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 (d = 1)
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 (N = 1)
The interface is described by a scalar height field:
,
where
is the internal coordinate and
represents time.
Examples: domain walls and propagating fronts
Again we neglect overhangs or pinch-off:
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
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
The symbol
indicates the average over the thermal noise and the diffusion constant is fixed by the Einstein relation
. We set
The potential energy of surface tension (
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
and
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
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
.
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
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
.
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
. 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} }