1. Introduction

Composite materials play extremely important role in modern technologies. Composite materials with different types of inclusions as structure-forming elements form a large class of such materials.

Studying of properties such materials, processes of their forming is a challenging problem [5-7]. In the present article we introduce a model that describes thermodynamics of interaction of a thread-like inclusion (for example, carbon thread) with a compound of the matrix of a composite material. We assume the thread is in thermodynamics equilibrium with the material of the matrix and describe dynamics of the thread by one-dimensional wave equation with appropriate boundary conditions. Compound of the matrix is considered as a thermostat. On the infinite-dimensional phase space of the thread, we introduce Gibbs measure with the Hamilton function, which describes the oscillation of the thread. Such approach allowed us to evaluate exactly the partition function for our dynamical system. Using the partition function, we calculate free energy, entropy, and heat capacity related to the interaction of the thread and the matrix. To the best of author’s knowledge such approach is new.

2. Gaussian Measure and Partition Function

In this section we provide necessary information about Gaussian measure on infinite-dimensional spaces. Detailed exposition can be found in [1].

We start with some definitions.

Definition 1. Let μ be a finite Borel measure on a real separable Hilbert space H with inner product ( , ) and norm . The measure μ is said to be Gaussian with mean value zero if the function defined on H has Gaussian distribution with mean zero and variance , i.e.

Definition 2. The characteristic functional of a finite Borel measure on a Hilbert space H is defined by the formulas

(1)

The following theorem is valid

Theorem 1. A finite Borel measure on a Hilbert space H is a Gaussian measure with mean value zero if and only if its characteristic functional is given by

(2)

where is a positive selfadjoint trace class operator on the space H.

The operator is called the covariance operator and .

Consider now a dynamical system whose phase space is the space H. We denote the energy of the system in the state (Hamiltonian function) by and assume that

(3)

Let be a sequence of positive numbers () such that . Without loss of generality we may assume that Select and orthonormal basis of the space H and define an operator on H as follows:

(4)

It is clear that B₀ is a positive selfadjoint operator of the trace class.

Let μ₀ be a Gaussian measure μ₀ on H with covariance operator B₀ and mean value zero.

Definition 3. The Gibbs distribution on the phase space H with inverse temperature β (β = 1/T) is a probability Borel measure μ on H which is absolutely continuous with respect to μ₀ with the density given by

(5)

where the partition function Ξ is defined as

(6)

Remark 1. We assume that the temperature T is measured in energy units.

Proposition 1. Suppose that condition (3) is fulfilled. Then

(7)

Remark 2. Since the operator B₀ belongs to the trace class, the infinite product in (7) converges, and Ξ (β) is an analytic function of β in the half-plane Re β > -1/(2λ₁). We will be interested only in values of β > 0.

Proof of the proposition is based on straight forward evaluation of the integral (7) as a product of Gaussian integrals.

Having the partition function Ξ (β) one easily obtains expression for the thermodynamics Helmholtz free energy of the system (see, for example, [2])

(8)

Now using thermodynamics relation one obtains the expression for entropy

(9)

and “heat capacity”

(10)

Below we give two examples on applications of formulas above.

3. Vibrating String

In this section we apply previous arguments to the dynamical system described by one-dimensional wave equation. We refer to [3, 4] for necessary information.

We consider a vibrating string of the length with the linear density , stretched along the horizontal axis with the force . Denote by the vertical coordinate of the point of the string with horizontal coordinate at moment . It is well known (see any text book on elementary PDE, for example [3, 4]) that the function satisfies the following partial differential equation

(11)

For definiteness we assume that

(12)

Put , where

The phase state of the system is completely determined by a vector

(13)

where

(14)

The Hamiltonian of the system is given by

(15)

The first term is the kinetic energy of the string, and the second term is its potential energy.

A covariance operator we select in the form , where ; are positive trace class operators in the spaces . We select them in the way that reflects boundary conditions (12).

Note that the function satisfies the same partial differential equation (11) and the same boundary conditions (12) that the function

(16)

Solving (16) by separation of variables , one obtains for the function a selfadjoint boundary value problem

(17)

Put

(18)

Where

(19)

is the Green's function associated with the boundary value problem (17). The coefficient τ ⁄ L has occurred from the requirement that the eigenvalues of the operator B₁ have to be in energy units. The operator B₁ is selfadjoint and positive, its eigenvalues λ⁽¹⁾_k are up to constant factor reciprocal to the eigenvalues of (17) and they are given by the formula

(20)

Sinc , the operator B₁ belongs to the trace class. Normalized eigenvectors e⁽¹⁾_k of the operator B₁ are given by

We select B₂ in the same form as B₁. Thus, the operator has eigenvalues

and each has multiplicity 2.

Formula (7) gives

,

where we used the following well-known formula

.

Formula (8) gives

For entropy S one obtains using (9)

and for heat capacity c using (10)

Note that . as

4. Conclusions

We simplified the analytic expressions for thermodynamics of interaction of thread-like inclusions with compound of a matrix in composite materials and our results could be used for modelling of processes of forming structure in such materials. With appropriate modifications, our method can be applied to the cases of two and three dimensional inclusions.