Intermediate Statistical Mechanics

Jayanta Bhattacharjee, Dhruba Banerjee

• Publisher: WSPC
• Year: 2016
• eBook ISBN:
  9789813201163
• Topic: Physical Sciences
• Subtopic: Condensed Matter

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In this new textbook, a number of unusual applications are discussed in addition to the usual topics covered in a course on Statistical Physics. Examples are: statistical mechanics of powders, Peierls instability, graphene, Bose-Einstein condensates in a trap, Casimir effect and the quantum Hall effect. Superfluidity and super-conductivity (including the physics of high-temperature superconductors) have also been discussed extensively.

The emphasis on the treatment of these topics is pedagogic, introducing the basic tenets of statistical mechanics, with extensive and thorough discussion of the postulates, ensembles, and the relevant statistics. Many standard examples illustrate the microcanonical, canonical and grand canonical ensembles, as well as the Bose-Einstein and Fermi-Dirac statistics.

A special feature of this text is the detailed presentation of the theory of second-order phase transitions and the renormalization group, emphasizing the role of disorder. Non-equilibrium statistical physics is introduced via the Boltzmann transport equation. Additional topics covered here include metastability, glassy systems, the Langevin equation, Brownian motion, and the Fokker-Planck equation.

Graduate students will find the presentation readily accessible, since the topics have been treated with great deal of care and attention to detail.

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0 Readership: Graduate students and academics focusing on statistical mechanics. -->Bose-Einstein Statistics, Fermi-Dirac Statistics, Boltzmann Transport Equation, Peierls Instability, Casimir Effect, Quantum Hall Effect, Langevin Equation, Brownian Motion, Phase Transitions0

1

Statistical Mechanics: The Basics

1.1From Micro to Macro

Thermodynamics describes the state of a system in terms of macroscopic variables like pressure, volume, temperature, energy, entropy etc. The laws of thermodynamics relate changes in these macroscopic variables. In the process thermodynamics requires some inputs from elsewhere. The best known example is that of an equation of state (i.e., relation between macroscopic variables) for an ideal gas, written as PV = nRT, where P is the pressure and V the volume of n moles of the gas. The temperature of the gas is T and R is the gas constant. Similarly for a non ideal gas the pressure is given by Van der Waals equation of state. One also needs to know that the molar specific heat of an ideal monatomic gas at constant volume is (3/2)R. Thermodynamics cannot calculate the pressure or the specific heat. This is because these calculations entail a microscopic picture that depends on the details of inter-molecular interactions. To calculate the pressure one needs to know what is the force exerted by the gas molecules on the walls of the container. This question has to do with knowledge of the dynamics of each of the particles. In principle this dynamics can be specified if one knows how a given molecule interacts with all the other molecules. All the other molecules mean an Avogadro number of them i.e. of the order of 10²³. In practice then, one would not dream of doing this calculation. How does one arrive at an answer?

Let us consider the wall which is parallel to the yz plane as shown in Fig.1.1. The container is rectangular and the surface shown is the right hand edge. The molecule which strikes it must have a positive x-component of velocity (v_x > 0). Collisions with the wall are elastic and hence the collision with the wall shown in Fig.(1.1) will result in a change of momentum of 2mv_x (m is the mass of the molecule) in the negative x-direction. This change of momentum of the molecule caused by the wall will result in an equal and opposite effect that produces the pressure. Since the gas is taken to be ideal (i.e., non interacting) the molecule will not change its momentum on its way to the wall. But constancy of velocity of a given molecule does not imply that velocities of all molecules are equal. In fact the gas will contain molecules with all possible velocities with the number of molecules in a given velocity range given by a definite distribution function. Interestingly enough, this distribution will be seen to be the outcome of interactions (however weak) between the gas molecules! Our ignorance of the exact dynamics of the gas molecules forces us to take recourse to the existence of this distribution function. This is the statistical description that we take recourse to whenever an exact description becomes impossible in practice (not in principle) and that sets up the subject of statistical mechanics.

Figure 1.1: Molecules colliding with a wall.

To calculate the force on the area δA of the wall, we need to how much momentum transfer occurred in unit time. This means we need to know how many collisions occurred in unit time. For the molecules with v_x as the x-component of the velocity, the number of collisions per unit time is the number of molecules with x-velocity of v_x that lie in a cylinder of cross-section δA and height v_x. This number is

where n(v_x) is the density of molecules with a x-velocity of v_x and the factor of

comes from the fact that we ne...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Preface
• Acknowledgments
• Contents
• 1 Statistical Mechanics: The Basics
• 2 The Microcanonical Ensemble
• 3 Other Ensembles
• 4 Non-interacting Systems
• 5 Interacting Classical Systems
• 6 Quantum Statistics
• 7 Fermi Distribution: Examples
• 8 Electrons in a Magnetic Field
• 9 Bose-Einstein Distribution: Examples
• 10 Superconductivity
• 11 Phase Transitions
• 12 Landau Theory and Related Models
• 13 The Renormalization Group
• 14 Disordered Systems
• 15 Transport Equation – I
• 16 Transport Equation – II
• 17 Transport Equation – III
• 18 Transport Equation – IV
• 19 Metastable States
• 20 Langevin Equations
• 21 Fokker-Planck Equations
• 22 Fluctuation Dissipation Theorem
• References
• Index