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.

Examples: domain walls and propagating fronts

Again we neglect overhangs or pinch-off: $h({\vec {r}},t)$ is single-valued

Note that using our notation the 1D front is both an interface and a directed polymer

Thermal Interfaces

• The dynamics is overdamped, so that we can neglect the inertial term.

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

$\langle \eta (r,t)\rangle =0,\;\langle \eta (r',t')\eta (r,t)\rangle =2dD\delta ^{d}(r-r')\delta (t-t')$

The symbol $\langle \ldots \rangle$ indicates the average over the thermal noise and the diffusion constant is fixed by the Einstein relation $D=\mu K_{B}T$. We set $\mu =K_{B}=1$

The potential energy of surface tension ($\nu$ is the stiffness) can be expanded at the lowest order in the gradient:

$E_{pot}=\nu \int d^{d}r{\sqrt {1+(\nabla h)^{2}}}\sim {\text{const.}}+{\frac {\nu }{2}}\int d^{d}r(\nabla h)^{2}$

Hence, we have the Edwards Wilkinson equation:

$\partial _{t}h(r,t)=\nu \nabla ^{2}h(r,t)+\eta (r,t)$

Scaling Invariance

The equation enjoys of a continuous symmetry because $h(r,t)$ and $h(r,t)+c$ 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

$b^{\alpha -z}\partial _{t}h(r,t)=b^{\alpha -2}\nabla ^{2}h(r,t)+b^{-d/2-z/2}\eta (r,t)$

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 $\alpha =(2-d)/2$.

Width of the interface

Consider a 1-dimensional line of size L with periodic boundary conditions. We consider the width square of the interface

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 w_2(t) = \left[\int_0^L \frac{d r}{L} \left(h(r,t) - \int_0^L \frac{dr}{L} h(r,t)\right)\right]^2 }

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 $q=2\pi n/L,n=\ldots ,-1,0,1,\ldots$ and recall 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 \int_0^L d r e^{iqr}= L \delta_{q,0} } . using de Parseval theorem for the Fourier series

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 w_2(t) = \sum_{q\ne 0} |\hat h_q(t)|^2 =\sum_{q\ne 0} \left(\hat h_q(t) \hat h_{-q}(t)\right) ^2 }

In the last step we used that ${\hat {h}}_{q}^{*}(t)={\hat {h}}_{-q}(t)$.

Solution in the Fourier space

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 ${\hat {h}}_{q}(0)=0$. Show that

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 \hat h_q(t) \hat h_{-q}(t) \rangle =\begin{cases} \dfrac{T(1 - e^{-2\nu q^{2}t})}{L \nu q^{2}}, & q \neq 0, \\[1.2em] \frac{2 T}{L} t, & q = 0. \end{cases} }

• The mean width square grows at short times and saturates at long times:

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 w_2(t)\rangle = \dfrac{T}{L \nu }\sum_{q\ne 0} \dfrac{1 - e^{-2\nu q^{2}t}}{q^{2}} =\begin{cases} \sqrt{t}, & t\ll L^2, \\[1.2em] \frac{T}{ \nu} \frac{L}{12} , & t\gg L^2. \end{cases} }