\displaystyle \frac{\partial h}{\partial t} = \nu\frac{\partial^2 h}{\partial x^2} + \frac{\lambda}{2}\left(\frac{\partial h}{\partial x}\right)^2 + \xi(x,t)

\displaystyle \frac{\partial h}{\partial t} = \nu\frac{\partial^2 h}{\partial x^2} + \frac{\lambda}{2}\left(\frac{\partial h}{\partial x}\right)^2 + \xi(x,t)

- \nu\partial_x^2 h describes
**diffusion**. - \xi(x,t) is
**noise**. - What about \frac{\lambda}{2}(\partial_x h)^2? Who ordered that?

Adding material at rate v to the surface causes it to rise by

\displaystyle v\delta t\sqrt{1+(\partial_x h)^2}

in time \delta t

Turbulent liquid crystals. Takeuchi & Sano, 2010

KPZ used to describe bacterial colony growth, flame propagation, etc.

The physics of each situation seems different.

What do we mean to say that each is

*described*by KPZ?

How do we understand

*macroscopic*matter in terms of*microscopic*constituents?Talking about a uniform

*state of matter*requires that widely separated regions are nearly independent.*Living matter*is quite different in this respect.

\displaystyle H = -J\sum_{\langle j\,k\rangle} \sigma_j \sigma_k + h \sum_j \sigma_j,\qquad \sigma_j = \pm 1

- System is
**scale invariant**(a fractal). - Correlation function is power law \displaystyle \langle \sigma(\mathbf{x})\sigma(\mathbf{y})\rangle = \frac{1}{|\mathbf{x}-\mathbf{y}|^\eta}, \qquad |\mathbf{x}-\mathbf{y}| \to \infty
- \eta is a
**critical exponent** - Exact solution in 2D: \eta=1/4.
- No exact solution in 3D.

- Ising model is clearly ridiculously oversimplified.
- Why don’t we study more complicated models?

- Near the critical point, the physics is the same!
- More precisely, critical exponents coincide.

\displaystyle \frac{\partial \phi}{\partial t} = D \frac{\partial^2 \phi}{\partial x^2}

Equation ‘smoothes’ \phi(x,t)

\displaystyle \phi(x,t)\longrightarrow \frac{\phi_0}{\sqrt{t}} \exp\left(-\frac{x^2}{4Dt}\right)

…no matter what you start with!

- Shows key property of diffusion that x\sim \sqrt{t}
- T \sim L^z with
**dynamical critical exponent**z=2. - This wouldn’t be changed if we added terms with more derivatives e.g. \frac{\partial^4 \phi}{\partial x^4}.

\displaystyle h(x,t) - vt \sim t^{1/3}

- Spatial scale of fluctuations is L\sim t^{2/3}, so z=3/2
- Different from diffusion, but
**universal**.

- From a theorist’s viewpoint!
- Particles on sites, one per site.
Particles hop to right with some rate, if they can.

From hops to heights…

\displaystyle \frac{\partial h}{\partial x} = \frac{1}{2}-\rho

Think in terms of local **density** \rho(x,t) and current j(x,t).

They obey the

**continuity equation**\displaystyle \frac{\partial \rho}{\partial t} = - \frac{\partial j}{\partial x}

Velocity v\propto (1-\rho), so

\displaystyle j = \rho v = \rho(1-\rho)

Putting it together gives

**Burgers’ equation**\displaystyle \frac{\partial \rho}{\partial t} = - \frac{\partial}{\partial x}\rho(1-\rho)

\displaystyle \frac{\partial \rho}{\partial t} = - \frac{\partial}{\partial x}\rho(1-\rho)

As we saw, \rho-\frac{1}{2}=\frac{\partial h}{\partial x}, so this is

\displaystyle \frac{\partial h}{\partial t} = \frac{\partial h}{\partial x} + \left(\frac{\partial h}{\partial x}\right)^2

Compare KPZ

\displaystyle \frac{\partial h}{\partial t} = \frac{\lambda}{2}\left(\frac{\partial h}{\partial x}\right)^2 + \nu\frac{\partial^2 h}{\partial x^2} + \xi(x,t)

Noise and diffusion are higher order effects

Polymer is stretched out in (1,1) direction, but attracted to points.

Wants to maximize number visited, subject to moving up and to the right.

This is the **directed polymer in a random potential**. It’s equivalent to KPZ!

Remember T \sim L^z. Here T = x+y, L=x-y, so z=3/2

- Model for propagation of light in random refractive index, or fluctuating spacetime geometry.

This means: calculate the statistical properties exactly

**Nope**Actually, it’s a struggle to

*define*it!

**Martin Hairer** won the Fields medal in 2014 for giving precise **meaning** to the KPZ equation.

Thus we resort to

**models**. If we believe that scaling behaviour is insensitive to details of system, we are free to choose!Such models have played a

**vital role**in our understanding of critical phenomena.

Back to directed polymer

Permutation (23154)

**Longest increasting subsequences** are (2,3,5) and (2,3,4).

The fluctuations of **height** of a KPZ surface, or of the energy of a **random polymer**, are equivalent to the statistics of the longest increasing subsequence of a **random permutation**!

- Statistical physics reduced to
**combinatorics**.

- Exact solution describe different probability distributions.
- Same h(x,t) - vt \sim t^{1/3} scaling.

Almost anything about higher dimensions (e.g. growth on a 2D interface). Critical exponents and scaling functions.

A rough-to-smooth phase transition?

Any generalization of the ‘tricks’ (e.g. random permutation) used in 1D.