12.4 Ensemble averages and distribution functions

Chapter 1: Understanding the physical universe
- 1.1 The programme of physics
- 1.2 The building blocks of matter
- 1.3 Matter in bulk
- 1.4 The fundamental interactions
- 1.5 Exploring the physical universe: the scientific method
- 1.6 The role of physics: its scope and applications
Chapter 2 Using mathematical tools in physics
- 2.1 Applying the scientific method
- 2.2 The use of variables to represent displacement and time
- 2.3 Representation of data
- 2.4 The use of differentiation in analysis: velocity and acceleration in linear motion
- 2.5 The use of integration in analysis
- 2.6 Maximum and minimum values of physical quantities: general linear motion
- 2.7 Angular motion: the radian
- 2.8 The role of mathematics in physics
- Chapter 2 Problems
  - Answers to Chapter 2 problems
Chapter 3 The causes of motion: dynamics
- 3.1 The concept of force
- 3.2 The first law of dynamics (Newton's first law)
- 3.3 The fundamental dynamical principle (Newton's second law)
- 3.4 Systems of units: SI
- 3.5 Time dependent forces: oscillatory motion
- 3.6 Simple harmonic motion
- 3.7 Mechanical work and energy
- 3.8 Plots of potential energy functions
- 3.9 Power
- 3.10 Energy in simple harmonic motion
- 3.11 Dissipative forces: damped harmonic motion
  - 3.11.1 Trial solution technique for solving the damped harmonic motion equation
- 3.12 Forced oscillations
- 3.13 Non-linear dynamics: chaos
- 3.14 Phase space representation of dynamical systems
- Chapter 3 Problems
  - Answers to Chapter 3 problems
Chapter 4 Motion in two and three dimensions
- 4.1 Vector physical quantities
- 4.2 Vector algebra
- 4.3 Velocity and acceleration vectors
- 4.4 Force as a vector quantity: vector form of the laws of dynamics
- 4.5 Constraint forces
- 4.6 Friction
- 4.7 Motion in a circle: centripetal force
- 4.8 Motion in a circle at constant speed
- 4.9 Tangential and radial components of acceleration
- 4.10 Hybrid motion: the simple pendulum
  - 4.10.1 Large angle corrections for the simple pendulum
- 4.11 Angular quantities as vectors: the cross product
- Chapter 4 Problems
  - Answers to Chapter 4 Problems
Chapter 5 Force fields
- 5.1 Newton's law of universal gravitation
- 5.2 Force fields
- 5.3 The concept of flux
- 5.4 Gauss' lew for gravitation
- 5.5 Applications of Gauss’ law
- 5.6 Motion in a constant uniform field: projectiles
- 5.7 Mechanical work and energy
- 5.8 Power
- 5.9 Energy in a constant uniform field
- 5.10 Energy in an inverse square law field
- 5.11 Moment of a force: angular momentum
- 5.12 Planetary motion: circular orbits
- 5.13 Planetary motion: elliptical orbits and Kepler's laws
  - 5.13.1 Conservation of the Runge-Lens vector
- Chapter 5 Problems
  - Answers to Chapter 5 problems
Chapter 6 Many-body interactions
- 6.1 Newton's third law
- 6.2 The principle of conservation of momentum
- 6.3 Mechanical energy of systems of particles
- 6.4 Particle decay
- 6.5 Particle collisions
- 6.6 The centre of mass of a system
- 6.7 The two-body problem: reduced mass
- 6.8 Angular momentum of systems of particles
- 6.9 Conservation principles in physics
- Chapter 6 Problems
  - Answers to Chapter 6 problems
Chapter 7 Rigid-body dynamics
- 7.1 Rigid bodies
- 7.2 Rigid bodies in equilibrium: statics
- 7.3 Torque
- 7.4 Dynamics of rigid bodies
- 7.5 Measurement of torque: the torsion balance
- 7.6 Rotation of a rigid body about a fixed axis: moment of inertia
- 7.7 Calculation of moments of inertia: the parallel axis theorem
- 7.8 Conservation of angular momentum of rigid bodies
- 7.9 Conservation of mechanical energy in rigid body systems
- 7.10 Work done by a torque: torsional oscillations: rotational power
- 7.11 Gyroscopic motion
  - 7.11.1 Precessional angular velocity of a top
- 7.12 Summary - connection between rotational and translational motions
- Chapter 7 problems
  - Answers to Chapter 7 problems
Chapter 8 Relative motion
- 8.1 Applicability of Newton's laws of motion: inertial reference frames
- 8.2 The Galilean transformation
- 8.3 The CM (centre-of-mass) reference frame
- 8.4 Example of a non-inertial frame: centrifugal force
- 8.5 Motion in a rotating frame: the Coriolis force
- 8.6 The Foucault pendulum
  - 8.6.1 Precession of a Foucault pendulum
- 8.7 Practical criteria for inertial frames: the local view
- Chapter 8 problems
  - Answers to Chapter 8 Problems
Chapter 9 Special relativity
- 9.1 The velocity of light
  - 9.1.1 The Michelson-Morley experiment
- 9.2 The Principle of Relativity
- 9.3 Consequences of the Principle of Relativity
- 9.4 The Lorentz transformation
- 9.5 The Fitzgerald-Lorentz contraction
- 9.6 Time dilation
- 9.7 Paradoxes in special relativity
  - 9.7.1 Simultaneity: quantitative analysis of the twin paradox
- 9.8 Relativistic transformation of velocity
- 9.9 Momentum in relativistic mechanics
- 9.10 Four-vectors: the energy-momentum 4-vector
- 9.11 Energy-momentum transformations: relativistic energy conservation
  - 9.11.1 The force transformations
- 9.12 Relativistic energy: mass-energy equivalence
- 9.13 Units in relativistic mechanics
- 9.14 Mass-energy equivalence in practice
- 9.15 General relativity
- Chapter 9 problems
  - Answers to Chapter 9 Problems
Chapter 10 Continuum mechanics: mechanical properties of materials: microscopic models of matter
- 10.1 Dynamics of continuous media
- 10.2 Elastic properties of solids
- 10.3 Fluids at rest
- 10.4 Elastic properties of fluids
- 10.5 Pressure in gases
- 10.6 Archimedes' principle
- 10.7 Fluid dynamics; the Bernoulli equation
- 10.8 Viscosity
- 10.9 Surface properties of liquids
- 10.10 Boyle's law (Mariotte's law)
- 10.11 A microscopic theory of gases
- 10.12 The SI unit of amount of matter; the mole
- 10.13 Interatomic forces: modifications to the kinetic theory of gases
- 10.14 Microscopic models of condensed matter systems
- Chapter 10 problems
  - Answers to Chapter 10 Problems
Chapter 11 Thermal physics
- 11.1 Friction and heating
- 11.2 The SI unit of thermodynamic temperature; the kelvin
- 11.3 Heat capacities of thermal systems
- 11.4 Comparison of specific heat capacities: calorimetry
- 11.5 Thermal conductivity
- 11.6 Convection
- 11.7 Thermal radiation
- 11.8 Thermal expansion
- 11.9 The first law of thermodynamics
- 11.10 Change of phase: latent heat
- 11.11 The equation of state of an ideal gas
- 11.12 Isothermal, isobaric and adiabatic processes: free expansion
- 11.13 The Carnot cycle
- 11.14 Entropy and the second law of thermodynamics
- 11.15 The Helmholtz and Gibbs functions
- Chapter 11 problems
  - Answers to Chapter 11 Problems
Chapter 12 Microscopic models of thermal systems: kinetic theory of matter
- 12.1 Microscopic interpretation of temperature
- 12.2 Polyatomic molecules: principle of equipartition of energy
- 12.3 Ideal gas in a gravitational field: the ‘law of atmospheres’
- 12.4 Ensemble averages and distribution functions
- 12.5 The distribution of molecular velocities in an ideal gas
- 12.6 Distribution of molecular speeds
- 12.7 Distribution of molecular energies; Maxwell-Boltzmann statistics
- 12.8 Microscopic interpretation of temperature and heat capacity in solids
- Chapter 12 problems
  - Answers to Chapter 12 Problems
Chapter 13 Wave motion
- 13.1 Characteristics of wave motion
- 13.2 Representation of a wave which is travelling in one dimension
- 13.3 Energy and power in a wave motion
- 13.4 Plane and spherical waves
- 13.5 Huygens’ principle: the laws of reflection and refraction
- 13.6 Interference between waves:
- 13.7 Interference of waves passing through openings: diffraction
- 13.8 Standing waves
  - 13.8.1 Standing waves in three dimensional cavity
- 13.9 The Doppler effect
- 13.10 The wave equation
- 13.11 Waves along a string
- 13.12 Waves in elastic media: longitudinal waves in a solid rod
- 13.13 Waves in elastic media: sound waves in gases
- 13.14 Superposition of two waves of slightly different frequencies: wave and group velocities
- 13.15 Other waveforms: Fourier analysis
- Chapter 13 problems
  - Answers to Chapter 13 Problems
Chapter 14 Introduction to quantum mechanics
- 14.1 Physics at the beginning of the twentieth century
- 14.2 The blackbody radiation problem; Planck’s quantum hypothesis
- 14.3 The specific heat capacity of gases
- 14.4 The specific heat capacity of solids
- 14.5 The photoelectric effect
  - 14.5.1 Example of an experiment to study the photoelectric effect
- 14.6 The X-ray continuum
- 14.7 The Compton effect: the photon model
- 14.8 The de Broglie hypothesis: wave-particle duality
- 14.9 Interpretation of wave-particle duality
- 14.10 The Heisenberg uncertainty principle
- 14.11 The wavefunction: expectation values
- 14.12 The Schrödinger (wave mechanical) method
  - 14.12.1 Expectation value of momentum
- 14.13 The free particle
- 14.14 The time-independent Schrödinger equation: eigenfunctions and eigenvalues
  - 14.14.1 Derivation of the Ehrenfest theorem
- 14.15 The infinite square potential well
- 14.16 Potential steps
- 14.17 Other potential wells and barriers
- 14.18 The simple harmonic oscillator
  - 14.18.1 Ground state of the simple harmonic oscillator
- 14.19 Further implications of quantum mechanics
- Chapter 14 problems
  - Answers to Chapter 14 Problems
Chapter 15 Electric currents
- 15.1 Electric currents
- 15.2 The electric current model; electric charge
- 15.3 The unit of electric current; the ampere
- 15.4 Heating effect revisited: electrical resistance
- 15.5 Strength of a power supply: emf
- 15.6 Resistance of a circuit
- 15.7 Potential difference
- 15.8 Effect of internal resistance
- 15.9 Comparison of emfs: the potentiometer
- 15.10 Multiloop circuits
- 15.11 Kirchhoff's rules
- 15.12 Comparison of resistances: the Wheatstone bridge
- 15.13 Power supplies connected in parallel
- 15.14 Resistivity and conductivity
- 15.15 Variation of resistance with temperature
- Chapter 15 problems
  - Answers to Chapter 15 Problems
Chapter 16 Electric fields
- 16.1 Electric charge at rest
- 16.2 Electric fields: electric field strength
- 16.3 Force between point charges: Coulomb's law
- 16.4 Electric flux and electric flux density
- 16.5 Electric fields due to systems of charges
- 16.6 The electric dipole
- 16.7 Gauss’ law for electrostatics
- 16.8 Applications of Gauss’s law
- 16.9 Potential difference in electric fields
- 16.10 Electric potential
- 16.11 Equipotential surfaces
- 16.12 Determination of electric field strength from electric potential
- 16.13 Acceleration of charged particles
- 16.14 The laws of electrostatics in differential form
- Chapter 16 problems
  - Answers to Chapter 16 Problems
Chapter 17 Electric fields in materials
- 17.1 Conductors in electric fields
- 17.2 Insulators in electric fields; polarisation
- 17.3 Electric susceptibility
- 17.4 Boundaries between dielectric media
- 17.5 Ferroelectricity and paraelectricity; permanently polarised materials
- 17.6 Uniformly polarised rod; the ‘bar electret’
- 17.7 Microscopic models of electric polarisation
- 17.8 Capacitors
- 17.9 Examples of capacitors with simple geometry
- 17.10 Energy stored in an electric field
- 17.11 Capacitors in series and in parallel
- 17.12 Charge and discharge of a capacitor through a resistance
- 17.13 Measurement of permittivity
- Chapter 17 problems
  - Answers to Chapter 17 Problems
Chapter 18 Magnetic fields
- 18.1 Magnetism
- 18.2 The work of Ampère, Biot and Savart
- 18.3 Magnetic pole strength
- 18.4 Magnetic field strength
- 18.5 Ampère's law
- 18.6 The Biot-Savart law
- 18.7 Applications of the Biot-Savart law
- 18.8 Magnetic flux and magnetic flux density
- 18.9 Magnetic fields of permanent magnets; magnetic dipoles
- 18.10 Forces between magnets; Gauss’ law for magnetism
- 18.11 The laws of magnetostatics in differential form
- Chapter 18 problems
  - Answers to Chapter 18 Problems
Chapter 19 Electric currents and moving charges in magnetic fields
- 19.1 Forces between currents magnets
- 19.2 The force between two long parallel wires
- 19.3 Current loop in a magnetic field
- 19.4 Magnetic fields due to moving charges
- 19.5 Force on a moving electric charge in a magnetic field
- 19.6 Applications of moving charges in uniform magnetic fields; the classical Hall effect
- 19.7 Charge in a combined electric and magnetic field; the Lorentz force
- 19.8 Magnetic dipole moments of charged particles in closed orbits
- 19.9 Polarisation of magnetic materials; magnetisation, magnetic susceptibility
- 19.10 Paramagnetism and diamagnetism
- 19.11 Boundaries between magnetic media
- 19.12 Ferromagnetism; the magnetic needle revisited
- 19.13 Moving coil meters and electric motors
- 19.14 Electric and magnetic fields in moving reference frames
- Chapter 19 problems
  - Answers to Chapter 19 Problems
Chapter 20 Electromagnetic induction: time-varying emfs
- 20.1 The principle of electromagnetic induction
- 20.2 Simple applications of electromagnetic induction
- 20.3 Self-inductance
- 20.4 The series L-R circuit
- 20.5 Discharge of a capacitor through an inductor and resistor
- 20.6 Time-varying emfs: mutual inductance: transformers
- 20.7 Alternating current (a.c.)
- 20.8 Alternating current transformers
- 20.9 Resistance, capacitance and inductance in a.c. circuits
- 20.10 The series L-C-R circuit: phasor diagrams
- 20.11 Power in an a.c. circuit
- Chapter 20 problems
  - Answers to Chapter 20 Problems
Chapter 21 Maxwell’s equations; electromagnetic radiation
- 21.1 Reconsideration of the laws of electromagnetism: Maxwell's equations
- 21.2 Plane electromagnetic waves
- 21.3 Experimental observation of electromagnetic radiation
- 21.4 The electromagnetic spectrum
- 21.5 Polarization of electromagnetic waves
- 21.6 Energy, momentum and angular momentum in electromagnetic waves
- 21.7 The photon model revisited
- 21.8 Reflection of electromagnetic waves at an interface between non-conducting media
- 21.9 Electromagnetic waves in a conducting medium
- 21.10 Invariance of electromagnetism under the Lorentz transformation
- 21.11 Maxwell’s equations in differential form
- Chapter 21 problems
  - Answer to Chapter 21 problems
Chapter 22 Wave optics
- 22.1 Electromagnetic nature of light
- 22.2 Coherence: the laser
- 22.3 Diffraction at a single slit
- 22.4 Two slit interference and diffraction: Young's double slit experiment
- 22.5 Multiple slit interference: the diffraction grating
- 22.6 Diffraction of X-rays: Bragg scattering
- 22.7 The SI unit of luminous intensity, the candela
- Chapter 22 problems
  - Answers to Chapter 22 Problems
Chapter 23 Geometrical optics
- 23.1 The ray model: geometric optics
- 23.2 Reflection of light
- 23.3 Image formation by spherical mirrors
- 23.4 Refraction of light
- 23.5 Refraction at successive plane interfaces
- 23.6 Image formation by spherical lenses
- 23.7 Image formation of extended objects: magnification
- 23.8 Dispersion of light
- Chapter 23 problems
  - Answers to Chapter 23 problems
Chapter 24 Atomic Physics
- 24.1 Atomic models
- 24.2 The spectrum of hydrogen: the Rydberg formula
- 24.3 The Bohr postulates
- 24.4 The Bohr theory of the hydrogen atom
- 24.5 The quantum mechanical (Schrödinger) solution of the one-electron atom
  - 24.5.1 The angular and radial equations for a one-electron atom
  - 24.5.2 The radial solutions of the lowest energy state of hydrogen
- 24.6 Interpretation of the one-electron atom eigenfunctions
- 24.7 Intensities of spectral lines: selection rules
  - 24.7.1 Radiation from an accelerated charge
  - 24.7.2 Expectation value of the electric dipole moment
- 24.8 Quantisation of angular momentum
  - 24.8.1 The angular momentum quantisation equations
- 24.9 Magnetic effects in one-electron atoms: the Zeeman effect
- 24.10 The Stern-Gerlach experiment: electron spin
  - 24.10.1 The Zeeman effect
- 24.11 The spin-orbit interaction
  - 24.11.1 The Thomas precession
- 24.12 Identical particles in quantum mechanics: the Pauli exclusion principle
- 24.13 The periodic table: multielectron atoms
- 24.14 The theory of multielectron atoms
- 24.15 Further uses of the solutions of the one-electron atom
- Chapter 24 problems
  - Answers to Chapter 24 problems
Chapter 25 Electrons in solids: quantum statistics
- 25.1 Bonding in molecules and solids
- 25.2 The classical free electron model of solids
- 25.3 The quantum mechanical free electron model of solids: Fermi energy
- 25.4 The electron energy distribution at zero K
- 25.5 Electron energy distributions at T > zero K
  - 25.5.1 The quantum distribution functions
- 25.6 Specific heat capacity and conductivity in the quantum free electron model
- 25.7 Quantum statistics: systems of bosons
- 25.8 Superconductivity
- Chapter 25 problems
  - Answers to Chapter 25 problems
Chapter 26 Semiconductors
- 26.1 The band theory of solids
- 26.2 Conductors, insulators and semiconductors
- 26.3 Intrinsic and extrinsic (doped) semiconductors
- 26.4 Junctions in conductors
- 26.5 Junction in semiconductors; the p-n junction
- 26.6 Biased p-n junctions; the semiconductor diode
- 26.7 Photodiodes, particle detectors and solar cells
- 26.8 Light emitting diodes; semiconductor lasers
- 26. 9 The tunnel diode
- 26.10 Transistors
- Chapter 26 problems
  - Answers to Chapter 26 problems
Chapter 27 Nuclear and particle physics
- 27.1 Properties of atomic nuclei
- 27.2 Nuclear binding energies
- 27.3 Nuclear models
- 27.4 Radioactivity
- 27.5 alpha-, beta- and gamma-decay
- 27.6 Detection of radiation: units of radioactivity
- 27.7 Nuclear reactions
- 27.8 Nuclear fission and nuclear fusion
- 27.9 Fission reactors
- 27.10 Thermonuclear fusion
- 27.11 Sub-nuclear particles
- 27.12 The quark model
- Chapter 27 problems
  - Answers to Chapter 27 problems
Errata

This section provides a very brief introduction to the applied mathematics of statistical systems. The material covered will be applied to ideal gases in the remaining sections of Chapter 12. Such techniques also have important applications in modeling any physical system comprising a large number of microscopic entities, such molecules, atoms, ions, electrons, etc. (Chapters 24 and 25), where probabilistic issues arise (Chapter 14) or in the analysis of experimental data.

A good overview of statistical techniques is to be found in the NIST/SEMATECH . In the context of Sections 12.5 to 12.7 of Understanding Physics, interested readers might study the sections and .

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12.4 Ensemble averages and distribution functions

Understanding Physics

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