By Mehran Kardar

Whereas many scientists are conversant in fractals, fewer are accustomed to scale-invariance and universality which underly the ubiquity in their shapes. those houses may perhaps emerge from the collective behaviour of easy basic components, and are studied utilizing statistical box theories. preliminary chapters attach the particulate standpoint built within the spouse quantity, to the coarse grained statistical fields studied the following. in line with lectures taught by means of Professor Kardar at MIT, this textbook demonstrates how such theories are formulated and studied. Perturbation conception, unique options, renormalization teams, and different instruments are hired to illustrate the emergence of scale invariance and universality, and the non-equilibrium dynamics of interfaces and directed paths in random media are mentioned. perfect for complex graduate classes in statistical physics, it includes an built-in set of difficulties, with options to chose difficulties on the finish of the ebook and a whole set to be had to academics at www.cambridge.org/9780521873413

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14) 2 The expansion coefficients can be regarded as phenomenological parameters which can be determined by comparing to experiments. In particular, matching the phase diagrams in Figs. 3 requires that t should be a monotonic function of temperature which vanishes at Tc , necessitating a0 = 0 and a1 = a > 0. Stability of the ferromagnetic phase in the vicinity of the critical point requires that u and K should be positive. g. a or u, to zero, and maintain the phase diagram and stability by appropriate choice of the higher order terms.

2) Here • indicates the thermal average of •, which can be used in place of the time average in most cases due to ergodicity. 2) indicates that a uniform density only leads to forward scattering (q = 0), while the long-wavelength fluctuations can be studied by working at small angles or with small k. If scattering is caused by the magnetization 35 36 Fluctuations density, we can use the Landau–Ginzburg Hamiltonian to compute its intensity. 3) As discussed earlier, the most probable configuration is uniform, with m x = √ mˆe1 , where eˆ 1 is a unit vector (m is zero for t > 0, and equal to −t/4u for t < 0).

E. f h = f h , where h= h. (a) Show that m = dd xm x /V = −h f h /h. (b) Relate the susceptibility tensor = m / h , to f h , m, and h. (c) Show that the transverse and longitudinal susceptibilities are given by t = m/h and = −f h ; where m is the magnitude of m. (d) Conclude that t diverges as h → 0, whenever there is a spontaneous magnetization. Is there any similar a priori reason for to diverge? 4. Superfluid He4 –He3 mixtures: The superfluid He4 order parameter is a complex number x . In the presence of a concentration c x of He3 impurities, the system has the following Landau–Ginzburg energy c = dd x K 2 2 + t 2 2 +u 4 +v 6 + cx 2 − cx 2 2 2 with positive K, u and v.