Abstract
In these lectures I want to discuss the recent developments in dynamic critical phenomena using renormalization group techniques. An attractive feature of this topic is that it brings together ideas from several areas of theoretical physics. We will discuss the renormalization group ideas which have their roots in quantum field theory, the statistical mechanics of phase transformations and the principles of non-equilibrium transport phenomena. I hope to show how these principles can be amalgamated into a single theory describing time dependent processes in systems near second order phase transitions. The theory I will discuss not only leads to a good description of dynamics of phase transitions but has suggested new ideas in treating the frontier problems of turbulence1 and spinodal decomposition. 2
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
D. Forster, D. R. Nelson and M. J. Stephen, Phys. Rev. Lett. 36, 867 (1976) and to be published.
E. D. Siggia, Phys. Rev. A4, 1730 (1977)
J. W. Cahn, Act. Met. 9, 795 (1961)
J. W. Cahn, Act. Met. 10, 179 (1962).
J. W. Cahn and J. E. Hilliard, Act. Met. 19, 151 (1971)
J. S. Langer, Ann. Phys. (N.Y.) 54, 258 (1969)
J. S. Langer, Ann. Phys. (N.Y.) 65, 53 (1971)
J. S. Langer and M. Baron, Ann. Phys. (N. Y.) 78, 421 (1973)
J. S. Langer and L. A. Turski, Phys. Rev. A8, 3230 (1973)
J. S. Langer, M. Bar-on and H. D. MiIler, Phys. Rev. A11, 1417 (1975)
K. Kawasaki, Prog. Theo. Phys. 57, 410 (1977).
K. Kawasaki, Prog. Theo. Phys. 57, 826 (1977).
K. G. Wilson and J. Kogut, Phys. Repts. 12C, 75 (1974).
M. E. Fisher, Rev. Mod. Phys. 46, 597(1974).
K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975).
S. Ma, Rev. Mod. Phys. 45, 589 (1973).
S. Ma, Modern Theory of Critical Phenomena, 1976 (Benjamin, New York).
P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys., to be published.
L. P. Landau and E. M. Lifshitz, Statistical Physics, 1969 (Pergamon, Oxford).
See, for example, C. Kittel, Quantum Theory of Solids, 1963 (Wiley, New York) for an elementary introduction.
H. E. Stanley, Phys. Rev. 176, 718 (1968).
H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, New York, 1971).
L P. Kadanoff et al., Rev. Mod. Phys. 39, 615 (1967).
L. P. Kadanoff, Physics (N.Y.) 2, 263 (1966).
B. Widom, J. Chem. Phys. 43, 3989 (1965).
K. G. Wilson, Phys. Rev. B4, 3174 (1971).
L. P. Kadanoff, Annals of Phys. (N.Y.), to be published. Th. Niemejer and J. M. J. van Leeuwen, in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green (Academie, New York, 1976), Vol. VI.
There are many examples of cross fertalization between different branches of physics via the renormalization group. One particularly interesting example is the recent introduction into quantum field theory of a lattice gauge theory. This theory of strong interactions includes guarks on lattice sites interacting through “strings” which are generalizations of exchange interactions between spins to include gauge invariance. There is hope that this model will lead to an understanding of guark confinement. See K. G. Wilson, Phys. Rev. D10, 2445 (1974).
L. P. Kadanoff, Rev. Mod. Phys. 49, 267 (1977).
The external field conjugate to the order parameter is also a relevant variable. We assume here for simplicity that the external field is fixed at zero.
K. G. Wilson and M. E. Fisher, Phys. Rev. Lett. 28, 240 (1972).
E. Brézin, J. C. LeGuillou, and J. Zinn-Justin, in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green (Akademie, New York, 1976), Vol. VI.
L. N. Lipatov, Zh. Eksp. Teor. Fiz. Pis’ma Red 25, 116 (1977).
E. Brézin, J. C. Le Guillou and J. Zinn-Justin, Phys. Rev. D15, 1544, 1558 (1977).
E. Brézin, private communication.
R. Kubo, in Lectures in Theoretical Physics, Vol. I, chapter 4 (Interscience, 1959).
L. P. Kadanoff and P. C. Martin, Ann. Phys. (N.Y.) 24, 419 (1963).
P. C. Martin, in Many Body Physics, edited by C. De Wilt and R. Balian, 1968 (Gordon and Breach, N. Y.).
R. M. White, Quantum Theory of Magnetism, (McGraw-Hill, New York, 1970)
D. Forster, Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions, 1975 (W. A. Benjamin Inc. Reading, Mass.).
If our first choice for ψi (1) has a non-zero average <ψi (1)> we then choose as our variable ψi = ψi (1) < ψi (1)> which does have zero average.
We can construct the particle density, momentum density and kinetic energy density from f(x,p) by multiplying by 1, p and p2/2m, respectively, and integrating over p.
To keep things simple I am assuming we have a classical theory where the ψ’s commute. This is not a necessary requirement but its elimination would necessitate additional, but essentially unilluminating, further formal development.
H. Mori, Progr. Theor. Phys. 33, 423 (1965).
G. F. Mazenko, Phys. Rev. A9, 360 (1974).
G. F. Mazenko and S. Yip, in Statistical Mechanics, Pt. B, edited by B. J. Berne (Plenum Press, New York, 1977).
P. Resibois and M. De Leener, Phys. Rev. 152, 305, 318 (1966) and 178, 806, 819 (1969).
K. Kawasaki, Ann. Phys. (N. Y.) 61, 1 (1970).
This result, of course, follows from the projection operator approach due to Mori33. We wish to show here that the projection operators, which can present some mathematical problems when trying to develop approximation schemes (See G. F. Mazenko, Phys. Rev. A7, 209 (1973) for discussion and further references), are not a necessary part of the development.
We define Xij -1 by (math)
In principle this equation is valid only for t > o. Of course we can change our time origin to to and then it is valid for t > to. We can then let to → -∞. We can then Fourier transform over all times.
This is not the actual long time behavior of the velocity autocorrelation function. There is a strong exponential decay for times of order two to three collision times. However for longer times there is a power law decay given by t-d/2 where d is the dimensionality for long times. This long time behavior is associated with non-linear couplings similar to those we will treat in the next section. See for example Y. Pomeau and P. Resibois, Phys. Rept. 19C, 64 (1975)
J. R. Dorfman and E. G. D. Cohen, Phys. Rev. Lett. 25, 1257 (1970)
J. R. Dorfman and E. G. D. Cohen, Phys. Rev. A6, 2247 (1972).
There is a very nice discussion of the Drude model in N. W. Ashcroft and N. D. Mermin, Solid State Physics, 1976 (Holt, Rinehart and Winston, New York).
These ideas are discussed extensively by Forster in ref. 29.
Ma7 has a nice discussion of this point.
H. Mori, Prog. Theo. Phys. 28, 763 (1962).
L Van Hove, Phys. Rev. 93, 1374 (1954).
I will, for convenience, usually choose units such that the constant C in the Ornstein-Zemike expression for X(q) (See (2. 23)) can be set equal to 1.
B. I. Halperin and P. C. Hohenberg, Phys. Rev. Lett. 19, 700 (1967)
B. I. Halperin and P. C. Hohenberg, Phys. Rev. 177, 952 (1969).
V. J. Minkewicz, M. F. Collins, R. Nathans, and G. Shirane, Phys. Rev. 182, 624 (1969).
For a list of experimental references see R. Freedman and G. F. Mazenko, Phys. Rev. B13, 4967 (1976).
J. V. Sengers, in Critical Phenomena, edited by M. S. Green (Academie, New York, 1971)
M. S. Green, J. Chem. Phys. 20, 1281 (1952)
M. S. Green, J. Chem. Phys. 22, 398 (1954).
R. W. Zwanzig, Phys. Rev. 124, 983 (1961).
H. Mori and H. Fujisaka, Prog. Theor. Phys. 49, 764 (1973).
K. Kawasaki, in Critical Phenomena, edited by M. S. Green (Academic, New York, 1971).
In this case ψ¡ does not satisfy (4.4).
Mi (t) can be identified with with those Fourier compo nents of M¡ (t) less than the cut-off Λ.
S. Ma and G. F. Mazenko, Phys. Rev. Lett. 33, 1383 (1974),
S. Ma and G. F. Mazenko, Phys. Rev. B11, 4077 (1975).
B. I. Halperin, P. C. Hohenberg and S. Ma, Phys. Rev. Lett. 29, 1548 (1972),
B. I. Halperin, P. C. Hohenberg and S. Ma Phys. Rev. B10, 139 (1974)
B. I. Halperin, P. C. Hohenberg and S. Ma Phys. Rev. B13, 4779 (1976).
See the discussion in ref. 8.
This follows from (4.14) by expressing Jij and Mi(t) in terms of their Fourier transforms, expanding in powers of the wavenumber, and transforming back to coordinate space.
K. Kawasaki, Phys. Rev. 150, 291 (1966), in Statistical Mechanics, edited by S. A. Rice, K. F. Freed, and J. C. Light (University of Chicago, 1972) and refs. 37 and 55.
M. Fixman, J. Chem. Phys. 36, 310 (1962)
L. P. Kadanoff and J. Swift, Phys. Rev. 166, 89 (1968).
R. A. Ferrell, Phys. Rev. Lett. 24, 1169 (1970).
R. Zwanzig, in Statistical Mechanics, edited by S. A. Rice, K. F. Freed, and J. C. Light, (University of Chicago Press, Chicago, 1972).
P. C. Martin, E. D. Siggia, and H. A. Rose, Phys. Rev. A8, 423 (1973).
H. K. Janssen, Z. Physik B23, 377(1976),
R. Bausch, H. J. Janssen and H. Wagner, Z. Physik, B24, 113 (1976).
K. Kawasaki, Prog. Theor. Phys. 52, 1527 (1974).
C. De Dominicis, Nuovo Cimento Lett. 12, 576 (1975)
C. De Dominicis, E. Brezin and J. Zinn-Justin, Phys. Rev. B12, 4945 (1975).
G. F. Mazenko, M. Nolan and R. Freedman, Phys. Rev. B, to be published, discuss situations where the fluctuation — dissipation theorem may have a slightly different form.
We have already investigated the behavior of u and ro under the RG in the static case. Since we generate exactly the same statics from our equation of motion we would obtain the same recursion relations.
M. J. Nolan and G. F. Mazenko, Phys. Rev. B15, 4471 (1977).
B. I. Halperin, P. C. Hohenberg, and E. D. Siggia, Phys. Rev. Lett. 32, 1289 (1974)
B. I. Halperin, P. C. Hohenberg, and E. D. Siggia, Phys. Rev. B13, 1299 (1976).
P. C. Hohenberg, B. I. Halperin, and E. D. Siggia, Phys. Rev. B14, 2865 (1976)
E. D. Siggia, Phys. Rev. B11, 4736 (1975).
B. I. Halperin, P. C. Hohenberg, and E. D. Siggia, Phys. Rev. Lett. 32, 1289 (1974)
B. I. Halperin, P. C. Hohenberg, and E. D. Siggia, Phys. Rev. B13, 1299 (1976).
R. Freedman and G. F. Mazenko, Phys. Rev. Lett. 34, 1575 (1975)
R. Freedman and G. F. Mazenko, Phys. Rev. B13, 4967 (1976).
Ref. 73 and E. D. Siggia, B. I. Halperin, and P. C. Hohenberg, Phys. Rev. B13, 2110 (1976).
I. M. Gel’fand and A. M. Yaglom, J. Math. Phys. 1, 18 (1960).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1978 Plenum Press, New York
About this chapter
Cite this chapter
Mazenko, G.F. (1978). Renormalization Group Approach to Dynamic Critical Phenomena. In: Halley, J.W. (eds) Correlation Functions and Quasiparticle Interactions in Condensed Matter. NATO Advanced Study Institutes Series, vol 35. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-3360-9_3
Download citation
DOI: https://doi.org/10.1007/978-1-4684-3360-9_3
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4684-3362-3
Online ISBN: 978-1-4684-3360-9
eBook Packages: Springer Book Archive