LBan-II: Difference between revisions

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Introduction: Interfaces and Directed Polymers
The physical properties of many materials are governed by manifolds embedded in them. Examples include: dislocations in crystals, domain walls in ferromagnets or vortex lines in superconductors. We fix the following notation:
- <math>D</math>: spatial dimension of the embedding medium
– <math>d</math>: internal dimension of the manifold
– <math>N</math>: dimension of the displacement (or height) field
These satisfy the relation:
<center><math>D = d + N</math></center>
We focus on two important cases:
=== Directed Polymers (<math>d = 1</math>)===
The configuration is described by a vector function:
<math>\vec{x}(t)</math>,
where <math>t</math> is the internal coordinate. The polymer lives in <math>D = 1 + N</math> dimensions.
Examples: vortex lines, DNA strands,  fronts.
Although polymers may form loops, we restrict to directed polymers, i.e., configurations without overhangs or backward turns.
===Interfaces (<math>N = 1</math>)===
The interface is described by a scalar height field:
<math>h(\vec{r}, t)</math>,
where <math>\vec{r} \in \mathbb{R}^d</math> is the internal coordinate and <math>t</math> represents time. Again, Introduction: Interfaces and Directed Polymers
=Interfaces: thermal shaking  =
=Interfaces: thermal shaking  =



Revision as of 15:12, 7 August 2025

Introduction: Interfaces and Directed Polymers

The physical properties of many materials are governed by manifolds embedded in them. Examples include: dislocations in crystals, domain walls in ferromagnets or vortex lines in superconductors. We fix the following notation: - : spatial dimension of the embedding medium – : internal dimension of the manifold – : dimension of the displacement (or height) field

These satisfy the relation:

We focus on two important cases:

Directed Polymers ()

The configuration is described by a vector function: , where is the internal coordinate. The polymer lives in dimensions. Examples: vortex lines, DNA strands, fronts. Although polymers may form loops, we restrict to directed polymers, i.e., configurations without overhangs or backward turns.

Interfaces ()

The interface is described by a scalar height field: , where is the internal coordinate and represents time. Again, Introduction: Interfaces and Directed Polymers



Interfaces: thermal shaking

Consider domain wall fluctuating at equilibrium at the temperature . Here is time, defines the d-dimensional coordinate of the interface and is the scalar height field. Hence, the domain wall separating two phases in a film has , in a solid instead .

Two assumptions are done:

  • Overhangs, pinch-off are neglected, so that is a scalar univalued function.
  • The dynamics is overdamped, so that we can neglect the inertial term.

Derivation

The Langevin equation of motion is

The first term is the elastic force trying to smooth the interface, the mobility is the inverse of the viscosity. The second term is the Langevin noise. It is Guassian and defined by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \eta(r,t) \rangle =0, \; \langle \eta(r',t')\eta(r,t) \rangle = 2 d D \delta^d(r-r') \delta(t-t') }

The symbol Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \ldots \rangle} indicates the average over the thermal noise and the diffusion constant is fixed by the Einstein relation . We set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu= K_B=1}

The potential energy of surface tension (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu } is the stiffness) can be expanded at the lowest order in the gradient:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{pot} = \nu \int d^d r\sqrt{1 +(\nabla h)^2} \sim \text{const.} + \frac{\nu}{2} \int d^d r (\nabla h)^2 }

Hence, we have the Edwards Wilkinson equation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial_t h(r,t)= \nu \nabla^2 h(r,t) + \eta(r,t) }

Scaling Invariance

The equation enjoys of a continuous symmetry because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(r,t) } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(r,t)+c } cannot be distinguished. This is a condition of scale invariance:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(b r, b^z t) \overset{in law}{\sim} b^{\alpha} h(r,t) }

Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z, \alpha } are the dynamic and the roughness exponent respectively. From dimensional analysis

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^{\alpha-z} \partial_t h(r,t)= b^{\alpha-2} \nabla^2 h(r,t) +b^{-d/2-z/2} \eta(r,t) }

From which you get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=2 } in any dimension and a rough interface below Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d=2 } with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha =(2-d)/2 } .

Explicit Solution

For simplicity, consider a 1-dimensional line of size L with periodic boundary conditions. It is useful to introduce the Fourier modes:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat h_q(t)= \frac{1}{L} \int_0^L e^{iqr} h(r,t), \quad h(r,t)= \sum_q e^{-iqr} \hat h_q(t) }

Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q=2 \pi n/L, n=\ldots ,-1,0,1,\ldots} and recall .

  • Show that the EW equation writes
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial_t \hat h_q(t)= -\nu q^2 \hat h_q(t) + \eta_q(t), \quad \text{with} \; \langle \eta_{q_1}(t') \eta_{q_2}(t)\rangle =\frac{2 T}{L} \delta_{q_1,-q_2}\delta(t-t') }

The solution of this first order linear equation writes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat h_q(t)= \hat h_q(0) e^{-\nu q^2 t} +\int_0^t d s e^{- \nu q^2 (t-s)} \eta_q(s) }

Assume that the interface is initially flat, namely Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat h_q(0) =0 } .

  • Compute the width Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle h(x,t)^2\rangle = \sum_q \langle h_q(t)h_{-q}(t) \rangle } . Comment about the roughness and the short times growth.