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: - : spatial dimension of the embedding medium – 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} : internal dimension of the manifold – 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 N} : dimension of the displacement (or height) field
These satisfy the relation:
We focus on two important cases:
Directed Polymers (d = 1)
The configuration is described by a vector function: 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 \vec{x}(t)} , where 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 t} 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 (N = 1)
The interface is described by a scalar height field: 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(\vec{r}, t)} , where 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 \vec{r} \in \mathbb{R}^d} is the internal coordinate 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 t} represents time.
Examples: domain walls and propagating fronts
Again we neglect overhangs or pinch-off: 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(\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
The first term 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 - \delta E_{pot}/\delta h(r,t) } is the elastic force trying to smooth the interface, the mobility 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 } is the inverse of the viscosity. The second term is the Langevin noise. It is Guassian and defined by
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 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= \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:
Hence, we have the Edwards Wilkinson equation:
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:
Here are the dynamic and the roughness exponent respectively. From dimensional analysis
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 } .
Width of the interface
Consider a 1-dimensional line of size L with periodic boundary conditions. We consider the width square of the interface
It is useful to introduce the Fourier modes:
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 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
In the last step we used 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 \hat h_q^*(t)= \hat h_{-q}(t) } .
Solution in the Fourier space
show that the EW equation writes
The solution of this first order linear equation writes
- 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 } . Show that
- The mean width square grows at short times and saturates at long times: