions of space occupied by the magnetic material of particle i and j, and M_{i}(r) is M_{i} for r inside v_{i} and zero outside. Write the Hamiltonian as

H_{S}(c_{N_{S}}) = \{ \begin{matrix}H_{S}^{M}  if no particles overlap each other or the boundaries of S \\ +∞ otherwise, \end{matrix}  (37)

where

H_{S}^{M} = ∑_{i<j=1}^{N} U_{S}^{ij}  (38)

is the magnetic interaction between the particles. Define the partition function Z_{S} for the box in the same manner as in (2). For superparamagnetic particles where the magnetization can rotate relative to the particle orientations, we include in Ω_{i}, in addition to the Euler angles, a discrete variable O_{i} = ±1 specifying that the magnetization is parallel (+1) or opposite (−1) to a direction fixed in the particle, and ∫ dΩ_{i} includes a sum over O_{i}. We may permit magnetization to rotate independently of the particle orientation by augmenting the integration in (2) with an integration over the direction of M. We require that the particle distribution satisfy the packing condition (5) and define the free energy F_{S} as in (3). The free energy, which depends on the choice of particles, trivially satisfies the translation invariance (6).

B. Subadditivity of the free energy

To show that the free energy satisfies subadditivity (7), we consider a container S composed of two adjacent nonoverlapping rectangular boxes I_{1}, I_{2} ∈ I. Define the interaction energy between the two boxes I_{1}, I_{2} by

H_{I_{1},I_{2}} ≡ H_{S} − H_{I_{1}} − H_{I_{2}}  (39)

Let F_{S}(λ) be the free energy of the combined system when the Hamiltonian is H_{I_{1}} + H_{I_{2}} plus a scaled interaction λH_{I_{1},I_{2}}. Because F_{S}(λ) is a concave function (that is, F′′_{S}(λ) ≤ 0), it is bounded above by

F_{S}(λ) ≤ F_{S}(0) + λF′_{S}(0),  (40)

where the right side is a line tangent to the graph of F_{S}(λ) at λ = 0; here F′_{S}(λ) and F′′_{S}(λ) are the first and second derivatives.

As a consequence of (40), the free energy F_{S}(1) of the fully interacting system satisfies the Gibbs inequality [22]

F_{S}(1) ≤ F_{S}(0) + [H_{I_{1},I_{2}}]_{λ=0},  (41)

where F_{S}(0) = F_{I_{1}}+F_{I_{2}} is the free energy of the non-interacting subsystems, and the classical ensemble average

[H_{I_{1},I_{2}}]_{λ=0} =  (42)

\frac{1}{Ω^{N_{I_{1}}+N_{I_{2}}}N_{I_{1}}!N_{I_{2}}!Z_{I_{1}}Z_{I_{2}}} ∫_{I_{1}} ∏_{i=1}^{N_{1}} d^{3}r_{i}dΩ_{i} ∫_{I_{2}} ∏_{j=1}^{N_{2}} d^{3}r_{j}dΩ_{j}H_{I_{1},I_{2}}e^{−(H_{I_{1}}+H_{I_{2}})/k_{B}T}.

Consider a θ operator [5] which acts on any given system and has the following properties. It leaves the center of mass positions r_{i} of particles unchanged, it maps the particle orientations Ω_{i} on to themselves in such a way that it leaves the integration measure ∏_{i} dΩ_{i} unchanged, and it leaves the Hamiltonian H invariant. In addition, it reverses the magnetization of every particle. For superparamagnetic particles, spontaneous magnetization reversal acts as a θ operator. For non-superparamagnetic particles, a 180^{∘} rotation of particles about an axis of two-fold symmetry suffices as θ operator.

Applying the θ operator to I_{1} but not I_{2} leaves both H_{I_{1}} and H_{I_{2}} invariant but reverses the sign of H_{I_{1},I_{2}}^{M} while leaving the integration measure unchanged. Hence the average in (42) vanishes and from (41) we get

F_{S} ≤ F_{I_{1}} + F_{I_{2}}.  (43)

The free energy, therefore, satisfies subadditivity (7).

C. Lower bound on the free energy

The lower bound (8) easily follows if the potential is stable [1,2]. For stable potentials

H_{S} ≥ −ωV_{S},  (44)

where ω is some positive constant independent of the number of particles. Substituting the lower bound (44) on H_{S} in (2) gives an upper bound on the partition function and thus a lower bound (8) on the free energy. Therefore, in this section we will prove the stability for our model.

Let M_{S}(r) be the magnetization distribution in a box S, so that M_{S}(r) = M_{i} if r is inside particle i and zero if r is outside any particle. Let H_{S}^{D}(r) be the magnetic field due to the particles in S. To prove stability we make use of the positivity of field energy. Adding the magnetic self energy of each particle to H_{S}^{M} gives the total energy of the whole system, considered as one magnetization distribution (see Ref. [23] for identities on magnetization distributions),

H_{S}^{T} = H_{S}^{M} + ∑_{i=1}^{N_{S}} E_{i}^{self} = −\frac{1}{2} ∫ d^{3}r H_{S}^{D}(r) · M_{S}(r).  (45)

Here H_{S}^{D}(r) is the field, due to all particles inside the container S and

E_{i}^{self} = −\frac{1}{2}M_{i} · ∫_{v_{i}} d^{3}r H_{i}^{D}(r),  (46)

where H_{i}^{D}(r) is the field from magnetization M_{i} of particle i with volume v_{i}.

We place a lower bound on H by a method similar to that of Griffiths [4]. For any magnetization distribution M(r) and the field H^{D}(r) caused by it

−\frac{1}{2} ∫ d^{3}r H^{D}(r) · M(r) = \frac{1}{8π} ∫ d^{3}r |H^{D}(r)|^{2} ≥ 0.  (47)

Hence

H_{S}^{M} + ∑_{i=1}^{N_{S}} E_{i}^{self} ≥ 0.  (48)

Brown [23] rewrites the self energy in (46) as

E_{i}^{self} = 2π ∑_{k,l} D_{i}^{kl}M_{i}^{k}M_{i}^{l} v_{i},  (49)

where D_{i} is the demagnetizing tensor of an “equivalent ellipsoid”; it exists for a particle of any shape [23], and k and l index the components of D_{i} and M. Since D_{i} is positive definite, with trace equal to 1,

E_{i}^{self} ≤ 2πM^{2}v_{i},  (50)

so that

H_{S}^{M} ≥ −2πM^{2} ∑_{i=1}^{N_{S}} v_{i}.  (51)

If the particles are unable to fit inside S without overlap then H_{S} = +∞ because of the hard core repulsion. Otherwise, if particles fit inside S without overlap, then ∑_{i=1}^{N_{S}} v_{i} ≤ V and

H_{S} = H_{S}^{M} ≥ −2πM^{2}V_{S}.  (52)

The free energy of a box I ∈ I therefore satisfies the lower bound (8), and we have previously shown that it satisfies the translation invariance (6) and subadditivity (7). The rest of the proof is identical to that of hard core particles polydisperse in size and shape and is omitted.

IV. CONCLUSION

We study the thermodynamic limit of hard core fluids consisting of particles chosen from a distribution that is polydisperse in size and shape. We show that a thermodynamic limit exists with probability 1 and is independent of the choice of particles. The existence of a thermodynamic limit implies shape independence of thermodynamic properties. In this section we discuss some models that are not covered by our proof, although we believe they do possess thermodynamic limits.

This paper addressed fluids of particles with hard core interactions and magnetic interactions. We believe the proof could be extended to incorporate interactions that fall off faster than 1/r^{3}. One example is a Lennard-Jones fluid with interaction

U_{ij} = \frac{A_{ij}}{r^{m}} − \frac{B_{ij}}{r^{n}}  (53)

between particles i and j, where m > n > 3 and A_{ij}, B_{ij} > 0 are some constants chosen randomly for every pair of particles from some distribution. A “random” Stockmayer fluid is another example, where in addition to the Lennard-Jones interaction described above, each particle i has a randomly chosen magnetic dipole moment μ_{i}. Another example is a polydisperse charged colloid in ionic solution [29,30] with an interaction such as

U_{ij} = \frac{q_{i}q_{j}e^{−κr}}{εr}  (54)

where κ > 0 is the Debye screening length.

For such systems the free energies for a container S composed of two adjacent nonoverlapping rectangular boxes I_{1}, I_{2} ∈ I separated by a distance R ≥ R_{0} > 0 satisfy

F_{S} ≤ F_{I_{1}} + F_{I_{2}} + Δ_{12}.  (55)

Here

Δ_{12} = \frac{N_{I_{1}}N_{I_{2}}δ_{12}}{R^{3+ϵ}}  (56)

with some ϵ > 0 and δ_{12} depending upon the probability distributions of the pair potentials. Our proof of thermodynamic limit uses a subadditive ergodic theorem which requires strict subadditivity (7) with Δ_{12} = 0. However, we suppose that provided that the probability distributions are sufficiently restricted, a subadditive ergodic theorem similar to that of Akcoglu and Krengel [10] but with the weaker subadditivity condition (55) will hold and a proof of a thermodynamic limit will follow.

Granular magnetic solids provide other examples of systems outside the realm of our present considerations. Consider a solid system with particles frozen in space but magnetic moments able to rotate with respect to the body of the particles [27] (or a solid containing rotationally free magnetic particles inside cavities [28]) One way to prepare such systems is to freeze the carrier fluid in a colloidal suspension of magnetic hard core particles so that the positions of particles are representative of an equilibrium configuration of the fluid. Our proof does not directly apply to such systems because the infinite system cannot be constructed from finite cubes of frozen particles. A system constructed from finite cubes will have no particles overlapping the boundaries of these cubes and thus the spatial arrangement of particles differs from an equilibrium configuration of the fluid. If the particles are “unfrozen” and frozen back again after their positions have relaxed, then subadditivity (7) no longer holds strictly, though it will hold on average.

ACKNOWLEDGMENTS

We acknowledge useful discussions with A. Pisztora and L. Chayes. This work was supported in part by NSF grant DMR-9732567 at Carnegie Mellon University and by NSF grant CHE-9981772 at University of Maryland.

APPENDIX A: EXAMPLES OF ALLOWED PARTICLE DISTRIBUTIONS

Here we present some examples of particle size and shape distributions which obey the packing condition. Our discussion hinges on the ability to pack a monodisperse collection of hard spheres with a hexagonal close packing fraction of ϕ^{hcp} = \frac{π}{3\sqrt{2}}. Our presentation is informal and primarily intended to motivate the form of our packing condition, rather than to achieve rigorous proofs.

Binary mixture of hard core spheres: Consider spherical hard cores of two types with radii r_{α} and r_{β} occurring with probabilities p_{α} and p_{β}, respectively. A given choice c_{N} ∈ C_{N} of N particles yields N_{α} spheres of type α and N_{β} = N − N_{α} of type β. Define the partition function in a volume V as

Z_{V} (c_{N}) ≡ \frac{1}{N_{α}!N_{β}!} ∫_{V} ∏_{i} dr_{i}e^{−H_{V} (c_{N})/k_{B}T} .  (A1)

The factor of N_{α}!N_{β}! is chosen instead of N! because it yields the usual entropy of mixing [24]. Given a choice c_{N}, a volume V (c_{N}) exists that is sufficiently large that the partition function exceeds a threshold

Z_{V} (c_{N}) ≥ z_{0}^{N}.  (A2)

We construct the volume V (c_{N}) satisfying the inequality (A2) by assigning particles to spherical shells of radius r′_{α} = r_{α} + r_{0} and r′_{β} = r_{β} + r_{0} where \frac{4π}{3}r_{0}^{3} = z_{0}. Shells of type α and β surround volumes v′_{α} = \frac{4π}{3}r_{α}^{′3} and v′_{β} = \frac{4π}{3}r_{β}^{′3}. If we choose the volume

V (c_{N}) = (N_{α}v′_{α} + N_{β}v′_{β})/ϕ^{hcp}  (A3)

then the inequality (A2) holds.

To verify this claim, divide the volume into subregions of volumes N_{α}v′_{α}/ϕ^{hcp} and N_{β}v′_{β}/ϕ^{hcp} respectively. Closely pack the α- and β-type shells into their respective subregions and evaluate the configurational integral in Eq. (A1) in two stages. First, confine each particle within its shell and integrate, yielding a contribution of z_{0}^{N}. Second, permute identical particles within identical shells, yielding a factor of N_{α}!N_{β}!. Since the combinatorial factor cancels against the normalization of the partition function in Eq. (A1), the partition function Z_{V} (c_{N}) is at least as large as z_{0}^{N}.

Now we show that, with probability 1, V (c_{N}) does not greatly exceed Nv̄′(z_{0})/ϕ^{hcp}, where we identify the mean volume per shell v̄′(z_{0}) ≡ p_{α}v′_{α} + p_{β}v′_{β}. Note that v̄′(z_{0}) approaches the mean particle volume v̄ as z_{0} → 0. Take a cube Γ of volume

V_{Γ} = \frac{Nv̄′(z_{0})}{ϕ^{hcp}}(1 + ϵ)  (A4)

where ϵ is an arbitrarily small positive number. Divide the set C_{N} of choices c_{N} into a “packing subset”

P_{Γ} = {c_{N} ∈ C_{N} | V (c_{N}) ≤ V_{Γ}}  (A5)

and a complementary “nonpacking subset”

Q_{Γ} = {c_{N} ∈ C_{N} | V (c_{N}) > V_{Γ}}.  (A6)

Since P_{Γ} and Q_{Γ} are complementary subsets, their measures obey μ_{N}(P_{Γ}) + μ_{N}(Q_{Γ}) = 1. By Chebyshev’s inequality [18] on the sum of independent random variables we bound the measure of the nonpacking subset by

μ_{N}(Q_{Γ}) ≤ \frac{p_{α}^{2}v_{α}^{′2} + p_{β}^{2}v_{β}^{′2}}{Nv̄′(z_{0})^{2}ϵ^{2}} .  (A7)

For sufficiently large N the probability that a choice does not pack into V_{Γ} vanishes. Thus V (c_{N}) ≤ V_{Γ} with probability 1 at density

ρ ≡ N/V_{Γ} = \frac{ϕ^{hcp}}{v̄′(z_{0})(1 + ϵ)}.  (A8)

Therefore, binary mixture of hard spheres obeys the packing condition (5) with ρ^{∗}(z_{0}) = ϕ^{hcp}/v̄′(z_{0}). Note that ρ^{∗}(z_{0})v̄ approaches ϕ^{hcp} as z_{0} → 0.

Finitely many types of hard spheres: Consider some finite number, t, of types of hard spheres, each type with a different finite radius and a certain probability of occurrence. The argument presented above for a mixture of two types of spheres generalizes easily to the case of t > 2. The combinatorial factor in the partition function (A1) generalizes to Π_{α}N_{α}! and the mean volume per shell generalizes to v̄′(z_{0}) ≡ ∑_{α} p_{α}v′_{α}. An argument similar to that for a binary mixture of hard spheres shows that this distribution satisfies the packing condition (5) with a critical packing density ρ^{∗}(z_{0}) = ϕ^{hcp}/v̄′(z_{0}). A packing fraction of ϕ^{hcp} is achievable in the limit z_{0} → 0, but for finitely many types of hard spheres a higher packing fraction should be achievable because smaller particles can occupy the interstitial sites of larger particles [25].

The distributions above satisfy the packing condition using the prefactor Π_{α}N_{α}! to define the partition function. However, the prefactor of (∑_{α} N_{α})! = N! may also be used to define the partition function as in Eq. (2). The logarithm of the ratio of the two prefactors is the entropy of mixing between the particles of various types. As long as the particles are chosen randomly from the distribution and not “unmixed”, the factor of N! suffices to give an extensive entropy [6,24]. To show that the distribution satisfies the packing condition using prefactor N!, enclose particles in shells so that they have free volume of at least z′_{0} inside the shell. The partition function is bounded from below by

Z_{V} ≥ \frac{Π_{α}N_{α}!}{N!}z′_{0}^{N} ≥ (\frac{z′_{0}}{t} )^{N}.  (A9)

Set z′_{0} = tz_{0} to show that the distribution satisfies the packing condition (5) as long as t is finite. Now the critical packing density is ρ^{∗}(z_{0}) = ϕ^{hcp}/v̄′(tz_{0}). As before, ρ^{∗}(z_{0})v̄ approaches ϕ^{hcp} as z_{0} → 0.

Continuous distribution of hard spheres with an upper size cut-off: Consider a continuous distribution of hard spheres with volumes 0 ≤ v < v_{max}. This distribution has essentially an infinite number of types of particles and so our discussion above does not directly apply to it, however, it still satisfies the packing condition (5). To show this, define the partition function with prefactor N! to make the free energy extensive. Break the distribution into a finite number, t, of bins with boundaries 0 < v_{1} < v_{2}... < v_{t} = v_{max}. Let the i^{th} bin B_{i} contain all the particles with v_{i−1} ≤ v < v_{i}. Enclose every particle in bin B_{i} within a shell of volume

v′_{B_{i}} = \frac{4π}{3} \{(\frac{3v_{i}}{4π} )^{1/3} + (\frac{3z_{0}}{4π} )^{1/3}\}^{3}  (A10)

so that the largest particle in B_{i} has a free volume of at least z′_{0} = tz_{0}, then apply the same argument as for finitely many types of spheres. The resulting ρ^{∗}(z_{0}) = ϕ^{hcp}/v̄′(tz_{0}) results in a suboptimal achievable packing fraction ϕ = ρ^{∗}(z_{0})v̄ = ϕ^{hcp}v̄/v̄′(tz_{0}) < ϕ^{hcp} due to the bin width and the choice of prefactor N!. By using a sufficiently large but finite number of bins and taking z_{0} → 0, a packing fraction arbitrarily close to ϕ^{hcp} can be achieved. As in the case of discrete distributions, for a continuous distribution of hard spheres one expects a higher packing fraction than that of a monodisperse system should be achievable [25].

Continuous distribution of hard spheres with no particle size cut-off: Now we consider a continuous distribution of hard core spheres such that the particle size has no upper limit. To ensure that the large sized particles occur sufficiently infrequently, we assume that there exists a particle volume v^{∗} such that the probability of a particle with volume greater than v_{0} for any v_{0} > v^{∗} falls off as

μ_{1}(v > v_{0}) ≤ \frac{γ}{v_{0}^{1+δ}} ,  (A11)

where γ and δ are positive constants. Note that this condition guarantees the existence of a mean particle volume but not the existence of higher moments.

To show that this distribution satisfies the packing condition (5) we break the distribution into a finite number, t + 1, of bins with boundaries 0 < v_{1} < v_{2}... < v_{t−1} < v_{t} < ∞. We presume that v_{t−1} > v^{∗}. The next-to-last bin B_{t} includes all the particles with v_{t−1} ≤ v < v_{t} and the last bin B_{t+1} contains all the particles with v_{t} ≤ v < ∞. We now send v_{t} to infinity in such a manner that B_{t+1} is empty with high probability but the maximum particle size in B_{t} grows in a controlled fashion. To show that the distribution satisfies the packing condition (5) we first show that the set of choices with no particle in B_{t+1} is a set of measure 1. Then we show that in this set there is a subset of measure 1 of choices which have a partition function of at least z_{0}^{N}.

The probability that a collection of N particles randomly chosen from the distribution contains at least one particle in the last bin B_{t+1} is bounded by

μ_{N}(c_{N} | at least one particle in B_{t+1}) ≤ \frac{Nγ}{v_{t}^{1+δ}} .  (A12)

We wish to avoid all choices with at least one particle in B_{t+1}. To ensure that the probability of such choices goes to zero for large N, v_{t} must grow faster than N^{1/(1+δ)}. At the same time v_{t} must grow slower than N to ensure that the largest particles in B_{t} are much smaller than the system size. Hence let v_{t} grow proportional to N^{1/(1+ \frac{δ}{2} )}, so that

μ_{N}(c_{N} | at least one particle in B_{t+1}) ≤ \frac{γ′}{N^{\frac{δ}{2+δ}}} → 0,  (A13)

where γ′ is some constant. For large N, the probability approaches 1 that no particle is in bin B_{t+1}.

Now consider the particles in bin B_{t}. The next-to-last bin B_{t} must be treated specially because the largest particle size in B_{t} diverges as the system size to goes infinity. Enclose each particle j (with volume v_{j}) in B_{t} inside its own shell with volume v′_{j} so that it has a free volume of z′_{0} = tz_{0}. Using condition (A11) the total volume of the shells corresponding to bin B_{t} (averaged over choices c_{N}) obeys

⟨ ∑_{v_{j}∈B_{t}} v′_{j}⟩ ≤ \frac{γ(1 + δ)}{δ} \frac{N}{v_{t−1}^{δ}} .  (A14)

Since the volume of the system is proportional to N, the volume fraction occupied by shells corresponding to bin B_{t} is proportional to v_{t−1}^{−δ}. This volume fraction can be made arbitrarily small by choosing a sufficiently large v_{t−1}. We assume that the N_{t} particles in bin B_{t} (enclosed in their shells) can be packed inside a small region of the system in N_{t}! different ways with high probability (tending to 1 as N goes to infinity).

The particles in the remaining bins B_{1}, ..., B_{t−1} have a partition function of at least z_{0}^{N−N_{t}} in the remaining volume of the system with high probability (tending to 1) as we showed previously for a continuous distribution of spheres with an upper size cut-off. Therefore, these particles together with the particles in B_{t} satisfy the packing condition (5).

Non-spherical particles: Finally, consider a distribution of non-spherical particles. To show that the distribution satisfies the packing condition (5), enclose each particle inside a spherical shell with a diameter equal to the largest dimension of the particle. If the volume of the shells satisfy condition (A11) then the rest of the argument is the same as that of a continuous distribution of hard spheres. Although this argument shows the existence of a finite particle number density ρ^{∗}(z_{0}), it does not guarantee the existence of a finite packing fraction. In fact, it is possible to have distributions with oddly shaped particles (fractals for example) so that a finite ρ^{∗}(z_{0}) exists but the particle packing fraction is zero.

APPENDIX B: VARIANCE OF FREE ENERGY DENSITY FOR CUBES

To show the variance relation (11) consider two adjacent nonoverlapping cubes Γ_{k}^{l} and Γ_{k}^{m} such that I_{k} = Γ_{k}^{l} ∪ Γ_{k}^{m} is a rectangular box. Using the subadditive relation (7) we get

f_{I_{k}} ≤ \frac{1}{2}(f_{Γ_{k}^{l}} + f_{Γ_{k}^{m}}).  (B1)

Adding ω_{A} (see lower bound (8)) to both sides of (B1) and squaring gives

f_{I_{k}}^{2} + 2ω_{A}f_{I_{k}} + ω_{A}^{2} ≤ \frac{1}{4}(f_{Γ_{k}^{l}} + f_{Γ_{k}^{m}})^{2} + ω_{A}^{2} + ω_{A}(f_{Γ_{k}^{l}} + f_{Γ_{k}^{m}}).  (B2)

Averaging both sides over all choices of particles we write

⟨f_{I_{k}}^{2}⟩ + 2ω_{A}⟨f_{I_{k}}⟩ + ω_{A}^{2} ≤ \frac{1}{2}⟨f_{Γ_{k}}^{2}⟩ + \frac{1}{2}⟨f_{Γ_{k}}⟩^{2} + 2ω_{A}⟨f_{Γ_{k}}⟩ + ω_{A}^{2}.  (B3)

Subtracting ω_{A}^{2} + ⟨f_{I_{k}}^{2}⟩ + 2ω_{A}⟨f_{I_{k}}⟩ and rearranging terms on the right-hand side gives

⟨(f_{I_{k}} − ⟨f_{I_{k}}⟩)^{2}⟩ ≤ \frac{1}{2}⟨(f_{Γ_{k}} − ⟨f_{Γ_{k}}⟩)^{2}⟩ + ⟨f_{Γ_{k}}⟩^{2} − ⟨f_{I_{k}}⟩^{2} + 2ω_{A}(⟨f_{Γ_{k}}⟩ − ⟨f_{I_{k}}⟩).  (B4)

Using the bound

⟨f_{Γ_{k}}⟩ + ⟨f_{I_{k}}⟩ ≤ −2k_{B}T ln z_{0} ≡ 2ω_{B},  (B5)

we get

⟨(f_{I_{k}} − ⟨f_{I_{k}}⟩)^{2}⟩ ≤ \frac{1}{2}⟨(f_{Γ_{k}} − ⟨f_{Γ_{k}}⟩)^{2}⟩ + 2(ω_{A} + ω_{B})(⟨f_{Γ_{k}}⟩ − ⟨f_{I_{k}}⟩).  (B6)

Both ⟨f_{I_{k}}⟩ and ⟨f_{Γ_{k}}⟩ go to the same limit f_{Γ} since ⟨f_{Γ_{k+1}}⟩ ≤ ⟨f_{I_{k}}⟩ ≤ ⟨f_{Γ_{k}}⟩. Therefore, we can bound the second term on the right in (B6) below any ϵ′ > 0 for sufficiently large k. It follows that

⟨(f_{I_{k}} − ⟨f_{I_{k}}⟩)^{2}⟩ ≤ \frac{1}{2}⟨(f_{Γ_{k}} − ⟨f_{Γ_{k}}⟩)^{2}⟩ + ϵ′.  (B7)

By stacking two rectangular boxes I_{k} together to form a square slab, then putting together two adjacent slabs, we construct a cube Γ_{k+1}. The variance of the cube Γ_{k+1} is related to that of Γ_{k} by

⟨(f_{Γ_{k+1}} − ⟨f_{Γ_{k+1}}⟩)^{2}⟩ ≤ \frac{1}{8}⟨(f_{Γ_{k}} − ⟨f_{Γ_{k}}⟩)^{2}⟩ + ϵ,  (B8)

where ϵ = (1 + \frac{1}{2} + \frac{1}{4})ϵ′ = \frac{7}{4}ϵ′.

APPENDIX C: CONVEXITY AND CONTINUITY OF THE FREE ENERGY DENSITY FOR CUBES

Consider a cube Γ_{k} with volume V_{Γ_{k}} that contains N_{k} particles. Since N_{k} is an integer, the free energy density f_{Γ_{k}}(ρ, c_{N_{k}}) is defined only for discrete values of ρ. Since f_{Γ_{k}}(ρ, c_{N_{k}}) is a random variable, a linear interpolation to include fractional number of particles is not possible. However, the average free energy density ⟨f_{Γ_{k}}(ρ)⟩ is not random and we define ⟨f(ρ)⟩ for all values of ρ by linear interpolation between values of ρ that correspond to integer values of N_{k}.

Now consider a cube Γ_{k+1} consisting of 4 cubes Γ_{k} each containing N′_{k} particles and 4 cubes Γ_{k} each containing N′′_{k} particles. From subadditivity (7) of free energy, and averaging over all possible choices of particles in the cubes Γ_{k}, it follows that

⟨f_{Γ_{k+1}}(ρ)⟩ ≤ \frac{1}{2}(⟨f_{Γ_{k}}(ρ′)⟩ + ⟨f_{Γ_{k}}(ρ′′)⟩),  (C1)

where ρ′ = N′_{k}/V_{Γ_{k}}, ρ′′ = N′′_{k}/V_{Γ_{k}} and ρ = (ρ′ + ρ′′)/2. By linear interpolation Eq. (C1) holds for any ρ′ and ρ′′. Taking the limit k → ∞ gives

⟨f_{Γ}(ρ)⟩ ≤ \frac{1}{2}(⟨f_{Γ}(ρ′)⟩ + ⟨f_{Γ}(ρ′′)⟩).  (C2)

Therefore, ⟨f_{Γ}(ρ)⟩ ≡ f̄_{Γ}(ρ) is a convex function of ρ. Since f̄_{Γ}(ρ) is a bounded function of ρ, it is also continuous function of ρ [26].

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FIGURES

FIG. 1. An example of a size and distribution of particles. The hard core particles A, B, C are chosen with fixed probabilities.

FIG. 2. One particular choice of particles from the distribution in Fig 1. into basic cubes, with N_{0} = 2 particles in each basic cube. The box I =[(l_{0}, l_{0}), (3l_{0}, 2l_{0})) in this example is assigned particles C, B, B, and A. For easy visualization we show the figure in two-dimensions instead of the actual case of three dimensions.

FIG. 3. (a) A cross-section of an arbitrary shape S_{j}. The space inside S_{j}, and the space between S_{j} and Γ_{K}, is maximally filled by cubes Γ_{k}. Dashed lines indicate the cubes that intersect the boundary of S_{j}. The volume of cubes Γ_{k} that intersect S_{j} becomes negligible in comparison to the volume of S_{j} as the size of S_{j} goes to infinity and the size of Γ_{k} grows less rapidly than S_{j}. (b) A cube Γ_{k} in d dimensions consists of 2^{kd} basic cubes Γ_{0}. In this case k = 1 and d = 2.

FIG. 4. (a) Particle shapes with a two-fold symmetry perpendicular to their magnetization. The arrow indicates the direction of magnetization. (b) Particle shapes lacking two-fold symmetry perpendicular to their magnetization.