Statistical Mechanics: An Introduction

P.K. Ahluwalia

Learning Outcomes

After studying this module, you shall be able to

• know the broad learning goals of this course on Statistical Mechanics for PG students

• know the overall place of studying Statistical Mechanics in the study of physics

• get an overview of representative models of statistical mechanics and their application in various areas of physics

• know the importance of learning statistical methodology beyond physics in the new and emerging areas in biology, geology, ecology, economics and financial markets

1. INTRODUCTION

1.1 Pre-requisites

Following are the Pre-requisites B.Sc. Level for appreciation of this course

(i) A feeling for thermodynamics and equation of state

(ii) A working knowledge of introductory quantum mechanics and classical mechanics

(iii) An intuitive knowledge of the following model systems

a. Classical ideal gas and kinetic theory of gases
b. Harmonic oscillator
c. Free electron gas (Fermi gas)
d. Photon gas (Bose gas)
e. Examples of phase transitions

Exposure to usage of spreadsheets and running of java applets of e-resources available on the internet.

1.2 Broad Learning Goals of this e-course on Statistical Mechanics

The broad learning goals of this course on Statistical Mechanics designed for an M.Sc. Physics student are

• To learn physical concepts and relevant methodology to understand a macroscopic physical system made up of a large number of entities (~1023) overcoming the limitation of inherent lack of information, a methodology which works without going through complete mechanical description of the system.

• To seek a statistical mechanical foundation to the subject of thermodynamics whose edifice is historically built on the foundation of phenomenological theories based on empirical facts.

• To understand a few selected applications of statistical mechanics to see how postulates of statistical mechanics work to explain variety of physical phenomena in different areas of physics.

• To have a holistic view of macroscopic world of physics and beyond from the window of statistical mechanics

• To put in perspective, the spread and sweep of the applications of statistical mechanics.

Scope of this e-course is limited to Equilibrium Statistical Mechanics

2. What is Statistical Mechanics?

2.1 Microscopic Route to Thermodynamics

To answer the question, what is statistical mechanics? One must appreciate the fact that emergence of Statistical Mechanics as one of the core knowledge domains of physics is a result of persistent efforts made by researchers to understand the microscopic origin of phenomenological theory of macroscopic natural phenomenon called thermodynamics,which has very well established laws based on experimental facts and successful technological applications. These laws are

1. Zeoth law of Themodynamics
2. First Law of Thermodynamics
3. Second Law of Thermodynamics
4. Third law of Thermodynamics

These laws have been known to us well before

(i) the revelation of the fact that matter is made up of atoms or molecules and

(ii) the development of quantum mechanics.

As it became clear that macroscopic bodies are made up of such microscopic particles (atoms, molecules, electrons etc.),it was natural to ask the question, can we explain the behavior of these bodies, which are typically made up of particles of the order of Avogadro’s number i.e. ~1023, starting from the laws governing motion of the microscopic constituents.

2.2 Complexity of the Task

The moment one recognizes that a typical macroscopic body say 18 gm of water, is made up of 6.023×1023 molecules of water, the problem indeed becomes very complex. Desiring to keep track of all these molecules (their positions and velocities) with time by setting up as many equations of motion and solving these an impossible task,by the fastest of the super computers available today in a typical life time of a person (it is worth estimating this amount of time required for such an effort by a present day computer!). Lack of means to gather such a huge information leaves us with an information crunch also. It is, therefore, natural to believe that it is a hopeless task which inherently has to begin with perhaps too meager information to arrive at worthwhile conclusions for an enormously complex system.

2.3 Way Out: Statistical Approach

Statistical approach offers a methodology to look at such a complex many body problem to make sense of its dynamical behavior, bypassing the need to solve equation of motion (Newton’s Equation of motion for a classical particle or Schrodinger’s Equation of motion for a quantum particle) for each of the constituting particles.

As is characteristic of the statistical approach, it may not provide us knowledge of the behavior of each individual particle, yet it is capable of providing average behavior of important physical quantities of interest in a physical system with certain rules of transaction on energy, number of particles and external conditions etc..

Therefore, statistical mechanics is a way to bring in new concepts besides fundamental laws of classical or quantum mechanics as the case may be. These new concepts involve probability theory via principle of equal prior probability and are powerful enough to deal with the complexity of the many body problems encountered in nature despite lack of information. The basic question which statistical mechanics tries to address is that can observed properties of a system be obtained from the average behavior of its constituents when these are brought together. Answer indeed leads us to an understanding of macroscopic behavior of a system from microscopic behavior of constituent particles.

2.4 Dilemmas of This Approach

To begin with there are two dilemmas in which one finds oneself when asked to apply this approach.

• Firstly, with ones’ training most of the time in deterministic thinking involved in the study of classical mechanics, electrodynamics, theory of relativity etc., it is not easy to comprehend the reasons to choose the statistical path.

• Secondly by looking at a macroscopic body it is not obvious that what is being observed is an average behavior which stays constant over long periods of time during equilibrium without much concern with regard to the individual dynamical state of individual particles.

There is only one way to overcome these dilemma and that comes from the success of this approach in explaining various natural phenomena encountered in macroscopic world through the window of microscopic constituents. And this is exactly what we shall explore while going through this course on Statistical Mechanics.

2.5 Entropy: A Bridge between Thermodynamics and Statistical Mechanics

Entropy of a macroscopic system, which was defined by Rudolf Clausiusin a paper published in the Annalen der Physik und Chemie in 1850, is an important measurable physical quantity which offers a bridge to link statistical mechanics (theory) with thermodynamics (experiment).

The meaning of entropy has been a matter of much discussion in thermodynamics in particular and physics in general. The best intuitive interpretation of entropy came from famous Physicist Freeman Dyson while answering the question, what is heat? He described heat of a macroscopic system as nothing but its disordered energy. He said that Its full knowledge requires two numbers, one is a measure of its quantity in say joules and the second a measure of its disorderedness called entropy.

Where Sis entropy, Q is heat, T is temperature,KB is Boltzmann constant and ω is thermo-dynamical probability. In nut shell statistical mechanics deals with finding the thermodynamic probability by statistical approach providing a link with other thermodynamic variables via free energy. In thermodynamics free energy, F , is a quantity which can be known by measurements alone, whereas statistical mechanics provides a mean to calculate free energy via Boltzmann formula linking theory with experiments.

3. Why Study Statistical Mechanics?

The success of statistical mechanics lies not only in understanding physical phenomenon but goes much beyond that. It has influenced many areas beyond physics across disciplines in science, engineering, mathematics and social sciences. It has a very wide reach and a rich variety of models/methods which are applicable in analogous situations.

3.1 Reach of the Statistical Mechanics in Physics

In physics, statistical mechanics has been applied to model understanding of many phenomena/laws. Following Table (Table 1) provides a bird’s eye view of applications of statistical mechanics in physics. In this course we shall study some of these models and their applications.

3.2 Statistical Mechanics Methodology beyond Physics

Statistical mechanics goes much beyond its usual link with thermodynamics The methodology of statistical mechanics has become a very important tool in the hands of scientists who truly study very complex systems like biologists, chemists, geologists, information scientists, computer scientist, ecologists, social scientists and economists etc..For All these disciplinesuse of statistical mechanics has been possible by drawing very interesting analogies between corresponding variables.

For example in economics a system of particles becomes a system of agents, energy (E) which is a conserved quantity becomes money (m) and temperature (T) becomes effective temperature of the  system measured in money currency ($) following Gibbs distribution law with similar rules for trading. In the case of ecology where ecologists are interested in collective behavior of populations consisting of constituent particles (e.g. insects) in the form of biological organisms, principles of statistical mechanics can be usefully applied.Biologists are applying tools of statistical mechanics to a host of problems like force extension curves for DNA, lattice models of protein folding, analysis of polymerization and molecular motors.In information theory, one deals with networks, to be precise communication networks, dealing with say people talking on phones or investors in financial markets etc. exchanging information with one another in a very complex manner. Shannon was the first person to define information quantitatively analogous to entropy. Insurance companies, financial markets, psephologists, bioinformatics experts, traffic model experts all make use of tools of statistical mechanics and find training in ideas of statistical mechanics a fundamental requirement to deal with the complexity in these systems. This has enriched their own field and the field of statistical mechanics. It will be worth quoting Noble laureate Prof. PW Anderson from his famous article entitled More is Different where he sums up the complexity encountered in nature, beautifully underlining its importance in many body theory and other systems:

There may well be no useful parallel to be drawn between the way in which complexity appears in the simplest cases of many body theory and chemistry and the way it appears in the truly complex cultural and biological ones, except perhaps to say that, in general the relationship between the system and its parts is intellectually a one way street. Synthesis is expected to be all but impossible; analysis on the other hand, may be not only possible but fruitful in all kind of ways.

This is precisely what this journey through statistical mechanics is expected to reveal, a remarkable way of analyzing complex problems.

4. Summary

In this module we have learnt

• about broad learning goals of this e-course in statistical mechanics and what are the pre-requisites for appreciating it.

• that despite the laws governing the constituent particles are well known, yet coming together of a large number of particles makes the understanding of such a system possible only by statistical means, revealing new laws unheard of in the realm of individual particle behavior.

• that new laws are probabilistic in nature and give an average behavior of the properties of the system.

• that how statistical approach relates to thermo-dynamical physical quantities via boltzmann’s famous relation for entropy and thus opens a pathway to link statistical mechanics with thermodynamics(a very well established phenomenological theory)?

• that statistical mechanics deals with a many particle system with a challenge of handling time.

• that the long list of its applications provide a wide sweep and utility of statistical mechanics not only in understanding physical phenomenon, but also phenomenon encountered in variety of diverse fields in biology, economics, ecology, information theory and computer science.

References:

1.  James Sethna, Statistical Mechanics: Entropy, Order Parameters and Complexity (Oxford University Press, 2006).
2. Landau L.D., Lifshitz E.M., “ Statistical Physics Part 1,” 3rdEdition, Oxford: Pergamon Press., 1982
3. Ma S.K., “ Statistical Mechanics,” Singapore: World Scientific Publishing Co. Pte. Ltd., 1985.
4. MatveevA.N., “ Statistical Physics,” Moscow: Mir Publishers, 1985.
5. Pathria R.K. and  Beale P. D., Statistical Mechanics,3rd ed. (Elsevier, 2011).