By Johannes Berg, Gerold Busch
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Extra info for Advanced Statistical Physics: Lecture Notes (Wintersemester 2011/12)
3: Growth of w with time for the BD-model. We can distinguish two regimes: For t we have a power law behavior, for t tx we have saturation. 2 An even simpler model: Random deposition Modify the BD rule so that a particle moves down until it touches a particle (or surface) below it. Then, there are no correlations of hi (t) accross the columns. hi grows by 1 when a particle is dropped above column i. The probability P (h, N ) that a given column has height h after N particles have been dropped follows a binomial distribution P (h, N ) = N h ph (1 − p)N −h with p = 1/L, the probability that a given particle is released above a certain column.
G. 1 ∆t dx(x − z)3 p(x, t + ∆t|z, t) ≤ |x−z|< < 1 ∆t ∆t ∆t→0 → dx|x − z|(x − z)2 p(x, t + ∆t|z, t) |x−z|< dx(x − z)2 p(x, t + ∆t|z, t) |x−z|< B(z, t) = O( ) Hence, knowing drift and diffusion are sufficient to characterize the system. To derive the equation of motion of p(x, t) consider the expectation value ∞ dx p(x, t|y, t )f (x) −∞ 53 of some function f (x) which is twice continuously differentiable. ∂t 1 ∆t→0 ∆t 1 = lim ∆t→0 ∆t dxf (x) p(x, t + ∆t|y, t ) − p(x, t|y, t ) dx f (x) p(x, t|y, t ) = lim − dx dzf (x)p(x, t + ∆t|z, t)p(z, t|y, t ) dzf (z)p(z, t|y, t ) Let us expand f (x) in a Taylor series around z f (x) = f (z) + 1 d2 f df (x − z) + (x − z)2 + |x − z|2 R(x, z) dz 2 dz 2 where we know that R(x, z) → 0 as |x − z| → 0.
26) b with Tab = 44 ∂ka ∂kb k=k∗ (Taylor expansion). 27) c b The eigenvalues are used to classify the parameters: • yi > 0, ui relevant • yi < 0, ui irrelevant • yi = 0, ui marginal Universality is the prediction of RG that the thermodynamical properties of a system near a fixed point depend only on a small number of features and are not sensitive to the microscopic properties of this system. All systems that flow onto the same fixed points are called to be in the same universality class. We will now define the reduced free energy (per spin): f = − N1 ln Z.