Interaction Model in Statistical Mechanics

Email: farida.kachapova@aut.ac.nz Abstract: Statistical mechanics considers several models such as Ising model, Potts model, Heisenberg model etc. A rigorous mathematical approach based on the axiomatic foundation of probability would benefit the study and applications of these models. In this paper we use this approach to generalize some of these models into one construction named an interaction model. We introduce a mathematically rigorous definition of the model on an integer lattice that describes a physical system with many particles interacting with an external force and with one another; a random field Xt ( )  t∈Z models some property of the system such as electric


Introduction
Statistical mechanics studies models of physical systems with many particles, which interact with an external force and with one another. Well-known models include Ising, Potts, Heisenberg, and n-vector models (see, for example, Duminil-Copin et al., 2017;Kashapov, 1977;Külske et al., 2014;Malyshev and Minlos, 1991;Malyshev, 1980). In these models a random field X t is used to model some property of the system such as electric charge, density etc.
Ising model is the simplest and most popular model. It describes a system with two states and models the phenomenon of ferromagnetism. It is also used in quantum field theory. Potts model is a generalization of Ising model to a system with a finite number of states.
An n-vector model represents classical spins by ndimensional vectors of unit length. This model can be used to describe many physical phenomena. Particular cases of this model include the Ising model for n = 1, XY-model for n = 2 and Heisenberg model for n = 3. Kachapova and Kachapov (2016) introduced the concept of interaction model as a generalization of some existing models; there we provided a proof based on this concept that the random field X t transformed by renormalization group converges to an independent random field with Gaussian distribution.
In this paper we generalize and improve the interaction model from (Kachapova and Kachapov, 2016). The new model does not have restrictions on the distribution of X t and the set of values of X t can be unbounded, which is an advantage of this model comparing to all aforementioned models, which have the values of X t bounded.
In this paper we use a rigorous mathematical approach based on the axiomatic foundation of probability. We introduce a mathematically precise definition of interaction model on an integer lattice: first as a finite model and then as the thermodynamic limit of the finite models with Gibbs probability measure.
We study properties of the interaction model and show how some well-known models are represented as particular cases of the interaction model.
In Section 1 we introduce main components of the interaction model of a physical system that include an integer lattice v ℤ , the set of configurations of the system, initial independent probability measure P 0 and a random field ( ) v t X t∈ ℤ that models a property of the system. Next we introduce three characteristics of the interaction model: a main parameter λ, radius of interaction r and potential Φ.
In Section 2 we study Gibbs modification of a probability measure. In particular, we split Gibbs modification of the initial independent probability measure into two steps reflecting the influence of an external field on the first step and the interaction between particles on the second step. The first-step modification is mathematically simple and leaves the field X t independent, therefore this construction simplifies mathematical computations.
In Section 3 we define a finite interaction model on an integer cube using Gibbs modification of the initial probability measure P 0 . We prove some properties of the finite model.
In Section 4 we define an infinite interaction model and in Section 5 we show that Ising and Potts models are particular cases of the interaction model. In Section 6 we discuss how to generalize our model, so that the n-vector model becomes its particular case too.

Main Components of Interaction Model
In Section 3 we will construct an interaction model to describe a physical system with many particles. Here we introduce its main components.
We call P 0 the initial probability measure.
For the rest of the paper we fix the objects , , , X ν Ω ∑ ℤ and P 0 from this definition.

Remark
There always exists P 0 satisfiying (1). For example, if F is any probability distribution function, we can take P 0 (ω(t) ≤ a) = F(a) for any t and define the rest by formula (1).
ℤ is an independent random field on this probability space.

Proof
The lemma immediately follows from the definitions. ■ Definition 1.3 1. Consider a graph (V, E), where the set of vertices V is a finite subset of ν ℤ and E is the set of edges; each edge can be regarded as a pair of distinct vertices (there are no loops). The length of each edge is the distance between its end vertices. 2. The graph (V, E) is called 1-connected if it is connected and the length of any of its edges equals 1. 3. For a finite set B ν ⊂ ℤ define its size S(B) as the minimum number of edges of 1-connected graphs (V, E) such that B V ⊆ .

Definition 1.4
Here we introduce three main characteristics of interaction: , , r λ Φ and a set B.

2.
, 1 r r ∈ ≥ ℝ . r is called the radius of interaction. Φ is called the potential of the system. Clearly, each set B∈B is finite. Φ B characterizes the interaction energy of the set B. If B consists of two or more points, then the random variable Φ B represents interaction between elements of the set B. If B = {t} is a singleton, then Φ B represents interaction of t with an external field and the influence of kinetic energy.

Lemma 1.5
If sets B, C∈B and B ∩C = ∅, then the random variables Φ B and Φ C are independent with respect to the probability measure P 0 , that is on the probability space (Ω, Σ, P 0 ).

Proof
This is proven using standard techniques of probability theory, first for the case of discrete Φ B and Φ C , and then for general random variables Φ B , Φ C as limits of discrete random variables. ■

Gibbs Modification
To describe interaction between particles, we modify the initial probability measure P 0 , so that the corresponding distribution of the random field is not independent any more. This section describes the modification in general.
For any probability measure P on (Ω, Σ) denote . P the expectation with respect to P.

Definition 2.1
Suppose P is a probability measure on (Ω, Σ) and U is a bounded random variable on (Ω, Σ).
Gibbs modification of the probability measure P by the random variable U is denoted P U and is defined as follows. For any event A∈Σ: where I A denotes the indicator of event A.

Remark
Since the random variable U is bounded, both expectations in formula (2) exist and is always defined.
The following lemma is used in literature without proof. In order to have a complete picture we provide an accurate proof here.

Lemma 2.2
In conditions of the previous definition: 1) P U is a probability measure on (Ω, Σ); 2) for any random variable Y on ( ) , , Proof 1) To complete the proof it remains to show that for any sequence of disjoint events A i ∈Σ (i = 1, 2, ...) the following holds: (3) U is a bounded random variable, so for some constant M, |U| ≤ M and 0 < e U ≤ e M .
Since P is a probability measure, we have: So for any ε>0 there is n ∈ ℕ such that   Proof of (3) • Case 2: It is a well known fact in probability theory that any random variable can be represented as a uniform limit of discrete random variables. Thus, for any ω∈Ω: where the convergence is uniform with respect to ω∈Ω and each V n is a discrete random variable with a finite number of values; this is a random variable from Case 2. Then: As n→ ∞, we have for any ω∈Ω: V n (ω) − Y(ω) → 0 uniformly on Ω, so |V n (ω)− Y(ω)| → 0 and: U is a bounded random variable, so for some constant M, |U| ≤ M and 0 < e U ≤ e M .

Definition 3.1.
A finite interaction model with characteristics λ, r, four objects defined as follows.

A function
: U Λ Ω → ℝ is called the interaction energy and is defined by the following: where the set B is defined in Definition 1.4. U Λ (ω) characterizes the energy of configuration ω in Λ .

Remark 1
The random variable U Λ is bounded because the sum (9) has a finite number of addends and each ( ) since 0 1 λ ≤ < . Therefore Lemma 2.2.1) holds when stated for P Λ instead of : U P P Λ is a probability measure on (Ω, Σ) and A Λ is indeed a probability space.

Remark 2
The random field { } t X t ν ∈ ℤ is independent on the base probability space (Ω, Σ, P 0 ) but it may not be independent on the probability space ( ) , , P Λ Ω Σ . The finite interaction model describes a physical system with many particles represented by points in an integer cube. The random field Xt describes some property of the physical system. The interaction model generalizes some well-known models in statistical mechanics (we give details in Sections 5 and 6). In those models the values of random variables X t are bounded. Here we have a more general case when the values of X t are not bounded.
For are independent with respect to the probability measure P 0 , that is on the probability space (Ω, Σ, P 0 ); this follows from the Definition 1.1. Based on that, similarly to Lemma 1.5 it is proven that for any B ⊂ Λ the random variables I A and Φ B are independent with respect to P 0 . Therefore I A and For the rest of this section we fix the finite interaction model from Definition 3.1.

Definition 3.3
Here we introduce random variables U′ and U″. For any ω∈Ω we define: Thus, U = U′+ U″. The function U is split into U′ and U″, where U′ is a sum over singleton sets B and U″ is a sum over sets B with two or more elements.
Consider consecutive Gibbs modifications P′= (P 0 ) U ′ and P″ = (P′) U″ . We call P′ the single modification and P″ the plural modification.

Lemma 3.4.
1) For any t ∈ Λ and x ∈ ℝ : 3) After the single modification the field { } t X t ν ∈ ℤ is still independent. That is, this field is independent on the probability space (Ω, Σ, P′).

Infinite Interaction Model
Definition 4.1 An infinite interaction model with characteristics λ, r, Φ is the ordered sequence of two objects (X, A) defined as follows.
1. X is the fixed random field on (Ω, Σ) introduced in Definition 1.1. Let P λ be a probability measure on (Ω, Σ) that is a limit of the probability measures P N in some sense, for example, their weak limit, as N→∞.

For any
A is defined by: Thus, A is a probability space. 3. The probability measure P λ is called Gibbs measure.
4. The infinite interaction model (X, A) is also called the thermodynamic limit or macroscopic limit of the finite interaction models ( ) , ,

Remark
The problem of existence of the probability measure P λ as a limit of the probability measures P N is studied in literature but not in a rigorous mathematical context; it needs further investigation.

Ising Model
Ising model is an important mathematical model of ferromagnetism in statistical mechanics. It can be described as a particular case of the interaction model: • P 0 (X s = 1) = P 0 (X s = −1) = 0.5; • r = 1; After the single modification (by U′) the distribution of X t is still independent and is given by: After the plural modification (by U″) we get the Ising model, that is the modification by U Λ , as proven in Theorem 2.3. Modification in two steps can simplify computations.
Some authors study the Ising model with four point interaction; see, for example, (Yang et al., 2017) and references in it. In that case we add to the above definition of Φ B a clause for a 4-point set B: where B consists of the vertices of a unit square or the vertices of a tetrahedron with three edges of length 1 and three other edges of length 2 .

Potts Model
Standard Potts model can be described as a particular case of the interaction model: The Ising model is a particular case of the standard Potts model when q = 2 (it can be reduced to the Ising model by linear transformation X t →2X t −3).

Discussion
In this study we develop a mathematically rigorous concept of interaction model for a physical system with many particles, which interact with an external force and with one another; a random field ( ) t X t ν ∈ ℤ models some property of the system such as electric charge, density etc. We introduce a finite model first and then define the thermodynamic limit of the finite models with Gibbs probability measure. Unlike most existing models, in our model the set of values of X t can be unbounded, which provides more generality.
We study properties of the interaction model. In particular, we split Gibbs modification of the initial independent probability measure into two steps reflecting the influence of an external field on the first step and the interaction between particles on the second step. The first-step modification is mathematically simple and leaves the field X t independent, therefore this construction simplifies mathematical computations.
Next we show that Ising and Potts models are particular cases of the interaction model. If we change the set of values of the random field X t from ℝ to n ℝ , then the generalized interaction model will also include the n-vector model and its special cases for n = 2 (XY-model) and n = 3 (Heisenberg model). We are planning to research further mathematical properties and physical applications of the interaction model.