LBan-II - Revision history

Rosso: /* Entropy and scaling relation */

2025-09-16T05:56:32Z

Entropy and scaling relation

← Older revision		Revision as of 07:56, 16 September 2025
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	<center>		<center>
	<math>		<math>
	\text{Entropy}= \ln\binom{t}{\frac{t-x}{2}} \approx t \ln 2 -\frac{x^2}{t} +O(x^4)		\text{Entropy}= \ln\binom{t}{\frac{t-x}{2}} \approx t \ln 2 -\frac{x^2}{2 t} +O(x^4)
	</math></center>		</math></center>
	From which one could guess from dimensional analysis		From which one could guess from dimensional analysis

Rosso: /* Directed Polymers on a lattice */

2025-09-16T02:55:05Z

Directed Polymers on a lattice

← Older revision		Revision as of 04:55, 16 September 2025
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	<center>		<center>
	<math>		<math>
	E_{\min} = c_\infty t + b_\~~infty~~ t^{\theta}\chi, \quad x_{\min}(t/2)) \sim a_\~~infty~~ t^{\zeta} \tilde \chi		E_{\min} = c_\infty t + \kappa_1 t^{\theta}\chi, \quad x_{\min}(t/2)) \sim \kappa_2 t^{\zeta} \tilde \chi
	</math></center>		</math></center>

	<strong>Universal exponents: </strong> Both <math> \theta, \zeta </math> are Independent of the lattice, the disorder distribution, the elastic constants, or the boudanry conditions.		<strong>Universal exponents: </strong> Both <math> \theta, \zeta </math> are Independent of the lattice, the disorder distribution, the elastic constants, or the boudanry conditions.

	<strong>Non-universal constants: </strong> <math> c_\infty,b_\~~infty~~, a_\~~infty~~ </math> are of order 1 and depend on the lattice, the disorder distribution, the elastic constants... However <math> c_\infty </math> is independent on the boudanry conditions!		<strong>Non-universal constants: </strong> <math> c_\infty,\kappa_1, \kappa_2 </math> are of order 1 and depend on the lattice, the disorder distribution, the elastic constants... However <math> c_\infty </math> is independent on the boudanry conditions!

	<strong>Universal distributions: </strong> <math> \chi, \tilde \chi </math> are instead universal, but depends on the boundary condtions. Starting from 2000 a magic connection has been revealed between this model and the smallest eigenvalues of random matrices. In particular I discuss two different boundary conditions:		<strong>Universal distributions: </strong> <math> \chi, \tilde \chi </math> are instead universal, but depends on the boundary condtions. Starting from 2000 a magic connection has been revealed between this model and the smallest eigenvalues of random matrices. In particular I discuss two different boundary conditions:

Rosso: /* Directed polymers in random media */

2025-09-16T02:46:45Z

Directed polymers in random media

← Older revision		Revision as of 04:46, 16 September 2025
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	This relation is actually exact also for the continuum model.		This relation is actually exact also for the continuum model.

	=Directed polymers in ~~random media~~=		=Directed polymers in the continuum=
	We now reanalyze the previous problem in the presence of quenched disorder.		We now reanalyze the previous problem in the presence of quenched disorder.
	Instead of discussing the case of interfaces, we will focus on directed polymers.		Instead of discussing the case of interfaces, we will focus on directed polymers.

Rosso: /* Introduction: Interfaces and Directed Polymers */

2025-09-16T02:45:30Z

Introduction: Interfaces and Directed Polymers

← Older revision		Revision as of 04:45, 16 September 2025
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	'''Note''' that using our notation the 1D front is both an interface and a directed polymer		'''Note''' that using our notation the 1D front is both an interface and a directed polymer

			=Directed Polymers on a lattice=



			[[File:SketchDPRM.png\|thumb\|left\|Sketch of the discrete Directed Polymer model. At each time the polymer grows either one step left either one step right. A random energy <math> V(\tau,x)</math> is associated at each node and the total energy is simply <math> E[x(\tau)] =\sum_{\tau=0}^t V(\tau,x)</math>. ]]


			We introduce a lattice model for the directed polymer (see figure). In a companion notebook we provide the implementation of the powerful Dijkstra algorithm. Dijkstra allows to identify the minimal energy among the exponential number of configurations <math> x(\tau)</math>
			<center> <math>
			E_{\min} = \min_{x(\tau)} E[x(\tau)].
			</math></center>

			We are also interested in the ground state configuration <math> x_{\min}(\tau) </math>.
			For both quantities we expect scale invariance with two exponents <math> \theta, \zeta</math> for the energy and for the roughness
			<center>
			<math>
			E_{\min} = c_\infty t + b_\infty t^{\theta}\chi, \quad x_{\min}(t/2)) \sim a_\infty t^{\zeta} \tilde \chi
			</math></center>

			<strong>Universal exponents: </strong> Both <math> \theta, \zeta </math> are Independent of the lattice, the disorder distribution, the elastic constants, or the boudanry conditions.

			<strong>Non-universal constants: </strong> <math> c_\infty,b_\infty, a_\infty </math> are of order 1 and depend on the lattice, the disorder distribution, the elastic constants... However <math> c_\infty </math> is independent on the boudanry conditions!

			<strong>Universal distributions: </strong> <math> \chi, \tilde \chi </math> are instead universal, but depends on the boundary condtions. Starting from 2000 a magic connection has been revealed between this model and the smallest eigenvalues of random matrices. In particular I discuss two different boundary conditions:

			* <strong>Droplet</strong>: <math> x(\tau=0) = x(\tau=t) = 0 </math>. In this case, up to rescaling, <math> \chi</math> is distributed as the smallest eigenvalue of a GUE random matrix (Tracy Widom distribution <math>F_2(\chi) </math>)

			* <strong> Flat</strong>: <math> x(\tau=0) = 0 </math> while the other end <math> x(\tau=t) </math> is free. In this case, up to rescaling, <math> \chi</math> is distributed as the smallest eigenvalue of a GOE random matrix (Tracy Widom distribution <math>F_1(\chi) </math>)
			===Entropy and scaling relation===

			It is useful to compute the entropy
			<center>
			<math>
			\text{Entropy}= \ln\binom{t}{\frac{t-x}{2}} \approx t \ln 2 -\frac{x^2}{t} +O(x^4)
			</math></center>
			From which one could guess from dimensional analysis
			<center>
			<math>
			\theta=2 \zeta-1
			</math></center>
			This relation is actually exact also for the continuum model.

	=Directed polymers in random media=		=Directed polymers in random media=

Rosso: /* Thermal Interfaces */

2025-09-16T02:42:43Z

Thermal Interfaces

← Older revision		Revision as of 04:42, 16 September 2025
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	'''Note''' that using our notation the 1D front is both an interface and a directed polymer		'''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~~
	~~<center> <math>~~
	~~\partial_t h(r,t)= - \mu \frac{\delta E_{pot}}{\delta h(r,t)} + \eta(r,t)~~
	~~</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 the inverse of the viscosity. The second term is the Langevin noise. It is Guassian and defined by
	~~<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')~~
	~~</math></center>~~
	~~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>~~
	~~D= \mu K_B T~~
	~~</math>. We set <math> \mu= K_B=1</math>~~

	~~The potential energy of surface tension (<math>\nu </math> is the stiffness) can be expanded at the lowest order in the gradient:~~
	~~<center> <math>~~
	~~E_{pot} \sim \text{const.} + \frac{\nu}{2} \int d^d r (\nabla h)^2~~
	~~</math></center>~~
	~~Hence, we have the Edwards Wilkinson equation:~~
	~~<center> <math>~~
	~~\partial_t h(r,t)= \nu \nabla^2 h(r,t) + \eta(r,t)~~
	~~</math></center>~~
	~~=== Scaling Invariance===~~
	~~The equation enjoys of a continuous symmetry because <math> h(r,t) </math> and <math> h(r,t)+c </math> cannot be distinguished. This is a condition of scale invariance:~~
	~~<center> <math>~~
	~~h(b r, b^z t) \overset{in law}{\sim} b^{\alpha} h(r,t)~~
	~~</math></center>~~
	~~Here <math>~~
	~~z, \alpha~~
	~~</math> are the dynamic and the roughness exponent respectively. From dimensional analysis~~
	~~<center> <math>~~
	~~b^{\alpha-z} \partial_t h(r,t)= b^{\alpha-2} \nabla^2 h(r,t) +b^{-d/2-z/2} \eta(r,t)~~
	~~</math></center>~~
	~~From which you get <math> z=2 </math> in any dimension and a rough interface below <math> d=2 </math> with <math> \alpha =(2-d)/2 </math>.~~

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

	~~Consider a 1-dimensional line of size L with periodic boundary conditions.~~
	~~We consider the width square of the interface~~
	~~<center> <math>~~
	~~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~~
	~~</math></center>~~
	~~It is useful to introduce the Fourier modes:~~
	~~<center> <math>~~
	~~\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)~~
	~~</math></center>~~
	~~Here <math> q=2 \pi n/L, n=\ldots ,-1,0,1,\ldots</math> and recall <math> \int_0^L d r e^{iqr}= L \delta_{q,0} </math>.~~
	~~using de Parseval theorem for the Fourier series~~
	~~<center> <math>~~
	~~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~~
	~~</math></center>~~
	~~In the last step we used that <math>~~
	~~\hat h_q^*(t)= \hat h_{-q}(t)~~
	~~</math>.~~

	~~=== Solution in the Fourier space===~~
	~~show that the EW equation writes~~
	~~<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{2 T}{L} \delta_{q_1,-q_2}\delta(t-t')~~
	~~</math></center>~~

	~~The solution of this first order linear equation writes~~
	~~<center> <math>~~
	~~\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>~~

	* Assume that the interface is initially flat, namely <math> \hat h_q(0) =0 </math>. Show that
	~~<center> <math>~~
	~~\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}~~
	~~</math></center>~~

	*The mean width square grows at short times and saturates at long times:
	~~<center> <math>~~
	~~\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}~~
	~~T \sqrt{\frac{2 t}{\pi \nu}}, & t\ll L^2, \\[1.2em]~~
	~~\frac{T}{ \nu} \frac{L}{12} , & t\gg L^2.~~
	~~\end{cases}~~
	~~</math></center>~~

	=Directed polymers in random media=		=Directed polymers in random media=

Rosso: /* Thermal Interfaces */

2025-09-15T15:32:23Z

Thermal Interfaces

← Older revision		Revision as of 17:32, 15 September 2025
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	The potential energy of surface tension (<math>\nu </math> is the stiffness) 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} \sim \text{const.} + \frac{\nu}{2} \int d^d r (\nabla h)^2
	</math></center>		</math></center>
	Hence, we have the Edwards Wilkinson equation:		Hence, we have the Edwards Wilkinson equation:
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	\end{cases}		\end{cases}
	</math></center>		</math></center>

	=Directed polymers in random media=		=Directed polymers in random media=
	We now reanalyze the previous problem in the presence of quenched disorder.		We now reanalyze the previous problem in the presence of quenched disorder.

Rosso: /* Feynman-Kac formula */

2025-09-15T09:34:18Z

Feynman-Kac formula

← Older revision		Revision as of 11:34, 15 September 2025
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	Z_p(x ,t) = \int_{-\infty}^\infty d A e^{-p A} P[A,x,t] =\int_{x(0)=0}^{x(t)=x} {\cal D} x(\tau) e^{-\frac{1}{T} \int_0^t d \tau \frac{1}2(\partial_\tau x)^2 -p \int_0^t d \tau V(x(\tau),\tau)}		Z_p(x ,t) = \int_{-\infty}^\infty d A e^{-p A} P[A,x,t] =\int_{x(0)=0}^{x(t)=x} {\cal D} x(\tau) e^{-\frac{1}{T} \int_0^t d \tau \frac{1}2(\partial_\tau x)^2 -p \int_0^t d \tau V(x(\tau),\tau)}
	</math></center>		</math></center>
	* Third, ~~the backward approach. Consider~~ free paths evolving up to <math>t+dt</math> and reaching <math>x</math> :		* Third, consider free paths evolving up to <math>t+dt</math> and reaching <math>x</math> :
	<center> <math>		<center> <math>
	Z_p(x,t+dt)= \left\langle e^{-p \int_0^{t+dt} d \tau V(x(\tau),\tau)}\right\rangle= \left\langle e^{-p \int_0^{t} d \tau V(x(\tau),\tau)}\right\rangle e^{-p V(x,t) d t } =[1 -p V(x,t) d t +\dots]\left\langle Z_p(x-\Delta x,t) \right\rangle_{\Delta x}		Z_p(x,t+dt)= \left\langle e^{-p \int_0^{t+dt} d \tau V(x(\tau),\tau)}\right\rangle= \left\langle e^{-p \int_0^{t} d \tau V(x(\tau),\tau)}\right\rangle e^{-p V(x,t) d t } =[1 -p V(x,t) d t +\dots]\left\langle Z_p(x-\Delta x,t) \right\rangle_{\Delta x}

Rosso: /* Remarks */

2025-08-27T16:19:47Z

Remarks

← Older revision		Revision as of 18:19, 27 August 2025
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	<Strong>Remark 1:</Strong>		<Strong>Remark 1:</Strong>

	This equation is a diffusive equation with multiplicative noise. Edwards Wilkinson is instead a diffusive equation with additive noise~~. The Cole Hopf transformation allows to map the diffusive equation with multiplicative noise in a non-linear equation with additive noise. We will apply this tranformation and have a surprise~~.		This equation is a diffusive equation with multiplicative noise <math>V(x,\tau)/T</math> . Edwards Wilkinson is instead a diffusive equation with additive noise.

	<Strong>Remark 2:</Strong>		<Strong>Remark 2:</Strong>
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	</math>		</math>
	</center>		</center>



	In absence of disorder, one can find the propagator of the free particle, that, in the original variables, writes:		In absence of disorder, one can find the propagator of the free particle, that, in the original variables, writes:
	<center> <math>		<center> <math>
	Z_{\text{free}}(x,t)=\frac{e^{-x^2/(2Tt)}}{\sqrt{2 \pi Tt}}		Z_{\text{free}}(x,t)=\frac{e^{-x^2/(2Tt)}}{\sqrt{2 \pi Tt}}
	</math></center>		</math></center>

Rosso: /* Directed polymers in random media */

2025-08-27T16:08:24Z

Directed polymers in random media

← Older revision		Revision as of 18:08, 27 August 2025
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	Let us consider polymers <math>x(\tau)</math> of length <math>t</math>.		Let us consider polymers <math>x(\tau)</math> of length <math>t</math>.
	The energy associated with a given polymer configuration can be written as		The energy associated with a given polymer configuration can be written as
			<center>
	<math>		<math>
	E[x(\tau)] = \int_0^t d\tau \, \left[ \frac{1}{2} \left( \frac{dx}{d\tau} \right)^2 + V(x(\tau), \tau) \right]		E[x(\tau)] = \int_0^t d\tau \, \left[ \frac{1}{2} \left( \frac{dx}{d\tau} \right)^2 + V(x(\tau), \tau) \right]
	</math>		</math>
			</center>
	The first term describes the elastic energy of the polymer,		The first term describes the elastic energy of the polymer,
	while the second one is the disordered potential, which we assume to be		while the second one is the disordered potential, which we assume to be
			<center>
	<math>		<math>
	\overline{ V(x,\tau) } = 0, \qquad		\overline{ V(x,\tau) } = 0, \qquad
	\overline{ V(x,\tau) V(x',\tau') } = D \, \delta(x-x') \, \delta(\tau-\tau') .		\overline{ V(x,\tau) V(x',\tau') } = D \, \delta(x-x') \, \delta(\tau-\tau') .
	</math>		</math>
			</center>
	where 'D' is the disorder strength.		where 'D' is the disorder strength.

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	== Polymer partition function and propagator of a quantum particle==		== Polymer partition function and propagator of a quantum particle==

	Let us consider polymers ~~<math>~~		Let us consider polymers starting in <math>0 </math>, ending in <math>x </math> and at thermal equilibrium at temperature <math>T</math>. The partition function of the model writes as
	~~x(\tau) </math> of length <math>~~
	~~t </math>,~~ starting in <math>0 </math> ~~and~~ ending in <math>x </math> and at thermal ~~equlibrium~~ at temperature <math>T</math>. The partition function of the model writes as
	<center> <math>		<center> <math>
	Z(x,t) =\int_{x(0)=0}^{x(t)=x} {\cal D} x(\tau) \exp\left[- \frac{1}{T} \int_0^t d \tau \frac{1}2(\partial_\tau x)^2 +V(x(\tau),\tau)\right]		Z(x,t) =\int_{x(0)=0}^{x(t)=x} {\cal D} x(\tau) \exp\left[- \frac{1}{T} \int_0^t d \tau \frac{1}2(\partial_\tau x)^2 +V(x(\tau),\tau)\right]

Rosso: /* Directed polymers in random media */

2025-08-27T15:22:09Z

Directed polymers in random media

@@ Line 113: / Line 113: @@
 </math></center>
 =Directed polymers in random media=
 Let us consider polymers <math>
@@ Line 120: / Line 138: @@
 Z(x,t) =\int_{x(0)=0}^{x(t)=x} {\cal D} x(\tau) \exp\left[- \frac{1}{T} \int_0^t d \tau \frac{1}2(\partial_\tau x)^2 +V(x(\tau),\tau)\right]
 </math></center>
-For simplicity, we assume a white noise, <math> \overline{V(x,\tau) V(x',\tau')} = D \delta(x-x') \delta(\tau-\tau') </math>. Here, the partition function is written as a sum over all possible paths, corresponding to all possible polymer configurations that start at <math>0</math> and end at <math>x</math>, weighted by the appropriate Boltzmann factor.
+Here, the partition function is written as a sum over all possible paths, corresponding to all possible polymer configurations that start at <math>0</math> and end at <math>x</math>, weighted by the appropriate Boltzmann factor.
 Let's perform the following change of variables: <math>\tau=i t' </math>. We also identifies <math>T</math> with <math>\hbar</math> and <math> \tilde t= -i t </math> as the time.
@@ Line 132: / Line 149: @@
 From the Feymann path integral formulation, <math> Z[x,\tilde t]</math>  is the propagator of the quantum particle.
-== Feynman-Kac formula==
+=== Feynman-Kac formula===
 Let's derive the Feyman Kac formula for  <math>Z(x,t)</math> in the general case:
 * First, focus on free paths and introduce the following probability
@@ Line 181: / Line 194: @@
 </math>
 </center>