1 Introduction ................................................. 1
2 Nonequilibrium thermodynamics ................................ 5
2.1 Local thermodynamic properties .......................... 5
2.1.1 Equilibrium thermodynamics ....................... 5
2.1.2 Continuum hypothesis and the assumption of
local equilibrium ................................ 7
2.2 Balance laws ............................................ 8
2.2.1 Mass balance ..................................... 9
2.2.2 Pressure tensor ................................. 11
2.2.3 Momentum balance ................................ 15
2.2.4 Energy balance .................................. 16
2.2.5 Additional remarks .............................. 19
2.3 Entropy balance, dissipative fluxes and thermodynamic
forces ................................................. 20
2.3.1 Phenomenological relations ...................... 22
2.3.2 Onsager's reciprocal relations .................. 23
2.4 Hydrodynamic equations ................................. 24
2.4.1 Hydrodynamic equations for a one-component
fluid ........................................... 24
2.4.2 Hydrodynamic equations for a binary mixture ..... 28
2.5 Boundary conditions .................................... 33
2.5.1 Conditions at the interface between two
continuous media ................................ 33
2.5.2 A flat horizontal fluid boundary ................ 36
3 Fluctuations in fluids in thermodynamic equilibrium ......... 39
3.1 Fluctuating hydrodynamics .............................. 40
3.2 Fluctuation-dissipation theorem for fluids and fluid
mixtures ............................................... 43
3.2.1 Fluctuation-dissipation theorem for a
one-component fluid ............................. 43
3.2.2 Fluctuation-dissipation theorem for a binary
mixture ......................................... 45
3.3 Hydrodynamic fluctuations in a one-component fluid ..... 46
3.3.1 Structure factor ................................ 51
3.3.2 Equal-time correlations ......................... 54
3.4 Equilibrium correlation functions and entropy
probability functional ................................. 56
3.5 Extension of fluctuating hydrodynamics to
nonequilibrium steady states ........................... 60
4 Thermal nonequilibrium fluctuations in one-component
fluids ...................................................... 63
4.1 Boussinesq approximation ............................... 64
4.2 Bulk structure factor in the presence of a stationary
temperature gradient ................................... 70
4.3 Nonequilibrium effects on the Brillouin doublet ........ 76
4.4 Nonequilibrium fluctuations in heat conduction ......... 82
5 Thermal nonequilibrium fluctuations in fluid mixtures ....... 89
5.1 Linearized fluctuating Boussinesq equations for
a binary liquid ........................................ 91
5.1.1 The linear stability problem .................... 94
5.2 Structure factor in a large-Lewis-number
approximation .......................................... 97
5.3 Comprehensive structure factor of a nonequilibrium
binary mixture ........................................ 102
5.4 Nonequilibrium fluctuations in isothermal
free-diffusion processes .............................. 110
6 Finite-size effects in hydrodynamic fluctuations ........... 113
6.1 The hydrodynamic operator ............................. 115
6.1.1 Structure factors .............................. 122
6.2 Hydrodynamic modes and decay rates for two free
boundaries ............................................ 125
6.3 Hydrodynamic modes and decay rates for two rigid
boundaries ............................................ 129
6.3.1 Decay rates for rigid boundaries in the limit
of small q‖ .................................... 131
6.3.2 Decay rates for rigid boundaries in the limit
of large q‖ .................................... 134
6.4 The slowest decay rate ................................ 136
7 Thermal nonequilibrium fluctuations in one-component-
fluid layers ............................................... 141
7.1 A fluid confined between two free boundaries .......... 142
7.2 A fluid confined between two rigid boundaries ......... 147
7.3 Limiting behavior of the structure factor for rigid
boundaries at small and large wave numbers ............ 152
7.4 A Galerkin approximation for two rigid boundaries ..... 155
7.5 Correlations in real space ............................ 162
7.6 Contribution of nonequilibrium fluctuations to heat
transfer .............................................. 166
8 Thermal fluctuations close to the Rayleigh-Benard insta
bility ..................................................... 167
8.1 Critical slowing down of nonequilibrium
fluctuations .......................................... 168
8.2 The most-unstable-mode approximation .................. 169
8.3 The Swift-Hohenberg approximation ..................... 172
8.4 Power of thermal fluctuations ......................... 175
8.5 Wave number of maximum enhancement of fluctuations .... 179
8.6 Wave number of fluctuations with maximum growth
rate .................................................. 181
9 Thermal nonequilibrium fluctuations in binary-fluid
layers ..................................................... 185
9.1 The hydrodynamic operator for binary Boussinesq ....... 187
9.2 A Galerkin approximation for rigid and impermeable
boundary conditions ................................... 189
9.3 Evaluation of nonequilibrium structure factors ........ 194
9.4 Structure factor in thermal equilibrium ............... 197
9.5 Nonequilibrium fluctuations for positive separation
ratio ................................................. 198
9.5.1 Temperature fluctuations ....................... 199
9.5.2 Concentration fluctuations ..................... 204
9.6 Large-Lewis-number approximation for positive
separation ratio ...................................... 207
9.7 Concluding remarks .................................... 211
10 Experiments on nonequilibrium fluctuations in the
Rayleigh-Benard problem .................................... 213
10.1 Experimental techniques ............................... 214
10.1.1 Small-angle light scattering ................... 214
10.1.2 Quantitative shadowgraph analysis .............. 222
10.1.3 Dependence of shadowgraph images on
collecting times ............................... 228
10.2 Nonequilibrium light-scattering experiments in
one-component fluids .................................. 229
10.2.1 Propagating modes .............................. 229
10.2.2 Brillouin doublet .............................. 230
10.2.3 Rayleigh scattering ............................ 232
10.3 Shadowgraph experiments in one-component fluids ....... 235
10.3.1 Divergence of critical fluctuations near the
onset of convection ............................ 235
10.3.2 Critical slowing down of fluctuations near
the onset of convection ........................ 237
10.4 Light scattering in binary systems .................... 239
10.4.1 Nonequilibrium fluctuations in liquid
mixtures ....................................... 240
10.4.2 Concentration fluctuations in dilute polymer
solutions ...................................... 243
10.5 Shadowgraphy in binary mixtures ....................... 246
11 Other nonequilibrium fluctuations .......................... 247
11.1 Nonequilibrium fluctuations in isothermal
one-component fluids under shear ...................... 248
11.2 Nonequilibrium interface fluctuations ................. 252
11.3 Nonequilibrium fluctuations in nematic liquid
crystals .............................................. 253
11.4 Concentration fluctuations in reaction-diffusion
problems .............................................. 256
11.4.1 Equilibrium concentration fluctuations in
reaction-diffusion ............................. 257
11.4.2 Nonequilibrium concentration fluctuations in
reaction-diffusion ............................. 263
12 Epilogue ................................................... 269
Bibliography .................................................. 273
List of symbols and corresponding SI units .................... 297
List of abbreviations ......................................... 303
Subject index ................................................. 305
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