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Chapter Three

Classical Statistical Mechanics

1. Introduction

In this chapter, we deal with the concepts and properties of statistical

ensemble. It also introduces the concept of phase space as a

representative of the statistical system. We also deal with the applications

of the micro canonical ensemble to study thermodynamic variables in the

case of the ideal single-atom gas .

Statistical mechanics deals with the properties of matter at equilibrium in

the empirical sense used in thermodynamics. Statistical mechanics aims

to derive all characteristics of the macroscopic molecular system in the

equilibrium state through molecular kinetic laws, as well as to derive the

thermodynamic qualitative functions of the system under study.

The system is considered conventional (classical) if it contains a large

number of particles (N) and occupies a large volume (V), where the

typical values for these quantities are as follows: N≅1023

, ,𝑉≅1023

, and

given that these values are very large, it is appropriate to consider the

limit case is as follows:

………...1

Where 𝑣 is specific volume.

If the interactions between the system and the external environment are

weak, then the energy of the system remains approximately constant,

meaning that the energy is constant in motion, and this is just an ideal

idea, as there is no truly and completely isolated system unless the walls

containing the system are completely reflective in an ideal way.

2. Statistical Ensemble

To describe any macroscopic system, the scientist Gibbs introduced the

idea of the statistical ensemble, and this idea is summarized as follows:

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It is clear that a very large (infinite) number of states of the system

correspond to a specific condition of the system. For example, the

requirement that a gas be contained in a box of a finite size corresponds

to an infinite number of ways to distribute its particles in that space.

Through macroscopic measurements, we cannot distinguish between two

gases present in different states, but which meet the same macroscopic

conditions. Therefore, when studying the behavior of a gas under certain

macroscopic conditions, we are not really referring to a single state, but

rather to an infinite number of states. In other words,

We refer to a grouping of systems that are similar in structure and

macroscopic condition (macroscopic state) but in different states. Gibbs

named this compilation of systems as Statistical Ensemble.

** That is, the statistical ensemble can be defined as follows:

It is a collection of a very large number of microscopic states with the

same macroscopic state.

The state of an assembly at a given instant of time may be defined by

specifying the position and either the momentum or velocity of each of

the component systems. (It will be seen later that, mathematically, the

definition in terms of the momentum is most convenient.)

The position and momentum may be specified in Cartesian coordinates

by taking the position as (x, y, z) in Euclidean space, while the

corresponding components of the momentum (px, py, pz) specify the

'position' of the system in momentum space.

The state of a system is thus precisely defined by the six coordinates x, y,

z, px, py, pz and it is, therefore, convenient to consider the system to be

moving in a six-dimensional space which is termed phase space or

𝛤 − space.

As it is convenient to define an element of volume in Euclidean space so

that a system with coordinates in the range x to x + dx, y to y+dy and

z to z+dz lies within the volume:

dV = dxdydz

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it is also convenient to define an element of volume in phase space

so that a system with its position and momentum coordinates in

the range x to x+dx, y to y+dy, z to z+dz, px to px+dpx, py to py+dpy,

pz to pz+dpz, lies within a volume:

The kinetic energy of a system which has its coordinates lying within this

volume d𝛤 will be:

Where m is the mass of the system and the velocities are assumed to be

non-relativistic.

As the state of a single system is defined in terms of six coordinates

so it is possible to define the state of an assembly of N systems in terms

of 6N coordinates which is -3N position coordinates and 3N momentum

coordinates.

It is sometimes convenient to allow these 6N coordinates to define

mathematically a 6N-dimensional phase space - a 𝛤6N - space.

The coordinates of the system i may be written as:

And the coordinates of the assembly are then made up from all such sets

with the suffix i running from 1 to N. If the coordinates are taken to be in

the range xi to xi+dxi , pxi to pxi +dpxi , and so on for each of the 6N

coordinates then the 'point' representing the assembly in 𝛤6N- space will

be within the element of volume:

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Where (d𝛤)i is the volume element of the six-dimensional phase space

for the ith system.

The kinetic energy of an assembly which has its coordinates within the

volume (d𝛤6N) will be given by:

** In classical systems, statistical ensemble is a framework for studying

the behavior of these systems in a space called gamma space 𝛤 −𝑠𝑝𝑎𝑐𝑒,

or sometimes called phase space.

3. Phase Space of classical System

In classical mechanics, particle motion can be studied in what are known

as generalized coordinates, and generalized momenta, which are

symbolized as follows:

Where the number of degrees of freedom of particle motion is 3, and

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N is the number of system particles.

By analogy, Gibbs showed that it is possible to determine or describe the

state of the system in the statistical mechanics under study by (3N) from

these canonical coordinates (𝑞1, 𝑞2,………3𝑁 ) and (3N) from the

momentum associated with each coordinate (𝑝1 ,𝑝2,……..𝑝3𝑁).

The vector space (6 N) specified by (𝑞𝑖, 𝑝𝑖) is called the phase space of

the system, and the point in this space represents the state of the total

system in which N of the particles and indicates a representative point.

Therefore, according to the idea of Gibbs, the statistical ensemble is

represented geometrically by the distribution of the points represented in

the phase space, and this distribution is usually continuous.

If these coordinates are a function of time (t), then the movement of this

system can be determined by the following equations:

……..2

Where the function (𝑞𝑖, 𝑝𝑖) is the Hamilton function.

These equations give a description of the movement of the point

represented in this space, meaning that the path of the point may be a

closed curve that never intersects with itself.

In addition, the geometric locus of two represented points two finite ones

that can never intersect.

With the passage of time, the coordinates of the point (𝑞𝑖, 𝑝𝑖) which

define the microscopic state, are constantly changing.

Conversely, the points represented in the phase space plot paths

(trajectories) that are in directions defined by the velocity vector

ν(�̇�𝑖,�̇�𝑖 ( given in equation 2.

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These paths remain within a bounded region in this space, because the

specified system size V is limited by the coordinate values of 𝑞𝑖

While the finite system energy E is constrained the values of both 𝑞𝑖, 𝑝𝑖

by Hamiltonian function.

In particular, if the total system power has a precise value ,E, the path

corresponding to the point represented remains bound at the hypersurface

of the phase space, that is to say, the total energy of the system within the

range {𝐸 – Δ/2, 𝐸 + Δ/2}, and the corresponding pathway is bound in the

super sphere defined by these energy values.

Let us consider systems of the ensemble consisting of mental copies that

do not interact between them, at any point in time, the elements of this

ensemble can be classified as possible microscopic states, and each

element corresponds to the physical state of the system and we assume

that this element is a general representative of all the elements of the

ensemble.

With the passage of time, each element in the ensemble undergoes a

constant change of states and moves through phase space paths, so this

situation can be likened to a function that depends on coordinates,

momentum, and time, and this is called the density function ρ, and it is

symbolized by ρ (𝑞, p, t) .

Suppose a volumetric component in the phase space (𝑑3𝑁𝑞 𝑑3𝑁𝑝) around

the point (𝑞, 𝑝), as in the figure below:

The figure shows a phase space cell

Where the number of points represented in this element is as follows:

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…….3

** Physically, the density function expresses how the ensemble elements

are distributed for all possible microscopic states at different time points.

Average Ensemble

Suppose that f (q, p) represents a measurable physical quantity in a

system, such as energy or momentum, whose values are different in

different systems of microscopic states. And when this system is in a

thermodynamic equilibrium, the observed value of this quantity is the

result of what we obtain through a process of taking its rate on all the

microscopic states of this system (all ensemble members).

Mathematically, the ensemble rate, ˂𝑓˃, is known as:

............4

Noting that the integration in equation (4) extends over all regions in the

phase space, and the fact that the occupied regions in this space:

(Where 𝜌 ≠ 0) contribute to this integration.

Generally, the ensemble rate is a function of time.

Definition: An ensemble that has a time-dependent density function is

called a stationary ensemble, meaning it is:

∂𝜌 / ∂t = 0 for all time

** In this case, the ensemble rate (equation 4) is time independent, and

the stable ensemble is worthy of representing the system in equilibrium.

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There are two types of physical quantitative modifier and they are:

1. the most probable value is the value or magnitude f (q, p) that has the

largest number of ensemble systems (as explained earlier).

2. Ensemble modifier, which is determined (with Equation 4(:

The ensemble rate is approximately equal to the most probable value if

the square of the fluctuations of the quantity is small, and this condition

is expressed mathematically as follows:

If this condition is not fulfilled, then there is no one way to find how to

measure the observed value of the physical quantity f (q, p).

** It can be said that for all physical states, the square rate of

fluctuations is in the order of (1 / N) and thus when N →∞, the

ensemble rate and the most probable value become the same.

4. Liouville’s Theorem

** Liouville theory states:

The local density of the points represented, as seen from the moving

observer with which point represented, is constant over time.

Mathematically, this text can be expressed as follows:

Where the magnitude [𝜌, 𝐻] is known as Poisson Bracket

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Physically, the movement of a swarm of points represented in phase

space can be likened to the incompressible fluid flowing through

physical space.

And so that the magnitude of the Poisson Bracket is zero, that is:

The following condition must be met: that is ρ not an explicitly

temporal function, and that ρ is a function of coordinates (q, p) only,

i.e.:

(𝑞,𝑝) = constant

That is, the density function is equal to a constant amount in the relevant

region in the phase space (and of course everywhere in the phase

vacuum), and that means that the ensemble systems are uniformly

distributed over all microscopic states and at all times.

In this case, the ensemble rate (equation 3) is as follows:

Where 𝜔 denotes the total volume in this region of the phase space, and

this can be generalized to the neighboring region.

** When a macroscopic system is in thermodynamic equilibrium, its state

is equally likely in whatever condition the macroscopic conditions of the

system are satisfied, for all the different microscopic states (or different

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volume elements) in the permissible phase space. This hypothesis is

known: Postulate of Equal a Priori Probability

This hypothesis includes that the system in thermodynamic equilibrium

is a component of a set called a micro canonical ensemble that has the

following density function:

Where all the elements of the ensemble have the same number of

particles (N) and the same volume (V).

To understand the equation (Poisson Bracket):

It should be assumed that ρ depends on the coordinates (q, p) through the

Hamiltonian function, i.e.:

𝜌(𝑞,𝑝)=𝜌[ 𝐻(𝑞,𝑝)]

This equation gives a row of the density functions at which the ensemble

is stable.

5. Properties of micro canonical ensemble

In this ensemble, the macroscopic system under study is known as the

number of particles (N), the volume (V), and the total energy (E). But

instead of a sharp value of energy, we define a range of energy values as

follows:

E−Δ / 2, E+Δ / 2

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The choice remains for the case of the macroscopic system if it is any one

of the many microscopic states. Likewise, in the phase space, the points

representing the ensemble have the option to fall through the super cortex

specified by the following condition:

The volume of the phase space enveloping this shell is as follows:

Where the indicated integral, it means the extension of the integral over

the part of the phase space subject to condition 1.

That is, ω is a function of the following parameters ω (N, V, E, Δ).

** Accordingly, the micro canonical ensemble is defined as:

It combines the following systems with a density function, (for all time):

Accordingly, the expected value of the number of points represented and

located within the volume component 𝑑𝜔 in the considered region of the

super cortex is directly proportional to 𝑑𝜔. In other words, the

preconceived probability of finding the point represented in the given

volume element 𝑑𝜔 is equal to finding the point represented in the same

equivalent volume 𝑑𝜔 as placed anywhere in this super cortex, and this is

consistent with the hypothesis of equal probabilities for any element in

this ensemble.

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** If the ensemble is stable, then the time rate for any measurable

physical quantity does not give new results, and this gives the following:

<𝑓> =𝑓𝑒𝑥𝑝

Where <𝑓> represents the time mean of the ensemble average over a

long period of time, while

𝑓𝑒𝑥𝑝 represents the expected value of the physical quantity when

conducting an experiment over a long period of time.

** That is, one of the characteristics of the microscope ensemble is:

1. The ensemble rate for any physical quantity is similar to the value

expected to be obtained by conducting an appropriate measurement on

the system.

2. That the ensemble members can be linked with the laws of

thermodynamics through direct correspondence between the microscopic

ensemble states and the location represented in the phase space.

To illustrate this, we assume that:

ω represents the permissible region in the phase space, and Γ represents

the number of microscopic states.

In order to find the relationship between ω and Γ we need to define a

fundamental volume ωo which is the equivalent volume of a single

microscopic state, where:

Therefore, the relationship between entropy and the number of

microscopic states is as follows:

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Where ω0 is analogous to the nature of angular momentum, i.e. with a

dimension (𝑞𝑝) 3𝑁

6. Applications of microscopic ensemble:

Some examples of classic systems will be illustrated, using the properties

of the microscopic ensemble. .

Among these systems: the ideal gas and the simple harmonic oscillating

in one dimension.

Monoatomic ideal gas:

The ideal gas is known as a gas composed of identical particles, where

there is a weak interaction between these particles because they are

divergent and the gas density is low, and this means that the potential

energy of these particles is small compared to their kinetic energy, and

the system energy is a result of the kinetic energy of the particles. The

inert gases, such as helium gas and neon ..., are ideal gases and are also

monoatomic gases.

Suppose that a system of ideal gas consists of N particles and is confined

to a constant volume V, in a stable (static) thermal equilibrium at

temperature T. The energy of this system (the Hamilton function) is as

follows:

Where 𝑝𝑖 = momentum of the linear particle

The permissible volume of the phase space for the presence of the points

representing the elements of the microscopic ensemble is as follows:

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And the integration in equation (1) is restricted by the following

conditions:

1. Confining the particles to the physical space of a volume of V.

2. Considering the total energy values of the system within the following

specified energy range:

In the case of an ideal gas, the Hamilton function is a function of linear

momentum pi, and therefore the integral can be performed on the spatial

coordinate qi directly, and the result of this integration is as follows:

ω =𝑉𝑁

While integration on the momentum:

It gives an amount equal to the volume of the hyper shell of dimensions

(3N), and is limited to radii of:

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In the case of Δ≪𝐸, this volume is approximately equal to the product

of the thickness of the shell, Δ(𝑚 /2𝐸) 1/2

by area of the shell has radius

(2mE½ (

Where N = the number of degrees of freedom for the particle, for

example in the case of a single particle this number is equal to 3.

Therefore, the number of microscopic states = the size of the permissible

region for particle movement in the phase space divided by ℎ3.

Suppose: Σ (P) = the number of available microscopic states of free

particles enclosed in the volume V of the physical space, which have a

linear momentum quantity p, such that: 𝑝≤𝑃

To find Σ (P), we use the integral in the phase space of all elements, or

From equation (4), the number of microscopic states having linear

momentum of value within the range {p + d p, p} can be obtained as

follows:

The function 𝑔 (𝑝) = density of momentum states

Returning to classical mechanics, the linear momentum is in terms of

energy: 𝑝 = √2𝑚𝐸, so we can find the number of microscopic states in

terms of energy E, i.e.:

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From equation (6), the number of microscopic states having energy

within the energy range (𝜀 + 𝑑𝜀, 𝜀) can be obtained as follows:

α(ε) energy state density