L2 ICFP - Revision history

Rosso at 14:41, 1 March 2026

2026-03-01T14:41:22Z

Rosso: /* Solution in Fourier space */

2026-01-25T15:58:51Z

Solution in Fourier space

← Older revision		Revision as of 17:58, 25 January 2026
Line 107:		Line 107:

	We now focus on a one-dimensional interface (<math>d=1</math>) of size <math>L</math>		We now focus on a one-dimensional interface (<math>d=1</math>) of size <math>L</math>
	with periodic boundary conditions.		with periodic boundary conditions. We use the Fourier decomposition

	We use the Fourier decomposition
	<center><math>		<center><math>
	\hat h_q(t)		\hat h_q(t)
Line 131:		Line 129:
	<center><math> \langle \eta_{q_1}(t')\eta_{q_2}(t)\rangle = \frac{2T}{L}\,\delta_{q_1,-q_2}\,\delta(t-t'). </math></center>		<center><math> \langle \eta_{q_1}(t')\eta_{q_2}(t)\rangle = \frac{2T}{L}\,\delta_{q_1,-q_2}\,\delta(t-t'). </math></center>

	~~The Fourier components of the noise satisfy~~
	~~<center><math>~~
	~~\langle \eta_{q_1}(t')\eta_{q_2}(t)\rangle~~
	~~= \frac{2T}{L}\,\delta_{q_1,-q_2}\,\delta(t-t').~~
	~~</math></center>~~

	With these definitions, the Edwards–Wilkinson equation becomes diagonal in		With these definitions, the Edwards–Wilkinson equation becomes diagonal in
Line 173:		Line 166:

	The modes with <math>q\neq 0</math> describe internal fluctuations of the		The modes with <math>q\neq 0</math> describe internal fluctuations of the
	interface.		interface. The relaxation time of a mode of wavevector <math>q</math> scales as
	~~Since <math>q</math> has the dimension of an inverse length, the~~ relaxation time
	of a mode of wavevector <math>q</math> scales as
	<center><math>		<center><math>
	\tau_q \sim \frac{1}{\nu q^2}.		\tau_q \sim \frac{1}{\nu q^2}.
	</math></center>		</math></center>

	~~This~~ suggests the existence of a growing dynamical length scale		Since <math>q</math> has the dimension of an inverse length, this relaxation time suggests the existence of a growing dynamical length scale
	<center><math>		<center><math>
	\ell(t) \sim t^{1/z},		\ell(t) \sim t^{1/z},
Line 188:		Line 179:
	(i.e. <math>q \gg 1/\ell(t)</math>) have already equilibrated, while at longer		(i.e. <math>q \gg 1/\ell(t)</math>) have already equilibrated, while at longer
	wavelengths the interface still retains memory of the initial flat condition.		wavelengths the interface still retains memory of the initial flat condition.
			Form dimensional analysis, the equilibrium decay <math>\sim 1/(L q^2)</math> is consistent with the
	~~Note that~~ the ~~dimension of the observable~~
	~~<math>\langle \|\hat h_q(t)\|^2 \rangle</math>~~
	~~is that of <math>h^2 \times \text{length}</math>.~~
	~~The~~ equilibrium decay <math>\sim 1/q^2</math> is ~~therefore~~ consistent with the
	roughness exponent <math>\alpha = 1/2</math>, as expected for the		roughness exponent <math>\alpha = 1/2</math>, as expected for the
	Edwards–Wilkinson universality class in one dimension.		Edwards–Wilkinson universality class in one dimension.

Rosso: /* Solution in Fourier space */

2026-01-25T15:36:53Z

Solution in Fourier space

← Older revision		Revision as of 17:36, 25 January 2026
Line 122:		Line 122:
	n=\ldots,-1,0,1,\ldots		n=\ldots,-1,0,1,\ldots
	</math></center>		</math></center>

			For these discrete wavevectors, the Fourier modes satisfy the orthogonality relation

			<center><math> \int_0^L dx\, e^{i(q_1+q_2)x} = L\,\delta_{q_1,-q_2}. </math></center>

			Assuming a spatially and temporally white noise, <math> \langle \eta(x,t)\eta(x',t')\rangle = 2T\,\delta(x-x')\,\delta(t-t'), </math> one finds that the Fourier cmponents of the noise satisfy

			<center><math> \langle \eta_{q_1}(t')\eta_{q_2}(t)\rangle = \frac{2T}{L}\,\delta_{q_1,-q_2}\,\delta(t-t'). </math></center>

	The Fourier components of the noise satisfy		The Fourier components of the noise satisfy

Rosso: /* Thermal interfaces: Langevin dynamics */

2026-01-25T15:23:06Z

Thermal interfaces: Langevin dynamics

← Older revision		Revision as of 17:23, 25 January 2026
Line 81:		Line 81:

	The EW equation is invariant under global height shifts		The EW equation is invariant under global height shifts
	<math>h(\vec r,t) \to h(\vec r,t) + c</math>.		<math>h(\vec r,t) \to h(\vec r,t) + c</math>. This symmetry forbids any local term depending on the absolute height and ensures the absence of a characteristic length scale, leading to scale invariance of the form
	This symmetry ~~leads~~ to scale invariance of the form
	<center><math>		<center><math>
	h(b\vec r, b^z t) \overset{\text{in law}}{\sim} b^\alpha h(\vec r,t),		h(b\vec r, b^z t) \overset{\text{in law}}{\sim} b^\alpha h(\vec r,t),
Line 186:		Line 185:
	is that of <math>h^2 \times \text{length}</math>.		is that of <math>h^2 \times \text{length}</math>.
	The equilibrium decay <math>\sim 1/q^2</math> is therefore consistent with the		The equilibrium decay <math>\sim 1/q^2</math> is therefore consistent with the
	roughness exponent <math>\~~zeta~~ = 1/2</math>, as expected for the		roughness exponent <math>\alpha = 1/2</math>, as expected for the
	Edwards–Wilkinson universality class in one dimension.		Edwards–Wilkinson universality class in one dimension.

Rosso: /* Width of the interface */

2026-01-23T20:24:45Z

Width of the interface

← Older revision		Revision as of 22:24, 23 January 2026
Line 221:		Line 221:
	</math></center>		</math></center>

	~~---~~

	=== Long-time behavior ===		=== Long-time behavior ===

Rosso: /* Symmetries and scaling invariance */

2026-01-23T20:24:32Z

Symmetries and scaling invariance

← Older revision		Revision as of 22:24, 23 January 2026
Line 102:		Line 102:

	Thus the interface is rough for <math>d<2</math> and marginal at <math>d=2</math>.		Thus the interface is rough for <math>d<2</math> and marginal at <math>d=2</math>.

	~~---~~

	== Solution in Fourier space ==		== Solution in Fourier space ==

Rosso: /* Short-time behavior */

2026-01-23T20:24:15Z

Short-time behavior

← Older revision		Revision as of 22:24, 23 January 2026
Line 256:		Line 256:
	\qquad t \ll L^2.		\qquad t \ll L^2.
	</math></center>		</math></center>

	~~---~~

	~~At short times the interface roughens algebraically, while at long times the~~
	~~width saturates due to the finite system size.~~

Rosso: /* Long-time behavior */

2026-01-23T20:24:03Z

Long-time behavior

← Older revision		Revision as of 22:24, 23 January 2026
Line 237:		Line 237:

	Thus the width saturates at a value proportional to the system size.		Thus the width saturates at a value proportional to the system size.

	~~---~~

	=== Short-time behavior ===		=== Short-time behavior ===

Rosso: /* Growth and saturation of the width */

2026-01-23T19:11:39Z

Growth and saturation of the width

@@ Line 191: / Line 191: @@
 Edwards–Wilkinson universality class in one dimension.
-== Growth and saturation of the width ==
+== Width of the interface ==
-The mean squared width evolves as
+The squared width of the interface is defined as
 <center><math>
 \langle w_2(t)\rangle
@@ Line 200: / Line 214: @@
 </math></center>
-In the continuum limit this gives
+For periodic boundary conditions, with <math>q = 2\pi n/L</math>, this can be
 <center><math>
 \langle w_2(t)\rangle
+\sim T\sqrt{\frac{2t}{\pi\nu}},
-\begin{cases}
+\qquad t \ll L^2.
 </math></center>
-At short times the interface roughens algebraically, while at long times the width
+---
 saturates due to the finite system size.

Rosso: /* Width of the interface */

2026-01-23T18:58:25Z

Width of the interface

← Older revision		Revision as of 20:58, 23 January 2026
Line 102:		Line 102:

	Thus the interface is rough for <math>d<2</math> and marginal at <math>d=2</math>.		Thus the interface is rough for <math>d<2</math> and marginal at <math>d=2</math>.

	~~---~~

	~~== Width of the interface ==~~

	~~We now focus on a one-dimensional interface (<math>d=1</math>) of size <math>L</math>~~
	~~with periodic boundary conditions.~~

	~~The squared width is defined as~~
	~~<center><math>~~
	~~w_2(t)~~
	~~= \int_0^L \frac{dr}{L}~~
	~~\left(~~
	~~h(r,t) - \int_0^L \frac{dr}{L} h(r,t)~~
	~~\right)^2.~~
	~~</math></center>~~

	~~Introduce Fourier modes~~
	~~<center><math>~~
	~~\hat h_q(t)~~
	~~= \frac{1}{L}\int_0^L dr\, e^{iqr} h(r,t),~~
	~~\qquad~~
	~~h(r,t)=\sum_q e^{-iqr}\hat h_q(t),~~
	~~</math></center>~~
	~~with <math>q = 2\pi n/L</math>.~~

	~~Using Parseval’s theorem one finds~~
	~~<center><math>~~
	~~w_2(t) = \sum_{q\neq 0} \|\hat h_q(t)\|^2.~~
	~~</math></center>~~

	---		---