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| Line 22: |
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| h \to x, \quad r\to t | | h \to x, \quad r\to t |
| </math></center> | | </math></center> |
|
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| =Directed polymers=
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|
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| ==Dijkstra Algorithm and transfer matrix==
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|
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| [[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>. ]]
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|
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| ==Back to the continuum model==
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|
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| Let us consider polymers <math>
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| x(\tau) </math> of length <math>
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| 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
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| <center> <math>
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| 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]
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| </math></center>
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| 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.
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|
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| === Polymer partition function and propagator of a quantum particle===
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|
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| 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.
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| <center> <math>
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| Z(x,\tilde t) =\int_{x(0)=0}^{x(\tilde t)=x} {\cal D} x(t') \exp\left[ \frac{i}{\hbar} \int_0^{\tilde t} d t' \frac{1}2(\partial_{t'} x)^2 -V(x(t'),t')\right]
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| </math></center>
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|
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| Note that <math> S[x]= \int_0^{\tilde t} d t' \frac{1}2(\partial_{t'} x)^2 -V(x(t'),t')</math> is the classical action of a particle with kinetic energy <math> \frac{1}2(\partial_\tau x)^2</math> and time dependent potential <math> V(x(\tau),\tau)</math>, evolving from time zero to time <math> \tilde t</math>.
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| From the Feymann path integral formulation, <math> Z[x,\tilde t]</math> is the propagator of the quantum particle.
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|
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| In absence of disorder, one can find the propagator of the free particle, that, in the original variables, writes:
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| <center> <math>
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| Z_{\text{free}}(x,t)=\frac{e^{-x^2/(2Tt)}}{\sqrt{2 \pi Tt}}
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| </math></center>
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|
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| == Feynman-Kac formula==
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| Let's derive the Feyman Kac formula for <math>Z(x,t)</math> in the general case:
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| * First, focus on free paths and introduce the following probability
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| <center> <math>
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| P[A,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 \right] \delta\left( \int_0^t d \tau V(x(\tau),\tau)-A \right)
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| </math></center>
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| * Second, the moments generating function
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| <center> <math>
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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)}
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| </math></center>
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| * Third, the backward approach. Consider free paths evolving up to <math>t+dt</math> and reaching <math>x</math> :
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| <center> <math>
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| 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}
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| </math></center>
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| Here <math> \langle \ldots \rangle</math> is the average over all free paths, while <math> \langle \ldots \rangle_{\Delta x}</math> is the average over the last jump, namely <math> \langle \Delta x \rangle=0
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| </math> and <math> \langle \Delta x^2 \rangle=T d t </math>.
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| * At the lowest order we have
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| <center> <math>
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| Z_p(x,t+dt)= Z_p(x,t) +dt \left[ \frac{T}{2} \partial_x^2 Z_p -p V(x,t) Z_p \right] +O(dt^2)
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| </math></center>
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| Replacing <math> p=1/T</math> we obtain the partition function is the solution of the Schrodinger-like equation:
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| <center> <math>
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| \partial_t Z(x,t) =- \hat H Z = - \left[ -\frac{T}{2}\frac{d^2 }{d x^2} + \frac{V(x,\tau)}{T}\right] Z(x,t)
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| </math>
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| <math>
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| Z[x,t=0]=\delta(x)
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| </math>
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| </center>
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|
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|
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|
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| ===Remarks===
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| <Strong>Remark 1:</Strong>
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|
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| 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.
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|
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| <Strong>Remark 2:</Strong>
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| This hamiltonian is time dependent because of the multiplicative noise <math>V(x,\tau)/T</math>. For a <Strong> time independent </Strong> hamiltonian, we can use the spectrum of the operator. In general we will have to parts:
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| * A discrete set of eigenvalues <math>E_n</math> with the eigenstates <math>\psi_n(x)</math>
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| * A continuum part where the states <math>\psi_E(x)</math> have energy <math>E</math>. We define the density of states (DOS) <math>\rho(E)</math>, such that the number of states with energy in <math>(E, E+dE)</math> is <math>\rho(E) dE </math>.
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| In this case <math> Z[x,t] </math> can be written has the sum of two contributions:
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| <center> <math>
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| Z[x,t] = \left( e^{- \hat H t} \right)_{0 \to x}= \sum_n \psi_n(0) \psi_n^*(x) e^{- E_n t} + \int_0^\infty dE \, \rho(E) \psi_E(0) \psi_E^*(x) e^{- E t}.
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| </math>
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| </center>
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|
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| ====Exercise: free particle in 1D====
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| * <Strong> Step 1: The spectrum</Strong>
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| For a free particle there a no discrete eigenvalue, but only a continuum spectrum. In 1D the hamiltonian is <math> \hat H =-(T/2) \partial_x^2</math>.
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| Show that the continuum spectrum has the form
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| <center><math> \psi_k(x) =\frac{1}{\sqrt{2 \pi}} e^{i k x} ,\quad \text{with} \; E_k =\frac{ T k^2}{2} </math></center>
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| Here <math>k </math> is a real number, while <math>E_k</math> is a positive number. Note that the states are delocalized on the entire real axis and they cannot be normalized to unit but you impose the Dirac delta normalization:
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| <center><math> \int_{-\infty}^\infty \, dx \, \psi_{k'}^*(x)\psi_k(x) = \delta(k-k') </math></center>
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| * <Strong> Step 2: the DOS </Strong>
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| For a given energy <math> E </math> there are two states
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| <center><math> \psi_E^{-}(x) =\frac{1}{\sqrt{2 \pi}} e^{-i \sqrt{2 E/T} x} ,\quad \text{and} ,\quad \psi_E^{+}(x) =\frac{1}{\sqrt{2 \pi}} e^{i \sqrt{2 E/T} x} </math></center>
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| Use the definition of DOS <center><math> \rho(E)= \int_{-\infty}^\infty \, dk \,\delta(E-E_k) </math></center> ans show that for both <math> \psi_E^{-}(x) , \psi_E^{-}(x) </math> you have :
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| <center><math> \rho(E) = \frac{1}{\sqrt{2 E T} } </math></center>
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| * <Strong> Step 3: the propagator </Strong>
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| Use the spectral decomposition of the propagator to recover <math> Z_{\text{free}}(x,t)</math>. <Strong>
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| Tip:</Strong> use <math> \int_{-\infty}^\infty \, dx \, \exp[-(a x^2 +b x)]= \sqrt{\pi/a} \exp[b^2/(4 a)] </math>.
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|
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| == Cole Hopf Transformation==
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| Replacing
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| * <math>T =2 \nu </math>
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| * <math>x = r </math>
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| * <math> Z(x,t) = \exp\left(\frac{\lambda}{2 \nu} h(r,t) \right) </math>
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| * <math>- V(x,t)=\lambda \eta(r,t) </math>
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| You get
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| <center> <math>
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| \partial_t h(r,t)= \nu \nabla^2 h(r,t)+ \frac{\lambda}{2} (\nabla h)^2 + \eta(r,t)
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| </math></center>
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| The KPZ equation!
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|
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| We can establish a KPZ/Directed polymer dictionary, valid in any dimension. Let us remark that the free energy of the polymer is
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| <center> <math>
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| F= - T \ln Z(x,t) = \frac{-1}{\lambda} h(r,t)
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| </math></center>
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| At low temperature, the free energy approaches the ground state energy, <math>E_{\min}</math>.
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|
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|
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|
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| {| class="wikitable"
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| |+ Dictionary
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| |-
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| ! KPZ !! KPZ exponents !! Directed polymer !! Directed polymer exponents
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| |-
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| | <math> r </math>|| <math> r \sim t^{1/z}</math>|| <math>x </math>|| <math>x\sim t^{\zeta}</math>
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| |-
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| | <math>t</math> ||<math>h(r,t) \sim t^{\alpha/z}</math> || <math>t</math>||
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| |-
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| | <math>h</math> || <math>h(r,t) \sim r^{\alpha}</math>|| <math>F, E_{\min}</math> || <math> \overline{(E_{\min} - \overline{E_{\min}})^2} \sim t^{2\theta} </math>
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| |}
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|
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| We conclude that
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| <center> <math>
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| \theta =\alpha/z, \quad \zeta=1/z
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| </math></center>
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| Moreover, the scaling relation <math>
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| \theta =2 \zeta- 1
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| </math> is a reincarnation of the Galilean invariance <math>
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| \alpha +z =2
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| </math>.
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Goal: This lecture is dedicated to a classical model in disordered systems: the directed polymer in random media. It has been introduced to model vortices in superconductur or domain wall in magnetic film. We will focus here on the algorithms that identify the ground state or compute the free energy at temperature T, as well as, on the Cole-Hopf transformation that map this model on the KPZ equation.
Polymers, interfaces and manifolds in random media
We consider the following potential energy
The first term represents the elasticity of the manifold and the second term is the quenched disorder, due to the impurities. In general, the medium is D-dimensional, the internal coordinate of the manifold is d-dimensional and the height filed is N-dimensional. Hence,the following equations always holds:
In practice, we will study two cases:
- Directed Polymers (), . Examples are vortices, fronts...
- Elastic interfaces (), . Examples are domain walls...
Today we restrict to polymers. Note that they are directed because their configuration is uni-valuated.
It is useful to study the model using the following change of variable