L-4

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Goal 1: final lecture on KPZ and directed polymers at finite dimension. We will show that for a "glass transition" takes place.

Goal 2: We will mention some ideas related to glass transition in true glasses.


Part 1: KPZ in finite dimension

  • In we found and a glassy regime present at all temperatures. Moreover, the stationary solution tell us that is a Brownian motion in . However this solution does not identify the actual distribution of for a given . In particular we have no idea from where Tracy Widom comes from.
  • In the exponents are not known. There is an exact solution for the Caley tree (infinite dimension) that predicts a freezing transition to an 1RSB phase ().

Let's do replica!

To make progress in disordered systems we have to go through the moments of the partition function. We recall 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 V(x,\tau)} is a Gaussian field with
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 \overline{V(x,\tau)}=0, \quad \overline{V(x,\tau) V(x',\tau')} = D \delta(x-x') \delta(\tau-\tau') }
  • From the Wick theorem, for a generic Gaussian field we have
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 \overline{\exp(W)} = \exp\left[\overline{W} +\frac{1}{2} (\overline{W^2}-\overline{W}^2)\right] }

The first moment of the partition function is

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 \overline{Z_t[x_1] } =\int {\cal D} x_1 \exp\left[- \frac{1}{T} \int_0^t d \tau (\partial_\tau x)^2\right] \overline{\exp\left[- \frac{1}{T} \int d \tau V(x,\tau ) \right]} }

Note that the term has a short distance divergence due to the delta-functiton. Hence we can write:

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 \overline{Z_t[x_1] } = \frac{1}{\sqrt{2 \pi t T}}\exp\left[ -\frac{x^2}{2t T} \right] \exp\left[ \frac{D t \delta(0)}{2T^2} \right] }

Exercise L4-A: the second moment

Show:

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 \overline{Z_t[x_1] Z_t[x_2]} = \exp\left[ \frac{D t \delta(0)}{T^2} \right] }

\int {\cal D} x_1\int {\cal D} x_2 \exp\left[- \int_0^t d \tau \frac{1}{2T}[(\partial_\tau x_1)^2+ (\partial_\tau x_2)^2 ]+ \frac{1}{T^2} \delta[x_1(\tau)-x_2(\tau)]\right]} </math>

Part 2: Structural glasses