Large Deviation Analysis of a Droplet Model Having a Poisson Equilibrium Distribution

1. Introduction

This paper is motivated by a natural question for a basic model of a droplet. Given $$ and c > b, K distinguishable particles are placed, each with equal probability 1/N, onto the N sites of a lattice Λ_N = {1,2, …, N}. Under the assumption that K/N = c and that each site is occupied by a minimum of b particles, what is the equilibrium distribution, as N → ∞, of the number of particles per site? We prove in Corollary 3 that this equilibrium distribution is a Poisson distribution, with mean c, restricted to the set of positive integers n satisfying n ≥ b. As we explain near the end of the Introduction, this equilibrium distribution has important applications to technologies using sprays and powders.

As in many other models in statistical mechanics, we can identify the equilibrium distribution by exhibiting it as the unique minimum point of a rate function in a large deviation principle (LDP). Other models for which this procedure can be implemented are discussed at the end of the Introduction.

For the droplet model we prove the LDP for a sequence of random probability measures, called number-density measures, which are the empirical measures of a sequence of dependent random variables that count the droplet sizes. This LDP is stated in Theorem 1. Our proof is self-contained and starts from first principles, using techniques that are familiar in applied mathematics and statistical mechanics. For example, the proof of the local large deviation estimate in Theorem 5, a key step in the proof of the LDP for the number-density measures, is based on combinatorics, Stirling’s formula, and Laplace asymptotics.

Our use of combinatorial methods goes back to Boltzmann in his work on the discrete ideal gas. He calculated the Maxwell-Boltzmann equilibrium distribution for this system by analyzing the asymptotic behavior of a particular multinomial coefficient [1]. Starting with Boltzmann’s work, combinatorial methods have remained an important tool in both statistical mechanics and in the theory of large deviations, offering insights into a wide variety of physical and mathematical phenomena via techniques that are elegant, powerful, and often elementary. In applications to statistical mechanics, this state of affairs is explained by the observation that “many fundamental questions … are inherently combinatorial, … including the Ising model, the Potts model, monomer-dimer systems, self-avoiding walks and percolation theory” [2]. For the two-dimensional Ising model and other exactly soluble models, [3, 4] are recommended.

A similar situation holds in the theory of large deviations. For example, Section 2.1 of [5] discusses combinatorial techniques for finite alphabets and points out that because of the concreteness of these applications the LDPs are proved under much weaker conditions than the corresponding results in the general theory, into which the finite-alphabet results give considerable insight. The text [6] devotes several early sections to large deviation results for i.i.d. random variables having a finite state space and proved by combinatorial methods, including a sophisticated, level-3 result for the empirical pair measure.

In order to formulate the LDP for the number-density measures in our droplet model, a standard probabilistic model is introduced. The configuration space is the set $$ consisting of all ω = (ω₁, ω₂, …, ω_K), where ω_i denotes the site in Λ_N occupied by the ith particle. The cardinality of Ω_N equals N^K. Denote by P_N the uniform probability measure that assigns equal probability 1/N^K to each of the N^K configurations ω ∈ Ω_N. The asymptotic analysis of the droplet model involves the two random variables, which are functions of the configuration ω ∈ Ω_N: for $$, $$ denotes the number of particles occupying the site $$ in the configuration ω; for $$, N_j(ω) denotes the number of sites $$ for which $$.

In order to carry out the asymptotic analysis of the droplet model, we introduce a quantity m = m(N) that converges to ∞ sufficiently slowly with respect to N; specifically, we require that m(N) ²/N → 0 as N → ∞. In terms of b and m we define the subset Ω_N,b,m of Ω_N consisting of all configurations ω for which every site of Λ_N is occupied by at least b particles and at most m of the quantities N_j(ω) are positive. This second condition is a useful technical device that allows us to control the errors in several estimates. In Appendix D of [7] we present evidence supporting the conjecture that this condition can be eliminated. The discussion in that appendix involves a number of interesting topics including Stirling numbers of the second kind (see [8, pp. 96-97] and [9, §5.4]) and their asymptotic behavior [10, Example 5.4].

The random quantities in the droplet model for which we formulate an LDP are the number-density measures Θ_N,b. For ω ∈ Ω_N,b,m these random probability measures assign to $$ the probability N_j(ω)/N, which is the number density of the jth droplet class. Because of the two conservation laws in (1) and because K/N = c, for ω ∈ Ω_N,b,m, Θ_N,b(ω) is a probability measure on $$ having mean c. Thus Θ_N,b takes values in $$, which is defined to be the set of probability measures on $$ having mean c.

The probability measure P_N,b,m defining the droplet model is obtained by restricting the uniform measure P_N to the set of configurations Ω_N,b,m. Thus P_N,b,m equals the conditional probability P_N(·∣Ω_N,b,m). In the language of statistical mechanics P_N,b,m defines a microcanonical ensemble that incorporates the conservation laws for number and mass expressed in (1).

The content of Theorem 1 is the following: as N → ∞, the sequence of number-density measures Θ_N,b satisfies the LDP on $$ with respect to the measures P_N,b,m. The rate function is the relative entropy R(θ∣ρ_b,α) of $$ with respect to the Poisson distribution ρ_b,α on $$ having components ρ_b,α;j = [Z_b(α)] ⁻¹ · α^j/j! for $$. In this formula Z_b(α) is the normalization that makes ρ_b,α a probability measure, and α equals the unique value α_b(c) for which $$ has mean c [Theorem A.2]. Using the fact that $$ equals 0 at the unique measure $$, we apply the LDP for Θ_N,b to conclude in Theorem 2 that $$ is the equilibrium distribution of Θ_N,b. Corollary 3 then implies that $$ is also the equilibrium distribution of $$.

The space $$ is the most natural space on which to formulate the LDP for Θ_N,b in Theorem 1. Not only is $$ the smallest convex set of probability measures containing the range of Θ_N,b for all $$, but also the union over $$ of the range of Θ_N,b is dense in $$. As we explain in part (a) of Theorem 4, $$ is not a complete, separable metric space, a situation that prevents us from directly applying general results in the theory of large deviations that require the setting of a complete, separable metric space.

The droplet model is defined in Section 2. Step 1 in the proof of the LDP for Θ_N,b is to derive the local large deviation estimate in part (b) of Theorem 5. This local estimate, one of the centerpieces of the paper, gives information not available in the LDP for Θ_N,b, which involves global estimates. Step 2 is to lift the local large deviation estimate to the large deviation limit for Θ_N,b lying in open balls and certain other subsets of $$ while Step 3 is to lift the large deviation limit for open balls and certain other subsets to the LDP for Θ_N,b stated in Theorem 1. Steps 2 and 3 are explained in Section 4.

Details of Steps 2 and 3 as well as other routine proofs are omitted from the present paper. They appear in the unpublished companion paper [7], which also contains additional background material. The paper [1] explores how our work on the droplet model was inspired by the work of Ludwig Boltzmann on a simple model of a discrete ideal gas. The main connection is via the local large deviation estimate in part (b) of Theorem 5. When b = 0, the LDP for a path version of Θ_n,0 with K = tN and t > 0 varying appears in [11, 12].

The main application of the results in this paper is to technologies using sprays and powders, which are ubiquitous in many fields, including agriculture, the chemical and pharmaceutical industries, consumer products, electronics, manufacturing, material science, medicine, mining, paper making, the steel industry, and waste treatment. In this paper we focus on sprays; our theory also applies to powders with only changes in terminology [13]. The behavior of sprays might be complex depending on various parameters including evaporation, temperature, and viscosity. Our goal here is to consider the simplest model where the only assumption is made on the average size of droplets in the spray. In many situations it is important to have good control over the sizes of the droplets, which can be translated into properties of probability distributions. The size distributions are important because they determine reliability and safety in each particular application.

Interestingly, there does not seem to be a rigorous theory that predicts the equilibrium distribution of droplet sizes, analogous to the Maxwell-Boltzmann distribution of energy levels in a discrete ideal gas [14, 15]. Our goal in the present paper is to provide such a theory. We do so by focusing on one aspect of the problem related to the relative entropy, an approach that characterizes the equilibrium distribution of droplet sizes as being a Poisson distribution restricted to $$. We expect that this distribution will be important in experimental observations. A full understanding of droplet behavior under dynamic conditions requires treating many other aspects and is beyond the scope of this paper. We plan to apply the ideas in this paper to understand the entropy of dislocation networks [16].

The importance of predicting droplet size can be seen from the wide range of applications utilizing sprays [17, 18]. Because of the importance of this problem, novel approaches for measuring size distribution of droplet size in sprays have been developed [19–23]. What makes the problem of predicting droplet size particularly interesting is the complexity of droplet-size distribution, which is attributed to many factors such as temperature and viscosity. As [24] shows, even the nozzle plays a significant role in the outcome. Many theoretical tools used to understand the distribution of droplet size in sprays include entropy [25], which also plays a key role in the present paper.

We end the Introduction by expanding on a comment made at the beginning of this section. This comment concerns one of the main applications of large deviation theory in statistical mechanics, which is to identify the equilibrium distribution or distributions of a model as the minimum point(s) of the rate function in an LDP for the model. This procedure is also useful to study phase transitions in the model, which concern how the structure of the set of equilibrium distributions changes as the parameters defining the model change. There are numerous other models for which this procedure has been used. They include the following three lattice spin models: the Curie-Weiss spin system, the Curie-Weiss-Potts model, and the mean-field Blume-Capel model, which is also known as the mean-field BEG model. As explained in the respective Sections 6.6.1, 6.6.2, and 6.6.3 of [26], the large deviation analysis shows that each of these three models has a different phase transition structure. Details of the analysis for the three models are given in the references [6, §IV.4], [27–29]. Section 9 of [30] outlines how large deviation theory can be applied to determine equilibrium structures in statistical models of two-dimensional turbulence. Details of this analysis are given in [31].

2. Definition of Droplet Model and Main Theorem

After defining the droplet model, we state the main theorem in the paper, Theorem 1. The content of this theorem is the LDP for the sequence of random, number-density measures, which are the empirical measures of a sequence of dependent random variables that count the droplet sizes in the model. As we show in Theorem 2 and in Corollary 3, the LDP enables us to identify a Poisson distribution as the equilibrium distribution both of the number-density measures and of the droplet-size random variables. In Theorem 4 we prove a number of properties of two spaces of probability measures in terms of which the LDP for the number-density measures is formulated.

We start by fixing parameters $$ and c ∈ (b, ∞). The droplet model is defined by a probability measure P_N,b parameterized by $$ and the nonnegative integer b. The measure depends on two other positive integers, K and m, where 2 ≤ m ≤ N < K. Both K and m are functions of N in the large deviation limit N → ∞. In this limit we take K → ∞ and N → ∞, where K/N, the average number of particles per site, equals c. Thus K = Nc. In addition, we take m → ∞ sufficiently slowly by choosing m to be a function m(N) satisfying m(N) → ∞ and m(N) ²/N → 0 as N → ∞; for example, m(N) = N^δ for some δ ∈ (0,1/2). Throughout this paper we fix such a function m(N). The parameter b and the function m = m(N) first appear in the definition of the set of configurations Ω_N,b,m in (3), where these quantities will be explained.

Because K and N are integers, c must be a rational number. This in turn imposes a restriction on the values of N and K. If c is a positive integer, then N → ∞ along the positive integers and K → ∞ along the subsequence K = cN. If c = x/y, where x and y are relatively prime, positive integers with y ≥ 2, then N → ∞ along the subsequence N = yn for $$ and K → ∞ along the subsequence K = cN = xn. Throughout this paper, when we write $$ or N → ∞, it is understood that N and K satisfy the restrictions discussed here.

In the droplet model K distinguishable particles are placed, each with equal probability 1/N, onto the sites of the lattice Λ_N = {1,2, …, N}. This simple description corresponds to a simple probabilistic model. The configuration space is the set $$ consisting of all sequences ω = (ω₁, ω₂, …, ω_K), where ω_i ∈ Λ_N denotes the site in Λ_N occupied by the ith particle. Let ρ^(N) be the measure on Λ_N that assigns equal probability 1/N to each site in Λ_N, and let P_N = (ρ^(N)) ^K be the product measure on Ω_N with equal one-dimensional marginals ρ^(N). Thus P_N is the uniform probability measure that assigns equal probability 1/N^K to each of the N^K configurations ω ∈ Ω_N; for subsets A of Ω_N we have P_N(A) = card⁡(A)/N^K, where card denotes cardinality.

The asymptotic analysis of the droplet model involves two random variables. For $$ and ω ∈ Ω_N, $$ denotes the number of particles occupying site $$ in the configuration ω. For $$ and ω ∈ Ω_N, N_j(ω) denotes the number of sites $$ for which $$. The dependence of $$ and N_j(ω) on N is not indicated in the notation. Because the distributions of both random variables depend on N, both $$ and N_j form triangular arrays.

The constraint restricting the number of positive components of N(ω) is a useful technical device that allows us to control the errors in several estimates. In Appendix D of [7] we give evidence supporting the conjecture that this restriction can be eliminated.

When b is a positive integer, for each ω ∈ Ω_N,b,m, each site in Λ_N is occupied by at least b particles. In this case it is useful to think of each particle as having one unit of mass and of the set of particles at each site $$ as defining a droplet. With this interpretation, for each configuration ω, $$ denotes the mass or the size of the droplet at site $$. The jth droplet class has N_j(ω) droplets and mass jN_j(ω). Because the number of sites in Λ_N equals N and the sum of the masses of all the droplet classes equals K, it follows that the quantities N_j(ω) satisfy the two conservation laws in (1) for all ω ∈ Ω_N,b,m.

We now consider the modifications that must be made in these definitions when b = 0. In this case the first constraint in the definition of Ω_N,b,m disappears because we allow sites to be occupied by 0 particles, and therefore N_j(ω) is indexed by $$. On the other hand, we retain the second constraint in the definition of Ω_N,0,m, which requires that for any configuration ω ∈ Ω_N,0,m at most m of the components N_j(ω) for $$ are positive. When b = 0, the definition of Ω_N,0,m becomes Ω_N,0,m = {ω ∈ Ω_N : |N(ω)|₊ ≤ m = m(N)}. Because the choice b = 0 allows sites to be empty, we lose the interpretation of the set of particles at each site as being a droplet. However, for ω ∈ Ω_N,0,m the two conservation laws in (1) continue to hold.

We next introduce several spaces of probability measures that arise in the large deviation analysis of the droplet model. $$ denotes the set of probability measures on $$. Thus $$ has the form $$, where the components θ_j satisfy θ_j ≥ 0 and $$. We say that a sequence of measures $$ in $$ converges weakly to $$, and write θ⁽ⁿ⁾⇒θ, if, for any bounded function f mapping $$ into $$, $$ as n → ∞. $$ is topologized by the topology of weak convergence. There is a standard technique for introducing a metric structure on $$ for which we quote the main facts. Because $$ is a complete, separable metric space with metric d(x, y) = |x − y|, there exists a metric π on $$ called the Prohorov metric with the following two properties: (1) convergence with respect to the Prohorov metric is equivalent to weak convergence [32, Thm. 3.3.1]; (2) with respect to the Prohorov metric, $$ is a complete, separable metric space [32, Thm. 3.1.7].

We denote by $$ the set of measures in $$ having mean c. Thus $$ has the form $$, where the components θ_j satisfy θ_j ≥ 0, $$, and $$. The number-density measures Θ_N,b defined in (5) take values in $$.

According to part (a) of Theorem 4, $$ is not a closed subset of $$. Hence it is natural to introduce the closure of $$ in $$. As we prove in part (b) of Theorem 4, the closure of $$ in $$ equals $$, which is the set of measures in $$ having mean lying in the closed interval [b, c]. Being the closure of the relatively compact, separable metric space $$, $$ is a compact, separable metric space with respect to the Prohorov metric. This space appears in the formulation of the large deviation upper bound in part (c) of Theorem 1.

As a consequence of the fact that $$ is not closed in $$, the large deviation upper bound takes two forms depending on whether the subset F of $$ is compact or whether F is closed. When F is compact, in part (b) we obtain the standard large deviation upper bound for F. When F is closed, in part (c) we obtain a variation of the standard large deviation upper bound, which, when F is compact, coincides with the upper bound in part (b). The refinement in part (c) is important. It is applied in the proof of Theorem 2 to show that $$ is the equilibrium distribution of the number-density measures Θ_N,b. In turn, Theorem 2 is applied in the proof of Corollary 3 to show that $$ is the equilibrium distribution of the droplet-size random variables $$.

In the next theorem we assume that m is the function m(N) appearing in the definition of Ω_N,b,m in (3) and satisfying m(N) → ∞ and m(N) ²/N → 0 as N → ∞. The assumption that m(N) ²/N → 0 is used to control error terms in Lemmas 6 and 7 in the present paper and in Lemma B.3 in [7]. This assumption on m(N) is optimal in the sense that it is a minimal assumption guaranteeing that error terms in parts (a) and (b) of Lemma B.3 in [7] converge to 0. In the next theorem, for A a subset of $$ or $$ we denote by $$ the infimum of $$ over θ ∈ A.

The properties of $$ in part (a) are proved in [33, Lem. 1.4.1] and part (a) of Theorem A.1. The basic step in proving the large deviation bounds in parts (b)–(d) is the local large deviation estimate in part (b) of Theorem 5. As explained in Section 4, this local estimate is lifted to large deviation limits involving open balls stated in Theorem 8, which in turn are used to derive the bounds in parts (b)–(d) of Theorem 1.

In the next theorem we use the large deviation upper bound in part (c) of Theorem 1 to prove that the Poisson distribution $$ is the equilibrium distribution of the number-density measures Θ_N,b. In this theorem $$ denotes the complement in $$ of the open ball $$. $$ denotes the complement in $$ of the open ball $$.

We now apply Theorem 2 to prove that $$ is also the equilibrium distribution of the random variables $$, which count the droplet sizes at the sites of Λ_N. This is the content of the next corollary. A fact needed in the proof is that Θ_N,b is the empirical measure of these random variables; that is, for ω ∈ Ω_N,b,m, Θ_N,b(ω) assigns to subsets A of $$ the probability $$. This representation is valid because both Θ_N,b(ω) and the empirical measure assign to j ∈ Λ_N the probability N_j(ω)/N.

The last theorem in this section proves several properties of $$ and $$ with respect to the Prohorov metric that are needed in the paper.

In the next section we present the local large deviation estimate that will be used in Section 4 to prove the LDP for Θ_N,b in Theorem 1.

3. Local Large Deviation Estimate Yielding Theorem 1

For ω ∈ Ω_N,b,m the components Θ_N,b;j(ω) of the number-density measure defined in (5) are N_j(ω)/N for $$, where N_j(ω) denotes the number of sites in Λ_N containing j particles in the configuration ω. We denote by N(ω) the sequence $$. By definition, for every ω ∈ Ω_N,b,m each site $$ is occupied by at least b particles, and |N(ω)|₊ ≤ m = m(N). It follows that A_N,b,m is the range of N(ω) for ω ∈ Ω_N,b,m; the two sums involving ν_j in (18) correspond to the two sums involving N_j(ω) in (1).

In part (b) of the next theorem we state the local large deviation estimate for the event {Θ_N,b = θ_N,b,ν}. In part (a) we introduce the Poisson distribution $$ that appears in the local estimate; $$ is defined in terms of a parameter α_b(c) guaranteeing that it has mean c.

In part (a) of Theorem C.2 in [7] we give the straightforward proof of the existence of α_b(c) for b = 1. The proof of the existence of α_b(c) for general $$ is much more subtle than the proof for b = 1. The proof for general $$ is given in Theorem A.2 in the present paper.

We now prove the local large deviation estimate in part (b) of Theorem 5. This proof is based on a combinatorial argument that is reminiscent of and is as natural as the combinatorial argument used to prove Sanov’s theorem for empirical measures defined in terms of i.i.d. random variables having a finite state space [1, §3]. Part (b) of Theorem 5 is proved by analyzing the asymptotic behavior of the product of two multinomial coefficients that we now introduce.

The next two steps in the proof of the local estimate given in part (b) of Theorem 5 are to prove the asymptotic formula for card⁡(Δ_N,b,m;ν) in Lemma 6 and the asymptotic formula for card⁡(Ω_N,b,m) in part (b) of Lemma 7. The proof of Lemma 6 is greatly simplified by a substitution in line 4 of (34). This substitution involves a parameter α ∈ (0, ∞), which, we emphasize, is arbitrary in this lemma. The substitution in line 4 of (34) allows us to express the asymptotic behavior of both card(Δ_N,b,m;ν) in Lemma 6 and card(Ω_N,b,m) in Lemma 7 directly in terms of the relative entropy R(θ_N,b,ν∣ρ_b,α), where ρ_b,α is the probability measure on $$ having the components defined in part (a) of Theorem 5. One of the major issues in the proof of part (b) of Theorem 5 is to show that the arbitrary parameter α appearing in Lemmas 6 and 7 must take the value α_b(c), which is the unique value of α guaranteeing that $$ [Theorem 5(a)]. We show that α must equal α_b(c) after the statement of Lemma 7.

The next step in the proof of the local large deviation estimate in part (b) of Theorem 5 is to prove the asymptotic formula for card⁡(Ω_N,b,m) stated in part (b) of the next lemma. The proof of this lemma uses Lemma 6 in a fundamental way. After the statement of this lemma we show how to apply it and Lemma 6 to prove part (b) of Theorem 5.

We now complete the proof of part (b) of Theorem 5 by proving Lemma 7.

In the next section we explain how the local large deviation estimate in part (b) of Theorem 5 yields the LDP in Theorem 1.

4. Proof of Theorem 1 from Part (b) of Theorem 5

In Theorem 1 we state the LDP for the sequence Θ_N,b of number-density measures. This sequence takes values in $$, which is the set of probability measures on $$ having mean c ∈ (b, ∞). The purpose of the present section is to explain how the local large deviation estimate in part (b) of Theorem 5 yields the LDP for Θ_N,b. All details appear in Section 4 of [7]. The basic idea is first to prove the large deviation limit for Θ_N,b lying in open balls in $$ and in other subsets defined in terms of open balls and then to use this large deviation limit to prove the LDP in Theorem 1.

In Theorem 8 we state the large deviation limit for open balls and other subsets defined in terms of open balls. Two types of open balls are considered. Let θ be a measure in $$, and take r > 0. Part (a) states the large deviation limit for open balls $$, where π denotes the Prohorov metric on $$. This limit is used to prove the large deviation upper bound for compact subsets of $$ in part (b) of Theorem 1 and the large deviation lower bound for open subsets of $$ in part (d) of Theorem 1. Now let θ be a measure in $$. Part (b) states the large deviation limit for sets of the form $$, where $$. This limit is used to prove the large deviation upper bound for closed subsets in part (c) of Theorem 1. If $$, then $$, and the conclusions of parts (a) and (b) of the next theorem coincide.

We prove Theorem 8 by applying the local large deviation estimate in part (b) of Theorem 5. A key step is to approximate probability measures in B_π(θ, ε) and in $$ by appropriate sequences of probability measures in the range of Θ_N,b. This procedure allows one to show in part (a) that the infimum $$ can be approximated by the infimum of $$ over θ lying in the intersection of B_π(θ, ε) and the range of Θ_N,b; a similar statement holds for the infimum in part (b). A set of hypotheses that allow one to carry out this approximation procedure is given in Theorem  4.2 in [7], a general formulation that yields Theorem 8 as a special case.

Theorem 1 states the LDP for the number-density measures Θ_N,b. In order to complete the proof of Theorem 1, we must lift the large deviation limits in Theorem 8 to the large deviation upper bound for compact sets and for closed sets and the large deviation lower bound for open sets. The large deviation lower bound for open sets is immediate from the limit in part (a). To prove the large deviation upper bound for compact sets, we cover the compact set by open balls and use the limit in part (a); the large deviation upper bound for closed sets follows by a similar procedure involving part (b). The details of this procedure are carried out as an application of general formulation in Theorem  4.3 in [7].

In the Appendix we prove two properties of the relative entropy and prove the existence of the quantity α_b(c) appearing in part (a) of Theorem 5.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.