L-2 - Revision history

Rosso: /* Edwards Wilkinson: an interface at equilibrium: */

2025-08-06T19:47:54Z

Edwards Wilkinson: an interface at equilibrium:

← Older revision		Revision as of 21:47, 6 August 2025
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	=~~Edwards Wilkinson: an interface at equilibrium~~: =		=Interfaces: thermal shaking =

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

	Two assumptions are done:		Two assumptions are done:
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	\partial_t h(r,t)= - \mu \frac{\delta E_{pot}}{\delta h(r,t)} + \eta(r,t)		\partial_t h(r,t)= - \mu \frac{\delta E_{pot}}{\delta h(r,t)} + \eta(r,t)
	</math></center>		</math></center>
	The first term <math> - \delta E_{pot}/\delta h(r,t) </math> is the elastic force trying to smooth the interface, the mobility <math> \mu </math> is ~~inversily proportional to~~ the viscosity. The second term is the Langevin ~~Gaussian~~ noise defined by ~~the correlations~~		The first term <math> - \delta E_{pot}/\delta h(r,t) </math> is the elastic force trying to smooth the interface, the mobility <math> \mu </math> is the inverse of the viscosity. The second term is the Langevin noise. It is Guassian and defined by
	<center> <math>		<center> <math>
	\langle \eta(r,t) \rangle =0, \; \langle \eta(r',t')\eta(r,t) \rangle = 2 d D \delta^d(r-r') \delta(t-t')		\langle \eta(r,t) \rangle =0, \; \langle \eta(r',t')\eta(r,t) \rangle = 2 d D \delta^d(r-r') \delta(t-t')
	</math></center>		</math></center>
	The symbol <math> \langle \ldots \rangle</math> indicates the average over the thermal noise.		The symbol <math> \langle \ldots \rangle</math> indicates the average over the thermal noise and the diffusion constant is fixed by the Einstein relation <math>
	~~The~~ diffusion constant is fixed by the ~~Eistein~~ relation ~~(fluctuation-dissipation theorem):~~
	~~<center>~~ <math>
	D= \mu K_B T		D= \mu K_B T
	</math>~~</center>~~		</math>. We set <math> \mu= K_B=1</math>
	We set <math> \mu= K_B=1</math>

	The potential energy of surface tension can be expanded at the lowest order in the gradient:		The potential energy of surface tension (<math>\nu </math> is the stiffness) can be expanded at the lowest order in the gradient:
	<center> <math>		<center> <math>
	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		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
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	\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)		\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)
	</math></center>		</math></center>
	Assume that the interface is ~~initialy~~ flat, namely <math> \hat h_q(0) =0 </math>.		Assume that the interface is initially flat, namely <math> \hat h_q(0) =0 </math>.
	* Compute the width <math> \langle h(x,t)^2\rangle = \sum_q \langle h_q(t)h_{-q}(t) \rangle </math>. Comment about the roughness and the short times growth.		* Compute the width <math> \langle h(x,t)^2\rangle = \sum_q \langle h_q(t)h_{-q}(t) \rangle </math>. Comment about the roughness and the short times growth.

Rosso: /* Derivation */

2025-01-26T14:05:59Z

Derivation

← Older revision		Revision as of 16:05, 26 January 2025
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	D= \mu K_B T		D= \mu K_B T
	</math></center>		</math></center>
	We set <math> \mu K_B=1</math>		We set <math> \mu= K_B=1</math>

	The potential energy of surface tension can be expanded at the lowest order in the gradient:		The potential energy of surface tension can be expanded at the lowest order in the gradient:

Rosso: /* The d=1 case */

2024-02-04T15:18:46Z

The d=1 case

← Older revision		Revision as of 17:18, 4 February 2024
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	\partial_t h(r,t)= \nu \partial_r^2 h(r,t)+ \frac{\lambda}{2} (\partial_r h)^2 + \eta(r,t)		\partial_t h(r,t)= \nu \partial_r^2 h(r,t)+ \frac{\lambda}{2} (\partial_r h)^2 + \eta(r,t)
	</math></center>		</math></center>
	The corresponding Fokker Planck equation for the probability <math>{\cal} P[h,t] </math> can be written as		The corresponding Fokker Planck equation for the probability <math>{\cal P}[h,t] </math> can be written as
	<center> <math>		<center> <math>
	\partial_t {\cal P}= - \int d r \frac{\delta}{\delta h_r} \left\{\left[\nu \partial_r^2 h +\frac{\lambda}{2} (\partial_r h)^2 \right] {\cal P} \right\} +T \int d r \frac{\delta^2 {\cal P}}{\delta h_r^2}		\partial_t {\cal P}= - \int d r \frac{\delta}{\delta h_r} \left\{\left[\nu \partial_r^2 h +\frac{\lambda}{2} (\partial_r h)^2 \right] {\cal P} \right\} +T \int d r \frac{\delta^2 {\cal P}}{\delta h_r^2}

Rosso: /* Exercise L2-A: Solve Edwards-Wilkinson */

2024-02-04T13:18:21Z

Exercise L2-A: Solve Edwards-Wilkinson

← Older revision		Revision as of 15:18, 4 February 2024
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	* Show that the EW equation writes		* Show that the EW equation writes
	<center> <math>		<center> <math>
	\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{T}{L} \delta_{q_1,-q_2}\delta(t-t')		\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')
	</math></center>		</math></center>

Rosso: /* Derivation */

2024-02-04T13:17:53Z

Derivation

← Older revision		Revision as of 15:17, 4 February 2024
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	The first term <math> - \delta E_{pot}/\delta h(r,t) </math> is the elastic force trying to smooth the interface, the mobility <math> \mu </math> is inversily proportional to the viscosity. The second term is the Langevin Gaussian noise defined by the correlations		The first term <math> - \delta E_{pot}/\delta h(r,t) </math> is the elastic force trying to smooth the interface, the mobility <math> \mu </math> is inversily proportional to the viscosity. The second term is the Langevin Gaussian noise defined by the correlations
	<center> <math>		<center> <math>
	\langle \eta(r,t) \rangle =0, \; \langle \eta(r',t')\eta(r,t) \rangle = 2 D \delta^d(r-r') \delta(t-t')		\langle \eta(r,t) \rangle =0, \; \langle \eta(r',t')\eta(r,t) \rangle = 2 d D \delta^d(r-r') \delta(t-t')
	</math></center>		</math></center>
	The symbol <math> \langle \ldots \rangle</math> indicates the average over the thermal noise.		The symbol <math> \langle \ldots \rangle</math> indicates the average over the thermal noise.

Rosso: /* Exercise L2-A: Solve Edwards-Wilkinson */

2024-02-04T13:09:58Z

Exercise L2-A: Solve Edwards-Wilkinson

← Older revision		Revision as of 15:09, 4 February 2024
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	* Show that the EW equation writes		* Show that the EW equation writes
	<center> <math>		<center> <math>
	\partial_t \hat h_q(t)= -\~~mu \sigma~~ q^2 \hat h_q(t)+ \eta_q(t), \quad \text{with} \; \langle \eta_{q_1}(t') \eta_{q_2}(t)\rangle =\frac{T}{L} \delta_{q_1,-q_2}\delta(t-t')		\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{T}{L} \delta_{q_1,-q_2}\delta(t-t')
	</math></center>		</math></center>

	The solution of this first order linear equation writes		The solution of this first order linear equation writes
	<center> <math>		<center> <math>
	\hat h_q(t)= \hat h_q(0) +\int_0^t d s e^{- \nu q^2 s} \eta_q(s)		\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)
	</math></center>		</math></center>
	Assume that the interface is initialy flat, namely <math> \hat h_q(0) =0 </math>.		Assume that the interface is initialy flat, namely <math> \hat h_q(0) =0 </math>.

Rosso: /* Exercise L2-A: Solve Edwards-Wilkinson */

2024-02-04T08:10:36Z

Exercise L2-A: Solve Edwards-Wilkinson

← Older revision		Revision as of 10:10, 4 February 2024
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	* Show that the EW equation writes		* Show that the EW equation writes
	<center> <math>		<center> <math>
	\partial_t \hat h_q(t)= -\mu \sigma q^2 \hat h_q(t)+ \eta_q(t), \quad \text{with} \; \langle \eta_{q_1}(t') \eta_{q_2}(t)\rangle =\frac{T}{L} \delta_{q_1,-q2}\delta(t-t')		\partial_t \hat h_q(t)= -\mu \sigma q^2 \hat h_q(t)+ \eta_q(t), \quad \text{with} \; \langle \eta_{q_1}(t') \eta_{q_2}(t)\rangle =\frac{T}{L} \delta_{q_1,-q_2}\delta(t-t')
	</math></center>		</math></center>

Rosso: /* An important symmetry */

2024-02-03T14:17:05Z

An important symmetry

← Older revision		Revision as of 16:17, 3 February 2024
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	-->		-->
	===An important symmetry===		===An important symmetry===
	Let us remark that if <math> h(r,t) </math> is a solution of KPZ,also <math> \tilde h(r,t)= h(r + \lambda v_0 t,t) +v_0 r +(v_0^2 \lambda /2) t </math> is a solution of KPZ, provided the change of variables <math>\tilde r =r+\lambda v_0 t</math>		Let us remark that if <math> h(r,t) </math> is a solution of KPZ,also <math> \tilde h(r,t)= h(r + \lambda v_0 t,t) +v_0 r +(v_0^2 \lambda /2) t </math> is a solution of KPZ, provided the change of variables <math>\tilde r =r+\lambda v_0 t</math> .
	You can check it, and you will obtain an equation with the statistically equivalent noise <math> \eta(\tilde{r} - \lambda v_0 t,t)</math>. The symmetry relies on two properties:		You can check it, and you will obtain an equation with the statistically equivalent noise <math> \eta(\tilde{r} - \lambda v_0 t,t)</math>. The symmetry relies on two properties:
	* The noise <math> \eta(r,t) </math> is delta correlated in time		* The noise <math> \eta(r,t) </math> is delta correlated in time

Rosso: /* An important symmetry */

2024-02-03T14:16:38Z

An important symmetry

← Older revision		Revision as of 16:16, 3 February 2024
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	-->		-->
	===An important symmetry===		===An important symmetry===
	Let us remark that if <math> h(r,t) </math> is a solution of KPZ,also <math> h(r + \lambda v_0 t,t) +v_0 r +(v_0^2 \lambda /2) t </math> is a solution of KPZ, provided .		Let us remark that if <math> h(r,t) </math> is a solution of KPZ,also <math> \tilde h(r,t)= h(r + \lambda v_0 t,t) +v_0 r +(v_0^2 \lambda /2) t </math> is a solution of KPZ, provided the change of variables <math>\tilde r =r+\lambda v_0 t</math>
			You can check it, and you will obtain an equation with the statistically equivalent noise <math> \eta(\tilde{r} - \lambda v_0 t,t)</math>. The symmetry relies on two properties:
	You can check it, and you will obtain an equation with the statistically equivalent noise <math> ~~\tilde~~ \eta(~~r,t)=~~\~~eta(~~r - \lambda v_0 t,t)</math>. The symmetry relies on two properties:
	* The noise <math> \eta(r,t) </math> is delta correlated in time		* The noise <math> \eta(r,t) </math> is delta correlated in time
	* Only sticked together the two terms <math> \partial_t h(r,t)</math> and <math> \frac{\lambda}{2} (\partial_r h(r,t))^2</math> enjoy the symmetry. Hence, under the rescaling		* Only sticked together the two terms <math> \partial_t h(r,t)</math> and <math> \frac{\lambda}{2} (\partial_r h(r,t))^2</math> enjoy the symmetry. Hence, under the rescaling

Rosso: /* An important symmetry */

2024-02-03T13:40:55Z

An important symmetry

← Older revision		Revision as of 15:40, 3 February 2024
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	-->		-->
	===An important symmetry===		===An important symmetry===
	Let us remark that if <math> h(r,t) </math> is a solution of KPZ,also <math> h(r - \lambda v_0 t,t) +v_0 r -(v_0^2 \lambda /2) t </math> is a solution of KPZ.		Let us remark that if <math> h(r,t) </math> is a solution of KPZ,also <math> h(r + \lambda v_0 t,t) +v_0 r +(v_0^2 \lambda /2) t </math> is a solution of KPZ, provided .

	You can check it, and you will obtain an equation with the statistically equivalent noise <math> \tilde \eta(r,t)=\eta(r - \lambda v_0 t,t)</math>. The symmetry relies on two properties:		You can check it, and you will obtain an equation with the statistically equivalent noise <math> \tilde \eta(r,t)=\eta(r - \lambda v_0 t,t)</math>. The symmetry relies on two properties:
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	Check that the important symmetry of the KPZ equation is nothing but the Galilean symmetry of the Burgers equation.		Check that the important symmetry of the KPZ equation is nothing but the Galilean symmetry of the Burgers equation.
	-->		-->

	===The d=1 case===		===The d=1 case===
	In the one dimensional case the KPZ equation writes		In the one dimensional case the KPZ equation writes