Nonequilibrium phase transitions : Noise and fluctuations

Pattern formation in physical systems is a non-equilibrium process that forms coherent structures in spite of thermal fluctuations [Cro88,CH93]. The statistical mechanics of extended non-equilibrium systems with thermal fluctuations is an outstanding open problem in physics since one has to account for nonlinearities, spatial interactions and random fluctuations [Cro88]. This often has to be treated outside a perturbation framework since the interactions between the fluctuations can give rise to large effects (fluctuation renormalization) [Kad00].

We will investigate this problem through pattern formation in the presence of thermal noise in the stochastic CCM

$$\xi_{{n+1}}^{} \xi_{n}^{} \eta_{n}^{}$$ (SCCM)

Here $\eta_{n}^{}$(x) is a Gaussian random process with mean zero and covariance given by

$$\langle$$$$\eta_{n}^{}$$(x)$$\eta_{m}^{}$$(x')$$\rangle$$ = C(|x - x'|)$$\delta$$(m - n),

where C(x) is a smooth function that decays rapidly for large x. The effective strength of the noise is $\sigma^{2}_{}$ = C(0) which can be identified as an effective ``temperature''.

Without added noise, (SCCM) reduces to (CCM) and gives coherent, asymptotically stable patterns that consist of a single domain (say, stripes with the same orientation throughout D). However, with the additional noise, the resulting fluctuations can destroy the long range order. A natural question is, at what value of the noise strength (``temperature'') is the long range order destroyed? This is analogous to a phase transition in equilibrium statistical mechanics (for instance, the temperature at which a crystal melts). The formulation (SCCM) therefore gives a computationally efficient way to study this question for pattern forming systems.

Brazovski [Bra75] and Swift & Hohenberg [SH77] have investigated the effect of fluctuations on the onset of a global stripe pattern using Renormalization Group (RG) techniques [Kad76,Wil83]. Recent experiments by Ahlers [SGAR00] have tested aspects of this theory. We are currently looking at the following questions -

Problem 9   Verify these results for stripe patterns in (SCCM).

Problem 10   Using our solution of (CCM) for patterns other than stripes, investigate the effects of noise on the onset of these patterns in (SCCM). Build a theory for this using RG.