MIT 8 334 - Generic scale invariance in equilibrium systems

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MIT OpenCourseWare http://ocw.mit.edu 8.334 Statistical Mechanics II: Statistical Physics of Fields Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.VIII.D Generic scale invariance in equilibrium systems We live in a world full of complex spatial patterns and structures such as coastlines and river networks. There are simil arly diverse temporal processes generically exhibiting “1/f”–noise, as in resistance fluctuations, sand flowing through an hour glass, and even in traffic and stock market movements. These phenomena l a ck natural length and time scales and exhibit scale invariance and self-similarity. The spacial aspects of scale invariant sys-tems can be characterized using fractal geometry[1]. In this section we explore dynamical processes that can naturally result in such scale invariant patterns. Let us assume that the system of interest is described by a scalar field m(x), distributed with a probability P[m]. Scale invariance can be probed by examining t he correlation func-tions of m(x), such as the two point correlator, C(|x−y|) ≡ �m(x)m(y)�−�m(x)��m(y)�. (It is assumed that the system has rotational and translational symmetry.) In a system with a characteristic length scale, correlations decay to zero for separations z = |x−y| ≫ ξ. By contrast, if the system possesses scale invariance, correlations are homogeneous at long distances, and li mz→∞ C(z) ∼ z2χ . As we have seen, in equilibrium statistical mechanics the probability is gi ven by Peq ∝ exp (−βH[m]) with β = (kBT )−1 . Clearly at infinite temperature there are no correlations for a finite Hamiltonian. As long as the interactions in H[m] are short ranged, it can be shown by high temperature expansions that correlations at small but finite β decay as C(z) ∝ exp(−z/ξ), indicating a characteristic length-scale† . The correla tion length usually increases upon reducing temperature, and may diverge if the system undergoes a continuous (critical) phase transition. At a critical transition the system i s scale invariant and C(z) ∝ z2−d−η . However, such scale invariance is non-generic in the sense that it can be obtained only by precise tuning of the system to the critical temperature. Most scale inva riant processes in nature do not require such precise tuning, and therefore the analogy to the critical point is not particularly instructive[2][3]. We shall frame our discussion of scale invariance by considering the dynamics of a surface, described by its height h(x, t). Specific examples are the distortions of a soap film or the fluctuations on the surface of water in a container. In both cases the minimum energy configuration is a flat surface (ignoring the small effects of gravity on the soap film). † It is of course possible to generate l ong-range correlations with long ranged interac-tions. However, it is most interesting to find out how long-range correlatio ns are generated from local, sho rt ranged interactions. 151� �The energy cost of small fluctuations for a soap film comes from the increased area and surface tension σ. Expanding t he area in powers of the slope results in ��� �� Hσ = σ dd x 1 + (∇h)2 − 1 ≈ σ 2 dd x (∇h)2 . (VIII.49) For the surface of water there is an additional gravitational potential energy, obtained by adding the contributions from all columns of water as � � h(x) � dd ρg ddHg = x ρgh(x) =2 xh(x)2 . (VIII.50) 0 The total (potential) energy of small fluctuations is thus given by H = dd x �σ (∇h)2 + ρg h2� , (VIII.51) 2 2 with t he second term a bsent for the soap film. The corresponding Langevin equation, ∂h(x, t) = −µρgh + µσ∇2h + η(x, t), (VIII.52) ∂t is linear, and can be solved by Fourier transforms. Starting with a flat interface, h(x, t = 0) = h(q, t = 0) = 0, the profile at time t is −iq·xh(x, t) = � (2dπdq )de � 0 t dτe−µ(ρg+σq2)(t−τ)η(q, t). (VIII.53) The average height of the surface, H¯= � ddx �h(x, t)�/Ld is zero, while its overall width is defined by w 2(t, L) ≡L1 d � dd x �h(x, t)2� = L1 d � (2dπdq )d � |h(q, t)|2� , (VIII.54) where L i s the linear size of the surface. Similar to Eq .(VIII.33) , we find that the width grows as � ddq D � −2γ(q)t � w 2(t, L) =(2π)d γ(q)1 − e . (VIII.55) There are a range of time scales in the problem, related to characteristic length scales a s in Eq. (VIII.31). The shortest time scale, tmin ∝ a2/(µσ), is set by an atomic size a. The longest time scale is set by either the capilla ry length (λc ≡ σ/ρg) or the system size (L). 152� � � � � � For simplicity we shall focus on the soap film where the effects of gravity are negligible and tmax ∝ L2/(µσ). We can now identify three different ranges of behavior in Eq.(VIII.55): (a) For t ≪ tmin, none of the mo des has relaxed since γ(q)t ≪ 1 for all q. Each mode grows diffusively, and ddq D 2Dt w 2(t, L) = 2γ(q)t = d . (VIII.56) (2π)d γ(q) a (b) For t ≫ tmax, all modes have relaxed to their equilibrium values since γ(q)t ≫ 1 for all q. The height fluctuations now saturate to a maximum value given by w 2(t, L)ddq D = . (2π)d µσq2 (VIII.57) The saturated value depends on the dimensionality of the surface, and in a general dimension d behaves as   a2−d for d > 2, (χ = 0) ln(L/a) for d = 2, (χ = 0+) , (VIII.58) L2−d for d < 2, (χ = 2−2 d) 2(t, L) ∝ D w µσ where we have defined a ro ughness exponent χ that governs the divergence of the width with sy stem size via limt→∞ w(t, L) ∝ Lχ . (The symbol 0+ is used to indicate a logarithmic divergence.) The exponent of χ = 1/ 2 in d = 1 indicates that the one dimensional interface fluctuates like a random walk. (c) For tmin ≪ t ≪ tmax only a fraction of the shorter


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