Nonequilibrium Statistical Mechanics

Nonequilibrium Statistical Mechanics

Volumn , Issue , 2008, Pages 1-478

  Mazenko, Gene F ^a  

^a UNIVERSITY OF CHICAGO   (United States)

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EID: 84889330859     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1002/9783527618958     Document Type: Book

References (457)

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2. As an extreme example we have the process of diffusion-limited aggregation, which leads to a highly tenuous fractal system. See T. A.Witten and L. S. Sander, Phys. Rev. Lett. 47, 1400 (1981).
3. See the review by M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993), where there is an effort to organize some of these strongly nonequilibrium situations.
4. See, for example, D. Ruelle, Phys. Today, May 2004, Conversations on Nonequilibrium Physics with an Extraterrestrial.
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10. The case of a random walk on a lattice is discussed in some detail in Section 7.2 in FOD.
11. It is more conventional to define the friction coefficient as γP = Mγ.
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18. A general overview is given in Appendix C of ESM.
19. This relaxation may take a very long time if there are degrees of freedom that get trapped and one has quenched impurities. Such behavior is very important in solid systems and leads to phenomena such as localization.
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28. In the GCE, we replace H→ H-μN where μ is the chemical potential and N the number operator.
29. The time dependence of the intrinsic Hamiltonian comes from the time evolution of the operators appearing in the Hamiltonian. When one turns off the external field the intrinsic Hamiltonian is time independent.
30. Thermodynamically B and h are conjugate variables.
31. For a discussion of the different representations see A. Messiah, Quantum Mechanics, Vol. I,Wiley, N.Y. (1966), pp. 312, 314.
32. For a discussion of the interplay of symmetry and invariance properties see the discussions in Section 1.13 in ESM and Section 1.3.3 in FOD.
33. The worry is that one is dealing with longrange interactions in charged systems and this requires considerable care.
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54. In the typical case the double prime indicates the imaginary part as discussed below.
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58. Ref. [5], p. 417.
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65. The quantum-mechanical formulation is set up in Chapter 5 in ESM.
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67. P. C. Martin, Phys. Rev. 161, 143 (1967)
68. See the discussion in Chapter 5 in ESM and Chapter 6 in FOD.
69. The case of a static set of charges is discussed in Chapter 6 of FOD for the case of a dielectric material.
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72. P. Drude, Ann. Phys. 1, 566 (1900)
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74. A realistic model treats the normal modes in the system.
75. See the discussion in N. Ashcroft, N. D. Mermin, Solid State Physics, Holt, Rinehart andWinston, N.Y. (1976), p. 543.
76. C. Kittel, Introduction to Solid State Physics, 7th edn.,Wiley, N.Y. (1996), p. 278.
77. The classic early numerical simulations on fluids were by B. J. Alder, T. E. Wainwright, Phys. Rev. Lett. 18, 988 (1968)
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79. B. J. Alder, D. M. Gass, T. E.Wainwright, J. Chem. Phys. 53, 3813 (1970)
80. For an introduction, summary of results, and references see J. P. Boon, S. Yip, Molecular Hydrodynamics, Mc Graw-Hill, N.Y. (1979). Definitive simulation for the magnetic models we introduced here are more recent.
81. See, for example, D. P. Landau, M. Krech, Spin Dynamics Simulations of Classical Ferro-and Antiferromagnetic Model Systems: Comparison with Theory and Experiment, Preprint.
82. There is an extensive discussion of the importance of conservation laws in equilibrium statistical mechanics in Chapter 1 of ESM.
83. Symmetry principles are discussed in some detail in Chapter 1 in ESM and FOD.
84. There is a detailed discussion of Nambu-Goldstone modes in Chapter 5 of FOD.
85. N. Bloembergen, Physica 15, 386 (1949), introduced the spin-diffusion coefficient.
86. Ohm's Law was introduced by George Ohm in Die Galvanische Kette Mathematisch Bearbeitet (1827).
87. See Ref. [1] in Chapter 1.
88. A. Fick, Ann. Phys.(Leipzig) 94, 59 (1855)
89. M. S. Green, J. Chem. Phys. 30, 1280 (1952)
90. J. Chem. Phys. 22, 398 (1954)
91. R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957)
92. P. A. M. Dirac, The Principles of Quantum Mechanics, 4th edn, Oxford University Press, Oxford (1958), Section 21.
93. B. O. Koopman, Proc. Natl. Acad. Sci. U. S. A 17, 315 (1931)
94. For a treatment of spin diffusion in the Heisenberg paramagnet see H. S. Bennett, P. C. Martin, Phys. Rev. 138, A608 (1965).
95. See Section 5.9.1 in ESM and Section 1.4.1 in FOD.
96. This trick is due to Professor P. C. Martin. See the discussion in Appendix F of ESM.
97. This is the statement, in keeping with causality, that the field does not depend on the noise at later times.
98. The quantum development is discussed in H. Mori, Prog. Theor. Phys. (Kyoto) 33, 423 (1965)
99. R.W. Zwanzig, in Lectures in Theoretical Physics, Vol. 3, Interscience, N.Y. (1961)
100. Phys. Rev. 24, 983 (1965)
101. H. Mori, Prog. Theor. Phys. (Kyoto) 33, 423 (1965)
102. G. F. Mazenko, Phys. Rev. A 7, 209 (1973)
103. Phys. Rev. A 9, 360 (1974)
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106. J. Phys. Chem. B 107, 10226 (2003)
107. J. Phys. Chem. B 107, 10234 (2003)
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109. Phys. Rev. 152, 318 (1969)
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111. We follow the proof of the second fluctuation dissipation theorem given by H. Mori in Ref. [22].
112. The use of the word Markoffian stems from the ideas associated with Markov process or chains [A. A. Markov, Izv. Akad. Nauk St. Petersburg 6, 61 (1907)]. These processes, roughly, correspond to sequences where the next step depends only on the previous step. In our context of the generalized Langevin equation the Markoffian approximation suggests that the memory function on a long enough time scale has no memory.We can replace K(d)(z) with its z = 0 limit.
113. The statistics governing the higherorder correlation functions are generally unknown. This point is discussed further in Chapter 9.
114. J. G. Kirkwood, J. Chem. Phys. 14, 180 (1946)
115. See p. 370 in ESM.
116. The nonlinear processes that lead to the breakdown of conventional hydrodynamics are discussed in Chapter 9.
117. See Chapter 6 in ESM.
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122. The Gibbs-Duhem relation is discussed in Section 2.3.3 in ESM.
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124. See Section 1.14.4 in ESM.
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127. Ann. Phys. 17, 88 (1922)
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135. See footnote 68 on p. 101 in ESM.
136. S. Brush, Kinetic Theory, Vol. 1, Pergamon, N.Y. (1965).
137. Seminal work in kinetic theory was carried out between 1859 and 1877 by J. C. Maxwell and L. Boltzmann. For a detailed discussion see S. Brush, Statistical Physics and the Atomic Theory of Matter, Princeton University Press, Princeton (1983).
138. J. Perin, Atoms, Ox Bow Press, Woodbridge, Conn. (1990).
139. This quantity was introduced in Chapter 5.
140. One of the advantages of this approach is that one need not be precise about the nature of the nonequilibrium average at this stage in the calculation. Eventually, when one wants precision, the definiteness of linear response theory becomes an asset.
141. Our development parallels that in J. Kestin, J. R. Dorfman, A Course in Statistical Thermodynamics, Academic, N.Y. (1971), Chapters 12, 13.
142. R. Clausius, Ann. Phys. ser. 2, 105, 239 (1865)
143. see the discussion on p. 54 in S. Brush, Statistical Physics and the Atomic Theory of Matter, Princeton University Press, Princeton (1983).
144. In a more careful treatment NT is proportional to a two-particle distribution f (x1, p1; x2, p2).We have followed Boltzmann and adopted the famous Stosszahlansatz or molecular chaos assumption where the two-particle distribution factorizes into a product of singlet-distribution function f (x1, p1; x2, p2) = f (x1, p1) f (x2, p2).
145. L. Boltzmann, Math.-Naturwiss. KI. 66, 275 (1872).
146. English translation in S. Brush, Kinetic Theory, Vol. 2, Pergamon, N.Y. (1966).
147. Boltzmann actually used the letter E rather than H. H was introduced later in S. H. Burbury, Philos. Mag. Ser. 5 30, 298 (1890)
148. A detailed discussion is given in S. Chapman, T. G. Cowling, The Mathematical Theory of Nonuniform Gases, Cambridge University Press, N.Y. (1953).
149. P. L. Bhatnagar, E. P. Gross, M. Krook, Phys. Rev. 94, 511 (1954)
150. E. P. Gross, E. A. Jackson, Phys. Fluids 2, 432 (1959)
151. G. F. Mazenko, T.Wei, S. Yip, Phys. Rev. A 6, 1981 (1972)
152. The exception is for so-called Maxwell molecules. In this case, where the interaction potential goes as r-4, one can solve for the eigenvalues and eigenfunctions explicitly. J. C. Maxwell, Collected Papers, Vol. 2, p. 42
153. Philos. Trans. R. Soc. 157, 49 (1867). See p. 175 in Ref. [13].
154. In this case the negative carriers move in a fixed background of positive charge, which provides overall change neutrality.
155. T. M. Reed, K. E. Gubbins, Applied Statistical Mechanics, Mc Graw-Hill, N.Y. (1973).
156. See the discussions in N. Ashcroft, N. D. Mermin, Solid State Physics, Holt, Rinehardt andWinston, N.Y. (1976), Chapters 1, 13.
157. L. Onsager, Phys. Rev. 37, 405 (1931)
158. Phys. Rev. 38, 2265 (1931)
159. For a discussion of a variety of thermoelectric, thermomagnetic and galvanomagnetic effects, like the Seebeck effect, see H. B. Callen, Phys. Rev. 73, 1349 (1948)
160. E. H. Hall, Am. J. Math. 2, 287 (1879)
161. See the discussion in Ashcroft and Mermin, Ref. [19], p. 15.
162. L. S. Ornstein, F. Zernike, Proc. Acad. Sci. Amsterdam 17, 793 (1914); see p. 305 in ESM.
163. E. E. Salpeter, Phys. Rev. 120, 1528 (1960)
164. D. C. Montgomery, D. A. Tidman, Plasma Kinetic Theory, McGraw-Hill, N.Y. (1964).
165. M. Nelkin, S. Ranganthan, Phys. Rev. 164, 222 (1967)
166. H. Gould, G. F. Mazenko, Phys. Rev. A 15, 1274 (1977)
167. Phys. Rev. Lett. 35, 1455 (1975)
168. See the discussion of the direct correlation function in Section 4.7.1 in ESM.
169. The direct correlation function for a plasma is given on p. 316 in ESM.
170. G. F. Mazenko, Phys. Rev. A 9, 360 (1974)
171. H. C. Anderson, J. Phys. Chem. B 107, 10234 (2003)
172. D. Forster, P. C. Martin, Phys. Rev. A 12, 1575 (1970)
173. A. Z. Akcasu, J. J. Duderstadt, Phys. Rev. 188, 479 (1969)
174. G. F. Mazenko, Phys. Rev. A 5, 2545 (1972)
175. J. R. Dorfmann, E. G. D. Cohen, Phys. Rev. Lett. 25, 1257 (1970)
176. Phys. Rev. A 6, 776 (1972)
177. In Chapter 1 of FOD the nature of the critical points in a variety of systems is discussed.
178. The identification of order parameter for a variety of systems is discussed in Chapter 1 of FOD.
179. For simplicity at the beginning we assume a scalar order parameter as one would find for an Ising model.We also assume for simplicity that the thermodynamically conjugate field is set to its critical value. For magnetic systems this means the externally applied magnetic field is set to zero.
180. For example, the average of the order parameter, the magnetization density, vanishes for a ferromagnet in the paramagnetic phase due to rotational invariance. This assumes zero external applied field.
181. See the discussion in Sections 4.1 and 4.2 in FOD.
182. See Table 4.4 in FOD.
183. L. S. Ornstein, F. Zernike, Proc. Acad. Sci. Amsterdam 17, 793 (1914)
184. L. P. Kadanoff, W. Götze, D. Hamblen, R. Hecht, E. A. S. Lewis, V.V. Palciauskas, 296 8 Critical Phenomena and Broken Symmetry M. Rayl, J. Swift, D. Aspnes, J. Kane, Rev. Mod. Phys. 39, 395 (1967)
185. Liquid crystals with biaxial symmetry are discussed in Chapter 10 of FOD.
186. B.Widom, J. Chem. Phys. 43, 3989 (1965)
187. L. P. Kadanoff, Physics (N.Y.) 2, 263 (1966)
188. P. Weiss, J. Phys. 6, 661 (1907); see Section 5.9.2 in ESM.
189. See Section 2.9 in ESM. Johannes Diderik Van der Waals, Over der Continuiteit van den Gas-en vloeisoftoestand, Leiden, Sijthoff (1873). [Transl.: The Continuity of the Liquid and Gaseous States of Matter], in Physical Memoirs, Vol. 1, Part 3. Taylor and Francis (The Physical Society), London (1890).
190. L. Onsager, Phys. Rev. 65, 117 (1944)
191. see also B. Kaufmann, Phys. Rev. 76, 1232 (1949)
192. B. M. Mc Coy, T. T.Wu, The Two-Dimensional Ising Model, Harvard University Press, Cambridge, (1973).
193. L. van Hove, Phys. Rev. 95, 1374 (1954)
194. P. C. Hohenberg, B. I. Halperin, Rev.Mod. Phys. 49, 435 (1977)
195. R. A. Ferrell, N. Menyh'ard, H. Schmidt, F. Schwable, P. Sz'epfalusy, Phys. Rev. Lett. 18, 891 (1967)
196. R. A. Ferrell, N. Menyh'ard, H. Schmidt, F. Schwable, P. Sz'epfalusy, Ann. Phys. (N.Y.) 47, 565 (1968)
197. B. I. Halperin, P. C. Hohenberg, Phys. Rev. Lett. 19, 700 (1967)
198. B. I. Halperin, P. C. Hohenberg, Phys. Rev. 177, 952 (1969)
199. The classical Poisson brackets are replaced by commutation relations in quantummechanical systems.
200. Remember that the Poisson brackets between the density and the momentum density lead to the sound modes in a simple fluid.
201. Spontaneous symmetry breaking is discussed in some detail in Section 1.3.2 in FOD.
202. The dynamic critical index for a ferromagnet ismeasured in O.W. Dietrich, J. Als-Nielsen, L. Passell, Phys. Rev. B 14, 4923 (1976)
203. In the low-temperature quantum analysis of such systems the NG modes are the lowest lying excitations. See the discussion in Section 5.6 in FOD.
204. This work was discussed in Chapters 1-3.
205. Corrections to the Boltzmann equation, introduced by D. Enskog [K. Svensk. Vet.-Akad. Handl. 63, no.4 (1921)] in the early 1920s allowed the description to be extended to higher densities. However, the basic physics still consisted of the uncorrelated binary collisions discussed in Chapter 7.
206. See the discussion in Chapter 16 in S. Chapman, T. G.Cowling, The Mathematical Theory of Nonuniform Gases, 3rd edn., Cambridge University Press, London (1970), Chapter 16.
207. After the establishment of the Green-Kubo equations for transport coefficients, R. W. Zwanzig, Phys. Rev. 129, 486 (1963)
208. J.Weinstock, Phys. Rev. 126, 341 (1962)
209. Phys. Rev. 132, 454, 470 (1963), proposed a low-density expansion for transport coefficients.
210. Similar work was carried out in K. Kawasaki, I. Oppenheim, Phys. Rev. 136, 1519 (1964)
211. Phys. Rev. 139, 689 (1965)
212. The divergence of the density expansion of transport coefficients was discovered by E. G. D. Cohen and J. R. Dorfmann, Phys. Lett. 16, 124 (1965)
213. J.Weinstock, Phys. Rev. 140, 460 (1965).
214. For some history of this discovery see J. R. Dorfmann, in Lectures in Theoretical Physics, Vol. IXC, Kinetic Theory, W. E. Brittin (ed),Gordon and Breach, N.Y. (1967), p443.
215. See also M. H. Ernst, L. K. Haines, J. R. Dorfmann, Rev. Mod. Phys. 41, 296 (1969)
216. The resummation of the expansion was carried out by K. Kawasaki and I. Oppenheim, Phys. Rev. 139, 1763 (1965)
217. B. J. Alder, T. E.Wainwright, Phys. Rev. Lett. 18, 988 (1967)
218. Phys. Rev. A 1, 18 (1970)
219. B. J. Alder, D. M. Gass, T. E. Wainwright, J. Chem. Phys. 53, 3813 (1970)
220. T. E.Wainwright, B. J. Alder, D. M. Gass, Phys. Rev. A 4, 233 (1971)
221. The connections between mode-coupling, divergence of transport coefficients and long-time tails were brought together in G. F. Mazenko, Phys. Rev. A 9, 360 (1974)
222. This choice of variables is discussed by M. S. Green, J. Chem. Phys. 20, 1281 (1952)
223. J. Chem. Phys. 22, 398 (1954)
224. There is extensive discussion of effective Hamiltonians or free energies in FOD in Chapters 2 and 3.
225. L. D. Landau and E. M. Lifshitz, Statistical Physics, 3rd edn., Pergamon, Oxford (1980).
226. This is discussed in detail in ESM (Appendix O) and FOD (Appendix B).
227. K.Wilson, Phys. Rev. B 4, 3174 (1971)
228. L. D. Landau, On the Theory of Phase Transitions, Phys. Z. SovietUn. 11, 26 (1937); reprinted in Collected Papers of L. D. Landau, D. terHaar (ed), Pergamon, New York (1965).
229. This model is treated in mean-field theory in Problem 8.8.
230. J. S. Langer, L. Turski, Phys. Rev. A 8, 3239 (1973)
231. See Section 10.3 on smectic A liquid crystals in FOD.
232. Static elastic theory is studied in Chapter 6 of ESM.
233. V. L.Ginzburg, L. D. Landau, Zh.Eksperim. Teor. Fiz. 20, 1064 (1950)
234. See the discussion in Chapter 9 on superconductivity in FOD.
235. This development builds on the work of M. S. Green [8] and R.W. Zwanzig, Phys. Rev. 124, 983 (1961)
236. J. Chem. Phys. 33, 1330 (1960). A direct use of Mori's method (discussed here for the linear case in Chapter 5)
237. the nonlinear case is given in H. Mori, H. Fujisaka, Prog. Theor. Phys. 49, 764 (1973)
238. H. Mori, H. Fujisaka, H. Shigematsu, Prog. Theor. Phys. 51, 109 (1974)
239. We follow the development of S. Ma and G. F. Mazenko, Phys. Rev. B 11, 4077 (1975)
240. A. D. Fokker, Ann. Phys. 43, 812 (1914)
241. M.Planck, Sitzungsber. Preuss. Akad.Wiss., p.324 (1917).
242. R. W. Zwanzig, K. S. J. Nordholm,W. C. Mitchell, Phys. Rev. A 5, 2680 (1972), cleared up some early confusion on this point.
243. L. D. Landau, I. M. Khalatnikov, Dokl. Akad. Nauk SSSR 96, 469 (1954); Reprinted in Collected Papers of L. D. Landau, D. terHaar (ed), Pergamon,New York (1965).
244. This development is the irreversible cousin to the reversible Gross-Pitaevskii equation. E. P. Gross, Nuovo Cimento 20, 454 (1961)
245. L. P. Pitaevskii, Sov. Phys. JETP 13, 451 (1961)
246. The TDGL equations for superconductors were first derived by A. Schmid, Phys. Konden.Mater. 5, 302 (1966)
247. See also M. Cyrot, Rep. Prog. Phys. 36, 103 (1973).
248. For further discussion and background see A. T. Dorsey, Phys. Rev. B 46, 8376 (1992)
249. P. C. Hohenberg, B. I. Halperin, Rev.Mod. Phys. 49, 435 (1977)
250. J.W. Cahn J. E. Hilliard, J. Chem. Phys. 28, 258 (1958)
251. H. E. Cook, Acta. Metall. 18, 297 (1970)
252. S. Ma, G. F. Mazenko, Phys. Rev. B 11, 4077 (1975)
253. R. Freedman, G. F. Mazenko, Phys. Rev. Lett. 34, 1575 (1975)
254. Phys. Rev. B 13, 1967 (1976)
255. B. S. Kim, G. F. Mazenko, J. Stat. Phys. 64, 631 (1991)
256. The role of multiplicative noise is discussed in Ref. [32].
257. G. Grinstein, R. Pelcovits, Phys. Rev. Lett. 47, 856 (1981)
258. discussed the equilibrium case, while dynamics were studied by G. F. Mazenko, S. Ramaswamy, J. Toner, Phys. Rev. Lett. 49, 51 (1982)
259. Phys. Rev. A 28, 1618 (1983)
260. S. Milner, P. C. Martin, Phys. Rev. Lett. 56, 77 (1986)
261. This is worked out in Appendix E of ESM.
262. Mode coupling and long-time tails go back to: M. H. Ernst, E. H. Hauge, J. M. J. van Leeuwen, Phys. Rev. Lett. 25, 1254 (1970)
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