LBan-II

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: - $D$: spatial dimension of the embedding medium – $d$: internal dimension of the manifold – $N$: dimension of the displacement (or height) field

These satisfy the relation:

$D=d+N$

We focus on two important cases:

Directed Polymers ($d=1$)

The configuration is described by a vector function: ${\vec {x}}(t)$, where $t$ is the internal coordinate. The polymer lives in $D=1+N$ 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 ($N=1$)

The interface is described by a scalar height field: $h({\vec {r}},t)$, where ${\vec {r}}\in \mathbb {R} ^{d}$ is the internal coordinate and $t$ represents time. Again, Introduction: Interfaces and Directed Polymers

Interfaces: thermal shaking

Consider domain wall $h(r,t)$ fluctuating at equilibrium at the temperature $T$. Here $t$ is time, $r$ defines the d-dimensional coordinate of the interface and $h$ is the scalar height field. Hence, the domain wall separating two phases in a film has $d=1,r\in {\cal {R}}$, in a solid instead $d=2,r\in {\cal {{R}^{2}}}$.

Two assumptions are done:

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

Derivation

The Langevin equation of motion is

$\partial _{t}h(r,t)=-\mu {\frac {\delta E_{pot}}{\delta h(r,t)}}+\eta (r,t)$

The first term $-\delta E_{pot}/\delta h(r,t)$ is the elastic force trying to smooth the interface, the mobility $\mu$ 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 $D=\mu K_{B}T$. 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 $\int _{0}^{L}dre^{iqr}=L\delta _{q,0}$.

• 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.