Approximations of the Sum of States by Laplace’s Method for a System of Particles with a Finite Number of Energy Levels and Application to Limit Theorems

We consider a generic system composed of a fixed number of particles distributed over a finite number of energy levels. We make only general assumptions about system’s properties and the entropy. System’s constraints other than fixed number of particles can be included by appropriate reduction of system’s state space. For the entropy we consider three generic cases. It can have a maximum in the interior of system’s state space or on the boundary. On the boundary we can have another two cases. There the entropy can increase linearly with increase of the number of particles and in the another case grows slower than linearly. The main results are approximations of system’s sum of states using Laplace’s method. Estimates of the error terms are also included. As an application, we prove the law of large numbers which yields the most probable state of the system. This state is the one with the maximal entropy. We also find limiting laws for the fluctuations. These laws are different for the considered cases of the entropy. They can be mixtures of Normal, Exponential and Discrete distributions. Explicit rates of convergence are provided for all the theorems.


Introduction
We consider a system which is composed of N particles distributed over m+1 energy levels. The energies of the levels are given by the vector ε = (ε 1 , ε 2 , . . . , ε m , ε m+1 ).
Tomasz M. Łapiński 84tomek@gmail.com 1 A single system's state is represented by the vector of the occupation numbers of the levels, denoted by (N 1 , N 2 , . . . , N m , N m+1 ). Since the number of particles in the system is fixed, the occupation numbers satisfy For each accessible state, let us introduce vectors of weights of the first m occupation numbers, that is, x = (x 1 , . . . , x m ) := N 1 N , . . . , N m N . Due to constraint (1), the last occupation number, i.e. N m+1 , is determined by the first m numbers. So, the vector x together with the number of particles N uniquely determines a single state represented by the vector (N 1 , . . . , N m+1 ). Now, let us define a set D ⊂ R m composed of vectors x ∈ R m such that the following constraints are valid and also define a set L N := { N N , N ∈ N m } for N N 0 , N 0 ∈ Z + . We assume that 0 ∈ N m . Then, the state space of the system is represented by the set D ∩ L N .
Initially, we assumed the system is constrained only by the number of particles. Constraints on the other properties of the system can also be included. Additional constraints can reduce the set D ∩ L N even further to a set A ∩ L N , where A ⊂ D. So, let A ∩ L N be the set of states of the system which are accessible under given generic constraints.
An example of additional constraint is bounded maximal average energy per particle, i.e. E m+1 i=1 ε i N i /N. When such exemplary system has a large number of particles, majority of its states are in the range (NE − , NE), for some small > 0. Therefore, such system could be considered as a microcanonical ensemble, i.e., a fixed number of particles and total energy. Let us consider system's entropy S defined on the set A. We assume the function S is sufficiently regular, has a unique maximum and can be represented as a product of two functions, that is S(x, N) = h(N)f (x, N), (2) where f, h are sufficiently regular, and h is also an increasing function.
Particular cases of the considered generic system might include features such as energy level degeneracy and indistinguishability of the particles. Such features might be reflected in the specific form of the functions f and h, as shows the example in the end of introduction.
The main results of this paper are approximations of the sum of states (N) defined by where the function g is sufficiently regular. First, we develop a result for two energy levels, i.e. m = 1, with the maximum of the entropy on the boundary of the domain of summation. The methodology of the proof is based on the analogous result for Laplace's integral in [7]. Although, this univariate case is rather insignificant in the physics context, we need a specific estimates for the further development.
Our main concern is with a finite number of energy levels. We prove the results for two cases of the function f in (3). The function f (·, N) can have a unique maximum in the interior of A or a unique non-critical maximum on the boundary {x : x 1 = 0}. For the integral instead of the sum, analogous results are proved in [7] and in a simplified form also in [5]. Here we use the same methodology as in [7], and also include an explicit remainder estimate.
An alternative way of approximating the sum of states for a similar class of systems is developed in [10].
Then we consider a discrete random vector X(N) := (X 1 , X 2 , . . . , X m ), where X(N) ∈ A ∩ L N . We use the fundamental postulate of statistical mechanics that system's microstates are equally probable, see e.g. [8] and [9], and define the pmf of X(N) to be where the entropy S(x, N) is given by (2).
As an application, we use the approximations of (N) to explicitely calculate the limit of X(N) as N → ∞. For that, we prove the law of large numbers, which can be interpreted as finding the most probable state of the system with a very large number of particles. This state is the point of maximum of the entropy (2). Our next results yields the distributions of the fluctuations from the most probable state. They are different for two cases of the entropy maximum. When the maximum is in the interior of the domain, the fluctuations have Normal distribution. When the maximum is on the boundary, there can be further two cases depending on the function h(N) in (2). If h(N) = N, then the fluctuations distribution is Exponential in the direction orthogonal to the boundary of the state space and Normal in other directions. When lim N→∞ h(N) N = 0, the fluctuations distribution is Discrete in direction orthogonal to the boundary and Normal in other directions. Explicit rate of convergence is provided for all the limit theorems.
Analogous limit theorems for the integrals instead of the sums are proved in [7]. For the integral with Gaussian measure, law of large numbers and central limit theorem are proved in [1,2] and [3]. Another application of Laplace's method to prove the limit theorems is presented in [4].
The results presented in this paper are applied in proofs of limit theorems in [6]. There the considered system is more specific. It consist of particles that are noninteracting and indistinguishable with the average energy per particle smaller or equal to some prescribed value. Furthermore, that system has degenerate energy levels, and the number of degeneracy G depends on the number of particles, that is, G = G(N). Moreover, three cases of the degeneracy function G are considered. The functions f , h and the point of maximum of the entropy were derived from the system's properties and are different for each case of G(N) where λ, ν are some constants obtained from the systems constraints. Then with use of the theorems developed in this work author proves that for the large enough system the points of maximum given above are the most probable states. Those states are well known Maxwell-Boltzmann statistics, Bose-Einstein statistics and Zipf-Mandelbrot Law, respectively. The fluctuations are also provided.

Approximation with Laplace's Method
First, let us make several technical assumptions about the sum of states (3) and the functions f, h and g. We consider a closed ball B ε ⊂ A with the center at the origin, radius ε > 0 and volume |B ε |. Then we specify the function f : The derivatives of f (·, N) up to third order exists on B ε and are uniformly bounded. For all N N 0 , where N 0 ∈ Z + , the function f (·, N) have a unique maximum at x * (N) ∈ B ε such that We choose the origin of our coordinate system to be the point x * = lim N→∞ x * (N). Further, we specify that the function h : R + → R is positive, increasing and lim N→∞ h(N) We also specify the function g : A → R. Assume g is differentiable in B ε and define constant where . is a max norm and D is a differential operator, that is, D := ( ∂ ∂x 1 , ∂ ∂x 2 , . . . , ∂ ∂x m ). Let us assume that the sum (3) is finite and specify the two cases of its function f (a) f (·, N) has a nondegenerate maximum in the interior of B ε and introduce constant (b) f (·, N) has a unique maximum on the boundary {x : x 1 = 0} and ∂f (x * (N),N) ∂x 1 < 0. We also introduce constant where y = (x 2 , . . . , x m ) and D y is a differential operator in that coordinates. Furthermore, we assume that on every section B ε (x 1 ) = {y : (x 1 , y) ∈ B ε }, x 1 ∈ [0, ε) we have a unique nondegenerate maximum of f.

Remark 1
The situation when the boundary of the domain is {x : x 1 = a} with a ∈ Q + can be reduced to the case of the boundary {x : x 1 = 0}, if we only consider N such that Na ∈ Z. This is because for those values, the structure of lattice L N is preserved after appropriate shift of the coordinate system.

One Dimensional Entropy
For the case (b) of the function f and A = [0, ∞) we define a set where the parameter δ ∈ (0, 1). Additionally, let us define N 1 := max h −1 ε

Theorem 1 For the case (b) of the function f and
where ω UB is defined in Proposition 1.
by the formula for the summation of the geometric series. Then, we estimate the last term by the simple approximation of the sum with an integral where I N > N h(N) 1−δ − 1 and the expression in the exponent is negative due to (12). Hence we get the result of the Proposition.

Proof of Theorem 1 Let us introduce S B (N), B (N) and using the Taylor's Theorem decompose (N)
Here and everywhere in the proofs x θ denotes a point between x and the point of the expansion. It might be different in a different instances. Now, we put together the above expressions For | 11 (N) − B (N)| we use the second order Taylor's Theorem to obtain The second term in the Taylor's Theorem can be bounded, that is |f (x θ , N)x 2 | F (2) x 2 , where F (2) is defined by (15). Next, using result (17) with the inequality |e t − 1| |t|e |t| and the fact that for any where G is defined by (6). We need the last term in the inequality (18) to be bounded as N → ∞, hence we set δ ∈ 0, 1 2 . Then, with use of Proposition 1 we obtain the estimate Next expression to approximate, | B (N) − S B (N)|, can be directly obtained from Proposition 1 Now, let us consider the sum 12 (N). Here again, we apply the second order Taylor's Theorem and since g has a bounded derivative in U N we obtain where G (1) is defined by (6). Further, applying Proposition 1 and using that x h(N) −1+δ we obtain For | 2 (N)| we apply the first order Taylor's Theorem which yields where F (1) is defined by (12). Then we substitute it into | 2 (N)| and get The number of elements in the set is bounded by εN + 1. Therefore In case of | 3 (N)| we have the following upper bound where the last inequality is due to assumption (7).
Then, we combine the above approximations

Multivariate Entropy with Maximum in the Interior
For the case (a) of the function f we define sets where the parameter δ ∈ 0, 1 2 . Further, let us define (2) h(N)

Theorem 2 For the case (a) of the function f , the following approximation holds
where ω UI is defined in Proposition 1.

Proof of Proposition 1 Let us define I (N) and (N)
and decompose I into four integrals with the indicator function Then decompose I 1 (N) into a smaller integrals and use the Taylor's Theorem We combine above decompositions into as the integral is equal to the volume of the hypercube V N,y , that is, dx.
Next, we estimate the derivative in (19) where F (2) is defined by (10). Since in the integration we included the points outside U N and within V N , hence we have Using that and the fact that the volume of V N,y is N −m we obtain where the last inequality is because D 2 f is negative definite in U N ⊂ B ε , hence occurring exponent can be bounded by 1. Now we estimate the size of the sum U N ∩L N 1. It is clear that this sum is bounded by the number of the hypercubes V N,y , y ∈ L N that intersects U N . The sphere of the radius h(N) − For the approximation of |I 2 (N) − I 3 (N)| let us introduce a set Since U N contains the domains of the integration of I 2 (N) and I 3 (N) we have where we used the fact that with F (2) defined by (8). Next, we calculate the integral in (21) by performing the change of coordinates system to the spherical Hence, we obtain the following estimate for |I G2 (N) − I G3 (N)| Finally, we approximate the last integral I 4 (N) where F (2) det is given by (9). Then we combine all approximations and for the error term to decrease as N → ∞, we set δ ∈ 0, 1 2(m+1) . N)f (x,N) .

Proof of Theorem 2 Let us introduce I G (N), G (N) and using the Taylor's Theorem decompose (N)
We combine the above decompositions into For | 11 (N) − G (N)| we use the third order Taylor's Theorem to obtain (N) is critical point. The third term in the Taylor's Theorem can be bounded where F (3) is defined by (11). Next, using the result (22) with the inequality |e t −1| |t|e |t| and the fact that for with G defined by (6). In order to bound the last term in the above estimate, we set δ ∈ 0, 1 6 . Then with use of Proposition 2 we obtain the following estimate with δ ∈ 0, 1 2(m+1) . For the estimate to be valid for all m ∈ Z + , we set δ ∈ 0, 1 3(m+1) . Next expression to approximate, | G (N) − I G (N)|, can be directly obtained from Proposition 2 Now, let us consider the sum 12 (N). Here again, we apply the third order Taylor's Theorem and since in U N the derivative of g is bounded by G (1) , we obtain where the constant G (1) is defined by (6). Further, applying Proposition 2 and using that |x − x * (N)| h(N) −1/2+δ in U N we obtain For | 2 (N)|, we apply the second order Taylor's Theorem to f to get and then estimate | 2 (N)| Since in the set B ε \U N , the function g is bounded by G and |x − x * (N) Now, we estimate the size of the sum U N ∩L N 1. It is clear that this sum is bounded by the number of the hypercubes V N,y , y ∈ L N that intersects B ε . The sphere of the radius ε + √ mN −1 and dimension m contains all such hypercubes. Therefore, this sphere volume divided by the volume of V N,y , that is is an upperbound for the number of V N,y that intersects B ε . Adding that to the estimate (23) yields In case of | 3 (N)| we have the following upper bound where the last inequality is due to assumptions (6) and (7).
Then we combine the above approximations

Theorem 3 For the case (b) of the function f , the following approximation holds
where ω UI is defined in Proposition 2, ω I in Theorem 2 and ω B1 , ω B2 , ω C are where ω UB inside ω B1 and ω B2 is defined in Proposition 1.

Remark 2
The situation when the boundary is an arbitrary hyperplane with rational coefficients can be reduced to the case with boundary {x : x 1 = 0}. This is because after appropriate rotation of the coordinate system, the structure of the lattice, which is essential for the application of Theorem 3 is preserved. That is, all the points of the domain are on the equally spaced hyperplanes parallel to the boundary {x : and approximate 2 (N) as the sum 3 (N) in the previous proof.
Then we express 1 (N) as where the sum is over the values of coordinate x 1 of the points in B ε ∩ L N . Further, we have where B ε ∩ L N (x 1 ) = {y : (x 1 , y) ∈ B ε ∩ L N }. Next, we apply Theorem 2 to 1 (x 1 , N) y, N). As the summation is over the set B ε ∩ L N (x 1 ), the constants which occurs as a result of application of Theorem 2 can be replaced by the appropriate constants for the larger set B ε , which are independent of x 1 , that is (13), (14), (15) and (16). Then, we apply Theorem 1 to 1 (N) where (0, y * (0, N)) = x * (N). Since Theorem 1 was applied on the curve y * (x 1 , N), the constants in the estimate of ω B1 (N) and ω B2 (N) can also be replaced by the constants for the larger set B ε i.e. (12), (13) and (15). (1) h(N) δ exp F (2) 2h(N) 1−δ × GF (2) 2 + G (1) Then we combine the above result with the estimate of 2 (N) to obtain the final result.

Limit Theorems
For the function f in (3), let us additionally assume

Weak Law of Large Numbers
Theorem 4 (Weak law of large numbers) As N → ∞, the random vector X(N) converges in distribution to the constant x * and the following estimate of the mgf holds where δ ∈ 0, 1 3(m+1) .
Remark 3 For this and the following limit theorems the convergence error term can be explicitly estimated with use of the results from the Section 2.
Proof To prove the convergence of X(N), it is sufficient to prove the convergence of its moment generating function A∩L N e h (N)f (x,N) , A∩{x:x 1 0}∩L N e h (N)f (x,N) , where |ξ | < h, for some h > 0. We approximate the denominator for the case (a) of f with use of Theorem 2, and for the case (b) using Theorem 3 Then we do the same for the numerator A∩{x: Dividing the approximation of denominator by the approximation of numerator yields for the both cases  Z m (N) . Furthermore, the following estimate of the mgf holds

Theorem 7 (Central limit theorem III) For X(N) with distribution (27) and assuming
) converges weakly to the discrete distribution with the pmf . . , Z m (N) . Furthermore, the following estimate of the mgf holds and substituting that into appropriate estimate of the mgf yields the final result.