Topics in Statistical Mechanics

by Cowan, Brian

• ISBN13: 9781860945694
• ISBN10: 1860945694
• Format: Paperback
• Copyright: 2005-11-30
• Publisher: Imperial College Pr

Table of Contents

Preface	v
The Methodology of Statistical Mechanics	1	(53)
Terminology and Methodology	1	(3)
Approaches to the subject	1	(2)
Description of states	3	(1)
Extensivity and the thermodynamic limit	3	(1)
The Fundamental Principles	4	(4)
The laws of thermodynamics	4	(2)
Probabilistic interpretation of the First Law	6	(1)
Microscopic basis for entropy	7	(1)
Interactions --- The Conditions for Equilibrium	8	(9)
Thermal interaction --- Temperature	8	(2)
Volume change --- Pressure	10	(2)
Particle interchange --- Chemical potential	12	(1)
Thermal interaction with the rest of the world --- The Boltzmann factor	13	(2)
Particle and energy exchange with the rest of the world --- The Gibbs factor	15	(2)
Thermodynamic Averages	17	(8)
The partition function	17	(1)
Generalised expression for entropy	18	(2)
Free energy	20	(1)
Thermodynamic variables	21	(1)
Fluctuations	21	(2)
The grand partition function	23	(1)
The grand potential	24	(1)
Thermodynamic variables	25	(1)
Quantum Distributions	25	(6)
Bosons and fermions	25	(3)
Grand potential for identical particles	28	(1)
The Fermi distribution	29	(1)
The Bose distribution	30	(1)
The classical limit --- The Maxwell distribution	30	(1)
Classical Statistical Mechanics	31	(11)
Phase space and classical states	31	(2)
Boltzmann and Gibbs phase spaces	33	(1)
The Fundamental Postulate in the classical case	34	(1)
The classical partition function	35	(1)
The equipartition theorem	35	(2)
Consequences of equipartition	37	(1)
Liouville's theorem	38	(2)
Boltzmann's H theorem	40	(2)
The Third Law of Thermodynamics	42	(12)
History of the Third Law	42	(1)
Entropy	43	(1)
Quantum viewpoint	44	(2)
Unattainability of absolute zero	46	(1)
Heat capacity at low temperatures	46	(2)
Other consequences of the Third Law	48	(2)
Pessimist's statement of the laws of thermodynamics	50	(4)
Practical Calculations with Ideal Systems	54	(66)
The Density of States	54	(7)
Non-interacting systems	54	(1)
Converting sums to integrals	54	(1)
Enumeration of states	55	(1)
Counting states	56	(2)
General expression for the density of states	58	(1)
General relation between pressure and energy	59	(2)
Identical Particles	61	(1)
Indistinguishability	61	(1)
Classical approximation	62	(1)
Ideal Classical Gas	62	(7)
Quantum approach	62	(2)
Classical approach	64	(1)
Thermodynamic properties	64	(2)
The 1/N! term in the partition function	66	(1)
Entropy of mixing	67	(2)
Ideal Fermi Gas	69	(18)
Methodology for quantum gases	69	(1)
Fermi gas at zero temperature	70	(2)
Fermi gas at low temperatures --- simple model	72	(3)
Fermi gas at low temperatures --- series expansion	75	(3)
Chemical potential	78	(2)
Internal energy	80	(1)
Thermal capacity	81	(1)
More general treatment of low temperature heat capacity	81	(3)
High temperature behaviour --- the classical limit	84	(3)
Ideal Bose Gas	87	(11)
General procedure for treating the Bose gas	87	(1)
Number of particles --- chemical potential	88	(1)
Low temperature behaviour of Bose gas	89	(2)
Thermal capacity of Bose gas --- below Tc	91	(2)
Comparison with superfluid 4He and other systems	93	(2)
Two-fluid model of superfluid 4He	95	(1)
Elementary excitations	96	(2)
Black Body Radiation --- The Photon Gas	98	(7)
Photons as quantised electromagnetic waves	98	(1)
Photons in thermal equilibrium --- black body radiation	99	(1)
Planck's formula	100	(2)
Internal energy and heat capacity	102	(1)
Black body radiation in one dimension	103	(2)
Ideal Paramagnet	105	(15)
Partition function and free energy	105	(1)
Thermodynamic properties	106	(4)
Negative temperatures	110	(2)
Thermodynamics of negative temperatures	112	(8)
Non-Ideal Gases	120	(23)
Statistical Mechanics	120	(4)
The partition function	120	(1)
Cluster expansion	121	(1)
Low density approximation	122	(1)
Equation of state	123	(1)
The Virial Expansion	124	(6)
Virial coefficients	124	(1)
Hard core potential	124	(2)
Square-well potential	126	(1)
Lennard-Jones potential	127	(3)
Second virial coefficient for Bose and Fermi gas	130	(1)
Thermodynamics	130	(4)
Throttling	130	(1)
Joule--Thomson coefficient	131	(1)
Connection with the second virial coefficient	132	(2)
Inversion temperature	134	(1)
Van der Waals Equation of State	134	(5)
Approximating the partition function	134	(1)
Van der Waals equation	135	(2)
Microscopic ``derivation'' of parameters	137	(1)
Virial expansion	138	(1)
Other Phenomenological Equations of State	139	(4)
The Dieterici equation	139	(1)
Virial expansion	139	(1)
The Berthelot equation	140	(3)
Phase Transitions	143	(100)
Phenomenology	143	(16)
Basic ideas	143	(2)
Phase diagrams	145	(2)
Symmetry	147	(1)
Order of phase transitions	148	(1)
The order parameter	149	(2)
Conserved and non-conserved order parameters	151	(1)
Critical exponents	152	(2)
Scaling theory	154	(4)
Scaling of the free energy	158	(1)
First-Order Transition --- An Example	159	(11)
Coexistence	159	(3)
Van der Waals fluid	162	(1)
The Maxwell construction	163	(2)
The critical point	165	(1)
Corresponding states	166	(2)
Dieterici's equation	168	(1)
Quantum mechanical effects	169	(1)
Second-Order Transition --- An Example	170	(10)
The ferromagnet	170	(2)
The Weiss model	172	(1)
Spontaneous magnetisation	173	(3)
Critical behaviour	176	(1)
Magnetic susceptibility	177	(1)
Goldstone modes	178	(2)
The Ising and Other Models	180	(11)
Ubiquity of the Ising model	180	(2)
Magnetic case of the Ising model	182	(2)
Ising model in one dimension	184	(1)
Ising model in two dimensions	185	(3)
Mean field critical exponents	188	(2)
The XY model	190	(1)
The spherical model	191	(1)
Landau Treatment of Phase Transitions	191	(10)
Landau free energy	191	(2)
Landau free energy for the ferromagnet	193	(3)
Landau theory -- second-order transitions	196	(2)
Thermal capacity in the Landau model	198	(1)
Ferromagnet in a magnetic field	199	(2)
Ferroelectricity	201	(9)
Description of the phenomenon	201	(1)
Landau free energy	202	(1)
Second-order case	203	(1)
First-order case	204	(4)
Entropy and latent heat at the transition	208	(1)
Soft modes	209	(1)
Binary Mixtures	210	(16)
Basic ideas	210	(1)
Model calculation	211	(1)
System energy	212	(1)
Entropy	213	(1)
Free energy	214	(1)
Phase separation --- the lever rule	215	(2)
Phase separation curve --- the binodal	217	(2)
The spinodal curve	219	(1)
Entropy in the ordered phase	220	(2)
Thermal capacity in the ordered phase	222	(1)
Order of the transition and the critical point	223	(2)
The critical exponent β	225	(1)
Quantum Phase Transitions	226	(10)
Introduction	226	(2)
The transverse Ising model	228	(1)
Revision of mean field Ising model	228	(2)
Application of a transverse field	230	(2)
Transition temperature	232	(1)
Quantum critical behaviour	233	(1)
Dimensionality and critical exponents	234	(2)
Retrospective	236	(7)
The existence of order	236	(1)
Validity of mean field theory	237	(1)
Features of different phase transition models	238	(5)
Fluctuations and Dynamics	243	(48)
Fluctuations	244	(10)
Probability distribution functions	244	(2)
Mean behaviour of fluctuations	246	(4)
The autocorrelation function	250	(3)
The correlation time	253	(1)
Brownian Motion	254	(6)
Kinematics of a Brownian particle	255	(2)
Short time limit	257	(1)
Long time limit	258	(2)
Langevin's Equation	260	(8)
Introduction	260	(1)
Separation of forces	261	(2)
The Langevin equation	263	(1)
Mean square velocity and equipartition	264	(1)
Velocity autocorrelation function	265	(2)
Electrical analogue of the Langevin equation	267	(1)
Linear Response --- Phenomenology	268	(13)
Definitions	268	(2)
Response to a sinusoidal excitation	270	(1)
Fourier representation	271	(1)
Response to a step excitation	272	(1)
Response to a delta function excitation	273	(1)
Consequence of the reality of X(t)	274	(1)
Consequence of causality	275	(2)
Energy considerations	277	(1)
Static susceptibility	278	(2)
Relaxation time approximation	280	(1)
Linear Response --- Microscopics	281	(10)
Onsager's hypothesis	281	(2)
Nyquist's theorem	283	(2)
Calculation of the step response function	285	(1)
Calculation of the autocorrelation function	286	(5)
Appendixes	291	(22)
Appendix 1 The Gibbs--Duhem Relation	291	(2)
Homogeneity of the fundamental relation	291	(1)
The Euler relation	291	(1)
A caveat	292	(1)
The Gibbs--Duhem relation	292	(1)
Appendix 2 Thermodynamic Potentials	293	(6)
Equilibrium states	293	(2)
Constant temperature (and volume): the Helmholtz potential	295	(1)
Constant pressure and energy: the Enthalpy function	296	(1)
Constant pressure and temperature: the Gibbs free energy	296	(1)
Differential expressions for the potentials	297	(1)
Natural variables and the Maxwell relations	298	(1)
Appendix 3 Mathematica Notebooks	299	(11)
Chemical potential of Fermi gas at low temperatures	299	(2)
Internal energy of the Fermi gas at low temperatures	301	(2)
Fugacity of the ideal gas at high temperatures --- Fermi, Maxwell and Bose cases	303	(4)
Internal energy of the ideal gas at high temperatures --- Fermi, Maxwell and Bose cases	307	(3)
Appendix 4 Evaluation of the Correlation Function Integral	310	(3)
Initial domain of integration	310	(1)
Transformation of variables	310	(1)
Jacobian of the transformation	311	(2)
Index	313