principal investigator paper coordinator

1

Physics

Statistical Mechanics

Study of Two Prototype Physical Models from Statistical Route:

an Ideal Gas and a Collection of N Three Dimensional Classical Harmonic

Oscillators

Paper No. : Statistical Mechanics

Module : Study of Two Prototype Physical Models from Statistical Route:


Oscillators

Prof. Vinay Gupta, Department of Physics and Astrophysics,

University of Delhi, Delhi

Development Team

Principal Investigator

Paper Coordinator

Content Writer

Content Reviewer

Prof. P.K. Ahluwalia, Physics Department, Himachal Pradesh University, Shimla 171005

Prof. P.K. Ahluwalia, Physics Department, Himachal Pradesh University, Shimla 171005

Prof. P. N. Kotru, Department of Physics, university of Jammu

Physics

2

Physics




Oscillators

Description of Module

Subject Name Physics

Paper Name Statistical Mechanics

Module Name/Title Equilibrium, Thermodynamic Parameters and Response Functions

Module Id M8

3

Physics




Oscillators

TABLE OF CONTENTS 1. Learning Outcomes

2. Introduction 3. Classical Monoatomic Ideal Gas and Derivation of Entropy from Statistical Route

4. Back of Stamp Estimation of Microstates & Ideal Gas Law 5. Quantum Mechanical approach to Estimate number of Microstates

6. Explicit Calculation of Number of Microstates of an Ideal Gas and derivation of its thermodynamic properties

Derivation of Specific Heat at Constant Volume

Derivation of Specific Heat at Constant Pressure

Change in Entropy of an Isothermal Process

Equation of State for a reversible adiabatic change in an adiabatic process

7. Entropy of a collection of N three Dimensional Harmonic Osci llators

9. Summary

Appendix

A1 Stirlings formula

A2 Spreadsheet for Stirlings formula in asymptotic limit.

A3 Spreadsheet to visualize two dimensional quantum space of ideal gas

4

Physics




Oscillators

1. Learning Outcomes

After studying this module, you shall be able to

Understand the physical importance of the prototype model of statistical physics, studied in this module, called monoatomic ideal gas model in the development of equilibrium statistical mechanics.

Apply the statistical approach to get thermodynamic properties of this prototype model in an exploratory manner without explicit calculation of the number of microstates and further derive equation of state, pressure as energy density and equation of state of an irreversible adiabatic process

Evaluate explicitily (a possibility only in ideal cases), the number of microstates for an ideal gas consisting of N particles in the asymptotic limit and get an expression for entropy.

See the violation of extensive nature of entropy desired by thermodynamics derived for the ideal gas.

Derive expression of entropy asymptotically for system of quasi particles resulting from the

case of a collection of distinguishable harmonic oscillators of same frequency .and ponder on the fact why here entropy turns out to be extensive in nature unlike the case of the monoatomic ideal gas..

5

Physics




Oscillators

Introduction

2. Introduction

In this module we embark upon calculation of number of microstates of an ideal monoatomic gas enclosed in a volume V, having total Energy E and number of gas particles N, so that entropy of the gas may be calculated from the statistical route. N is an enormously large

number typically of the order of Avogadro’s number, making it a fit system to apply statistical methods to understand its behavior.

This gas is ideal in the sense that there is negligible interaction among the particles of the gas. In other words, particles of the gas are free to move such that compared to their kinetic energy mutual interaction potential energy between the particles can be neglected. To say

that there is negligible interaction is as good as saying no interaction and is, therefore, an idealization. In real gases there is always an interaction.

There is however a dilemma, if there is no interaction the speed of the atoms of the each gas atom shall be conserved. Therefore, in this case system can not go through all possible microstates violating equal apriori probability axiom studied in module 7. So allowing a weak

interaction is desirable, howsoever small it may be. One may prefer to describe such a gas as an ideal gas or a real gas in a dilute limit.

3. Classical Monoatomic Ideal Gas and Derivation of Entropy from Statistical

Route

We begin by taking a monoatomic gas of particles, particles are non interacting so that the

Hamiltonian of the system has kinetic energy only. We further assume particles to be non-relativistic. The Hamiltonian of the system can then be written as

(1)

The ideal gas is completely isolated from the surroundings enclosed in a container with non-coducting walls having a volume . The energy of the gas system. and volume stays

constant. Since system is ideal, i.e. noninteracting, the total energy E is the sum of the energies of

each atom of the monoatomic gas:

(2)

Such that

6

Physics




Oscillators

(3)

(a) Back of the Stamp Estimation of Number of Microstates & Ideal Gas Law:

Before we evaluate explicitly the number of microstates, , which an ideal gas can

have, it is possible to estimate this number through some probabilistic arguments. Since gas

is ideal and the gas atoms have no knowledge of how other gas atoms are distributed in the container, any gas atom can go anywhere and is independent of the presence of other atoms there. This implies statistical independence, i.e. the state of one atom does not affect the

probabilities of other atoms to be in different states. So now we can ask the question, how many possible ways are there that the N monoatomic atoms composing the system may

distribute themselves spatially in volume V. This shall be equal to the product of the number of ways in which each particle can be placed in the volume V. Obviously the number of ways in which these atoms can be placed depends on the spatial extent in which they are allowed

to go and this is proportional to the volume of the container. So the numer of possible ways in which N particles can be distributed is directly proportional to the product of the volume

taken N times: (4)

(5)

So that entropy of the system becomes (6)

Where C is a proportionality constant. Recalling

(7)

We have

(8)

Which is nothing but equation of state of an ideal gas. If Avogadro’s number,

gives us the gas constant . So knowledge of dependence of has yielded

the ideal gas equation , the famous ideal gas law of thermodynamics.

(b) Quantum Mechanical approach to Estimate number of Microstates :

(a) According to quantum mechanics, the single particle energies confined to move in a cubical box of edge are discrete and the energy of these particles must satisfy

7

Physics




Oscillators

equations (1) and (2). The permissible energy states for which wave function

must vanish on the boundaries are

(9)

Such that , h is Plancks constant, m mass of the particle.

Interestingly the number of microstates which we are looking for is a count of

allowed points in three dimensional quantum space of allowed positive integral values of , satisfying the condition for a single particle:

(10)

For particles, it amounts to number of independent positive integral solutions of

(11)

At this point, we do not need to explicitly estimate this number (which we shall do later) and

still we can do some estimation, One point is obvious, the count must depend on

and , through a form . Therefore, This further implies that entropy

should be of the form

(12)

From (12), a few familiar thermodynamic results follow immediately.

For a reversible adiabatic process, which requires and to be constant, it is possible only if

(13)

Recalling that , we have

(14)

This proves that pressure is two third of energy density. Results (13) and (14) obtained above are true for a system of non-relativistic, ideal classical as well as quantum monoatomic gas.

(c) Explicit Calculation of Number of Microstates of an Ideal Gas

After having a feel of enumeration of microstates of an ideal gas and its possible functional

form on and . We proceed further to explicitly count the number of microststes of an ideal monoatomic gas.

8

Physics




Oscillators

First of all for a single particle in a cubical box, let us try to visulalize the possible microstates in three dimensional number space with ( ) as co-ordinates. It is a discrete space

with allowed points represented by the positive integer co -ordinates Let us rewrite equation (10) as

where

(15)

Equation (15) represents surface of a sphere of radius of this dotted quantum number space, with only those values allowed which lie in the positive octant of the sphere. A two

dimensional counter part of the same is shown in the figure 1 below.

Figure 1 Visualization of the quantum number space, arc represents a sphere of radius R

Equation (11) represents equation of a sphere in 3N dimensional space. The surface of the sphere

corresponds to surface of constant energy , with a radius . is the number of points

lying on the surface of the positive compartment of this 3N dimensional sphere.

However, there is a word of caution about this number , though we will not prove it here,

but note that it is not a smoothly varying function of R or in other words . This is exhibited by the

vast difference in number of dots on two surfaces of 3N dimensional spheres whose value of radius lies

very close to each other. However, the function which corresponds to the number of

R

9

Physics




Oscillators

microstates in the positive part of the 3N dimensional sphere having energy less than E, turns out to be

a smooth function of radius and asymptotically shall be equal to the volume of the positive

compartment of the 3N dimensional sphere.

As we derived the volume of a sphere in 3N dimensional space while discussing the properties of

hyperspaces in module (???),

(16)

(17)

(18)

Taking the logarithm and applying stirling’s formula, since we have

(19)

Once again it is emphasized is not the same as , and the crucial part is how we

can connect the two, since we need to count the number of microstates . So once again we

revisit the constraint that the system has a precise value . Is it physically possible? And the answer is

10

Physics




Oscillators

no. Because system can not be completely isolated, we shall always find that energy shall lie in an

interval about with in the limits , with . So from (19), we can always find

the number of microstates around in the interval , which can be treated as equal to .

Therefore,

(20)

(21)

(22)

(23)

In equation (23), let us focus on the last two terms. is very large, asymptotically and third

term and fourth term approach zero reducing (23) to

(24)

11

Physics




Oscillators

Look and behold, (24) is the same as (19). So it can be stated that whether we count number of

microstates from 0 to or count them in a very small nterval around , it hardly matters. We are ready

to write down the entropy of an ideal gas using as follows

(25)

This is a result of great consequence, which has a problem that it is not extensive as it should be,

resolution of which we shall take up later in the next module. But, it takes us much further in deriving

thermodynamic properties of an ideal gas.

4.Thermodynamic properties

Equation (25) can be solved for E as function of (S,N,V) as given below

(26)

Knowing that we find

(27)

Or (28)

Where is Avogadro’s number, R is gas constant and n is the number of moles of the gas.

From equation (28), we can get specific heat at constant volume , equation of state of an ideal gas, specific heat at constant pressure. Specific heat at constant volume ( ):

(29)

Equation of State:

12

Physics




Oscillators

(30)

Specific Heat at Constant Pressure( ):

(31)

From equation (29) and equation (31), we get ratio of and

(32)

Once again a correct result. Change in entropy under isothermal condition:

Under isothermal condition stays constant, stays constant, stays constant. It is volume which changes say from intial volume .

Therefore

(33)

And

(34)

And hence, change in entropy, from equation (33) and (34) is

(35)

Reversible adiabatic change:

A reversible adiabatic change implies constant and constant. Then according to equation (26)

(36)

And, therefore, according to (28)

(37)

Then according to (30)

13

Physics




Oscillators

(38)

Or

(39)

A result matching with thermodynamics.

6. Entropy of N Three Dimensional Classical Harmonic Oscillators

Let us now discuss a problem of N three dimensional classical harmonic oscillators each with

frequency and total energy treated as distinguishable, i.e. as if each particle executing harmonic oscillations is sitting on a lattice identifiable with the lattice point tag. The Hamiltonian of this set is given by

(40)

This problem involves 3N position co-ordinates and 3N momentum co-ordinates. Which can be transformed into a problem of combined 6N co-ordinates by a suitable change of variables as given below:

(41)

(42)

Equation (40) then becomes

(43)

In terms of the new variables, energy condition becomes

(44)

To count the number of microstates we need to go in two steps

(i) Calculate the volume in phase space of dimensionality equal to over a region satisfying the energy constraint. We call it, . This is a volume of 6N dimensional sphere with

radius .

14

Physics




Oscillators

(ii) Divide this volume by volume of quantum cells in phase space allowed by uncertainty principle equal to to calculate the number of all possible microstates,

The volume then is given by

(45)

Since and

(46)

So that

(47)

As was done in the case of ideal gas, we need to compute the volume corresponding to an energy

shell of thickness , which is

(48)

For large N, and by using stirlings approximation we obtain entropy S(E,N)

(49)

An interesting observation about this result is that unlike entropy in the case of an ideal gas, entropy for this system of N classical harmonic oscillators, entropy is extensive. We will come to this discussion in the next module 8, where the question of extensivity of entropy shall be discussed at length. Let us also calculate temperature of such a system of harmonic oscillators

(50)

This result is in accordance with law of equipartition of energy, giving energy per oscillator equal to

.

9. Summary

In this module we have learnt

The application of statistical physics to a monoatomic ideal gas and discussed the thermodynamic properties of the ideal gas derived from the calculated entropy using the methodology of counting the microstates in asymptotical limit.

15

Physics




Oscillators

Monoatomic Ideal gas is a system of negligibly interacting particles, with kinetic energy overwhelming the negligible interaction among the particles.

That in the case of a monoatomic gas of particles, the number of microstates .

Quantum mechanically microstates of a monoatomic ideal gas can be visualized as dots in points in 3N dimensional quantum number space with each dot having quantum number co-ordinate space which need to be counted subject to

two constraints and .

That because of the property of dimensional hyper space, the total number of

microstates within a radius of of the 3N dimensional quantum number space is asymptotically equal to the number of microstates in a thin shell of thickness around and is enough for the calculation of accessible microstates.

That the entropy of an ideal gas calculated by this procedure has a physical flaw of entropy no longer turning out to be an extensive quantity, which needs to be fixed.

That entropy calculated by statistical method correctly reproduces thermodynamic properties of a monoatomic ideal gas listed below:

Property of the ideal gas As obtained from entropy

calculated statistically

Specific heat at constant volume ( ) Equation of State

Specific Heat at Constant

Pressure( ) Ratio of and

Change in entropy under isothermal condition Reversible adiabatic change

That the same procedure as followed in the case of monoatomic ideal gas can be applied to a collection of N independent harmonic oscillators in an analogous manner with suitable transformation of co=ordinates yielding an extensively correct form of entropy and yielding energy per oscillator equal to in accordance with law of equipartition of energy.

Bibliography

16

Physics




Oscillators

1. Pathria R.K. and Beale P. D., Statistical Mechanics, 3rd ed. (Elsevier, 2011).

2. Landau L.D., Lifshitz E.M., “ Statistical Physics Part 1,” 3rd Edition, Oxford: Pergamon Press.,

1982

3. Pal P.B., “An Introductory Course of Statistical Mechanics”, New Delhi: Narosa Publishing House

Pvt. Ltd., 2008.

4. Panat P.V., “Thermodynamics and Statistical Mechanics,” New Delhi: Narosa Pub lishing House

Pvt. Ltd., 2008

5. Yoshioka D., “Statistical Physics An Introduction,” Berlin Heidelberg: Springer-Verlag, 2007

Appendices

A1 Stirling’s Approximation

In statistical physics factorial function very often appears in enumeration of microstate in

the form . Furthermore, when it is to be applied, it is desirable to have the value of for large N.

The approximations to for large leads to well known Stirling’s formula.

Ist Method (By definition of Factorial function)

We know

(51)

(52)

This sum can be approximated by the integral

(53)

17

Physics




Oscillators

IInd Method (From integral form of factorial function)

A simple integral form of the factorial function called gamma function form is

(54)

This can be easily checked by integrating this integral by parts N times that it is indeed true.

We note that the integrand , is such that is rapidly increasing function of and is rapidly decreasing function of x. This function, therefore, is negligibly small except in the neighborhood of its maxima at some (see spreadsheet) where it has

appreciable value.

So to know where the maxima of lies, we take as given below and find its

maxima:

(55)

At the maximum

i.e.

(56)

implying or .

(57)

To find an expression for near the maximum, put , , and expand

in taylor series in at .

(58)

(59)

18

Physics




Oscillators

(60)

(61)

Where . Neglecting higher order terms

(62)

The term shows that it is maximum at , i.e. . It becomes negligibly small

when . If N is a very large number,

Substituting (48) in (40) and noting that we get,

(63)

Since is very small, for , lower limit of the integral can be replaced by .

Therefore,

(64)

The integral in (14) is a standard integral equal to , hence

(65)

A2 Spreadsheet for Stirlings formula in asymptotic limit.

A3 Spreadsheet to visualize two dimensional quantum space of ideal gas

stirling_app.xlsx

two%20dimensional%20quantum%20number%20space.xlsx

principal investigator paper coordinator

Documents