phys212ln30 - Physics 212 Statistical mechanics II Fall...

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Physics 212: Statistical mechanics II, Fall 2006 Lecture XXIX In this lecture we introduce the theory of dynamical critical phenomena, which generalizes the previous examples of Glauber and Kawasaki dynamics on the d = 1 Ising model. We concluded from that simple case that one static universality class can have multiple dynamical universality classes, and that conservation laws play a key role in long-time dynamics. We would like to understand the problem of near-equilibrium dynamics near critical points (“dynamical critical phenomena”) in a more general way. This challenge will involve combining the Langevin dynamics introduced in the theory of Brownian motion with many ideas from static critical phenomena, including the Landau free energy and the notion of a scaling form. (This lecture will use the notation of the classic Rev. Mod. Phys. article of Hohenberg and Halperin; I will attempt to point out differences from our previous notation.) Start with a theory of a single scalar field φ ( x, t ), representing for example the coarse-grained magnetization density in an Ising model. If H is a static energy function on φ , consider the dissipative dynamical equation ∂φ ∂t = - Γ ∂H ∂φ + ζ ( x, t ) . (1) Here the first term on the right side describes overdamped relaxation to the minimum of energy. The rate Γ is assumed to be real and positive. The term ζ ( x, t ) is a Langevin random force, with correlations we assume to be described by ζ ( x, t ) ζ ( x , t ) = 2Γ ( x - x ) δ ( t - t ) . (2) The average here is an average over realizations of the random force. This model for critical dynamics of a single nonconserved field is referred to as “Model A”. As in our discussion of the Langevin theory of Brownian motion, we would like this average over random forcing to reproduce thermal equilibrium; actually this fixes the constant A to be just the temperature, no matter whether the energy function H is simple or complicated.
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